diff --git a/README.md b/README.md
index e69c369eb..1a66beaac 100644
--- a/README.md
+++ b/README.md
@@ -308,6 +308,7 @@ Recommended prereading: Chapters 1-2 (linear algebra) and chapter 3 (statistics)
### Week 38 September 20-24
- Lab Wednesday: Work on Project 1
- Lecture Thursday: Classification problems and Logistic Regression, from binary cases to several categories
+ - Video of Lecture at https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureSeptember23.mp4?vrtx=view-as-webpage
- Lecture Friday: Logistic Regression and start discussions of gradient optimization
- Reading recommendations:
diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt
index f83d7feb4..bc4fc936d 100644
--- a/doc/BookChapters/chapter3.do.txt
+++ b/doc/BookChapters/chapter3.do.txt
@@ -1211,6 +1211,967 @@ plt.show()
!ec
+===== More on Rescaling data =====
+
+We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.
+
+When you are comparing your own code with for example _Scikit-Learn_'s
+library, there are some technicalities to keep in mind. The examples
+here demonstrate some of these aspects with potential pitfalls.
+
+The discussion here focuses on the role of the intercept, how we can
+set up the design matrix, what scaling we should use and other topics
+which tend confuse us.
+
+The intercept can be interpreted as the expected value of our
+target/output variables when all other predictors are set to zero.
+Thus, if we cannot assume that the expected outputs/targets are zero
+when all predictors are zero (the columns in the design matrix), it
+may be a bad idea to implement a model which penalizes the intercept.
+Furthermore, in for example Ridge and Lasso regression, the default solutions
+from the library _Scikit-Learn_ (when not shrinking $\beta_0$) for the unknown parameters
+$\bm{\beta}$, are derived under the assumption that both $\bm{y}$ and
+$\bm{X}$ are zero centered, that is we subtract the mean values.
+
+
+If our predictors represent different scales, then it is important to
+standardize the design matrix $\bm{X}$ by subtracting the mean of each
+column from the corresponding column and dividing the column with its
+standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
+the results may differ.
+
+The
+"Standadscaler":"https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html"
+function in _Scikit-Learn_ does this for us. For the data sets we
+have been studying in our various examples, the data are in many cases
+already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
+survey of your data, with a critical assessment of them in case you need to scale the data.
+
+If you need to scale the data, not doing so will give an *unfair*
+penalization of the parameters since their magnitude depends on the
+scale of their corresponding predictor.
+
+Suppose as an example that you
+you have an input variable given by the heights of different persons.
+Human height might be measured in inches or meters or
+kilometers. If measured in kilometers, a standard linear regression
+model with this predictor would probably give a much bigger
+coefficient term, than if measured in millimeters.
+This can clearly lead to problems in evaluating the cost/loss functions.
+
+
+
+Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
+on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows
+
+!bc pycod
+"""
+#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+"""
+!ec
+
+
+Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as *zero centering*. For
+simplicity, we will focus on ordinary regression, as done in the above example.
+
+The cost/loss function for regression is
+!bt
+\[
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+\]
+!et
+Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.
+
+What we have done is to single out the $\beta_0$ term in the definition of the mean squared error (MSE).
+The design matrix
+$X$ does in this case not contain any intercept column.
+When we take the derivative with respect to $\beta_0$, we want the derivative to obey
+!bt
+\[
+\frac{\partial C}{\partial \beta_j} = 0,
+\]
+!et
+
+for all $j$. For $\beta_0$ we have
+
+!bt
+\[
+\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
+\]
+!et
+Multiplying away the constant $2/n$, we obtain
+!bt
+\[
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
+\]
+!et
+
+
+We assume
+that every column of $\bm{X}$ is centered, which we can do by subtracting the mean,
+!bc pycod
+X = X - np.mean(X,axis=0)
+!ec
+
+This means that we need to rewrite $X_{ij}$ as $\tilde{X}_{ij}=X_{ij}-\mu_j$, where
+!bt
+\[
+\mu_j = \frac{1}{n}\sum_{i=0}^{n-1}X_{ij}.
+\]
+!et
+
+Let us special first to the case where we have only two parameters $\beta_0$ and $\beta_1$.
+Our result for $\beta_0$ simplifies then to
+!bt
+\[
+n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
+\]
+!et
+Assuming that the matrix elements $X_{i1}$ are centered, what we have is
+!bt
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} \left(X_{i1}-\mu_{1}\right),
+\]
+!et
+where
+!bt
+\[
+\mu_1=\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1},
+\]
+!et
+and if we define the mean value of the outputs as
+!bt
+\[
+\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
+\]
+!et
+we have
+!bt
+\[
+\beta_0 = \mu_y - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}-\mu_{1}),
+\]
+!et
+and it is easy to see that the last sum equals zero! This means that we have
+!bt
+\[
+\beta_0 = \mu_y,
+\]
+!et
+if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of $\beta$.
+We have thus
+!bt
+\[
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\bm{y}},
+\]
+!et
+the average value of $\bm{y}$.
+
+Replacing $y_i$ with $y_i - \beta_0 = y_i - \overline{\bm{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)
+!bt
+\[
+C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
+\]
+!et
+
+
+If we minimize with respect to $\bm{\beta}$ we have then
+
+!bt
+\[
+\hat{\bm{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
+\]
+!et
+
+where $\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\bm{y}}$
+and $\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}$.
+
+For Ridge regression we need to add $\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}$ to the cost function and get then
+!bt
+\[
+\hat{\bm{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
+\]
+!et
+
+What does this mean? And why do we insist on all this? Let us look at some examples.
+
+
+This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.
+Note also that we do not split the data into training and test.
+
+!bc pycod
+import numpy as np
+import matplotlib.pyplot as plt
+
+from sklearn.linear_model import LinearRegression
+
+
+np.random.seed(2021)
+
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+def fit_beta(X, y):
+ return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+
+true_beta = [2, 0.5, 3.7]
+
+x = np.linspace(0, 1, 11)
+y = np.sum(
+ np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0
+) + 0.1 * np.random.normal(size=len(x))
+
+degree = 3
+X = np.zeros((len(x), degree))
+
+# Include the intercept in the design matrix
+for p in range(degree):
+ X[:, p] = x ** p
+
+beta = fit_beta(X, y)
+
+# Intercept is included in the design matrix
+skl = LinearRegression(fit_intercept=False).fit(X, y)
+
+print(f"True beta: {true_beta}")
+print(f"Fitted beta: {beta}")
+print(f"Sklearn fitted beta: {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column")
+print(MSE(y,ypredictOwn))
+print(f"MSE with intercept column from SKL")
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x, y, label="Data")
+plt.plot(x, X @ beta, label="Fit")
+plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
+
+
+# Do not include the intercept in the design matrix
+X = np.zeros((len(x), degree - 1))
+
+for p in range(degree - 1):
+ X[:, p] = x ** (p + 1)
+
+# Intercept is not included in the design matrix
+skl = LinearRegression(fit_intercept=True).fit(X, y)
+
+# Use centered values for X and y when computing coefficients
+y_offset = np.average(y, axis=0)
+X_offset = np.average(X, axis=0)
+
+beta = fit_beta(X - X_offset, y - y_offset)
+intercept = np.mean(y_offset - X_offset @ beta)
+
+print(f"Manual intercept: {intercept}")
+print(f"Fitted beta (wiothout intercept): {beta}")
+print(f"Sklearn intercept: {skl.intercept_}")
+print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept")
+print(MSE(y,ypredictOwn+intercept))
+print(f"MSE with Sklearn intercept")
+print(MSE(y,ypredictSKL))
+
+plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
+plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
+plt.grid()
+plt.legend()
+
+plt.show()
+
+!ec
+
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case).
+
+Printing the MSE, we see first that both methods give the same MSE, as
+they should. However, when we move to for example Ridge regression,
+the way we treat the intercept may give a larger or smaller MSE,
+meaning that the MSE can be penalized by the value of the
+intercept. Not including the intercept in the fit, means that the
+regularization term does not include $\beta_0$. For different values
+of $\lambda$, this may lead to differeing MSE values.
+
+To remind the reader, the regularization term, with the intercept in Ridge regression, is given by
+!bt
+\[
+\lambda \vert\vert \bm{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
+\]
+!et
+but when we take out the intercept, this equation becomes
+!bt
+\[
+\lambda \vert\vert \bm{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
+\]
+!et
+
+For Lasso regression we have
+!bt
+\[
+\lambda \vert\vert \bm{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
+\]
+!et
+
+It means that, when scaling the design matrix and the outputs/targets,
+by subtracting the mean values, we have an optimization problem which
+is not penalized by the intercept. The MSE value can then be smaller
+since it focuses only on the remaining quantities. If we however bring
+back the intercept, we will get a MSE which then contains the
+intercept.
+
+
+Armed with this wisdom, we attempt first to simply set the intercept equal to _False_ in our implementation of Ridge regression for our well-known vanilla data set.
+
+!bc pycod
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree))
+#We include explicitely the intercept column
+for degree in range(Maxpolydegree):
+ X[:,degree] = x**degree
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+p = Maxpolydegree
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 2, nlambdas)
+for i in range(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+ # Note: we include the intercept column and no scaling
+ RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+ RegRidge.fit(X_train,y_train)
+ # and then make the prediction
+ ytildeOwnRidge = X_train @ OwnRidgeBeta
+ ypredictOwnRidge = X_test @ OwnRidgeBeta
+ ytildeRidge = RegRidge.predict(X_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta)
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+!ec
+
+The results here agree when we force _Scikit-Learn_'s Ridge function to include the first column in our design matrix.
+We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
+What happens if we do not include the intercept in our fit?
+Let us see how we can change this code by zero centering.
+
+!bc pycod
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+from sklearn.preprocessing import StandardScaler
+
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(315)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree-1))
+
+for degree in range(1,Maxpolydegree): #No intercept column
+ X[:,degree-1] = x**(degree)
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean = np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
+X_test_scaled = X_test - X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler = np.mean(y_train)
+y_train_scaled = y_train - y_scaler
+
+p = Maxpolydegree-1
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 2, nlambdas)
+for i in range(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
+ intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
+ RegRidge = linear_model.Ridge(lmb)
+ RegRidge.fit(X_train,y_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta) #Intercept is given by mean of target variable
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print('Intercept from own implementation:')
+ print(intercept_)
+ print('Intercept from Scikit-Learn Ridge implementation')
+ print(RegRidge.intercept_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+!ec
+We see here, when compared to the code which includes explicitely the
+intercept column, that our MSE value is actually smaller. This is
+because the regularization term does not include the intercept value
+$\beta_0$ in the fitting. This applies to Lasso regularization as
+well. It means that our optimization is now done only with the
+centered matrix and/or vector that enter the fitting procedure. Note
+also that the problem with the intercept occurs mainly in these type
+of polynomial fitting problem.
+
+The next example is indeed an example where all these discussions about the role of intercept are not present.
+
+===== More complicated Example: The Ising model =====
+
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant $J$ is given by
+
+!bt
+\begin{align}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\end{align}
+!et
+
+where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins
+in the system is determined by $L$. For the one-dimensional system
+there is no phase transition.
+
+We will look at a system of $L = 40$ spins with a coupling constant of
+$J = 1$. To get enough training data we will generate 10000 states
+with their respective energies.
+
+
+!bc pycod
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+ energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+!ec
+
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+A more general form for the one-dimensional Ising model is
+
+!bt
+\begin{align}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\end{align}
+!et
+
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+!bt
+\begin{align}
+ \bm{H} = \bm{X} J,
+\end{align}
+!et
+
+where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the
+elements $-J_{jk}$. This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+!bt
+\begin{align}
+ \bm{y} = \bm{X}\bm{\beta} + \bm{\epsilon},
+\end{align}
+!et
+
+We split the data in training and test data as discussed in the previous example
+
+!bc pycod
+X = np.zeros((n, L ** 2))
+for i in range(n):
+ X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+!ec
+
+
+In the ordinary least squares method we choose the cost function
+
+!bt
+\begin{align}
+ C(\bm{X}, \bm{\beta})= \frac{1}{n}\left\{(\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y})\right\}.
+\end{align}
+!et
+
+We then find the extremal point of $C$ by taking the derivative with respect to $\bm{\beta}$ as discussed above.
+This yields the expression for $\bm{\beta}$ to be
+
+!bt
+\[
+ \bm{\beta} = \frac{\bm{X}^T \bm{y}}{\bm{X}^T \bm{X}},
+\]
+!et
+
+which immediately imposes some requirements on $\bm{X}$ as there must exist
+an inverse of $\bm{X}^T \bm{X}$. If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of $\bm{X}$ consists of $1$. We do this here
+
+!bc pycod
+X_train_own = np.concatenate(
+ (np.ones(len(X_train))[:, np.newaxis], X_train),
+ axis=1
+)
+X_test_own = np.concatenate(
+ (np.ones(len(X_test))[:, np.newaxis], X_test),
+ axis=1
+)
+!ec
+
+
+
+Doing the inversion directly turns out to be a bad idea since the matrix
+$\bm{X}^T\bm{X}$ is singular. An alternative approach is to use the _singular
+value decomposition_. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for $\bm{\beta}$ as
+
+!bt
+\[
+ \bm{\beta} = \bm{X}^{+}\bm{y},
+\]
+!et
+
+where the pseudoinverse of $\bm{X}$ is given by
+
+!bt
+\[
+ \bm{X}^{+} = \frac{\bm{X}^T}{\bm{X}^T\bm{X}}.
+\]
+!et
+
+Using singular value decomposition we can decompose the matrix $\bm{X} = \bm{U}\bm{\Sigma} \bm{V}^T$,
+where $\bm{U}$ and $\bm{V}$ are orthogonal(unitary) matrices and $\bm{\Sigma}$ contains the singular values (more details below).
+where $X^{+} = V\Sigma^{+} U^T$. This reduces the equation for
+$\omega$ to
+!bt
+\begin{align}
+ \bm{\beta} = \bm{V}\bm{\Sigma}^{+} \bm{U}^T \bm{y}.
+\end{align}
+!et
+
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as $\mathcal{O}(n^3)$, where $n$ is the number of elements in a
+general matrix. Instead, doing $QR$-factorization and solving the
+linear system as an equation would reduce this down to
+$\mathcal{O}(n^2)$ operations.
+
+
+!bc pycod
+def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+ u, s, v = scl.svd(x)
+ return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+!ec
+
+!bc pycod
+beta = ols_svd(X_train_own,y_train)
+!ec
+
+When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here
+
+!bc pycod
+J = beta[1:].reshape(L, L)
+!ec
+
+A way of looking at the coefficients in $J$ is to plot the matrices as images.
+
+
+!bc pycod
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J, **cmap_args)
+plt.title("OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+!ec
+It is interesting to note that OLS
+considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as
+valid matrix elements for $J$.
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+
+
+
+
+Let us now
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data. The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant $J$ is
+given by
+
+!bt
+\begin{align}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\end{align}
+!et
+where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.
+
+We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies.
+
+
+!bc pycod
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+ energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+!ec
+
+A more general form for the one-dimensional Ising model is
+
+!bt
+\begin{align}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\end{align}
+!et
+
+Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+!bt
+\begin{align}
+ H = X J,
+\end{align}
+!et
+
+where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the
+elements $-J_{jk}$. This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
+!bt
+\begin{align}
+ \bm{y} = \bm{X}\bm{\beta} + \bm{\epsilon}.
+\end{align}
+!et
+We organize the data as we did above
+!bc pycod
+X = np.zeros((n, L ** 2))
+for i in range(n):
+ X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+ (np.ones(len(X_train))[:, np.newaxis], X_train),
+ axis=1
+)
+
+X_test_own = np.concatenate(
+ (np.ones(len(X_test))[:, np.newaxis], X_test),
+ axis=1
+)
+!ec
+
+We will do all fitting with _Scikit-Learn_,
+
+!bc pycod
+clf = skl.LinearRegression().fit(X_train, y_train)
+!ec
+When extracting the $J$-matrix we make sure to remove the intercept
+!bc pycod
+J_sk = clf.coef_.reshape(L, L)
+!ec
+And then we plot the results
+!bc pycod
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+!ec
+The results agree perfectly with our previous discussion where we used our own code.
+
+
+Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a _regularizer_. This
+involves a new cost function which leads to a new estimate for the
+weights $\bm{\beta}$. This results in a penalized regression problem. The
+cost function is given by
+
+!bt
+\begin{align}
+ C(\bm{X}, \bm{\beta}; \lambda) = (\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y}) + \lambda \bm{\beta}^T\bm{\beta}.
+\end{align}
+!et
+!bc pycod
+_lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+!ec
+
+In the _Least Absolute Shrinkage and Selection Operator_ (LASSO)-method we get a third cost function.
+
+!bt
+\begin{align}
+ C(\bm{X}, \bm{\beta}; \lambda) = (\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y}) + \lambda \sqrt{\bm{\beta}^T\bm{\beta}}.
+\end{align}
+!et
+
+Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from _Scikit-Learn_.
+
+!bc pycod
+clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+!ec
+
+It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+$J_{j, j + 1} = -1$.
+
+
+
+
+We see how the different models perform for a different set of values for $\lambda$.
+
+
+!bc pycod
+lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+ "ols_sk": np.zeros(lambdas.size),
+ "ridge_sk": np.zeros(lambdas.size),
+ "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+ "ols_sk": np.zeros(lambdas.size),
+ "ridge_sk": np.zeros(lambdas.size),
+ "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+ for key, method in zip(
+ ["ols_sk", "ridge_sk", "lasso_sk"],
+ [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+ ):
+ method = method.fit(X_train, y_train)
+
+ train_errors[key][i] = method.score(X_train, y_train)
+ test_errors[key][i] = method.score(X_test, y_test)
+
+ omega = method.coef_.reshape(L, L)
+
+ plt.subplot(10, 5, plot_counter)
+ plt.imshow(omega, **cmap_args)
+ plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+ plot_counter += 1
+
+plt.show()
+!ec
+
+We see that LASSO reaches a good solution for low
+values of $\lambda$, but will "wither" when we increase $\lambda$ too
+much. Ridge is more stable over a larger range of values for
+$\lambda$, but eventually also fades away.
+
+
+To determine which value of $\lambda$ is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
+
+
+!bc pycod
+fig = plt.figure(figsize=(20, 14))
+
+colors = {
+ "ols_sk": "r",
+ "ridge_sk": "y",
+ "lasso_sk": "c"
+}
+
+for key in train_errors:
+ plt.semilogx(
+ lambdas,
+ train_errors[key],
+ colors[key],
+ label="Train {0}".format(key),
+ linewidth=4.0
+ )
+
+for key in test_errors:
+ plt.semilogx(
+ lambdas,
+ test_errors[key],
+ colors[key] + "--",
+ label="Test {0}".format(key),
+ linewidth=4.0
+ )
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+!ec
+
+From the above figure we can see that LASSO with $\lambda = 10^{-2}$
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of $\lambda$.
+
+
+
+
+
===== Exercises and Projects =====
@@ -1499,6 +2460,7 @@ Here is a simple part of a Python code which reads and plots the data
from such files
!bc pycod
+"""
import numpy as np
from imageio import imread
import matplotlib.pyplot as plt
@@ -1514,6 +2476,7 @@ plt.imshow(terrain1, cmap='gray')
plt.xlabel('X')
plt.ylabel('Y')
plt.show()
+"""
!ec
If you should have problems in downloading the digital terrain data,
@@ -1537,3 +2500,5 @@ of data presented here (either the terrain data we propose or other data sets).
+
+
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index 413c54051..18cd038f8 100644
--- a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb
@@ -1598,6 +1598,1604 @@
"cell_type": "markdown",
"metadata": {},
"source": [
+ "## More on Rescaling data\n",
+ "\n",
+ "We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.\n",
+ "\n",
+ "When you are comparing your own code with for example **Scikit-Learn**'s\n",
+ "library, there are some technicalities to keep in mind. The examples\n",
+ "here demonstrate some of these aspects with potential pitfalls.\n",
+ "\n",
+ "The discussion here focuses on the role of the intercept, how we can\n",
+ "set up the design matrix, what scaling we should use and other topics\n",
+ "which tend confuse us.\n",
+ "\n",
+ "The intercept can be interpreted as the expected value of our\n",
+ "target/output variables when all other predictors are set to zero.\n",
+ "Thus, if we cannot assume that the expected outputs/targets are zero\n",
+ "when all predictors are zero (the columns in the design matrix), it\n",
+ "may be a bad idea to implement a model which penalizes the intercept.\n",
+ "Furthermore, in for example Ridge and Lasso regression, the default solutions\n",
+ "from the library **Scikit-Learn** (when not shrinking $\\beta_0$) for the unknown parameters\n",
+ "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
+ "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
+ "\n",
+ "\n",
+ "If our predictors represent different scales, then it is important to\n",
+ "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
+ "column from the corresponding column and dividing the column with its\n",
+ "standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,\n",
+ "the results may differ. \n",
+ "\n",
+ "The\n",
+ "[Standadscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)\n",
+ "function in **Scikit-Learn** does this for us. For the data sets we\n",
+ "have been studying in our various examples, the data are in many cases\n",
+ "already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a\n",
+ "survey of your data, with a critical assessment of them in case you need to scale the data.\n",
+ "\n",
+ "If you need to scale the data, not doing so will give an *unfair*\n",
+ "penalization of the parameters since their magnitude depends on the\n",
+ "scale of their corresponding predictor.\n",
+ "\n",
+ "Suppose as an example that you \n",
+ "you have an input variable given by the heights of different persons.\n",
+ "Human height might be measured in inches or meters or\n",
+ "kilometers. If measured in kilometers, a standard linear regression\n",
+ "model with this predictor would probably give a much bigger\n",
+ "coefficient term, than if measured in millimeters.\n",
+ "This can clearly lead to problems in evaluating the cost/loss functions.\n",
+ "\n",
+ "\n",
+ "\n",
+ "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
+ "on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\"\"\"\n",
+ "#Model training, we compute the mean value of y and X\n",
+ "y_train_mean = np.mean(y_train)\n",
+ "X_train_mean = np.mean(X_train,axis=0)\n",
+ "X_train = X_train - X_train_mean\n",
+ "y_train = y_train - y_train_mean\n",
+ "\n",
+ "# The we fit our model with the training data\n",
+ "trained_model = some_model.fit(X_train,y_train)\n",
+ "\n",
+ "\n",
+ "#Model prediction, we need also to transform our data set used for the prediction.\n",
+ "X_test = X_test - X_train_mean #Use mean from training data\n",
+ "y_pred = trained_model(X_test)\n",
+ "y_pred = y_pred + y_train_mean\n",
+ "\"\"\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let us try to understand what this may imply mathematically when we\n",
+ "subtract the mean values, also known as *zero centering*. For\n",
+ "simplicity, we will focus on ordinary regression, as done in the above example.\n",
+ "\n",
+ "The cost/loss function for regression is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.\n",
+ "\n",
+ "What we have done is to single out the $\\beta_0$ term in the definition of the mean squared error (MSE).\n",
+ "The design matrix\n",
+ "$X$ does in this case not contain any intercept column.\n",
+ "When we take the derivative with respect to $\\beta_0$, we want the derivative to obey"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "for all $j$. For $\\beta_0$ we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Multiplying away the constant $2/n$, we obtain"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We assume \n",
+ "that every column of $\\boldsymbol{X}$ is centered, which we can do by subtracting the mean,"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = X - np.mean(X,axis=0)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This means that we need to rewrite $X_{ij}$ as $\\tilde{X}_{ij}=X_{ij}-\\mu_j$, where"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_j = \\frac{1}{n}\\sum_{i=0}^{n-1}X_{ij}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let us special first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
+ "Our result for $\\beta_0$ simplifies then to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Assuming that the matrix elements $X_{i1}$ are centered, what we have is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} \\left(X_{i1}-\\mu_{1}\\right),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_1=\\frac{1}{n}\\sum_{i=0}^{n-1} (X_{i1},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "and if we define the mean value of the outputs as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\mu_y - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} (X_{i1}-\\mu_{1}),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "and it is easy to see that the last sum equals zero! This means that we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\mu_y,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of $\\beta$.\n",
+ "We have thus"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1} y_i = \\overline{\\boldsymbol{y}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "the average value of $\\boldsymbol{y}$.\n",
+ "\n",
+ "Replacing $y_i$ with $y_i - \\beta_0 = y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
+ "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
+ "\n",
+ "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
+ "\n",
+ "\n",
+ "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
+ "Note also that we do not split the data into training and test."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "from sklearn.linear_model import LinearRegression\n",
+ "\n",
+ "\n",
+ "np.random.seed(2021)\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "\n",
+ "\n",
+ "def fit_beta(X, y):\n",
+ " return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
+ "\n",
+ "\n",
+ "true_beta = [2, 0.5, 3.7]\n",
+ "\n",
+ "x = np.linspace(0, 1, 11)\n",
+ "y = np.sum(\n",
+ " np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0\n",
+ ") + 0.1 * np.random.normal(size=len(x))\n",
+ "\n",
+ "degree = 3\n",
+ "X = np.zeros((len(x), degree))\n",
+ "\n",
+ "# Include the intercept in the design matrix\n",
+ "for p in range(degree):\n",
+ " X[:, p] = x ** p\n",
+ "\n",
+ "beta = fit_beta(X, y)\n",
+ "\n",
+ "# Intercept is included in the design matrix\n",
+ "skl = LinearRegression(fit_intercept=False).fit(X, y)\n",
+ "\n",
+ "print(f\"True beta: {true_beta}\")\n",
+ "print(f\"Fitted beta: {beta}\")\n",
+ "print(f\"Sklearn fitted beta: {skl.coef_}\")\n",
+ "ypredictOwn = X @ beta\n",
+ "ypredictSKL = skl.predict(X)\n",
+ "print(f\"MSE with intercept column\")\n",
+ "print(MSE(y,ypredictOwn))\n",
+ "print(f\"MSE with intercept column from SKL\")\n",
+ "print(MSE(y,ypredictSKL))\n",
+ "\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.scatter(x, y, label=\"Data\")\n",
+ "plt.plot(x, X @ beta, label=\"Fit\")\n",
+ "plt.plot(x, skl.predict(X), label=\"Sklearn (fit_intercept=False)\")\n",
+ "\n",
+ "\n",
+ "# Do not include the intercept in the design matrix\n",
+ "X = np.zeros((len(x), degree - 1))\n",
+ "\n",
+ "for p in range(degree - 1):\n",
+ " X[:, p] = x ** (p + 1)\n",
+ "\n",
+ "# Intercept is not included in the design matrix\n",
+ "skl = LinearRegression(fit_intercept=True).fit(X, y)\n",
+ "\n",
+ "# Use centered values for X and y when computing coefficients\n",
+ "y_offset = np.average(y, axis=0)\n",
+ "X_offset = np.average(X, axis=0)\n",
+ "\n",
+ "beta = fit_beta(X - X_offset, y - y_offset)\n",
+ "intercept = np.mean(y_offset - X_offset @ beta)\n",
+ "\n",
+ "print(f\"Manual intercept: {intercept}\")\n",
+ "print(f\"Fitted beta (wiothout intercept): {beta}\")\n",
+ "print(f\"Sklearn intercept: {skl.intercept_}\")\n",
+ "print(f\"Sklearn fitted beta (without intercept): {skl.coef_}\")\n",
+ "ypredictOwn = X @ beta\n",
+ "ypredictSKL = skl.predict(X)\n",
+ "print(f\"MSE with Manual intercept\")\n",
+ "print(MSE(y,ypredictOwn+intercept))\n",
+ "print(f\"MSE with Sklearn intercept\")\n",
+ "print(MSE(y,ypredictSKL))\n",
+ "\n",
+ "plt.plot(x, X @ beta + intercept, \"--\", label=\"Fit (manual intercept)\")\n",
+ "plt.plot(x, skl.predict(X), \"--\", label=\"Sklearn (fit_intercept=True)\")\n",
+ "plt.grid()\n",
+ "plt.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The intercept is the value of our output/target variable\n",
+ "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
+ "\n",
+ "Printing the MSE, we see first that both methods give the same MSE, as\n",
+ "they should. However, when we move to for example Ridge regression,\n",
+ "the way we treat the intercept may give a larger or smaller MSE,\n",
+ "meaning that the MSE can be penalized by the value of the\n",
+ "intercept. Not including the intercept in the fit, means that the\n",
+ "regularization term does not include $\\beta_0$. For different values\n",
+ "of $\\lambda$, this may lead to differeing MSE values. \n",
+ "\n",
+ "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "but when we take out the intercept, this equation becomes"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For Lasso regression we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It means that, when scaling the design matrix and the outputs/targets,\n",
+ "by subtracting the mean values, we have an optimization problem which\n",
+ "is not penalized by the intercept. The MSE value can then be smaller\n",
+ "since it focuses only on the remaining quantities. If we however bring\n",
+ "back the intercept, we will get a MSE which then contains the\n",
+ "intercept.\n",
+ "\n",
+ "\n",
+ "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known vanilla data set."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn import linear_model\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "\n",
+ "\n",
+ "# A seed just to ensure that the random numbers are the same for every run.\n",
+ "# Useful for eventual debugging.\n",
+ "np.random.seed(3155)\n",
+ "\n",
+ "n = 100\n",
+ "x = np.random.rand(n)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+ "\n",
+ "Maxpolydegree = 20\n",
+ "X = np.zeros((n,Maxpolydegree))\n",
+ "#We include explicitely the intercept column\n",
+ "for degree in range(Maxpolydegree):\n",
+ " X[:,degree] = x**degree\n",
+ "# We split the data in test and training data\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+ "\n",
+ "p = Maxpolydegree\n",
+ "I = np.eye(p,p)\n",
+ "# Decide which values of lambda to use\n",
+ "nlambdas = 6\n",
+ "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+ "MSERidgePredict = np.zeros(nlambdas)\n",
+ "lambdas = np.logspace(-4, 2, nlambdas)\n",
+ "for i in range(nlambdas):\n",
+ " lmb = lambdas[i]\n",
+ " OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+ " # Note: we include the intercept column and no scaling\n",
+ " RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+ " RegRidge.fit(X_train,y_train)\n",
+ " # and then make the prediction\n",
+ " ytildeOwnRidge = X_train @ OwnRidgeBeta\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta\n",
+ " ytildeRidge = RegRidge.predict(X_train)\n",
+ " ypredictRidge = RegRidge.predict(X_test)\n",
+ " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+ " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+ " print(\"Beta values for own Ridge implementation\")\n",
+ " print(OwnRidgeBeta)\n",
+ " print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+ " print(RegRidge.coef_)\n",
+ " print(\"MSE values for own Ridge implementation\")\n",
+ " print(MSEOwnRidgePredict[i])\n",
+ " print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+ " print(MSERidgePredict[i])\n",
+ "\n",
+ "# Now plot the results\n",
+ "plt.figure()\n",
+ "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')\n",
+ "plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n",
+ "\n",
+ "plt.xlabel('log10(lambda)')\n",
+ "plt.ylabel('MSE')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
+ "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
+ "What happens if we do not include the intercept in our fit?\n",
+ "Let us see how we can change this code by zero centering."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn import linear_model\n",
+ "from sklearn.preprocessing import StandardScaler\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "# A seed just to ensure that the random numbers are the same for every run.\n",
+ "# Useful for eventual debugging.\n",
+ "np.random.seed(315)\n",
+ "\n",
+ "n = 100\n",
+ "x = np.random.rand(n)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+ "\n",
+ "Maxpolydegree = 20\n",
+ "X = np.zeros((n,Maxpolydegree-1))\n",
+ "\n",
+ "for degree in range(1,Maxpolydegree): #No intercept column\n",
+ " X[:,degree-1] = x**(degree)\n",
+ "\n",
+ "# We split the data in test and training data\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+ "\n",
+ "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
+ "X_train_mean = np.mean(X_train,axis=0)\n",
+ "#Center by removing mean from each feature\n",
+ "X_train_scaled = X_train - X_train_mean \n",
+ "X_test_scaled = X_test - X_train_mean\n",
+ "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)\n",
+ "#Remove the intercept from the training data.\n",
+ "y_scaler = np.mean(y_train) \n",
+ "y_train_scaled = y_train - y_scaler \n",
+ "\n",
+ "p = Maxpolydegree-1\n",
+ "I = np.eye(p,p)\n",
+ "# Decide which values of lambda to use\n",
+ "nlambdas = 6\n",
+ "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+ "MSERidgePredict = np.zeros(nlambdas)\n",
+ "\n",
+ "lambdas = np.logspace(-4, 2, nlambdas)\n",
+ "for i in range(nlambdas):\n",
+ " lmb = lambdas[i]\n",
+ " OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)\n",
+ " intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data\n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ \n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n",
+ " RegRidge = linear_model.Ridge(lmb)\n",
+ " RegRidge.fit(X_train,y_train)\n",
+ " ypredictRidge = RegRidge.predict(X_test)\n",
+ " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+ " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+ " print(\"Beta values for own Ridge implementation\")\n",
+ " print(OwnRidgeBeta) #Intercept is given by mean of target variable\n",
+ " print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+ " print(RegRidge.coef_)\n",
+ " print('Intercept from own implementation:')\n",
+ " print(intercept_)\n",
+ " print('Intercept from Scikit-Learn Ridge implementation')\n",
+ " print(RegRidge.intercept_)\n",
+ " print(\"MSE values for own Ridge implementation\")\n",
+ " print(MSEOwnRidgePredict[i])\n",
+ " print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+ " print(MSERidgePredict[i])\n",
+ "\n",
+ "\n",
+ "# Now plot the results\n",
+ "plt.figure()\n",
+ "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')\n",
+ "plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')\n",
+ "plt.xlabel('log10(lambda)')\n",
+ "plt.ylabel('MSE')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see here, when compared to the code which includes explicitely the\n",
+ "intercept column, that our MSE value is actually smaller. This is\n",
+ "because the regularization term does not include the intercept value\n",
+ "$\\beta_0$ in the fitting. This applies to Lasso regularization as\n",
+ "well. It means that our optimization is now done only with the\n",
+ "centered matrix and/or vector that enter the fitting procedure. Note\n",
+ "also that the problem with the intercept occurs mainly in these type\n",
+ "of polynomial fitting problem.\n",
+ "\n",
+ "The next example is indeed an example where all these discussions about the role of intercept are not present.\n",
+ "\n",
+ "## More complicated Example: The Ising model\n",
+ "\n",
+ "The one-dimensional Ising model with nearest neighbor interaction, no\n",
+ "external field and a constant coupling constant $J$ is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+ "\\label{_auto1} \\tag{1}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
+ "in the system is determined by $L$. For the one-dimensional system\n",
+ "there is no phase transition.\n",
+ "\n",
+ "We will look at a system of $L = 40$ spins with a coupling constant of\n",
+ "$J = 1$. To get enough training data we will generate 10000 states\n",
+ "with their respective energies."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+ "import seaborn as sns\n",
+ "import scipy.linalg as scl\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "import tqdm\n",
+ "sns.set(color_codes=True)\n",
+ "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+ "\n",
+ "L = 40\n",
+ "n = int(1e4)\n",
+ "\n",
+ "spins = np.random.choice([-1, 1], size=(n, L))\n",
+ "J = 1.0\n",
+ "\n",
+ "energies = np.zeros(n)\n",
+ "\n",
+ "for i in range(n):\n",
+ " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we use ordinary least squares\n",
+ "regression to predict the energy for the nearest neighbor\n",
+ "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n",
+ "around. We will use linear regression to fit a value for\n",
+ "the coupling constant to achieve this.\n",
+ "\n",
+ "A more general form for the one-dimensional Ising model is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+ "\\label{_auto2} \\tag{2}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
+ "coupling constant. This latter expression can be formulated as\n",
+ "a matrix-product"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{H} = \\boldsymbol{X} J,\n",
+ "\\label{_auto3} \\tag{3}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
+ "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+ "with the form utilized in linear regression, that is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n",
+ "\\label{_auto4} \\tag{4}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We split the data in training and test data as discussed in the previous example"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = np.zeros((n, L ** 2))\n",
+ "for i in range(n):\n",
+ " X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+ "y = energies\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In the ordinary least squares method we choose the cost function"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n",
+ "\\label{_auto5} \\tag{5}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
+ "This yields the expression for $\\boldsymbol{\\beta}$ to be"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
+ "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
+ "intercept, i.e., a constant term, we must make sure that the\n",
+ "first column of $\\boldsymbol{X}$ consists of $1$. We do this here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X_train_own = np.concatenate(\n",
+ " (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+ " axis=1\n",
+ ")\n",
+ "X_test_own = np.concatenate(\n",
+ " (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+ " axis=1\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Doing the inversion directly turns out to be a bad idea since the matrix\n",
+ "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
+ "value decomposition**. Using the definition of the Moore-Penrose\n",
+ "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Using singular value decomposition we can decompose the matrix $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
+ "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
+ "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n",
+ "$\\omega$ to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n",
+ "\\label{_auto6} \\tag{6}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Note that solving this equation by actually doing the pseudoinverse\n",
+ "(which is what we will do) is not a good idea as this operation scales\n",
+ "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n",
+ "general matrix. Instead, doing $QR$-factorization and solving the\n",
+ "linear system as an equation would reduce this down to\n",
+ "$\\mathcal{O}(n^2)$ operations."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
+ " u, s, v = scl.svd(x)\n",
+ " return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "beta = ols_svd(X_train_own,y_train)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "J = beta[1:].reshape(L, L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A way of looking at the coefficients in $J$ is to plot the matrices as images."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J, **cmap_args)\n",
+ "plt.title(\"OLS\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It is interesting to note that OLS\n",
+ "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
+ "valid matrix elements for $J$.\n",
+ "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n",
+ "this problem can be removed, partly and only with Lasso regression. \n",
+ "\n",
+ "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "Let us now \n",
+ "focus on Ridge and Lasso regression as well. We repeat some of the\n",
+ "basic parts of the Ising model and the setup of the training and test\n",
+ "data. The one-dimensional Ising model with nearest neighbor\n",
+ "interaction, no external field and a constant coupling constant $J$ is\n",
+ "given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+ "\\label{_auto7} \\tag{7}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n",
+ "\n",
+ "We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+ "import seaborn as sns\n",
+ "import scipy.linalg as scl\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "import sklearn.linear_model as skl\n",
+ "import tqdm\n",
+ "sns.set(color_codes=True)\n",
+ "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+ "\n",
+ "L = 40\n",
+ "n = int(1e4)\n",
+ "\n",
+ "spins = np.random.choice([-1, 1], size=(n, L))\n",
+ "J = 1.0\n",
+ "\n",
+ "energies = np.zeros(n)\n",
+ "\n",
+ "for i in range(n):\n",
+ " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A more general form for the one-dimensional Ising model is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+ "\\label{_auto8} \\tag{8}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we allow for interactions beyond the nearest neighbors and a more\n",
+ "adaptive coupling matrix. This latter expression can be formulated as\n",
+ "a matrix-product on the form"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = X J,\n",
+ "\\label{_auto9} \\tag{9}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n",
+ "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+ "with the form utilized in linear regression, viz."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon}.\n",
+ "\\label{_auto10} \\tag{10}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We organize the data as we did above"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = np.zeros((n, L ** 2))\n",
+ "for i in range(n):\n",
+ " X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+ "y = energies\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)\n",
+ "\n",
+ "X_train_own = np.concatenate(\n",
+ " (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+ " axis=1\n",
+ ")\n",
+ "\n",
+ "X_test_own = np.concatenate(\n",
+ " (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+ " axis=1\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We will do all fitting with **Scikit-Learn**,"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "clf = skl.LinearRegression().fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When extracting the $J$-matrix we make sure to remove the intercept"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "J_sk = clf.coef_.reshape(L, L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "And then we plot the results"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_sk, **cmap_args)\n",
+ "plt.title(\"LinearRegression from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The results agree perfectly with our previous discussion where we used our own code.\n",
+ "\n",
+ "\n",
+ "Having explored the ordinary least squares we move on to ridge\n",
+ "regression. In ridge regression we include a **regularizer**. This\n",
+ "involves a new cost function which leads to a new estimate for the\n",
+ "weights $\\boldsymbol{\\beta}$. This results in a penalized regression problem. The\n",
+ "cost function is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "6\n",
+ "0\n",
+ " \n",
+ "<\n",
+ "<\n",
+ "<\n",
+ "!\n",
+ "!\n",
+ "M\n",
+ "A\n",
+ "T\n",
+ "H\n",
+ "_\n",
+ "B\n",
+ "L\n",
+ "O\n",
+ "C\n",
+ "K"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "_lambda = 0.1\n",
+ "clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)\n",
+ "J_ridge_sk = clf_ridge.coef_.reshape(L, L)\n",
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_ridge_sk, **cmap_args)\n",
+ "plt.title(\"Ridge from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\sqrt{\\boldsymbol{\\beta}^T\\boldsymbol{\\beta}}.\n",
+ "\\label{_auto12} \\tag{12}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n",
+ "J_lasso_sk = clf_lasso.coef_.reshape(L, L)\n",
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_lasso_sk, **cmap_args)\n",
+ "plt.title(\"Lasso from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It is quite striking how LASSO breaks the symmetry of the coupling\n",
+ "constant as opposed to ridge and OLS. We get a sparse solution with\n",
+ "$J_{j, j + 1} = -1$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "We see how the different models perform for a different set of values for $\\lambda$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "lambdas = np.logspace(-4, 5, 10)\n",
+ "\n",
+ "train_errors = {\n",
+ " \"ols_sk\": np.zeros(lambdas.size),\n",
+ " \"ridge_sk\": np.zeros(lambdas.size),\n",
+ " \"lasso_sk\": np.zeros(lambdas.size)\n",
+ "}\n",
+ "\n",
+ "test_errors = {\n",
+ " \"ols_sk\": np.zeros(lambdas.size),\n",
+ " \"ridge_sk\": np.zeros(lambdas.size),\n",
+ " \"lasso_sk\": np.zeros(lambdas.size)\n",
+ "}\n",
+ "\n",
+ "plot_counter = 1\n",
+ "\n",
+ "fig = plt.figure(figsize=(32, 54))\n",
+ "\n",
+ "for i, _lambda in enumerate(tqdm.tqdm(lambdas)):\n",
+ " for key, method in zip(\n",
+ " [\"ols_sk\", \"ridge_sk\", \"lasso_sk\"],\n",
+ " [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]\n",
+ " ):\n",
+ " method = method.fit(X_train, y_train)\n",
+ "\n",
+ " train_errors[key][i] = method.score(X_train, y_train)\n",
+ " test_errors[key][i] = method.score(X_test, y_test)\n",
+ "\n",
+ " omega = method.coef_.reshape(L, L)\n",
+ "\n",
+ " plt.subplot(10, 5, plot_counter)\n",
+ " plt.imshow(omega, **cmap_args)\n",
+ " plt.title(r\"%s, $\\lambda = %.4f$\" % (key, _lambda))\n",
+ " plot_counter += 1\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see that LASSO reaches a good solution for low\n",
+ "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n",
+ "much. Ridge is more stable over a larger range of values for\n",
+ "$\\lambda$, but eventually also fades away.\n",
+ "\n",
+ "\n",
+ "To determine which value of $\\lambda$ is best we plot the accuracy of\n",
+ "the models when predicting the training and the testing set. We expect\n",
+ "the accuracy of the training set to be quite good, but if the accuracy\n",
+ "of the testing set is much lower this tells us that we might be\n",
+ "subject to an overfit model. The ideal scenario is an accuracy on the\n",
+ "testing set that is close to the accuracy of the training set."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "\n",
+ "colors = {\n",
+ " \"ols_sk\": \"r\",\n",
+ " \"ridge_sk\": \"y\",\n",
+ " \"lasso_sk\": \"c\"\n",
+ "}\n",
+ "\n",
+ "for key in train_errors:\n",
+ " plt.semilogx(\n",
+ " lambdas,\n",
+ " train_errors[key],\n",
+ " colors[key],\n",
+ " label=\"Train {0}\".format(key),\n",
+ " linewidth=4.0\n",
+ " )\n",
+ "\n",
+ "for key in test_errors:\n",
+ " plt.semilogx(\n",
+ " lambdas,\n",
+ " test_errors[key],\n",
+ " colors[key] + \"--\",\n",
+ " label=\"Test {0}\".format(key),\n",
+ " linewidth=4.0\n",
+ " )\n",
+ "plt.legend(loc=\"best\", fontsize=18)\n",
+ "plt.xlabel(r\"$\\lambda$\", fontsize=18)\n",
+ "plt.ylabel(r\"$R^2$\", fontsize=18)\n",
+ "plt.tick_params(labelsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n",
+ "achieves a very good accuracy on the test set. This by far surpasses the\n",
+ "other models for all values of $\\lambda$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## Exercises and Projects\n",
"\n",
"\n",
@@ -1980,6 +3578,7 @@
},
"outputs": [],
"source": [
+ "\"\"\"\n",
"import numpy as np\n",
"from imageio import imread\n",
"import matplotlib.pyplot as plt\n",
@@ -1994,7 +3593,8 @@
"plt.imshow(terrain1, cmap='gray')\n",
"plt.xlabel('X')\n",
"plt.ylabel('Y')\n",
- "plt.show()"
+ "plt.show()\n",
+ "\"\"\""
]
},
{
diff --git a/doc/LectureNotes/_build/html/_sources/schedule.md b/doc/LectureNotes/_build/html/_sources/schedule.md
index a33d7bf3e..8fb9c6ab3 100644
--- a/doc/LectureNotes/_build/html/_sources/schedule.md
+++ b/doc/LectureNotes/_build/html/_sources/schedule.md
@@ -78,6 +78,7 @@ For the reading assignments we use the following abbreviations:
### Week 38 September 20-24
- Lab Wednesday: Work on Project 1
- Lecture Thursday: Classification problems and Logistic Regression, from binary cases to several categories
+ - Video of Lecture at https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureSeptember23.mp4?vrtx=view-as-webpage
- Lecture Friday: Logistic Regression and gradient optimization
- Reading recommendations:
diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html
index 71542c6b8..43a2d40ef 100644
--- a/doc/LectureNotes/_build/html/chapter3.html
+++ b/doc/LectureNotes/_build/html/chapter3.html
@@ -321,39 +321,49 @@
5.5. Cross-validation
+
The bias-variance tradeoff summarizes the fundamental tension in
@@ -1367,84 +1373,82 @@ Mean squared error on training data: 3.66204648
Mean squared error on test data: 8.14812206
Degree of polynomial: 7
Mean squared error on training data: 0.47075725
+Mean squared error on test data: 2.00607783
-
Mean squared error on test data: 2.00607783
-Degree of polynomial: 8
+
Degree of polynomial: 8
Mean squared error on training data: 0.04912436
Mean squared error on test data: 0.21596432
Degree of polynomial: 9
Mean squared error on training data: 0.02522069
Mean squared error on test data: 0.08576932
-
-
-
Degree of polynomial: 10
+Degree of polynomial: 10
Mean squared error on training data: 0.02511518
Mean squared error on test data: 1.20015436
Degree of polynomial: 11
Mean squared error on training data: 0.01640891
Mean squared error on test data: 1.35533773
-Degree of polynomial: 12
-Mean squared error on training data: 0.00813803
-Mean squared error on test data: 0.17446471
-
Degree of polynomial: 13
+
Degree of polynomial: 12
+Mean squared error on training data: 0.00813803
+Mean squared error on test data: 0.17446471
+Degree of polynomial: 13
Mean squared error on training data: 0.00759119
Mean squared error on test data: 1.08131003
Degree of polynomial: 14
Mean squared error on training data: 0.00472199
Mean squared error on test data: 0.81333793
-Degree of polynomial: 15
-Mean squared error on training data: 0.00410478
-Mean squared error on test data: 92.09145189
-
Degree of polynomial: 16
+
Degree of polynomial: 15
+Mean squared error on training data: 0.00410478
+Mean squared error on test data: 92.09145189
+Degree of polynomial: 16
Mean squared error on training data: 0.00315593
Mean squared error on test data: 234.39716546
Degree of polynomial: 17
Mean squared error on training data: 0.00242998
Mean squared error on test data: 1271.05295709
-Degree of polynomial: 18
-Mean squared error on training data: 0.00228740
-Mean squared error on test data: 108.42208194
-
Degree of polynomial: 19
+
Degree of polynomial: 18
+Mean squared error on training data: 0.00228740
+Mean squared error on test data: 108.42208194
+Degree of polynomial: 19
Mean squared error on training data: 0.00156372
Mean squared error on test data: 1388.41078073
Degree of polynomial: 20
Mean squared error on training data: 0.00137982
Mean squared error on test data: 1761.43341615
-Degree of polynomial: 21
-Mean squared error on training data: 0.00118170
-Mean squared error on test data: 15061.31603087
-
Degree of polynomial: 22
+
Degree of polynomial: 21
+Mean squared error on training data: 0.00118170
+Mean squared error on test data: 15061.31603087
+Degree of polynomial: 22
Mean squared error on training data: 0.00092354
Mean squared error on test data: 890.63488525
Degree of polynomial: 23
Mean squared error on training data: 0.00085887
Mean squared error on test data: 5483.16796929
-Degree of polynomial: 24
-Mean squared error on training data: 0.00084589
-Mean squared error on test data: 1695.57143061
-
Degree of polynomial: 25
+
Degree of polynomial: 24
+Mean squared error on training data: 0.00084589
+Mean squared error on test data: 1695.57143061
+Degree of polynomial: 25
Mean squared error on training data: 0.00078806
Mean squared error on test data: 131343.30655001
Degree of polynomial: 26
Mean squared error on training data: 0.00076916
Mean squared error on test data: 17709.14370264
-Degree of polynomial: 27
-Mean squared error on training data: 0.00068970
-Mean squared error on test data: 2975.38903780
-
Degree of polynomial: 28
+
Degree of polynomial: 27
+Mean squared error on training data: 0.00068970
+Mean squared error on test data: 2975.38903780
+Degree of polynomial: 28
Mean squared error on training data: 0.00062588
Mean squared error on test data: 3848.64522721
Degree of polynomial: 29
@@ -1458,7 +1462,7 @@ Mean squared error on test data: 2988.64001211
plt.plot(polynomial, np.log10(testerror), label='Test Error')
We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.
+
When you are comparing your own code with for example Scikit-Learn’s
+library, there are some technicalities to keep in mind. The examples
+here demonstrate some of these aspects with potential pitfalls.
+
The discussion here focuses on the role of the intercept, how we can
+set up the design matrix, what scaling we should use and other topics
+which tend confuse us.
+
The intercept can be interpreted as the expected value of our
+target/output variables when all other predictors are set to zero.
+Thus, if we cannot assume that the expected outputs/targets are zero
+when all predictors are zero (the columns in the design matrix), it
+may be a bad idea to implement a model which penalizes the intercept.
+Furthermore, in for example Ridge and Lasso regression, the default solutions
+from the library Scikit-Learn (when not shrinking \(\beta_0\)) for the unknown parameters
+\(\boldsymbol{\beta}\), are derived under the assumption that both \(\boldsymbol{y}\) and
+\(\boldsymbol{X}\) are zero centered, that is we subtract the mean values.
+
If our predictors represent different scales, then it is important to
+standardize the design matrix \(\boldsymbol{X}\) by subtracting the mean of each
+column from the corresponding column and dividing the column with its
+standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
+the results may differ.
+
The
+Standadscaler
+function in Scikit-Learn does this for us. For the data sets we
+have been studying in our various examples, the data are in many cases
+already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
+survey of your data, with a critical assessment of them in case you need to scale the data.
+
If you need to scale the data, not doing so will give an unfair
+penalization of the parameters since their magnitude depends on the
+scale of their corresponding predictor.
+
Suppose as an example that you
+you have an input variable given by the heights of different persons.
+Human height might be measured in inches or meters or
+kilometers. If measured in kilometers, a standard linear regression
+model with this predictor would probably give a much bigger
+coefficient term, than if measured in millimeters.
+This can clearly lead to problems in evaluating the cost/loss functions.
+
Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
+on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows
+
+
+
"""
+#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+"""
+
+
+
+
+
'\n#Model training, we compute the mean value of y and X\ny_train_mean = np.mean(y_train)\nX_train_mean = np.mean(X_train,axis=0)\nX_train = X_train - X_train_mean\ny_train = y_train - y_train_mean\n\n# The we fit our model with the training data\ntrained_model = some_model.fit(X_train,y_train)\n\n\n#Model prediction, we need also to transform our data set used for the prediction.\nX_test = X_test - X_train_mean #Use mean from training data\ny_pred = trained_model(X_test)\ny_pred = y_pred + y_train_mean\n'
+
+
+
+
+
Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as zero centering. For
+simplicity, we will focus on ordinary regression, as done in the above example.
Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.
+
What we have done is to single out the \(\beta_0\) term in the definition of the mean squared error (MSE).
+The design matrix
+\(X\) does in this case not contain any intercept column.
+When we take the derivative with respect to \(\beta_0\), we want the derivative to obey
Replacing \(y_i\) with \(y_i - \beta_0 = y_i - \overline{\boldsymbol{y}}\) and centering also our design matrix results in a cost function (in vector-matrix disguise)
What does this mean? And why do we insist on all this? Let us look at some examples.
+
This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (code example thanks to Øyvind Sigmundson Schøyen). Here our scaling of the data is done by subtracting the mean values only.
+Note also that we do not split the data into training and test.
+
+
+
importnumpyasnp
+importmatplotlib.pyplotasplt
+
+fromsklearn.linear_modelimportLinearRegression
+
+
+np.random.seed(2021)
+
+defMSE(y_data,y_model):
+ n=np.size(y_model)
+ returnnp.sum((y_data-y_model)**2)/n
+
+
+deffit_beta(X,y):
+ returnnp.linalg.pinv(X.T@X)@X.T@y
+
+
+true_beta=[2,0.5,3.7]
+
+x=np.linspace(0,1,11)
+y=np.sum(
+ np.asarray([x**p*bforp,binenumerate(true_beta)]),axis=0
+)+0.1*np.random.normal(size=len(x))
+
+degree=3
+X=np.zeros((len(x),degree))
+
+# Include the intercept in the design matrix
+forpinrange(degree):
+ X[:,p]=x**p
+
+beta=fit_beta(X,y)
+
+# Intercept is included in the design matrix
+skl=LinearRegression(fit_intercept=False).fit(X,y)
+
+print(f"True beta: {true_beta}")
+print(f"Fitted beta: {beta}")
+print(f"Sklearn fitted beta: {skl.coef_}")
+ypredictOwn=X@beta
+ypredictSKL=skl.predict(X)
+print(f"MSE with intercept column")
+print(MSE(y,ypredictOwn))
+print(f"MSE with intercept column from SKL")
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x,y,label="Data")
+plt.plot(x,X@beta,label="Fit")
+plt.plot(x,skl.predict(X),label="Sklearn (fit_intercept=False)")
+
+
+# Do not include the intercept in the design matrix
+X=np.zeros((len(x),degree-1))
+
+forpinrange(degree-1):
+ X[:,p]=x**(p+1)
+
+# Intercept is not included in the design matrix
+skl=LinearRegression(fit_intercept=True).fit(X,y)
+
+# Use centered values for X and y when computing coefficients
+y_offset=np.average(y,axis=0)
+X_offset=np.average(X,axis=0)
+
+beta=fit_beta(X-X_offset,y-y_offset)
+intercept=np.mean(y_offset-X_offset@beta)
+
+print(f"Manual intercept: {intercept}")
+print(f"Fitted beta (wiothout intercept): {beta}")
+print(f"Sklearn intercept: {skl.intercept_}")
+print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
+ypredictOwn=X@beta
+ypredictSKL=skl.predict(X)
+print(f"MSE with Manual intercept")
+print(MSE(y,ypredictOwn+intercept))
+print(f"MSE with Sklearn intercept")
+print(MSE(y,ypredictSKL))
+
+plt.plot(x,X@beta+intercept,"--",label="Fit (manual intercept)")
+plt.plot(x,skl.predict(X),"--",label="Sklearn (fit_intercept=True)")
+plt.grid()
+plt.legend()
+
+plt.show()
+
The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the \(y\)-axis (for a one-dimensional case).
+
Printing the MSE, we see first that both methods give the same MSE, as
+they should. However, when we move to for example Ridge regression,
+the way we treat the intercept may give a larger or smaller MSE,
+meaning that the MSE can be penalized by the value of the
+intercept. Not including the intercept in the fit, means that the
+regularization term does not include \(\beta_0\). For different values
+of \(\lambda\), this may lead to differeing MSE values.
+
To remind the reader, the regularization term, with the intercept in Ridge regression, is given by
It means that, when scaling the design matrix and the outputs/targets,
+by subtracting the mean values, we have an optimization problem which
+is not penalized by the intercept. The MSE value can then be smaller
+since it focuses only on the remaining quantities. If we however bring
+back the intercept, we will get a MSE which then contains the
+intercept.
+
Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known vanilla data set.
+
+
+
importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimporttrain_test_split
+fromsklearnimportlinear_model
+
+defMSE(y_data,y_model):
+ n=np.size(y_model)
+ returnnp.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+n=100
+x=np.random.rand(n)
+y=np.exp(-x**2)+1.5*np.exp(-(x-2)**2)
+
+Maxpolydegree=20
+X=np.zeros((n,Maxpolydegree))
+#We include explicitely the intercept column
+fordegreeinrange(Maxpolydegree):
+ X[:,degree]=x**degree
+# We split the data in test and training data
+X_train,X_test,y_train,y_test=train_test_split(X,y,test_size=0.2)
+
+p=Maxpolydegree
+I=np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas=6
+MSEOwnRidgePredict=np.zeros(nlambdas)
+MSERidgePredict=np.zeros(nlambdas)
+lambdas=np.logspace(-4,2,nlambdas)
+foriinrange(nlambdas):
+ lmb=lambdas[i]
+ OwnRidgeBeta=np.linalg.pinv(X_train.T@X_train+lmb*I)@X_train.T@y_train
+ # Note: we include the intercept column and no scaling
+ RegRidge=linear_model.Ridge(lmb,fit_intercept=False)
+ RegRidge.fit(X_train,y_train)
+ # and then make the prediction
+ ytildeOwnRidge=X_train@OwnRidgeBeta
+ ypredictOwnRidge=X_test@OwnRidgeBeta
+ ytildeRidge=RegRidge.predict(X_train)
+ ypredictRidge=RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i]=MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i]=MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta)
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas),MSEOwnRidgePredict,'r',label='MSE own Ridge Test')
+plt.plot(np.log10(lambdas),MSERidgePredict,'g',label='MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
The results here agree when we force Scikit-Learn’s Ridge function to include the first column in our design matrix.
+We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
+What happens if we do not include the intercept in our fit?
+Let us see how we can change this code by zero centering.
+
+
+
importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimporttrain_test_split
+fromsklearnimportlinear_model
+fromsklearn.preprocessingimportStandardScaler
+
+defMSE(y_data,y_model):
+ n=np.size(y_model)
+ returnnp.sum((y_data-y_model)**2)/n
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(315)
+
+n=100
+x=np.random.rand(n)
+y=np.exp(-x**2)+1.5*np.exp(-(x-2)**2)
+
+Maxpolydegree=20
+X=np.zeros((n,Maxpolydegree-1))
+
+fordegreeinrange(1,Maxpolydegree):#No intercept column
+ X[:,degree-1]=x**(degree)
+
+# We split the data in test and training data
+X_train,X_test,y_train,y_test=train_test_split(X,y,test_size=0.2)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean=np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled=X_train-X_train_mean
+X_test_scaled=X_test-X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler=np.mean(y_train)
+y_train_scaled=y_train-y_scaler
+
+p=Maxpolydegree-1
+I=np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas=6
+MSEOwnRidgePredict=np.zeros(nlambdas)
+MSERidgePredict=np.zeros(nlambdas)
+
+lambdas=np.logspace(-4,2,nlambdas)
+foriinrange(nlambdas):
+ lmb=lambdas[i]
+ OwnRidgeBeta=np.linalg.pinv(X_train_scaled.T@X_train_scaled+lmb*I)@X_train_scaled.T@(y_train_scaled)
+ intercept_=y_scaler-X_train_mean@OwnRidgeBeta#The intercept can be shifted so the model can predict on uncentered data
+ #Add intercept to prediction
+ ypredictOwnRidge=X_test@OwnRidgeBeta+intercept_
+ #Add intercept to prediction
+ ypredictOwnRidge=X_test_scaled@OwnRidgeBeta+y_scaler
+ RegRidge=linear_model.Ridge(lmb)
+ RegRidge.fit(X_train,y_train)
+ ypredictRidge=RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i]=MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i]=MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta)#Intercept is given by mean of target variable
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print('Intercept from own implementation:')
+ print(intercept_)
+ print('Intercept from Scikit-Learn Ridge implementation')
+ print(RegRidge.intercept_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas),MSEOwnRidgePredict,'b--',label='MSE own Ridge Test')
+plt.plot(np.log10(lambdas),MSERidgePredict,'g--',label='MSE SL Ridge Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+
+
+
Beta values for own Ridge implementation
+[ 3.43579948e-02 -5.43330971e-01 -3.10141414e-03 2.47116868e-01
+ 2.18613217e-01 1.02054837e-01 -4.25617662e-04 -5.90475506e-02
+ -7.68534263e-02 -6.68929213e-02 -4.24906604e-02 -1.40927184e-02
+ 1.11482289e-02 2.88529063e-02 3.67047975e-02 3.38135733e-02
+ 2.02198702e-02 -3.46383924e-03 -3.63025821e-02]
+Beta values for Scikit-Learn Ridge implementation
+[ 3.43579948e-02 -5.43330971e-01 -3.10141413e-03 2.47116868e-01
+ 2.18613217e-01 1.02054837e-01 -4.25617658e-04 -5.90475506e-02
+ -7.68534263e-02 -6.68929213e-02 -4.24906604e-02 -1.40927184e-02
+ 1.11482289e-02 2.88529063e-02 3.67047975e-02 3.38135733e-02
+ 2.02198702e-02 -3.46383925e-03 -3.63025821e-02]
+Intercept from own implementation:
+1.0330308045181225
+Intercept from Scikit-Learn Ridge implementation
+1.033030804518383
+MSE values for own Ridge implementation
+3.139255958275475e-06
+MSE values for Scikit-Learn Ridge implementation
+3.139255958572018e-06
+Beta values for own Ridge implementation
+[-0.05807125 -0.29822833 -0.08551306 0.08156108 0.13679863 0.12333649
+ 0.08251519 0.03815288 0.00111756 -0.02498832 -0.04010697 -0.04566964
+ -0.04355837 -0.03562355 -0.02348765 -0.00848904 0.00831018 0.0260906
+ 0.04423486]
+Beta values for Scikit-Learn Ridge implementation
+[-0.05807125 -0.29822833 -0.08551306 0.08156108 0.13679863 0.12333649
+ 0.08251519 0.03815288 0.00111756 -0.02498832 -0.04010697 -0.04566964
+ -0.04355837 -0.03562355 -0.02348765 -0.00848904 0.00831018 0.0260906
+ 0.04423486]
+Intercept from own implementation:
+1.0411487294305548
+Intercept from Scikit-Learn Ridge implementation
+1.0411487294305266
+MSE values for own Ridge implementation
+1.9601304850163794e-05
+MSE values for Scikit-Learn Ridge implementation
+1.9601304850085328e-05
+Beta values for own Ridge implementation
+[-0.1416398 -0.14021063 -0.05383795 0.01367553 0.04784395 0.05796251
+ 0.05447415 0.044613 0.03267527 0.02098261 0.01066519 0.00217499
+ -0.00440346 -0.00917248 -0.01231917 -0.01405935 -0.0146081 -0.01416528
+ -0.01290947]
+Beta values for Scikit-Learn Ridge implementation
+[-0.1416398 -0.14021063 -0.05383795 0.01367553 0.04784395 0.05796251
+ 0.05447415 0.044613 0.03267527 0.02098261 0.01066519 0.00217499
+ -0.00440346 -0.00917248 -0.01231917 -0.01405935 -0.0146081 -0.01416528
+ -0.01290947]
+Intercept from own implementation:
+1.0495569966278282
+Intercept from Scikit-Learn Ridge implementation
+1.0495569966278269
+MSE values for own Ridge implementation
+5.4959161509370406e-05
+MSE values for Scikit-Learn Ridge implementation
+5.4959161509366834e-05
+Beta values for own Ridge implementation
+[-0.13535942 -0.08593216 -0.03568439 -0.0036367 0.01397146 0.02229529
+ 0.02503753 0.0245528 0.02228115 0.01908936 0.01549377 0.01179792
+ 0.00817631 0.00472512 0.00149311 -0.00149956 -0.00424967 -0.00676387
+ -0.00905423]
+Beta values for Scikit-Learn Ridge implementation
+[-0.13535942 -0.08593216 -0.03568439 -0.0036367 0.01397146 0.02229529
+ 0.02503753 0.0245528 0.02228115 0.01908936 0.01549377 0.01179792
+ 0.00817631 0.00472512 0.00149311 -0.00149956 -0.00424967 -0.00676387
+ -0.00905423]
+Intercept from own implementation:
+1.039967668952797
+Intercept from Scikit-Learn Ridge implementation
+1.0399676689527975
+MSE values for own Ridge implementation
+7.571105947979326e-05
+MSE values for Scikit-Learn Ridge implementation
+7.57110594797945e-05
+Beta values for own Ridge implementation
+[-0.05100875 -0.04063602 -0.02723445 -0.01713366 -0.0100706 -0.00517114
+ -0.00174276 0.00068734 0.00243186 0.00369758 0.00462287 0.0053018
+ 0.00579953 0.006162 0.00642221 0.00660427 0.00672607 0.0068011
+ 0.00683964]
+Beta values for Scikit-Learn Ridge implementation
+[-0.05100875 -0.04063602 -0.02723445 -0.01713366 -0.0100706 -0.00517114
+ -0.00174276 0.00068734 0.00243186 0.00369758 0.00462287 0.0053018
+ 0.00579953 0.006162 0.00642221 0.00660427 0.00672607 0.0068011
+ 0.00683964]
+Intercept from own implementation:
+0.999955585168597
+Intercept from Scikit-Learn Ridge implementation
+0.999955585168597
+MSE values for own Ridge implementation
+0.0007698473260556339
+MSE values for Scikit-Learn Ridge implementation
+0.000769847326055633
+Beta values for own Ridge implementation
+[-0.00834567 -0.00803064 -0.00673407 -0.00554552 -0.00458878 -0.0038335
+ -0.00323332 -0.00274989 -0.0023548 -0.00202756 -0.00175331 -0.00152117
+ -0.001323 -0.0011526 -0.00100519 -0.00087697 -0.00076495 -0.00066668
+ -0.00058016]
+Beta values for Scikit-Learn Ridge implementation
+[-0.00834567 -0.00803064 -0.00673407 -0.00554552 -0.00458878 -0.0038335
+ -0.00323332 -0.00274989 -0.0023548 -0.00202756 -0.00175331 -0.00152117
+ -0.001323 -0.0011526 -0.00100519 -0.00087697 -0.00076495 -0.00066668
+ -0.00058016]
+Intercept from own implementation:
+0.9637117593816477
+Intercept from Scikit-Learn Ridge implementation
+0.9637117593816477
+MSE values for own Ridge implementation
+0.0023813163025848865
+MSE values for Scikit-Learn Ridge implementation
+0.002381316302584886
+
+
+
+
+
+
We see here, when compared to the code which includes explicitely the
+intercept column, that our MSE value is actually smaller. This is
+because the regularization term does not include the intercept value
+\(\beta_0\) in the fitting. This applies to Lasso regularization as
+well. It means that our optimization is now done only with the
+centered matrix and/or vector that enter the fitting procedure. Note
+also that the problem with the intercept occurs mainly in these type
+of polynomial fitting problem.
+
The next example is indeed an example where all these discussions about the role of intercept are not present.
where \(s_i \in \{-1, 1\}\) and \(s_{N + 1} = s_1\). The number of spins
+in the system is determined by \(L\). For the one-dimensional system
+there is no phase transition.
+
We will look at a system of \(L = 40\) spins with a coupling constant of
+\(J = 1\). To get enough training data we will generate 10000 states
+with their respective energies.
Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
A more general form for the one-dimensional Ising model is
Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
where \(X_{jk} = s_j s_k\) and \(J\) is a matrix which consists of the
+elements \(-J_{jk}\). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
We then find the extremal point of \(C\) by taking the derivative with respect to \(\boldsymbol{\beta}\) as discussed above.
+This yields the expression for \(\boldsymbol{\beta}\) to be
which immediately imposes some requirements on \(\boldsymbol{X}\) as there must exist
+an inverse of \(\boldsymbol{X}^T \boldsymbol{X}\). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \(\boldsymbol{X}\) consists of \(1\). We do this here
Doing the inversion directly turns out to be a bad idea since the matrix
+\(\boldsymbol{X}^T\boldsymbol{X}\) is singular. An alternative approach is to use the singular
+value decomposition. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \(\boldsymbol{\beta}\) as
Using singular value decomposition we can decompose the matrix \(\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T\),
+where \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are orthogonal(unitary) matrices and \(\boldsymbol{\Sigma}\) contains the singular values (more details below).
+where \(X^{+} = V\Sigma^{+} U^T\). This reduces the equation for
+\(\omega\) to
Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \(\mathcal{O}(n^3)\), where \(n\) is the number of elements in a
+general matrix. Instead, doing \(QR\)-factorization and solving the
+linear system as an equation would reduce this down to
+\(\mathcal{O}(n^2)\) operations.
<ipython-input-22-6f7a6bd7d79f>:7: UserWarning: FixedFormatter should only be used together with FixedLocator
+ cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+
+
+
+
+
It is interesting to note that OLS
+considers both \(J_{j, j + 1} = -0.5\) and \(J_{j, j - 1} = -0.5\) as
+valid matrix elements for \(J\).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
Let us now
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data. The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant \(J\) is
+given by
where \(s_i \in \{-1, 1\}\) and \(s_{N + 1} = s_1\). The number of spins in the system is determined by \(L\). For the one-dimensional system there is no phase transition.
+
We will look at a system of \(L = 40\) spins with a coupling constant of \(J = 1\). To get enough training data we will generate 10000 states with their respective energies.
Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+
+
+
+\[
+\begin{equation}
+ H = X J,
+\label{_auto9} \tag{9}
+\end{equation}
+\]
+
where \(X_{jk} = s_j s_k\) and \(J\) is the matrix consisting of the
+elements \(-J_{jk}\). This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
<ipython-input-27-5dd54edf2138>:7: UserWarning: FixedFormatter should only be used together with FixedLocator
+ cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+
+
+
+
+
The results agree perfectly with our previous discussion where we used our own code.
+
Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a regularizer. This
+involves a new cost function which leads to a new estimate for the
+weights \(\boldsymbol{\beta}\). This results in a penalized regression problem. The
+cost function is given by
<ipython-input-28-fe5b9d300cc0>:10: UserWarning: FixedFormatter should only be used together with FixedLocator
+ cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+
+
+
+
+
In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function Lasso from Scikit-Learn.
<ipython-input-29-25845e8df859>:9: UserWarning: FixedFormatter should only be used together with FixedLocator
+ cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+
+
+
+
+
It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+\(J_{j, j + 1} = -1\).
+
We see how the different models perform for a different set of values for \(\lambda\).
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/linear_model/_coordinate_descent.py:529: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations. Duality gap: 3.924197515789051, tolerance: 1.796796
+ model = cd_fast.enet_coordinate_descent(
+
+ 10%|█ | 1/10 [00:00<00:04, 2.06it/s]
+
+
+
20%|██ | 2/10 [00:00<00:03, 2.39it/s]
+
+
+
30%|███ | 3/10 [00:00<00:02, 3.03it/s]
+
+
+
40%|████ | 4/10 [00:00<00:01, 3.75it/s]
+
+
+
50%|█████ | 5/10 [00:01<00:01, 4.51it/s]
+
+
+
60%|██████ | 6/10 [00:01<00:00, 5.17it/s]
+
+
+
70%|███████ | 7/10 [00:01<00:00, 5.94it/s]
+
+
+
80%|████████ | 8/10 [00:01<00:00, 6.59it/s]
+
+
+
90%|█████████ | 9/10 [00:01<00:00, 7.10it/s]
+
+
+
100%|██████████| 10/10 [00:01<00:00, 7.07it/s]
+
+
+
100%|██████████| 10/10 [00:01<00:00, 5.83it/s]
+
+
+
+
+
+
+
+
+
We see that LASSO reaches a good solution for low
+values of \(\lambda\), but will “wither” when we increase \(\lambda\) too
+much. Ridge is more stable over a larger range of values for
+\(\lambda\), but eventually also fades away.
+
To determine which value of \(\lambda\) is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
From the above figure we can see that LASSO with \(\lambda = 10^{-2}\)
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of \(\lambda\).
The main aim of this project is to study in more detail various
regression methods, including the Ordinary Least Squares (OLS) method,
The total score is 100 points. Each subtask has its own final score.
@@ -1825,11 +3097,11 @@ which polynomial fits the data best.
-
+
-
5.6.1. Exercise: Ordinary Least Square (OLS) on the Franke function¶
+
5.8.1. Exercise: Ordinary Least Square (OLS) on the Franke function¶
We will generate our own dataset for a function
\(\mathrm{FrankeFunction}(x,y)\) with \(x,y \in [0,1]\). The function
\(f(x,y)\) is the Franke function. You should explore also the addition
@@ -1874,7 +3146,7 @@ approximately \(2/3\) to You can easily reuse the solutions to your exercises from week 35 and week 36.
-
5.6.2. Exercise: Bias-variance trade-off and resampling techniques¶
+
5.8.2. Exercise: Bias-variance trade-off and resampling techniques¶
Our aim here is to study the bias-variance trade-off by implementing the bootstrap resampling technique.
With a code which does OLS and includes resampling techniques,
we will now discuss the bias-variance trade-off in the context of
@@ -1923,7 +3195,7 @@ of data points, and possibly also your training and test data using the
Note also that when you calculate the bias, in all applications you don’t know the function values \(f_i\). You would hence replace them with the actual data points \(y_i\).
-
5.6.3. Exercise: Cross-validation as resampling techniques, adding more complexity¶
+
5.8.3. Exercise: Cross-validation as resampling techniques, adding more complexity¶
The aim here is to write your own code for another widely popular
resampling technique, the so-called cross-validation method. Again,
before you start with cross-validation approach, you should scale your
@@ -1938,7 +3210,7 @@ folds. You can also compare your own cross-validation code with the
one provided by Scikit-Learn.
-
5.6.4. Exercise: Ridge Regression on the Franke function with resampling¶
+
5.8.4. Exercise: Ridge Regression on the Franke function with resampling¶
Write your own code for the Ridge method, either using matrix
inversion or the singular value decomposition as done in the previous
exercise. Perform the same bootstrap analysis as in the
@@ -1949,7 +3221,7 @@ dependence on \(\lambda\).
the parameter \(\lambda\). For the bias-variance trade-off, use the bootstrap resampling method. Comment your results.
-
5.6.5. Exercise: Lasso Regression on the Franke function with resampling¶
+
5.8.5. Exercise: Lasso Regression on the Franke function with resampling¶
This exercise is essentially a repeat of the previous two ones, but now
with Lasso regression. Write either your own code (difficult and optional) or, in this case,
you can also use the functionalities of Scikit-Learn (recommended).
@@ -1958,7 +3230,7 @@ critical discussion of the three methods and a judgement of which
model fits the data best. Perform here as well an analysis of the bias-variance trade-off using the bootstrap resampling technique and an analysis of the mean squared error using cross-validation.
With our codes functioning and having been tested properly on a
simpler function we are now ready to look at real data. We will
essentially repeat in this exercise what was done in exercises 1-5. However, we
@@ -1982,7 +3254,7 @@ Python program using
---------------------------------------------------------------------------NameErrorTraceback (most recent call last)
-<ipython-input-12-d985fb40c43d>in<module>
+<ipython-input-33-d985fb40c43d>in<module>----> 1scipy.misc.imreadNameError: name 'scipy' is not defined
@@ -1994,21 +3266,23 @@ Python program using
from such files
Lecture Friday: Logistic Regression and gradient optimization
Reading recommendations:
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 8d6889083..3510073c9 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
+++ b/doc/LectureNotes/_build/html/searchindex.js
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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
index f83b4276c..a5fdd1e14 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter3.ipynb
@@ -397,10 +397,10 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "Runtime: 0.13859 sec\n",
+ "Runtime: 0.139992 sec\n",
"Jackknife Statistics :\n",
"original bias std. error\n",
- " 99.9054 99.8954 0.149328\n"
+ " 99.9142 99.9042 0.148517\n"
]
}
],
@@ -774,7 +774,7 @@
"text": [
"Bootstrap Statistics :\n",
"original bias std. error\n",
- " 99.7442 15.0015 99.7465 0.148127\n"
+ " 100.028 14.8941 100.026 0.148225\n"
]
}
],
@@ -828,7 +828,7 @@
"outputs": [
{
"data": {
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\n",
+ "image/png": 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/A1R1ROBrp/DEMcYf778PJyVvomVCsesoEWdwah5zdvUmO/Uj11GMI7V1FvesaQtXSGPqa/Zsb3Uu81Plw0hN7KqtYfChGh5TYGAIsxjjm9mz4aUYn1aiOn2br2XT/jZ8eyiVTNdhjBO1NQ3Z5CMm+lSaM+eLA5ns2PAkPft/6ihQZEuIKyUn9UPmFvbChgLGphoLgYgMVNX3ROTCqh5X1ak1PPc5YASwTVW7V/F4Nt6w1M8Cu6aq6v1B5jYmaHMKezMoNZ84sUtoq1M+jNQKQWyqrWnoTOA9vGsHKlOg2kIAvAA8BrxUwzHvl3dIG+OX2YW9uSBjkesYEW1I2nLu+fxaysogzpagijm1NQ3dE/h6XV1fWFUXikjHI8xlTEiUlMXz3q6ePHn8I66jRLROTbbSolExH32Uxik2IWnMEQ1ixikRSQfuAU7DOxNYBNyvqjtreV5HYEYNTUNvAJuBb4DfqOqaal5nFDAKIDMzs9ekSZNqzey3oqIiUlJSXMeos5jIXVDw/c2PCo7m8X/1Y/ydb/qUrHZFaWmkFBY6e/9gPTqxPxmdm3L55d4UHDHxWYkgfufOycnJV9Wsqh4LthDMBRYCrwR2XQFkq+ovanleR6ovBM2BMlUtEpGzgX+oaufasmRlZWlenvtr2XJzc8nOznYdo85iIneFzuI7Nl2PoDx47LP+BAtC7siRZE+c6Oz9gzVjR1/+nv5ncnO9+zHxWYkgfucWkWoLQbCtga1V9U+q+llgewDqN9JMVfeqalHg9iwgQUQy6vOaxlRmq5EFLyd1Jfn5sHev6yQm3IItBO+KyGUiEhfYLgHqteKpiBwt4k3/JSJ9AllqbGoypi6+PZTKxv1t6Nt8resoUSE5/gD9+sF8W2Ek5tQ2fPQ7vD4BAf6HH5qG4oAi4Dc1PHcikA1kiMhmvD6GBABVfQq4CLhJRA4D+4HLNJh2KmOC9G5hFmelriAhrtR1lKgxbBjMmgUXXOA6iQmn2kYNNTvSF1bVkbU8/hje8FJjfDG7sI81C9XRsGHw0EPYqmUxJui5Z0UkFegMJJXvq7x8pTGRolTjeHdXFn8+9hnXUaJKly6QmAirV7tOYsIpqD4CEbkeb9TQHOC+wNd7/YtlTP2s+K4zRyXspn3SNtdRooqId1bwzjuuk5hwCrazeCzQG/giMP/QKcBuv0IZU1/WLHTkyvsJTOwIthAcUNUDACLSWFU/Abr4F8uY+rFCcORyciA/H4qL411HMWESbCHYLCItgWnAXBF5C/jCr1DG1MeukhQ+Lj6W01t87DpKVEpOhv79IT8/1XUUEyZBdRaravlgsntFZAHQApjtWypj6mHerl6c3uJjkuJLXEeJWsOGwbx5tix5rAh6nsHAqmS3ACcBm1X1kH+xjDlyXrOQLVJfH2efDUuXptsw0hgR7Kihu4EXgXQgA3heRP7oZzBjjoSqN+209Q/UT+fOkJhYxqpVrpOYcAj2jOAKoLeq3hOYmrovcJV/sYw5MmvWQOO4Ejo32ew6SlQTgT59Cm0YaYwIthB8Q4ULyYDGwNehj2NM/cycCcPSluHNYmXqo0+fnVYIYkSNhUBE/ikijwJ7gDUi8oKIPA+sxq4jMBHo7bfhnPQPXMdoEE45ZTf5+bBnj+skxm+1jRoqn/g/H6i4skeuL2mMqYcdO2DVKsjusdJ1lOhVYS2HpJEjGZCQxLwBM/nl6vschjJ+q23SuRfLb4tIInB84O56VbWxeSaizJoFAwdC0m77aIbKsPRlvLOzD790HcT4KthRQ9nAp8DjwBNAgYic4V8sY+ru7bfhnHNcp2hYzk5bwjuFfWwYaQMXbGfxQ8BgVT1TVc8AhgD/518sY+rm0CGYOxeGD3edpGHp3PRrmsQd4mO7SLtBC7YQJKjq+vI7qlpAYJEZYyLBwoVwwgmQWa8FVE1VhqUvtdFDDVywhSBfRCaISHZge4YfOpKNcc6ahfwzLG2ZFYIGLthCcCOwFrglsK0FbvIrlDF1oWqFwE85LT9k5UrYaSuKN1i1TjonIvHAR6p6AvCw/5GMCVJgqOPa4o6UfvNnTrxlpLe6tgmpJvGHOOss72K9q692ncb4odYzAlUtBdaLSPsw5DGmzt7e0Y8R6UvsamIfnX8+TJvmOoXxS7BrFqfiXVm8DCgu36mq5/qSypg6mLGzH3/s8LLrGA3a8PHnMWbJq+w74yKaxh/84YEFC9yFMiETbCG4y9cUxhyhHYeas6q4E9ktV7qO0qClJ+ylV7MC5u3qxbkZNoVHQ1NjIRCRJLyO4p8Bq4BnVfVwOIIZE4xZhX05K3WFLUITBudn/IdpO06zQtAA1dZH8CKQhVcEhuFdWGZMxHh7Zz/OSV/sOkZMOC/jP7y9sx+lGvR6ViZK1PY/2lVVr1TVp4GLgNPDkMmYoBwqa8TcwiyGpy9xHSUmdEj6lnaNt/HBnm6uo5gQq60QfH++bU1CJtL8e3cPfp78BUcl7nYdJWaUNw+ZhqW2QtBDRPYGtu+Ak8pvi8jecAQ0pjozdvZjhDULhdV5Gf9h2o4BNgldA1NjIVDVeFVtHtiaqWqjCrebhyukMZWpWv+ACyclb6SMOFYXd3IdxYSQ9fqYqLR2LZRqHCcmb3IdJaaIwPkZi6x5qIHxrRCIyHMisk1EVlfzuIjIoyKyQUQ+FpGefmUxDc9bb8G5GR/Y1cQOWD9Bw+PnGcELwNAaHh8GdA5so4AnfcxiGpgpU+DiVv92HSMmDWi+ii8PHsWXB45yHcWEiG+FQFUXAoU1HHIe8JJ6lgAtRaS1X3lMw7F+PWzbBgNaVHmyaXzWKK6MEemLeWvHANdRTIiI+tj9LyIdgRmq2r2Kx2YA41R1UeD+fOB3qvqTdQ5EZBTeWQOZmZm9Jk2a5FvmYBUVFZGSkuI6Rp01hNwvvdSBPXsSGDNkluNUwSlKSyOlsKa/iSJTTbkXrezA1Pe68/BTn4Y5Ve0awmfcDzk5OfmqmlXVY8HONeSUqo4HxgNkZWVpdna220BAbm4ukZCjrhpC7tGj4emnYcAfJ7oNFaTckSPJnhgdWSuqKXef0sb8bcPr9OiRTWpqmIPVoiF8xsPN5aihr4F2Fe63Dewzplpr1sCePdCvn+sksa1p/EEGpn7IzJmuk5hQcFkIpgNXB0YP9QX2qOoWh3lMFJgyBS6+GOJs4LNz52f8x9YoaCB8axoSkYlANpAhIpuBewgseK+qTwGzgLOBDcA+4Dq/spiGQdUrBC+84DqJARiRvpix86C4GJKTXacx9eFbIVDVkbU8rsDNfr2/aXhWrYL9+6FPH9dJDHhrFPTrB9Onw8gaf9pNpLMTbBM1pkyBSy7BLiKLIFdeCa+84jqFqa+oGDVkTHmz0GuvuU5iKjrvqWHcvPhfbB9wJa0S9/z4QVvGMmrYGYGJChs2pFBaCr16uU5iKkppdIAR6YuZvC3HdRRTD1YITFTIzW1lzUIR6orMeby67ReuY5h6sEJgIp4qLFhwFJdc4jqJqcqg1Dw27W/Nhn1tXEcxR8gKgYl4+fkQH6+cfLLrJKYqjeLKuPSoXF6zs4KoZZ3FJuJNvnAS2Scejwy83XUUU40rMudx9brfc1eHl6z5LgrZGYGJaKowZVs2Ob03uo5iatCn2TrKiCPvuy6uo5gjYIXARLSlS715bTq12eU6iqmBCFxx1Dxe/daah6KRFQIT0V57DS5ttcCaG6LAFZnzmLRtIIfL7NdKtLH/MROx9u/3CsF1rWe7jmKC0Lnp13RI+pb5u+1ij2hjhcBErDfegN69oUPSt66jmCBdkTmPV6x5KOpYITARa/x4GDXKdQpTF5cetYC3d/SnuDTJdRRTB1YITERatw4+/RRGjHCdxNRFZuIu+rVYY+sZRxkrBCYiPfMMXHcdJCS4TmLqykYPRR+7oMxEjhxv4rIDpQm8vGQKS3v+N+TYonXR5vyMRYz+dCxbtkDr1q7TmGDYGYGJOG/uOJ2TUzZwbBMrAtEopdEBLjvqPZ5+2nUSEywrBCbijN8yglGtZ7iOYephTNs3efppOHTIdRITDCsEJqIU7GvL2uIOnJfxH9dRTD10S/6crl3h9dddJzHBsEJgIsqELcO55uh3SYw77DqKqacxY+DRR12nMMGwQmAixqGyRry4dTDXt57pOooJgXPOga1bYfly10lMbawQmIjx1o4BdEv+nOObbnYdxYRAfDzcfDP885+uk5ja2PBREzHGbxnBDXY20KD8+tdw3HHw7beQmek6jamOnRGYiLBxI6ws+hkXZLzvOooJobQ0uPhib7oQE7msEJiI8MwzcFXmXJLiS1xHMSE2Zgw8+aQNJY1kVgiMc7t2eYVgzDFTXUcxPjjxROjSBabaf2/EskJgnHv0UTj3XOjUZKvrKMYnY8ZYp3Eks0JgnNq7Fx57DO64w3US46dzz4WvvoL8fNdJTFWsEBinHn8cBg+Gzp1dJzF+atTIhpJGMl8LgYgMFZH1IrJBRH5fxePXish2EVkZ2K73M4+JLMXF8MgjcOedrpOYcLj+enjrLe8iMxNZfLuOQETigceBQcBmYLmITFfVtZUOnayqo/3KYSLXU0/BGWdA166ukxhfBKYVL5cOXJN8Mw8+eJGdGUQYP88I+gAbVHWTqh4CJgHn+fh+Jors3w9//7udDcSaOzu8wmuvwaZNrpOYikRV/XlhkYuAoap6feD+VcCpFf/6F5FrgT8D24EC4FZV/aqK1xoFjALIzMzsNWnSJF8y10VRUREpKSmuY9RZROQuKGDqe93IX9eWB2+eE9RTitLSSCks9DmYP6I1u1+5X1w8iM2bm3LnnetC/toQIZ/xI+B37pycnHxVzarqMddTTLwNTFTVgyLyX8CLwMDKB6nqeGA8QFZWlmZnZ4c1ZFVyc3OJhBx1FQm5D97zv1y19Hymdrub3hPXB/Wc3JEjyZ440edk/ojW7H7l7jV9FJ07Q2pqJj16hPzlI+IzfiRc5vazaehroF2F+20D+76nqjtV9WDg7gSgl495TIR4YetQujX9nN7NgysCpmFpdm4OdzT7J3cMXOL1I5Rvxhk/C8FyoLOIdBKRROAyYHrFA0Sk4oqm5wL+nCuaiFFSAuO+HMldHV92HcU49F9t3mbtvg4s3H2S6ygGHwuBqh4GRgNz8H7BT1HVNSJyv4icGzjsFhFZIyIfAbcA1/qVx0SGV16BTklbGNBitesoxqHGcSXc3/F5/rDpBnzqpjR14Ot1BKo6S1WPV9XjVPXBwL67VXV64PYfVLWbqvZQ1RxV/cTPPMatoiK49164r+MLrqOYCHB55ny+K23C2zv7u44S8+zKYhM2d9/tNQWf3nKV6ygmAsRLGf/baQJ3bLqeUrVfRS7Zd9+ExYoV8Oqr3rUDxpQbnr6Elo2KeOXbQa6jxDQrBMZ3paUwahSMGwcZGa7TmEgiAuOOHc/dn13LwYO1H2/8YYXA+O7xxyE5Ga691nUSE4lOa7marGYF3H+/6ySxy/UFZaYhy8lh84EM7s+bwKJTxiADf3LRuDEAPHH8I5z83BkMHw79re847OyMwPjqlg1juPmYaZyQbEXAVC8zcRdPPglXX+2NLjPhZYXA+Gb6jv6sLu7EH9q/6jqKiQLnn+/NRnv77a6TxB5rGjK+KCqCMZ+O4fkT/moL0pvg5OTwyOGm9MibwMzFjzI8fYm3f8ECt7ligJ0RGF/88Y+Q3fIjBqZ+6DqKiSLNG+3jxRPGccP629l+qIXrODHDCoEJuYkTYdo0eOi4J1xHMVHojJYfc2XmPG4suM2mnwgTKwQmpJYtg1tugenTISNxr+s4Jkr9qdNzfLr/GF7+drDrKDHBCoEJma+/hgsvhAkT4CSbVNLUQ+O4El7++Z+5feNNtppZGFghMCGxbx+cdx6MHu19Naa+eqRs5P6OzzNkiC147zcrBKbeysq8q4a7doXf/c51GtOQ3HTMdK66CoYMgV27XKdpuGz4qKm3+499ga929WZBj1uRgTZU1ITWXXfB7t0wfDi8+y5E4XLEEc/OCEy9TJkCz28dypvd7rLrBYwvROChh+CEE7w+KJucLvSsEJgj9sorXp/AtO53cXRjO283/hGB8eOheXO4/HI4fNh1oobFCoGpM1V44AG480547z04pdkG15FMDGjUyFvToqgIbrjB65syoWF9BKZOSkrgppvgww9h8WJo08Z1ItPg5eR8f7MxMLU0icFT/8aVB7szfrz1GYSCnRGYoO09fTgjMpez5c0l/LvpMNpckfOjH1JjwiE5/gBze/yGpCTo0wfWrnWdKPpZITBB2bwZTv/wUY5t8g1vdb+TlEYHXEcyMaxp/EGe+yyH/1f2F848eTev/PxB748S+8PkiFghMDVShX/9C049Fa7MnMsTnR+hUZw1zprIcF3r2czr8Rvu/+Jqblx/KwdKE1xHikpWCEy1Cgpg6FC4/36YNAn+X/vJiLhOZcyP9UjZSF6vG9l5uDkDPnyML79s4jpS1LFCYH5i3z7vIp7+/WHwYFixAk4/3XUqY6rXvNE+pnS9j+uOfocxY3oyapTXnGmCY4XAfE/VmzW0WzfvbOCjj7zVohLsbNtEAREY3XYaL7+8lLQ06NHD+/zu2OE6WeSz4aOGXbvg5Zfhqacg/stNjD/uCQZty4crXSczpu6aNz/MuHEwdiw8+KB3RfLo0XDrrdDC1rqpkp0RxChVb+2AX/0Kjj0WlizxCsHHWb9mUFq+63jGHLmCAsjJofXlOTy2JoflPxvJZ0/NoX17uOgieOMNOGCD3n7ECkEMKSmBVatacN990Kt5AZed+Q1d/v0067tewGtbcjjjnhzrDDYNTqcmW3nx5+P47DNv8MMTT0Dr1nDNNTB7tvdzEeusaagBKy2FNWtg/nxve/99yMz8GeefD3877mlyWn5InNhagCY2pP0yh+uB64EtP09jygc53PvmQC6VrvTuDf36eVvfvpCW5jptePlaCERkKPAPIB6YoKrjKj3eGHgJ6AXsBC5V1c/9zNQQHTwIX5xxFWuKO7J2XwfWFHdibXEHCva35ZjGOzir5QquSV3BC90/ZPU1w8meOBFSXac2xp3WjQsZ2/YNxrZ9g50lzVmyrSuLX+rKQ491Y/l3XWiTuJOsC9vTuTMcd9wPW6tWNMizZt8KgYjEA48Dg4DNwHIRma6qFS8I/zWwS1V/JiKXAX8BLvUrUyQrLYX9+722ywMHvIm19uz56bZjB2zZAt9888PXoiJoGzeOrslf0K3p5wxNW8Ztbafw8+QvSY63xlBjapKesJfh6UsYnr4EgFKNY3VxJ1acNYGNG2HmTNi40dtKSqBj2UaOSthNq8TdtErYQ6uE3bRK2E3G/95Gs2be3EcpKXx/OzkZkpIgPt7xP7QGfp4R9AE2qOomABGZBJwHVCwE5wH3Bm6/DjwmIqKqIW+v2LPHm762KtW9W/l+1Z/eLizsQcuW3gyI5ftUvV/oZWXeVn67tNSbNvfwYe+DVFLyw+1Dh7wCUFoKTZp4H5gmTbwPT4sWP93Sp03gzMSdtGm8k9ZJO2lzwg7SEr6zJh5jQiReyuiRspEeL1WYriIZOAl2lyTz+YGj2VaSyvaSlmw/1ILtJS1ZUXQ8O17z/iirajtwwDuTaNz4hy0x0RuaHR/vzax64EAWLVp49+ML1hEnZcShP/r6lzmn0KdP6P/N4sPvXO+FRS4Chqrq9YH7VwGnquroCsesDhyzOXB/Y+CYHZVeaxQwKnC3C7Del9B1kwFE4whlyx1+0ZrdcoeX37k7qGqrqh6Iis5iVR0PjHedoyIRyVPVLNc56spyh1+0Zrfc4eUyt5/DR78G2lW43zawr8pjRKQR0AKv09gYY0yY+FkIlgOdRaSTiCQClwHTKx0zHbgmcPsi4D0/+geMMcZUz7emIVU9LCKjgTl4w0efU9U1InI/kKeq04FngZdFZANQiFcsokVENVXVgeUOv2jNbrnDy1lu3zqLjTHGRAebYsIYY2KcFQJjjIlxVggqEZGxIrJaRNaIyP8E9vUQkcUiskpE3haR5jU8P15EPhSRGWELTf1yi0hLEXldRD4RkXUi0i+Kst8aeN5qEZkoIkk+5nxORLYFrn8p35cmInNF5NPA19TAfhGRR0Vkg4h8LCI9q3nNXoF/44bA8b5MYBDq7CLSVERmBj4za0RkXOVjIjF3pdeeXvF1Iz23iCSKyHgRKQh8338ZssCqaltgA7oDq4GmeB3p84Cf4Y2AOjNwzK+AP9XwGrcBrwEzoiU38CJwfeB2ItAyGrIDxwCfAU0C96cA1/qY9QygJ7C6wr6/Ar8P3P498JfA7bOBdwAB+gJLq3nNZYHHJXD8sGjIHvj/yqnwmXnfj+x+fM8Dx14Y+DldHerMPn5W7gMeCNyOAzJCltePb0K0bsDFwLMV7t8F/BbYww8d6+2AtdU8vy0wHxhIeAvBEefGu3bjs/Ljoul7jlcIvgLS8IrIDGCwz3k7VvrhXg+0DtxuDawP3H4aGFnVcRX2tQY+qXB/JPB0NGSv4rX/AdwQDbmBFGAR0BWfCoFPub8Ckv3Iak1DP7YaOF1E0kWkKV6lbgeswZsXCbxfXO2qef4jeL/EynzOWVl9cncCtgPPB5q0JohIcjhCBxxxdlX9Gvg78CWwBdijqu+GJfUPMlV1S+D2ViAzcLu8SJXbHNhX0TGB/TUd46f6ZP+eiLQEzsH7Iygc6pv7T8BDwD7fElbtiHMHvscAfxKRFSLyLxHJJESsEFSgquvwZkB9F5gNrARK8Zom/ltE8oFmwKHKzxWREcA2VQ378l71yY33l3RP4ElVPQUoxjttDYt6fs9T8YpFJ6ANkCwizhbYVO/Ptqgcj32k2cWbEWAi8KgGJpgMp7rmFpGTgeNU9U3fQgXhCL7fjfBaHD5Q1Z7AYrw/gkLCCkElqvqsqvZS1TOAXUCBqn6iqoNVtRfeh35jFU8dAJwrIp8Dk4CBIvJKFOTeDGxW1aWB+6/jFYawqUf2XwCfqep2VS0BpgL9w5ccgG9FpDVA4Ou2wP5gp1hpW8sxfqpP9nLjgU9V9RG/QlahPrn7AVmBn9NFwPEikutr2h/UJ/dOvDOYqYH7/yKEP6dWCCoRkaMCX9sT6FCqsC8O+CPwVOXnqeofVLWtqnbEu0L6PVUN21+n9ci9FfhKRLoEdp3Fj6cK992RZsdrEuobGMEieNnXhSf19ypOk3IN8FaF/VcHRoT0xWu22lLxiYH7e0WkbyD/1RWeHw5HnB1ARB7A62P6nzBkrag+3/MnVbVN4Of0NLw/OrLDE7teuRV4GyjPGtqfU786SqJ1wxv9sBb4CDgrsG8sUBDYxvFDJ2YbYFYVr5FNGDuL65sbOBnIAz4GpgGpUZT9PuATvL6Gl4HGPuaciNcXUYJ3JvVrIB2vbfxTvBFPaYFjBW9hpo3AKiCrwuusrHA7K5B9I/AYPnXahzo73l+tild4Vwa26yM9d6XX7oh/o4b8+Kx0ABYGfk7nA+1DldemmDDGmBhnTUPGGBPjrBAYY0yMs0JgjDExzgqBMcbEOCsExhgT46wQGGNMjLNCYIwxMc4KgTH1JCK9A/PIJ4lIcmB+/u6ucxkTLLugzJgQCEy3kAQ0wZu76c+OIxkTNCsExoSAiCTiLaZzAOivqqWOIxkTNGsaMiY00vEWPGmGd2ZgTNSwMwJjQkBEpuNNP94Jb3Wp0Y4jGRO0Rq4DGBPtRORqoERVXxOReOADERmoqu+5zmZMMOyMwBhjYpz1ERhjTIyzQmCMMTHOCoExxsQ4KwTGGBPjrBAYY0yMs0JgjDExzgqBMcbEuP8PZeN2FLzTAvAAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
@@ -1088,18 +1088,17 @@
"Bias^2: 0.3123314713548606\n",
"Var: 0.009164545680330616\n",
"0.32149601703519126 >= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912\n",
- "Polynomial degree:"
+ "Polynomial degree: 1\n",
+ "Error: 0.08426840630693411\n",
+ "Bias^2: 0.07968918676726028\n",
+ "Var: 0.004579219539673833\n",
+ "0.08426840630693411 >= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
- " 1\n",
- "Error: 0.08426840630693411\n",
- "Bias^2: 0.07968918676726028\n",
- "Var: 0.004579219539673833\n",
- "0.08426840630693411 >= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411\n",
"Polynomial degree: 2\n",
"Error: 0.10398646080125035\n",
"Bias^2: 0.10077114273548986\n",
@@ -1130,18 +1129,18 @@
"Error: 0.03781367141738898\n",
"Bias^2: 0.03365768507152761\n",
"Var: 0.004155986345861379\n",
- "0.03781367141738898 >= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899\n",
- "Polynomial degree: 7\n",
- "Error: 0.027609773491022498\n",
- "Bias^2: 0.02299949826036597\n",
- "Var: 0.004610275230656537\n",
- "0.027609773491022498 >= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505\n"
+ "0.03781367141738898 >= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Polynomial degree: 7\n",
+ "Error: 0.027609773491022498\n",
+ "Bias^2: 0.02299949826036597\n",
+ "Var: 0.004610275230656537\n",
+ "0.027609773491022498 >= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505\n",
"Polynomial degree: 8\n",
"Error: 0.017355848195591973\n",
"Bias^2: 0.010331721306655588\n",
@@ -1173,14 +1172,7 @@
"Bias^2: 0.01628578269590588\n",
"Var: 0.09919198949386163\n",
"0.1154777721897675 >= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751\n",
- "Polynomial degree:"
- ]
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- " 13\n",
+ "Polynomial degree: 13\n",
"Error: 0.22842468702166951\n",
"Bias^2: 0.01975416527163567\n",
"Var: 0.20867052175003387\n",
@@ -1196,7 +1188,7 @@
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@@ -1450,117 +1442,110 @@
"Mean squared error on training data: 3.66204648\n",
"Mean squared error on test data: 8.14812206\n",
"Degree of polynomial: 7\n",
- "Mean squared error on training data: 0.47075725"
+ "Mean squared error on training data: 0.47075725\n",
+ "Mean squared error on test data: 2.00607783\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
- "\n",
- "Mean squared error on test data: 2.00607783\n",
"Degree of polynomial: 8\n",
"Mean squared error on training data: 0.04912436\n",
"Mean squared error on test data: 0.21596432\n",
"Degree of polynomial: 9\n",
"Mean squared error on training data: 0.02522069\n",
- "Mean squared error on test data: 0.08576932\n"
- ]
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
+ "Mean squared error on test data: 0.08576932\n",
"Degree of polynomial: 10\n",
"Mean squared error on training data: 0.02511518\n",
"Mean squared error on test data: 1.20015436\n",
"Degree of polynomial: 11\n",
"Mean squared error on training data: 0.01640891\n",
- "Mean squared error on test data: 1.35533773\n",
- "Degree of polynomial: 12\n",
- "Mean squared error on training data: 0.00813803\n",
- "Mean squared error on test data: 0.17446471\n"
+ "Mean squared error on test data: 1.35533773\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Degree of polynomial: 12\n",
+ "Mean squared error on training data: 0.00813803\n",
+ "Mean squared error on test data: 0.17446471\n",
"Degree of polynomial: 13\n",
"Mean squared error on training data: 0.00759119\n",
"Mean squared error on test data: 1.08131003\n",
"Degree of polynomial: 14\n",
"Mean squared error on training data: 0.00472199\n",
- "Mean squared error on test data: 0.81333793\n",
- "Degree of polynomial: 15\n",
- "Mean squared error on training data: 0.00410478\n",
- "Mean squared error on test data: 92.09145189\n"
+ "Mean squared error on test data: 0.81333793\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Degree of polynomial: 15\n",
+ "Mean squared error on training data: 0.00410478\n",
+ "Mean squared error on test data: 92.09145189\n",
"Degree of polynomial: 16\n",
"Mean squared error on training data: 0.00315593\n",
"Mean squared error on test data: 234.39716546\n",
"Degree of polynomial: 17\n",
"Mean squared error on training data: 0.00242998\n",
- "Mean squared error on test data: 1271.05295709\n",
- "Degree of polynomial: 18\n",
- "Mean squared error on training data: 0.00228740\n",
- "Mean squared error on test data: 108.42208194\n"
+ "Mean squared error on test data: 1271.05295709\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Degree of polynomial: 18\n",
+ "Mean squared error on training data: 0.00228740\n",
+ "Mean squared error on test data: 108.42208194\n",
"Degree of polynomial: 19\n",
"Mean squared error on training data: 0.00156372\n",
"Mean squared error on test data: 1388.41078073\n",
"Degree of polynomial: 20\n",
"Mean squared error on training data: 0.00137982\n",
- "Mean squared error on test data: 1761.43341615\n",
- "Degree of polynomial: 21\n",
- "Mean squared error on training data: 0.00118170\n",
- "Mean squared error on test data: 15061.31603087\n"
+ "Mean squared error on test data: 1761.43341615\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Degree of polynomial: 21\n",
+ "Mean squared error on training data: 0.00118170\n",
+ "Mean squared error on test data: 15061.31603087\n",
"Degree of polynomial: 22\n",
"Mean squared error on training data: 0.00092354\n",
"Mean squared error on test data: 890.63488525\n",
"Degree of polynomial: 23\n",
"Mean squared error on training data: 0.00085887\n",
- "Mean squared error on test data: 5483.16796929\n",
- "Degree of polynomial: 24\n",
- "Mean squared error on training data: 0.00084589\n",
- "Mean squared error on test data: 1695.57143061\n"
+ "Mean squared error on test data: 5483.16796929\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Degree of polynomial: 24\n",
+ "Mean squared error on training data: 0.00084589\n",
+ "Mean squared error on test data: 1695.57143061\n",
"Degree of polynomial: 25\n",
"Mean squared error on training data: 0.00078806\n",
"Mean squared error on test data: 131343.30655001\n",
"Degree of polynomial: 26\n",
"Mean squared error on training data: 0.00076916\n",
- "Mean squared error on test data: 17709.14370264\n",
- "Degree of polynomial: 27\n",
- "Mean squared error on training data: 0.00068970\n",
- "Mean squared error on test data: 2975.38903780\n"
+ "Mean squared error on test data: 17709.14370264\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
+ "Degree of polynomial: 27\n",
+ "Mean squared error on training data: 0.00068970\n",
+ "Mean squared error on test data: 2975.38903780\n",
"Degree of polynomial: 28\n",
"Mean squared error on training data: 0.00062588\n",
"Mean squared error on test data: 3848.64522721\n",
@@ -1588,7 +1573,7 @@
},
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@@ -2057,6 +2042,2121 @@
"cell_type": "markdown",
"metadata": {},
"source": [
+ "## More on Rescaling data\n",
+ "\n",
+ "We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.\n",
+ "\n",
+ "When you are comparing your own code with for example **Scikit-Learn**'s\n",
+ "library, there are some technicalities to keep in mind. The examples\n",
+ "here demonstrate some of these aspects with potential pitfalls.\n",
+ "\n",
+ "The discussion here focuses on the role of the intercept, how we can\n",
+ "set up the design matrix, what scaling we should use and other topics\n",
+ "which tend confuse us.\n",
+ "\n",
+ "The intercept can be interpreted as the expected value of our\n",
+ "target/output variables when all other predictors are set to zero.\n",
+ "Thus, if we cannot assume that the expected outputs/targets are zero\n",
+ "when all predictors are zero (the columns in the design matrix), it\n",
+ "may be a bad idea to implement a model which penalizes the intercept.\n",
+ "Furthermore, in for example Ridge and Lasso regression, the default solutions\n",
+ "from the library **Scikit-Learn** (when not shrinking $\\beta_0$) for the unknown parameters\n",
+ "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
+ "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
+ "\n",
+ "\n",
+ "If our predictors represent different scales, then it is important to\n",
+ "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
+ "column from the corresponding column and dividing the column with its\n",
+ "standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,\n",
+ "the results may differ. \n",
+ "\n",
+ "The\n",
+ "[Standadscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)\n",
+ "function in **Scikit-Learn** does this for us. For the data sets we\n",
+ "have been studying in our various examples, the data are in many cases\n",
+ "already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a\n",
+ "survey of your data, with a critical assessment of them in case you need to scale the data.\n",
+ "\n",
+ "If you need to scale the data, not doing so will give an *unfair*\n",
+ "penalization of the parameters since their magnitude depends on the\n",
+ "scale of their corresponding predictor.\n",
+ "\n",
+ "Suppose as an example that you \n",
+ "you have an input variable given by the heights of different persons.\n",
+ "Human height might be measured in inches or meters or\n",
+ "kilometers. If measured in kilometers, a standard linear regression\n",
+ "model with this predictor would probably give a much bigger\n",
+ "coefficient term, than if measured in millimeters.\n",
+ "This can clearly lead to problems in evaluating the cost/loss functions.\n",
+ "\n",
+ "\n",
+ "\n",
+ "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
+ "on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "'\\n#Model training, we compute the mean value of y and X\\ny_train_mean = np.mean(y_train)\\nX_train_mean = np.mean(X_train,axis=0)\\nX_train = X_train - X_train_mean\\ny_train = y_train - y_train_mean\\n\\n# The we fit our model with the training data\\ntrained_model = some_model.fit(X_train,y_train)\\n\\n\\n#Model prediction, we need also to transform our data set used for the prediction.\\nX_test = X_test - X_train_mean #Use mean from training data\\ny_pred = trained_model(X_test)\\ny_pred = y_pred + y_train_mean\\n'"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "\"\"\"\n",
+ "#Model training, we compute the mean value of y and X\n",
+ "y_train_mean = np.mean(y_train)\n",
+ "X_train_mean = np.mean(X_train,axis=0)\n",
+ "X_train = X_train - X_train_mean\n",
+ "y_train = y_train - y_train_mean\n",
+ "\n",
+ "# The we fit our model with the training data\n",
+ "trained_model = some_model.fit(X_train,y_train)\n",
+ "\n",
+ "\n",
+ "#Model prediction, we need also to transform our data set used for the prediction.\n",
+ "X_test = X_test - X_train_mean #Use mean from training data\n",
+ "y_pred = trained_model(X_test)\n",
+ "y_pred = y_pred + y_train_mean\n",
+ "\"\"\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let us try to understand what this may imply mathematically when we\n",
+ "subtract the mean values, also known as *zero centering*. For\n",
+ "simplicity, we will focus on ordinary regression, as done in the above example.\n",
+ "\n",
+ "The cost/loss function for regression is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.\n",
+ "\n",
+ "What we have done is to single out the $\\beta_0$ term in the definition of the mean squared error (MSE).\n",
+ "The design matrix\n",
+ "$X$ does in this case not contain any intercept column.\n",
+ "When we take the derivative with respect to $\\beta_0$, we want the derivative to obey"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "for all $j$. For $\\beta_0$ we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Multiplying away the constant $2/n$, we obtain"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We assume \n",
+ "that every column of $\\boldsymbol{X}$ is centered, which we can do by subtracting the mean,"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = X - np.mean(X,axis=0)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This means that we need to rewrite $X_{ij}$ as $\\tilde{X}_{ij}=X_{ij}-\\mu_j$, where"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_j = \\frac{1}{n}\\sum_{i=0}^{n-1}X_{ij}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let us special first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
+ "Our result for $\\beta_0$ simplifies then to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Assuming that the matrix elements $X_{i1}$ are centered, what we have is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} \\left(X_{i1}-\\mu_{1}\\right),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_1=\\frac{1}{n}\\sum_{i=0}^{n-1} (X_{i1},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "and if we define the mean value of the outputs as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\mu_y - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} (X_{i1}-\\mu_{1}),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "and it is easy to see that the last sum equals zero! This means that we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\mu_y,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of $\\beta$.\n",
+ "We have thus"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1} y_i = \\overline{\\boldsymbol{y}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "the average value of $\\boldsymbol{y}$.\n",
+ "\n",
+ "Replacing $y_i$ with $y_i - \\beta_0 = y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
+ "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
+ "\n",
+ "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
+ "\n",
+ "\n",
+ "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
+ "Note also that we do not split the data into training and test."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "True beta: [2, 0.5, 3.7]\n",
+ "Fitted beta: [2.08376632 0.19569961 3.97898392]\n",
+ "Sklearn fitted beta: [2.08376632 0.19569961 3.97898392]\n",
+ "MSE with intercept column\n",
+ "0.004113634617443137\n",
+ "MSE with intercept column from SKL\n",
+ "0.0041136346174431284\n",
+ "Manual intercept: 2.083766322923905\n",
+ "Fitted beta (wiothout intercept): [0.19569961 3.97898392]\n",
+ "Sklearn intercept: 2.0837663229239025\n",
+ "Sklearn fitted beta (without intercept): [0.19569961 3.97898392]\n",
+ "MSE with Manual intercept\n",
+ "0.004113634617443136\n",
+ "MSE with Sklearn intercept\n",
+ "0.004113634617443135\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_110_1.png"
+ },
+ "needs_background": "light"
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "from sklearn.linear_model import LinearRegression\n",
+ "\n",
+ "\n",
+ "np.random.seed(2021)\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "\n",
+ "\n",
+ "def fit_beta(X, y):\n",
+ " return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
+ "\n",
+ "\n",
+ "true_beta = [2, 0.5, 3.7]\n",
+ "\n",
+ "x = np.linspace(0, 1, 11)\n",
+ "y = np.sum(\n",
+ " np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0\n",
+ ") + 0.1 * np.random.normal(size=len(x))\n",
+ "\n",
+ "degree = 3\n",
+ "X = np.zeros((len(x), degree))\n",
+ "\n",
+ "# Include the intercept in the design matrix\n",
+ "for p in range(degree):\n",
+ " X[:, p] = x ** p\n",
+ "\n",
+ "beta = fit_beta(X, y)\n",
+ "\n",
+ "# Intercept is included in the design matrix\n",
+ "skl = LinearRegression(fit_intercept=False).fit(X, y)\n",
+ "\n",
+ "print(f\"True beta: {true_beta}\")\n",
+ "print(f\"Fitted beta: {beta}\")\n",
+ "print(f\"Sklearn fitted beta: {skl.coef_}\")\n",
+ "ypredictOwn = X @ beta\n",
+ "ypredictSKL = skl.predict(X)\n",
+ "print(f\"MSE with intercept column\")\n",
+ "print(MSE(y,ypredictOwn))\n",
+ "print(f\"MSE with intercept column from SKL\")\n",
+ "print(MSE(y,ypredictSKL))\n",
+ "\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.scatter(x, y, label=\"Data\")\n",
+ "plt.plot(x, X @ beta, label=\"Fit\")\n",
+ "plt.plot(x, skl.predict(X), label=\"Sklearn (fit_intercept=False)\")\n",
+ "\n",
+ "\n",
+ "# Do not include the intercept in the design matrix\n",
+ "X = np.zeros((len(x), degree - 1))\n",
+ "\n",
+ "for p in range(degree - 1):\n",
+ " X[:, p] = x ** (p + 1)\n",
+ "\n",
+ "# Intercept is not included in the design matrix\n",
+ "skl = LinearRegression(fit_intercept=True).fit(X, y)\n",
+ "\n",
+ "# Use centered values for X and y when computing coefficients\n",
+ "y_offset = np.average(y, axis=0)\n",
+ "X_offset = np.average(X, axis=0)\n",
+ "\n",
+ "beta = fit_beta(X - X_offset, y - y_offset)\n",
+ "intercept = np.mean(y_offset - X_offset @ beta)\n",
+ "\n",
+ "print(f\"Manual intercept: {intercept}\")\n",
+ "print(f\"Fitted beta (wiothout intercept): {beta}\")\n",
+ "print(f\"Sklearn intercept: {skl.intercept_}\")\n",
+ "print(f\"Sklearn fitted beta (without intercept): {skl.coef_}\")\n",
+ "ypredictOwn = X @ beta\n",
+ "ypredictSKL = skl.predict(X)\n",
+ "print(f\"MSE with Manual intercept\")\n",
+ "print(MSE(y,ypredictOwn+intercept))\n",
+ "print(f\"MSE with Sklearn intercept\")\n",
+ "print(MSE(y,ypredictSKL))\n",
+ "\n",
+ "plt.plot(x, X @ beta + intercept, \"--\", label=\"Fit (manual intercept)\")\n",
+ "plt.plot(x, skl.predict(X), \"--\", label=\"Sklearn (fit_intercept=True)\")\n",
+ "plt.grid()\n",
+ "plt.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The intercept is the value of our output/target variable\n",
+ "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
+ "\n",
+ "Printing the MSE, we see first that both methods give the same MSE, as\n",
+ "they should. However, when we move to for example Ridge regression,\n",
+ "the way we treat the intercept may give a larger or smaller MSE,\n",
+ "meaning that the MSE can be penalized by the value of the\n",
+ "intercept. Not including the intercept in the fit, means that the\n",
+ "regularization term does not include $\\beta_0$. For different values\n",
+ "of $\\lambda$, this may lead to differeing MSE values. \n",
+ "\n",
+ "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "but when we take out the intercept, this equation becomes"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For Lasso regression we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It means that, when scaling the design matrix and the outputs/targets,\n",
+ "by subtracting the mean values, we have an optimization problem which\n",
+ "is not penalized by the intercept. The MSE value can then be smaller\n",
+ "since it focuses only on the remaining quantities. If we however bring\n",
+ "back the intercept, we will get a MSE which then contains the\n",
+ "intercept.\n",
+ "\n",
+ "\n",
+ "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known vanilla data set."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Beta values for own Ridge implementation\n",
+ "[ 1.03032441e+00 6.28336218e-02 -6.24175744e-01 5.21169159e-02\n",
+ " 2.80847477e-01 2.12552073e-01 8.13220609e-02 -1.69634577e-02\n",
+ " -6.50846112e-02 -7.38962192e-02 -5.94226022e-02 -3.50227564e-02\n",
+ " -9.80609614e-03 1.08299273e-02 2.41882037e-02 2.93492130e-02\n",
+ " 2.64742912e-02 1.63249532e-02 -5.01831250e-05 -2.15098090e-02]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[ 1.03032441e+00 6.28336218e-02 -6.24175744e-01 5.21169159e-02\n",
+ " 2.80847477e-01 2.12552073e-01 8.13220608e-02 -1.69634577e-02\n",
+ " -6.50846112e-02 -7.38962192e-02 -5.94226022e-02 -3.50227564e-02\n",
+ " -9.80609615e-03 1.08299273e-02 2.41882037e-02 2.93492130e-02\n",
+ " 2.64742912e-02 1.63249532e-02 -5.01831152e-05 -2.15098090e-02]\n",
+ "MSE values for own Ridge implementation\n",
+ "4.3632959273186007e-07\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "4.363295916523824e-07\n",
+ "Beta values for own Ridge implementation\n",
+ "[ 1.03630548 -0.01963611 -0.37900111 -0.07062318 0.12182967 0.16343471\n",
+ " 0.13003291 0.07490892 0.02365049 -0.01449782 -0.03814292 -0.04909093\n",
+ " -0.05009826 -0.04389027 -0.03279636 -0.01866537 -0.00289724 0.01348565\n",
+ " 0.02976145 0.04543942]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[ 1.03630548 -0.01963611 -0.37900111 -0.07062318 0.12182967 0.16343471\n",
+ " 0.13003291 0.07490892 0.02365049 -0.01449782 -0.03814292 -0.04909093\n",
+ " -0.05009826 -0.04389027 -0.03279636 -0.01866537 -0.00289724 0.01348565\n",
+ " 0.02976145 0.04543942]\n",
+ "MSE values for own Ridge implementation\n",
+ "5.194042826640948e-06\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "5.194042826840599e-06\n",
+ "Beta values for own Ridge implementation\n",
+ "[ 1.04220758 -0.10931453 -0.17641709 -0.06020587 0.02208512 0.05789007\n",
+ " 0.06491736 0.05785343 0.04537385 0.03196357 0.01969145 0.00934499\n",
+ " 0.00107405 -0.00526348 -0.00992331 -0.01318643 -0.01531845 -0.01655318\n",
+ " -0.01708852 -0.01708781]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[ 1.04220758 -0.10931453 -0.17641709 -0.06020587 0.02208512 0.05789007\n",
+ " 0.06491736 0.05785343 0.04537385 0.03196357 0.01969145 0.00934499\n",
+ " 0.00107405 -0.00526348 -0.00992331 -0.01318643 -0.01531845 -0.01655318\n",
+ " -0.01708852 -0.01708781]\n",
+ "MSE values for own Ridge implementation\n",
+ "2.094082198961287e-05\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "2.0940821989631478e-05\n",
+ "Beta values for own Ridge implementation\n",
+ "[ 1.01219292 -0.06043581 -0.10391807 -0.05651951 -0.01898855 0.00312361\n",
+ " 0.01463049 0.01975848 0.02123176 0.02068067 0.01905883 0.01691985\n",
+ " 0.01458337 0.01223198 0.00996754 0.00784393 0.00588657 0.00410387\n",
+ " 0.00249435 0.00105081]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[ 1.01219292 -0.06043581 -0.10391807 -0.05651951 -0.01898855 0.00312361\n",
+ " 0.01463049 0.01975848 0.02123176 0.02068067 0.01905883 0.01691985\n",
+ " 0.01458337 0.01223198 0.00996754 0.00784393 0.00588657 0.00410387\n",
+ " 0.00249435 0.00105081]\n",
+ "MSE values for own Ridge implementation\n",
+ "0.00031535148309579146\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "0.00031535148309581185\n",
+ "Beta values for own Ridge implementation\n",
+ "[ 8.38916861e-01 1.31276579e-01 8.97497404e-03 -1.72271878e-02\n",
+ " -2.11744554e-02 -1.91492986e-02 -1.57201944e-02 -1.23002365e-02\n",
+ " -9.30466214e-03 -6.81048318e-03 -4.78184120e-03 -3.15130074e-03\n",
+ " -1.84923989e-03 -8.13661243e-04 7.46984697e-06 6.56636616e-04\n",
+ " 1.16805821e-03 1.56912044e-03 1.88168312e-03 2.12318726e-03]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[ 8.38916861e-01 1.31276579e-01 8.97497404e-03 -1.72271878e-02\n",
+ " -2.11744554e-02 -1.91492986e-02 -1.57201944e-02 -1.23002365e-02\n",
+ " -9.30466214e-03 -6.81048318e-03 -4.78184120e-03 -3.15130074e-03\n",
+ " -1.84923989e-03 -8.13661243e-04 7.46984697e-06 6.56636616e-04\n",
+ " 1.16805821e-03 1.56912044e-03 1.88168312e-03 2.12318726e-03]\n",
+ "MSE values for own Ridge implementation\n",
+ "0.01507238889517716\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "0.015072388895177083\n",
+ "Beta values for own Ridge implementation\n",
+ "[0.37396662 0.14174745 0.0764924 0.04892055 0.03447512 0.02586427\n",
+ " 0.02024962 0.01633913 0.01347916 0.0113104 0.0096208 0.00827728\n",
+ " 0.00719176 0.00630331 0.00556826 0.0049544 0.00443743 0.0039987\n",
+ " 0.0036237 0.003301 ]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[0.37396662 0.14174745 0.0764924 0.04892055 0.03447512 0.02586427\n",
+ " 0.02024962 0.01633913 0.01347916 0.0113104 0.0096208 0.00827728\n",
+ " 0.00719176 0.00630331 0.00556826 0.0049544 0.00443743 0.0039987\n",
+ " 0.0036237 0.003301 ]\n",
+ "MSE values for own Ridge implementation\n",
+ "0.2640931530791003\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "0.26409315307910036\n"
+ ]
+ },
+ {
+ "data": {
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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_118_1.png"
+ },
+ "needs_background": "light"
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn import linear_model\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "\n",
+ "\n",
+ "# A seed just to ensure that the random numbers are the same for every run.\n",
+ "# Useful for eventual debugging.\n",
+ "np.random.seed(3155)\n",
+ "\n",
+ "n = 100\n",
+ "x = np.random.rand(n)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+ "\n",
+ "Maxpolydegree = 20\n",
+ "X = np.zeros((n,Maxpolydegree))\n",
+ "#We include explicitely the intercept column\n",
+ "for degree in range(Maxpolydegree):\n",
+ " X[:,degree] = x**degree\n",
+ "# We split the data in test and training data\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+ "\n",
+ "p = Maxpolydegree\n",
+ "I = np.eye(p,p)\n",
+ "# Decide which values of lambda to use\n",
+ "nlambdas = 6\n",
+ "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+ "MSERidgePredict = np.zeros(nlambdas)\n",
+ "lambdas = np.logspace(-4, 2, nlambdas)\n",
+ "for i in range(nlambdas):\n",
+ " lmb = lambdas[i]\n",
+ " OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+ " # Note: we include the intercept column and no scaling\n",
+ " RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+ " RegRidge.fit(X_train,y_train)\n",
+ " # and then make the prediction\n",
+ " ytildeOwnRidge = X_train @ OwnRidgeBeta\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta\n",
+ " ytildeRidge = RegRidge.predict(X_train)\n",
+ " ypredictRidge = RegRidge.predict(X_test)\n",
+ " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+ " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+ " print(\"Beta values for own Ridge implementation\")\n",
+ " print(OwnRidgeBeta)\n",
+ " print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+ " print(RegRidge.coef_)\n",
+ " print(\"MSE values for own Ridge implementation\")\n",
+ " print(MSEOwnRidgePredict[i])\n",
+ " print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+ " print(MSERidgePredict[i])\n",
+ "\n",
+ "# Now plot the results\n",
+ "plt.figure()\n",
+ "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')\n",
+ "plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n",
+ "\n",
+ "plt.xlabel('log10(lambda)')\n",
+ "plt.ylabel('MSE')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
+ "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
+ "What happens if we do not include the intercept in our fit?\n",
+ "Let us see how we can change this code by zero centering."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Beta values for own Ridge implementation\n",
+ "[ 3.43579948e-02 -5.43330971e-01 -3.10141414e-03 2.47116868e-01\n",
+ " 2.18613217e-01 1.02054837e-01 -4.25617662e-04 -5.90475506e-02\n",
+ " -7.68534263e-02 -6.68929213e-02 -4.24906604e-02 -1.40927184e-02\n",
+ " 1.11482289e-02 2.88529063e-02 3.67047975e-02 3.38135733e-02\n",
+ " 2.02198702e-02 -3.46383924e-03 -3.63025821e-02]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[ 3.43579948e-02 -5.43330971e-01 -3.10141413e-03 2.47116868e-01\n",
+ " 2.18613217e-01 1.02054837e-01 -4.25617658e-04 -5.90475506e-02\n",
+ " -7.68534263e-02 -6.68929213e-02 -4.24906604e-02 -1.40927184e-02\n",
+ " 1.11482289e-02 2.88529063e-02 3.67047975e-02 3.38135733e-02\n",
+ " 2.02198702e-02 -3.46383925e-03 -3.63025821e-02]\n",
+ "Intercept from own implementation:\n",
+ "1.0330308045181225\n",
+ "Intercept from Scikit-Learn Ridge implementation\n",
+ "1.033030804518383\n",
+ "MSE values for own Ridge implementation\n",
+ "3.139255958275475e-06\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "3.139255958572018e-06\n",
+ "Beta values for own Ridge implementation\n",
+ "[-0.05807125 -0.29822833 -0.08551306 0.08156108 0.13679863 0.12333649\n",
+ " 0.08251519 0.03815288 0.00111756 -0.02498832 -0.04010697 -0.04566964\n",
+ " -0.04355837 -0.03562355 -0.02348765 -0.00848904 0.00831018 0.0260906\n",
+ " 0.04423486]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[-0.05807125 -0.29822833 -0.08551306 0.08156108 0.13679863 0.12333649\n",
+ " 0.08251519 0.03815288 0.00111756 -0.02498832 -0.04010697 -0.04566964\n",
+ " -0.04355837 -0.03562355 -0.02348765 -0.00848904 0.00831018 0.0260906\n",
+ " 0.04423486]\n",
+ "Intercept from own implementation:\n",
+ "1.0411487294305548\n",
+ "Intercept from Scikit-Learn Ridge implementation\n",
+ "1.0411487294305266\n",
+ "MSE values for own Ridge implementation\n",
+ "1.9601304850163794e-05\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "1.9601304850085328e-05\n",
+ "Beta values for own Ridge implementation\n",
+ "[-0.1416398 -0.14021063 -0.05383795 0.01367553 0.04784395 0.05796251\n",
+ " 0.05447415 0.044613 0.03267527 0.02098261 0.01066519 0.00217499\n",
+ " -0.00440346 -0.00917248 -0.01231917 -0.01405935 -0.0146081 -0.01416528\n",
+ " -0.01290947]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[-0.1416398 -0.14021063 -0.05383795 0.01367553 0.04784395 0.05796251\n",
+ " 0.05447415 0.044613 0.03267527 0.02098261 0.01066519 0.00217499\n",
+ " -0.00440346 -0.00917248 -0.01231917 -0.01405935 -0.0146081 -0.01416528\n",
+ " -0.01290947]\n",
+ "Intercept from own implementation:\n",
+ "1.0495569966278282\n",
+ "Intercept from Scikit-Learn Ridge implementation\n",
+ "1.0495569966278269\n",
+ "MSE values for own Ridge implementation\n",
+ "5.4959161509370406e-05\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "5.4959161509366834e-05\n",
+ "Beta values for own Ridge implementation\n",
+ "[-0.13535942 -0.08593216 -0.03568439 -0.0036367 0.01397146 0.02229529\n",
+ " 0.02503753 0.0245528 0.02228115 0.01908936 0.01549377 0.01179792\n",
+ " 0.00817631 0.00472512 0.00149311 -0.00149956 -0.00424967 -0.00676387\n",
+ " -0.00905423]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[-0.13535942 -0.08593216 -0.03568439 -0.0036367 0.01397146 0.02229529\n",
+ " 0.02503753 0.0245528 0.02228115 0.01908936 0.01549377 0.01179792\n",
+ " 0.00817631 0.00472512 0.00149311 -0.00149956 -0.00424967 -0.00676387\n",
+ " -0.00905423]\n",
+ "Intercept from own implementation:\n",
+ "1.039967668952797\n",
+ "Intercept from Scikit-Learn Ridge implementation\n",
+ "1.0399676689527975\n",
+ "MSE values for own Ridge implementation\n",
+ "7.571105947979326e-05\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "7.57110594797945e-05\n",
+ "Beta values for own Ridge implementation\n",
+ "[-0.05100875 -0.04063602 -0.02723445 -0.01713366 -0.0100706 -0.00517114\n",
+ " -0.00174276 0.00068734 0.00243186 0.00369758 0.00462287 0.0053018\n",
+ " 0.00579953 0.006162 0.00642221 0.00660427 0.00672607 0.0068011\n",
+ " 0.00683964]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[-0.05100875 -0.04063602 -0.02723445 -0.01713366 -0.0100706 -0.00517114\n",
+ " -0.00174276 0.00068734 0.00243186 0.00369758 0.00462287 0.0053018\n",
+ " 0.00579953 0.006162 0.00642221 0.00660427 0.00672607 0.0068011\n",
+ " 0.00683964]\n",
+ "Intercept from own implementation:\n",
+ "0.999955585168597\n",
+ "Intercept from Scikit-Learn Ridge implementation\n",
+ "0.999955585168597\n",
+ "MSE values for own Ridge implementation\n",
+ "0.0007698473260556339\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "0.000769847326055633\n",
+ "Beta values for own Ridge implementation\n",
+ "[-0.00834567 -0.00803064 -0.00673407 -0.00554552 -0.00458878 -0.0038335\n",
+ " -0.00323332 -0.00274989 -0.0023548 -0.00202756 -0.00175331 -0.00152117\n",
+ " -0.001323 -0.0011526 -0.00100519 -0.00087697 -0.00076495 -0.00066668\n",
+ " -0.00058016]\n",
+ "Beta values for Scikit-Learn Ridge implementation\n",
+ "[-0.00834567 -0.00803064 -0.00673407 -0.00554552 -0.00458878 -0.0038335\n",
+ " -0.00323332 -0.00274989 -0.0023548 -0.00202756 -0.00175331 -0.00152117\n",
+ " -0.001323 -0.0011526 -0.00100519 -0.00087697 -0.00076495 -0.00066668\n",
+ " -0.00058016]\n",
+ "Intercept from own implementation:\n",
+ "0.9637117593816477\n",
+ "Intercept from Scikit-Learn Ridge implementation\n",
+ "0.9637117593816477\n",
+ "MSE values for own Ridge implementation\n",
+ "0.0023813163025848865\n",
+ "MSE values for Scikit-Learn Ridge implementation\n",
+ "0.002381316302584886\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_120_1.png"
+ },
+ "needs_background": "light"
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn import linear_model\n",
+ "from sklearn.preprocessing import StandardScaler\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "# A seed just to ensure that the random numbers are the same for every run.\n",
+ "# Useful for eventual debugging.\n",
+ "np.random.seed(315)\n",
+ "\n",
+ "n = 100\n",
+ "x = np.random.rand(n)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+ "\n",
+ "Maxpolydegree = 20\n",
+ "X = np.zeros((n,Maxpolydegree-1))\n",
+ "\n",
+ "for degree in range(1,Maxpolydegree): #No intercept column\n",
+ " X[:,degree-1] = x**(degree)\n",
+ "\n",
+ "# We split the data in test and training data\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+ "\n",
+ "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
+ "X_train_mean = np.mean(X_train,axis=0)\n",
+ "#Center by removing mean from each feature\n",
+ "X_train_scaled = X_train - X_train_mean \n",
+ "X_test_scaled = X_test - X_train_mean\n",
+ "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)\n",
+ "#Remove the intercept from the training data.\n",
+ "y_scaler = np.mean(y_train) \n",
+ "y_train_scaled = y_train - y_scaler \n",
+ "\n",
+ "p = Maxpolydegree-1\n",
+ "I = np.eye(p,p)\n",
+ "# Decide which values of lambda to use\n",
+ "nlambdas = 6\n",
+ "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+ "MSERidgePredict = np.zeros(nlambdas)\n",
+ "\n",
+ "lambdas = np.logspace(-4, 2, nlambdas)\n",
+ "for i in range(nlambdas):\n",
+ " lmb = lambdas[i]\n",
+ " OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)\n",
+ " intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data\n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ \n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n",
+ " RegRidge = linear_model.Ridge(lmb)\n",
+ " RegRidge.fit(X_train,y_train)\n",
+ " ypredictRidge = RegRidge.predict(X_test)\n",
+ " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+ " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+ " print(\"Beta values for own Ridge implementation\")\n",
+ " print(OwnRidgeBeta) #Intercept is given by mean of target variable\n",
+ " print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+ " print(RegRidge.coef_)\n",
+ " print('Intercept from own implementation:')\n",
+ " print(intercept_)\n",
+ " print('Intercept from Scikit-Learn Ridge implementation')\n",
+ " print(RegRidge.intercept_)\n",
+ " print(\"MSE values for own Ridge implementation\")\n",
+ " print(MSEOwnRidgePredict[i])\n",
+ " print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+ " print(MSERidgePredict[i])\n",
+ "\n",
+ "\n",
+ "# Now plot the results\n",
+ "plt.figure()\n",
+ "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')\n",
+ "plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')\n",
+ "plt.xlabel('log10(lambda)')\n",
+ "plt.ylabel('MSE')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see here, when compared to the code which includes explicitely the\n",
+ "intercept column, that our MSE value is actually smaller. This is\n",
+ "because the regularization term does not include the intercept value\n",
+ "$\\beta_0$ in the fitting. This applies to Lasso regularization as\n",
+ "well. It means that our optimization is now done only with the\n",
+ "centered matrix and/or vector that enter the fitting procedure. Note\n",
+ "also that the problem with the intercept occurs mainly in these type\n",
+ "of polynomial fitting problem.\n",
+ "\n",
+ "The next example is indeed an example where all these discussions about the role of intercept are not present.\n",
+ "\n",
+ "## More complicated Example: The Ising model\n",
+ "\n",
+ "The one-dimensional Ising model with nearest neighbor interaction, no\n",
+ "external field and a constant coupling constant $J$ is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+ "\\label{_auto1} \\tag{1}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
+ "in the system is determined by $L$. For the one-dimensional system\n",
+ "there is no phase transition.\n",
+ "\n",
+ "We will look at a system of $L = 40$ spins with a coupling constant of\n",
+ "$J = 1$. To get enough training data we will generate 10000 states\n",
+ "with their respective energies."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+ "import seaborn as sns\n",
+ "import scipy.linalg as scl\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "import tqdm\n",
+ "sns.set(color_codes=True)\n",
+ "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+ "\n",
+ "L = 40\n",
+ "n = int(1e4)\n",
+ "\n",
+ "spins = np.random.choice([-1, 1], size=(n, L))\n",
+ "J = 1.0\n",
+ "\n",
+ "energies = np.zeros(n)\n",
+ "\n",
+ "for i in range(n):\n",
+ " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we use ordinary least squares\n",
+ "regression to predict the energy for the nearest neighbor\n",
+ "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n",
+ "around. We will use linear regression to fit a value for\n",
+ "the coupling constant to achieve this.\n",
+ "\n",
+ "A more general form for the one-dimensional Ising model is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+ "\\label{_auto2} \\tag{2}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
+ "coupling constant. This latter expression can be formulated as\n",
+ "a matrix-product"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{H} = \\boldsymbol{X} J,\n",
+ "\\label{_auto3} \\tag{3}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
+ "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+ "with the form utilized in linear regression, that is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n",
+ "\\label{_auto4} \\tag{4}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We split the data in training and test data as discussed in the previous example"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = np.zeros((n, L ** 2))\n",
+ "for i in range(n):\n",
+ " X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+ "y = energies\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In the ordinary least squares method we choose the cost function"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n",
+ "\\label{_auto5} \\tag{5}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
+ "This yields the expression for $\\boldsymbol{\\beta}$ to be"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
+ "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
+ "intercept, i.e., a constant term, we must make sure that the\n",
+ "first column of $\\boldsymbol{X}$ consists of $1$. We do this here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X_train_own = np.concatenate(\n",
+ " (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+ " axis=1\n",
+ ")\n",
+ "X_test_own = np.concatenate(\n",
+ " (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+ " axis=1\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Doing the inversion directly turns out to be a bad idea since the matrix\n",
+ "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
+ "value decomposition**. Using the definition of the Moore-Penrose\n",
+ "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Using singular value decomposition we can decompose the matrix $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
+ "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
+ "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n",
+ "$\\omega$ to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n",
+ "\\label{_auto6} \\tag{6}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Note that solving this equation by actually doing the pseudoinverse\n",
+ "(which is what we will do) is not a good idea as this operation scales\n",
+ "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n",
+ "general matrix. Instead, doing $QR$-factorization and solving the\n",
+ "linear system as an equation would reduce this down to\n",
+ "$\\mathcal{O}(n^2)$ operations."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
+ " u, s, v = scl.svd(x)\n",
+ " return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 20,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "beta = ols_svd(X_train_own,y_train)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "J = beta[1:].reshape(L, L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A way of looking at the coefficients in $J$ is to plot the matrices as images."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ ":7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+ " cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_151_1.png"
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J, **cmap_args)\n",
+ "plt.title(\"OLS\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It is interesting to note that OLS\n",
+ "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
+ "valid matrix elements for $J$.\n",
+ "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n",
+ "this problem can be removed, partly and only with Lasso regression. \n",
+ "\n",
+ "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "Let us now \n",
+ "focus on Ridge and Lasso regression as well. We repeat some of the\n",
+ "basic parts of the Ising model and the setup of the training and test\n",
+ "data. The one-dimensional Ising model with nearest neighbor\n",
+ "interaction, no external field and a constant coupling constant $J$ is\n",
+ "given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+ "\\label{_auto7} \\tag{7}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n",
+ "\n",
+ "We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+ "import seaborn as sns\n",
+ "import scipy.linalg as scl\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "import sklearn.linear_model as skl\n",
+ "import tqdm\n",
+ "sns.set(color_codes=True)\n",
+ "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+ "\n",
+ "L = 40\n",
+ "n = int(1e4)\n",
+ "\n",
+ "spins = np.random.choice([-1, 1], size=(n, L))\n",
+ "J = 1.0\n",
+ "\n",
+ "energies = np.zeros(n)\n",
+ "\n",
+ "for i in range(n):\n",
+ " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A more general form for the one-dimensional Ising model is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+ "\\label{_auto8} \\tag{8}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we allow for interactions beyond the nearest neighbors and a more\n",
+ "adaptive coupling matrix. This latter expression can be formulated as\n",
+ "a matrix-product on the form"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = X J,\n",
+ "\\label{_auto9} \\tag{9}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n",
+ "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+ "with the form utilized in linear regression, viz."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon}.\n",
+ "\\label{_auto10} \\tag{10}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We organize the data as we did above"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = np.zeros((n, L ** 2))\n",
+ "for i in range(n):\n",
+ " X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+ "y = energies\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)\n",
+ "\n",
+ "X_train_own = np.concatenate(\n",
+ " (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+ " axis=1\n",
+ ")\n",
+ "\n",
+ "X_test_own = np.concatenate(\n",
+ " (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+ " axis=1\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We will do all fitting with **Scikit-Learn**,"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "clf = skl.LinearRegression().fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When extracting the $J$-matrix we make sure to remove the intercept"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "J_sk = clf.coef_.reshape(L, L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "And then we plot the results"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ ":7: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+ " cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_169_1.png"
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_sk, **cmap_args)\n",
+ "plt.title(\"LinearRegression from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The results agree perfectly with our previous discussion where we used our own code.\n",
+ "\n",
+ "\n",
+ "Having explored the ordinary least squares we move on to ridge\n",
+ "regression. In ridge regression we include a **regularizer**. This\n",
+ "involves a new cost function which leads to a new estimate for the\n",
+ "weights $\\boldsymbol{\\beta}$. This results in a penalized regression problem. The\n",
+ "cost function is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "6\n",
+ "0\n",
+ " \n",
+ "<\n",
+ "<\n",
+ "<\n",
+ "!\n",
+ "!\n",
+ "M\n",
+ "A\n",
+ "T\n",
+ "H\n",
+ "_\n",
+ "B\n",
+ "L\n",
+ "O\n",
+ "C\n",
+ "K"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 28,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ ":10: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+ " cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_172_1.png"
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "_lambda = 0.1\n",
+ "clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)\n",
+ "J_ridge_sk = clf_ridge.coef_.reshape(L, L)\n",
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_ridge_sk, **cmap_args)\n",
+ "plt.title(\"Ridge from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\sqrt{\\boldsymbol{\\beta}^T\\boldsymbol{\\beta}}.\n",
+ "\\label{_auto12} \\tag{12}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ ":9: UserWarning: FixedFormatter should only be used together with FixedLocator\n",
+ " cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_176_1.png"
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n",
+ "J_lasso_sk = clf_lasso.coef_.reshape(L, L)\n",
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_lasso_sk, **cmap_args)\n",
+ "plt.title(\"Lasso from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It is quite striking how LASSO breaks the symmetry of the coupling\n",
+ "constant as opposed to ridge and OLS. We get a sparse solution with\n",
+ "$J_{j, j + 1} = -1$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "We see how the different models perform for a different set of values for $\\lambda$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ "\r",
+ " 0%| | 0/10 [00:00"
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_178_13.png"
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "lambdas = np.logspace(-4, 5, 10)\n",
+ "\n",
+ "train_errors = {\n",
+ " \"ols_sk\": np.zeros(lambdas.size),\n",
+ " \"ridge_sk\": np.zeros(lambdas.size),\n",
+ " \"lasso_sk\": np.zeros(lambdas.size)\n",
+ "}\n",
+ "\n",
+ "test_errors = {\n",
+ " \"ols_sk\": np.zeros(lambdas.size),\n",
+ " \"ridge_sk\": np.zeros(lambdas.size),\n",
+ " \"lasso_sk\": np.zeros(lambdas.size)\n",
+ "}\n",
+ "\n",
+ "plot_counter = 1\n",
+ "\n",
+ "fig = plt.figure(figsize=(32, 54))\n",
+ "\n",
+ "for i, _lambda in enumerate(tqdm.tqdm(lambdas)):\n",
+ " for key, method in zip(\n",
+ " [\"ols_sk\", \"ridge_sk\", \"lasso_sk\"],\n",
+ " [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]\n",
+ " ):\n",
+ " method = method.fit(X_train, y_train)\n",
+ "\n",
+ " train_errors[key][i] = method.score(X_train, y_train)\n",
+ " test_errors[key][i] = method.score(X_test, y_test)\n",
+ "\n",
+ " omega = method.coef_.reshape(L, L)\n",
+ "\n",
+ " plt.subplot(10, 5, plot_counter)\n",
+ " plt.imshow(omega, **cmap_args)\n",
+ " plt.title(r\"%s, $\\lambda = %.4f$\" % (key, _lambda))\n",
+ " plot_counter += 1\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see that LASSO reaches a good solution for low\n",
+ "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n",
+ "much. Ridge is more stable over a larger range of values for\n",
+ "$\\lambda$, but eventually also fades away.\n",
+ "\n",
+ "\n",
+ "To determine which value of $\\lambda$ is best we plot the accuracy of\n",
+ "the models when predicting the training and the testing set. We expect\n",
+ "the accuracy of the training set to be quite good, but if the accuracy\n",
+ "of the testing set is much lower this tells us that we might be\n",
+ "subject to an overfit model. The ideal scenario is an accuracy on the\n",
+ "testing set that is close to the accuracy of the training set."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_180_0.png"
+ }
+ },
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "\n",
+ "colors = {\n",
+ " \"ols_sk\": \"r\",\n",
+ " \"ridge_sk\": \"y\",\n",
+ " \"lasso_sk\": \"c\"\n",
+ "}\n",
+ "\n",
+ "for key in train_errors:\n",
+ " plt.semilogx(\n",
+ " lambdas,\n",
+ " train_errors[key],\n",
+ " colors[key],\n",
+ " label=\"Train {0}\".format(key),\n",
+ " linewidth=4.0\n",
+ " )\n",
+ "\n",
+ "for key in test_errors:\n",
+ " plt.semilogx(\n",
+ " lambdas,\n",
+ " test_errors[key],\n",
+ " colors[key] + \"--\",\n",
+ " label=\"Test {0}\".format(key),\n",
+ " linewidth=4.0\n",
+ " )\n",
+ "plt.legend(loc=\"best\", fontsize=18)\n",
+ "plt.xlabel(r\"$\\lambda$\", fontsize=18)\n",
+ "plt.ylabel(r\"$R^2$\", fontsize=18)\n",
+ "plt.tick_params(labelsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n",
+ "achieves a very good accuracy on the test set. This by far surpasses the\n",
+ "other models for all values of $\\lambda$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## Exercises and Projects\n",
"\n",
"\n",
@@ -2121,7 +4221,7 @@
},
{
"cell_type": "code",
- "execution_count": 11,
+ "execution_count": 32,
"metadata": {
"collapsed": false,
"editable": true
@@ -2129,16 +4229,15 @@
"outputs": [
{
"data": {
- "image/png": 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\n",
+ "image/png": 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VMupDQdSr/H4RzemWVSXuHMF0TsBYr2WZNVZk/BZT0zGRdhZzHa0VjRVde7vcumDBAixevBizZs1yPeacc87BRRddhGOPPRb3L78dhx9+eHsPNoHB6E1+ngYGcmO5lgY4taAUCiVPPVE6nUA2m6/1xkWhaZqvXbyurgRGRvKGa0Gh4E0y8XgEsqxAkgToOlAolHx9KCMRYuDvtAUej5MvBt2tCwLmc9L6EDXZa098STB5ci/Wr984qrWZze1EkdgXk37GfCDCSzNaeW3NCnEycEG31PXcSF0QePT0pDE4mPe1pmYFcb/48xe/jvJH3nUpv4hsPQX7PnF/YOdrBxM2chJFFhzHQNdR63FrpQWFFDyjUck1jXNDMhkzRob7A4NoNIJKxZ+9L617EEJQHGtEnYZTfUiSxHGrVzmZ2/X3dxsz+ciA0bpKfDTN1K2ki40KcftMPmdSZ9nO7wjaIQ9tgDwYHDlxkfEvrk84cqLREvnC8LVoyU+Pm+UsiEalltIrQSAvBf2w+UEkIkIQOJTLsi9iMssXSqUKZLnqWCPieQ6qqhof/k6PHqfiy8a1kMbcsfYT13Uduq4jkyGRR50U7H5RVTgNiGgGhmHaTmOdRZeiZXLMunXr8Mwza7D33rPR1TXFV3qaz+dx3HHH4Uc/+hGmTZtmue3pp5/GrbfeCl3XMW3aNFx77bVIp9N48cUXcc0116C7uxt33nknuAgLLhocoYTkZAMtejMMg0hEaGpV4nx/pha2A9ms/+17WivSdfgSsjEMjMfxW7Ohv7huRGuuEcXjUQiCAJ7nDU0TjRza2ZZvFU6NudRP3Fyv6tSum11GYCeFupYpinSab8nDKkhFu5Oift269Vi5cgWuuGIpenv7MG/esfj6109zPUczhXg+n8fll1+Ohx9+GJMnT8Ytt9yCW2+9FUuWLMHFF1+M5cuXY8cddwQA8FEWQiw4MSwfING1vYbxXgBFXbtEdtVkWW3JcMy8SyZJgi8hZl3AqSGXKyCVisNL/Ge2RSmVKohGJc/HovUy/83EOlRVNcY41SOH+ra82Ra3k7CnXOZ6VSoVB8OQmXjB6quavwfNtUxWH3H7DwGJZDpD7rKsYJtttsXjj6/Cxo2D+N3vnkM8nmh6H6oQv/DCCx3OJ+Pyyy83hJW77LILVq1aBQB44oknIAhkQKogCGB5DqwQHDmxY9SF0AzjTk7mord5t4tET6Kvc9htdyXJe0vVScDphVbV3nUpgupbUe4Ep8ih3vPG1Xa6mqddQe1+meswDMOgv78bmqYFWq9qNbpx87ByGhBh9nPqFFiWxbRp03DkkV/xPLaZQry7uxsHH3wwAKBcLmPFihX4+te/DoDsdr755pu4+OKL8cgjj4CXOGiRACMnaQsnp8YWFBLB6Dp89Y+52e56iRjdtvztrpZmxGIRcJybYZ2TropGcv4V5X5hNW9zT7taGYDQDuhrVSgEXa9qP7pplAdYB0SwLGu0KnWq7SdojVYul8OiRYswffp0fOUrdcLbZZdd8MgjjwAAWJ4NOHLagtO6un2us57Ii2Da0SHVd8p0gwStoILK+g1mAnSKfuiMOTNGq3dqZeKvc9rV2NbCssyYaK/81Ku8+u+CrAvZDeR6elJQFHWUHlbNQchp1KcBQHrozjjjDOy333645JJLAACVSgXPPfecEVUBACew4IQAC+IBnqtdjDk50TSOmLS5t4U0+4J6tZM4ERvdKWuWxtkfs1W1t3marzP5dR72tpb6eG9S9xIE3rUe0zqaRzhu9SpRFGsuDcETg581l8v1tNxNRjGael5QUgJVVbFw4UIcfvjhWLRokfF3nuexdOlSTJkyBTNnziSPKXDgxOC+zkFGYe1iTMnJbp+rKEqT3rPGgQN+WlecQBt98/my7235VqMfu8uBFziOA8dJNQdF+we58bm3C5oC0lRDURSIomjUY8y7gJ12NvCrr1JVtWNEZf/Bc5NR2Ivrbm02zo/BYDQvJVWIr1u3Dv/4xz+gqiqeeuopAMDMmTNx9dVX46abbsJ3v/td9Pb2YuXKleB4FlqQkdMESOvGTCFeH/0t+PLaJi0XMWQypJbkPT2ljlgsAkVRIMuqEckUCmXPD3wiEUWlIiMSEX2ryiMRMliS4/zrsehrUK3KEEWhYfeNEKOAbNafwtgPEokYdF03voQA+ZEgqmyy+9ZqJMNxLLq70xgcHA5kjfVhDGKtXiV7FvpbRW9vFzKZnK/zkbSUN1paqL+5V+9dd3cKiqKhWPQXeQWhEP/Poq9DHlg/6vNQCP2TscNtm7lC3KxdomOW/FjamlOz+vQUv9olHRzHIRqVfEcyAFljvTnY330EgQfL+vcPp88lny+iXK5A0/SG3TdV1aDr5Dl0UoBpd5l0imSaNecGDXM6FY9HUSyWW65XeaGVYrWbh5XdU8u+prHwjLIjTOtaPTnPIJEgX/ZWUh4zUql4S+ZuAEmZaB+e319c8mvNolSq+JIW0EI56bdSPYnJbKNify723bdEIgZJEtDTkwIwdgJMpxRHkurNueYteZLqdkYzRFOvTtSrRlNspx5WVk8t85p0/OxnP0OhkMenPjUbn/jEDp4eS4VCHqeddryjOvyNN97AkiVLkM/nMXv2bCxduhSZTAann366cUwsFsMDDzwATmChb2YF8Y6sgNrnArpRiykUyi0RE9UqlUoVFIv+RnnT3TiGYXwXfM1WvbKs+Ip+BIFHMhlFuSz7kgnwPIdkMoZqtT4V1w26rhtkNTAwjKGhTM1E3zp0gLbbdBLEWqSIjRtHMDg4jHK5ClHkjUEDyWQMnRmB1Bh50HpVJuM8jKGnx9/rEmRU07imLIrFIh599FGcdNLxOOKIQ/Dmm/90vf/rr7+GRYvmu/qHX3DBBbj00kvx1FNPQdd1PPTQQ+jt7cXjjz+Oxx9/HI8++iimTJkCAGA5jggxg7q0Yb3cbFDDG2+8gaOPPhqHHXYYvvOd7xjzCZshcHKqF72Z2oeXjFnyG8GQ1pAIRJF0+ft5EgAhgFQqZhRV/YDjWKRSMaiq6ttDPBqVjL498ovdfNs/EiHTVvL5cksNyBSqSmpfIyNZbNgwZOi5Uqm44eNNNVidBIkaKshk8jXSzEJRVLAs0+KkFm/4iW6IV1QJw8PkdaEuA8kkeV26u1OIxaINerlOKsQVRcURRxyFX//61/jVr1bjggsuwVZbbe16/KpVj+Jb3/q2o3/4hx9+iHK5jD333BMA8NWvfhWrV6+2HPPwww8jEiE1VZrWBXVpNa175ZVXcPzxx7dEtF4I9OfXyT43GpV8/1LZJ+am097tJEDjEExJEuHlWFovzNfn2jUjGneZgPvOWqttK3W4n5NGVVRHZPYVJwMr6z5NnUwBVVVFuVyBJInYuHGkLdW6O1onECeVOKlXpSz1KqBT02LqIMr5yfjc5w5qetxFF13qepvZIxwA+vv7sX59veCtqiqWL1+OlStXksfkODABDpqg5/r4448b6p6pVAqpVMryt4ceesi1DceJaH/wgx/ghBNOaLqGwHMDeytJJCL6CqWddvGaKbYBK2FYCUBHs6DQfTy5MylwHIdEwn/NjA5CaHXScKuw+4pTa1yngnZnUCeR5qp1WOpVXqnzaEWYzfRVANDX19VRfVUQqaPT/c3p83PPPYftttsOO+ywAwCA5VjoAfbDsbVI/MQTT8SHH35oue3ss8/GN7/5Tcvfrr76atdzeRGtGwIlp2SS+nObW0m8CAaIxaKOO171HbvGOzcjDLcIqJ1eNy83AXfhZvttK60oxM2wW+PabVgURTXmvAWhFnd7X/2p1usRnsOZEWTqRWtD5XIVktSNkZGcxeokCOElRd0Fc3Trnzx5MgYHB43rAwMDlvTv6aefxhe/+EXjOivyQJC7dbVz/eQnP3GMnFqBF9G6IVByyufLaPxQNbaEUFCCIUZy/oreAJmaG4kITXRFjRGQn+kpdlJwj7CcEY2KEATBl3CTbBpwgffdmWG2YUkmY2BZzjBwowQyFupsN9W6m4tAkO0rZtDz2q1OnISX7Q49GK1FL8XUqVMhSRJeeuklfPrTn8Zjjz2GAw44wLj95ZdfxoIFC+qPy3FgAoycaFq31VZbjfpcXkTrhkDJSdP0hlqPWxRA60Tmmo8d9jYUc5TVrI5jf0x7qtkcDFiWqc2nc+6nc7pPIkF0L82GZlJwHGv0+LEsMyYqbbq5QGUCbpqmSkX2vQnRLqx+4lYXAYYhwxg0TQPLsoG+Hm5RuHe9auxU9Gb/8GXLlmHJkiUoFAqYMWMGTj75ZOO4999/39ipA2ppWIA1JzbADRYvonVDx/ejGwmGijHtdSKn+9ZJxlos9xdlmd0O/NjuEuEji0gk1kI/HfGgqlSqKJW861E0gisWKygWiwDqRe1kkqQ8qqo2nRIbBMyaJjpwQJJEpNNE0+SHMIOqrdhdBKijQV9fV8D9d97rdbIMpip68zw+t02HdtK63/72t8b/aYEbAKZPn45f/OIXjvd55ZVXLNdZIeC0LgCpil+idUOg7StUDW5GPB41FMat9p9R73CWZXy1vFDwPFGHsyzTkl9TIhEFx3HI54v4w5RPQ5frL81nNrzYcDwhmkjN6sU7wjL362maBlluLA6TmlUEkkSKt9RypFKpjqpO5NS+4ob6F5JEt9TZwF5cF0UB8XgUw8PZttflBPMQAhrhiaIAQeBHpVrneQ7pdBIbN460vTZqzStJ9fWY61U8z6Ovr8vXd2fNmtW47747oWkqTj31VJx44omW25999lksW7YMALDzzjvjiiuuQDweRzabxfnnn4/3338fO++8M2655RZkblsCLTPU9vOyg033IL3oqsDO1w7GoPGXWIrYt/t93VOHYTjnt+4DwOiDyuVKvto/GIbBC1vPBgBwUQ66rFmICQBemERupyRFiaZUKntqe1pxK5BlBeVyBQzDIpPJGX1vYykVaGxr4Wt+SNYaUadSHHNE5qxaF5uo1v2dt100q1d98MH7uPDCCzFr1izsttun8OlP741EwtkJc2BgA1auvA133nk/ttqqB8cddxz23Xdfw3Y3m83ioosuwv33348dd9wRK1euxE033YQlS5bg5ptvxuzZs7FixQqsWbOGPDdeABvg3DqGH/8ZeIGSk9vODWmkbW3KLtnhIb9MfgWS5v49mh41w1923K/+eFGSr+uyVrtOQkC1ZP3AvzBpNrr3SGH2Cy8gmy2A47imOw80Ha1W/bfu1Hc4zSmGu1Sg03Ui88BOc40oFiMbBul0IlBzO4aBa1e/uT5E58xR4vSTcgVN6Ob1SFIchx56KP7yl7/gkUcewbbbboc773Runn3xxb9gr71mI5VKIxaL4bDDDsPq1atx9tlnAwDeffddbL311gZZHXjggZg/fz6WLFmCtWvX4ic/+QkA4POf/zx5bjwLBFkQnwCuBB3urSNfplYIBqhv38uy4jvKMqeMskxGLjXDO/MahxBSYjKDi7INBDX8ShZ/m/NZ7PzE02i25W2uLwXROOskFZAkEV1dCTAMOyYOmGbCrFZlxGISqlXFIhMYvb85A8A7KqOqdWok50TeZsvgTu0CUnAch2OOOQ5nnHEG1q/PoFRy/8wPDg6gt7fPuD5p0iS8+uqrxvVPfvKTWLduHf75z39i+vTpePLJJ40dL7NuiK/ZfTAsF2hBnGE348Zfmsa1OmfMvH0vioIvPYRdi0SHMrrhnXmHY+Bl/zYfNIrq3s2q7/jXFw/Grk/91ukuRtrn30mhdVACoIVk80y1sRqCoOtokAmMVineLom467xIgZ1OEu6k2wPxctIhCELT8eBe2p9UKoXrr78el156KTRNwzHHHNP0fIzAAQH2WzKboyuBXbUtioKvnSea/rTqQNCKFolGS07E5BQ1mdH3qS4AgFq1HvfGYZ9Heloak1c8DKBVN8zgxIb2OpGdJDRNg6Ko4Di2o3a9bkrx1kZcBfO62C2Dnd0ego00/TZB9/dPwiuv/M24vmHDBov2R1VVTJkyBT//+c8BAK+//jq22WYbACTKGhwcxJQpU6AoCnie3ywjp0ATS55njebbfL5U+/B5v+mkoBhFuWx1IGjmI06aduM1P3C7srxRW9WMmLxAiQkAONH5JVt/5tHgOBbJZL2R2OvX38tOYzQg/XdFDA1lMDAwXIsoWfT0pNHX141kMg5JEkflKOAV4dAUMJst2BwWpKYOC51Iv6gLaLVqd3sQ0dfXjd7erkAal2nk5IXZs/fBSy/9FcPDwyiVSlizZo1F+8MwDE4//XSsX78euq7jrrvuMhThc+bMwWOPPQYA+N3vfkeOr/XWBXkZbwQaOSmK3mCF6zWooB0fca9ajv0xzfWlvt27mj6HgZe8t2M5kbVEUJkPMkhPS+OjM76CaXc/7llfIr/ikZrI0KzUrkLXvV+zVkHn4FWrCorFUsOk2mZz3pqjtQiHOizYR1zVJ7WQSIZh2JbO63u1poK421pGqxL3W3Tv75+EBQsWYfHis6DrGubNm4fdd9/dog264oorMH/+fFSrVey///4444wzAADnnHMOLrroInzpS1/C9ttvj0MOOQQMzwOB7taN+9S44G167c+JtCrwKBSsPkbmqSZuHkeiyIPnecvtRO3NIZ8vN63ldHUlMDKSR/mqs/Dxy+8hOSWF7EcZX8/BTFDmqMkMe3oHwJLeuYGqwysVGblcwbLjJAi8sQkgijw2bvS3Xj8gEZ1ubIFTkAk4dU1TK4X1SESCJAnG2PDRwOywEI1KNdKuBCqbMOunvNZC61U0uvTrwkkGjXLI5/23YwVh01t+/DboheA+L0w8jciXF3kf2EGMuUIc8NfnRu5bj5ysam9/NSkzMbWC/k/3kMdv8iG0R08AiaBw5tGuBEWbgs0Rn1k3Q4iCfDl5nkd/f3fNJrYaiDraKRrRdRhWtLkcHJt06e6cPSIMMv0yOyyQwQsVwzY5uGGd/qJRq2q9mQsnfV/sj9FhTxYHMGyw7StMB0sOfjEmsVtjnxvnaxeLEls703nXX/h15NeNGMTkN2oyo2e7Xgy9s9H1dieCAkj9yU5QtCnY/rzvXNvdcP+Fh+TAskSE6dz/FsRYJ2c4NenaU6/69NzOgGFgqNKdHBZaFV+a0Q7BN06N4Q07mHQ6abwvGzcOQdcVpNNdnuek6nBZlnHGGadb1OFvvPEGLrroIuP60NAQ0uk0fvWrX+HFF1/ENddcA1mWMXXqVNxwww3EJYAXyCUoTAARZsfTOo5jEYtFUCiUDA9tL6va+rk4Q69UKPgfUBmLRfDe6V+2/K0dcurelhBHM4ICGlO89LQ0AGDyiodrzp5kp6pQqBfJVzwTB0DqdK7nVXWcedCIcZ2mGdb0y5+3OO3bs6d1rcCcglJvJE3TkM8XGiKI0aCnJ418vugqgbCvw6/DAnkNVN+fPz8w9yVef/21eOihh7DLLtMxe/a+OPbYE9Dd3dNwn4GBDVi0aD7uvPN+CIKIs8+ejxtvvNEQXJpRKpXwta99DZdffjlmz56NQw45BMuXL8eOO+6IZcuWoaurC/Pnz0dlzd3Qi8FNS2JiSUiHnhbY+dpB4JGTvYit67ph2dvqdN5oVALDMMhm874++DT1y1zzjTZXXwclJsA7gqLYZr/tjf9nP9iI9WcejZ0eXA1ZVoxfXaBOTAAZAgFYSUpVddOxXcb/zzxoxNA11dMvCamU37RndOxhFz3S3S0aQYzGaqRhpU3ecCfxpR+HhSAn8dbXWZcsnHnmIsydOxd/+MPz+OtfX8S///0v7L33fg33MavDATSow824/fbbsffee2P2bNI+9cQTT0AQBMiyjPXr19eHIjAcEOT2P7OZ7dY5IRIhBJPLFX1HPlTtXa0qEAR/dQ16n+GrFmLoTavBejtRkx3NCGrbz+4EzeXLWC7X62p3PUv7rBqfEM8zFlJywopnuqCqOr5xaKYh/bKnPfVaVefU4qR5WUEuV7BZn6RM7petT41ptZbl1n9nd1igLUGdAs/z2GuvvbDrrrPw9a+718W81OEU2WwWDz30EFatWmX8TRAEvPnmmzjttNPA8zwuuOAC+uABp3Xjv1vXsRWYC9gAfBOTuUFY0zQIQsz3fYavWjiqNVOYoyYvTNtnB8e/p6b1ItrXherN38Jdu91muY1GS0A9YjITk9fty9ekjf9/41BCvGZjOaoWp7UiOgevk4VapyIyMQVsJbKjGF1R2dp/V3dYiETq7gZkHcGlohR+pAR+nSFXrVqFgw8+GL29vZa/77LLLnjhhRfw4IMP4sorr8Stt95KoqYgtUkTQITZEXKyF7C7upw7s81wMpJjGKapXa25nkOJyR41xfqSiPU1btWyPIeP/t9/W3hWzaMnludco6fTX1+E+3a/DU430+fH84xj/YmhmyYuWdLyNWmomo6zv1C3LXFSiyeTcUQiEqLRiCWiGV1rjTuJ2HVEzgVt5zUEuQtofi0YJlWLoBhXF87Rwg85eanDKZ5++mmcddZZxvVKpYLnnnsOBx98MADgyCOPxH333UduDCMnb8Ridc8iv9GSm5Gcl0LcuM8Njbl69w7EXrTSRIOz9V7bAkBLJGUmKLeoiaI0OIJoXxcA4OuvLMT9e/wIAKCocEzh7PUn1ZSO8YIpmpIbb//h6rpcwkxUQL2lpFwmE1NoMZkWyZ28mvygFRKxRnZmN4E4dF2zSCaC9hA3r5dGcI0unPUpLaNpnvZT15o9ex/cddcKDA8PIxqNYs2aNbjyyistx+i6jtdffx2f+tSnjL/xPI+lS5diypQpmDlzJp588knMnDmTHM9y0IOMdjbHyElRyBfA/AZRknH6RRFFAdGofyM5+30qN55TfxxFNUipGVibtcTWe21rEJSflK5nu17E+rscz+sWPTG1EEjVqH94YyG8fiz5l+cYKE4kJjAGQTmBEpWdpACnYrLVqykoc7tmcCto0zUwDINIJIJKpRKoZML+GXTTM/kfxOD2GM2PMavDZVnBcccd06AOHxoaqu0CSsb9OI7DTTfdhO9+97tQVRWTJ082pp7oTMDkNAEK4oFLCZzcMFOpOPL5xsZc2rpCnCGdl0GV3vX7EJ1U5d4rwIgSim+/CwCQ+npQXjdgua9b1GQnJzNKG/25OjqRE4WdoGj0pOsa7pn1I9f7VSruZEBJyo2UzFGU4tDEfOkJuuc2ulmlLUlCTaBZdRQbAv4V162AYRj09XWjWq1CEAQwTHDj2Ht7u5DJ5HwTHh3EIEmiL4cFhmEweXIvhobynpsbZgShEC+9+CT0SoDvgxRDdHajrdBYYowSSx3mUN08QMBL7U2lCaTbnxTYqw9cbxCT1Ed0JHZiagfRST2ITurB0BvvNj2udyZJ50rrveUFZjAMi0XFG3BbrHHwoKbqEGrRlNxE++QWNXE154dKxfmLd+VPGQA8zj3CfX1Oc/DIFr1ZKkBIQlXVjpi3kSgbyGYLhqc7KaxH2iisW9Fqu6LTIAbzLD430uykZ5QbwrSuTZi1T/W5bn5rHDp4vl5gZx65EQAsxOSEdqImip5dP+lJUM1gT+/MtafS2+/izIN/hXs2zgUAlMuNn2Q7SdlTO1p/MpOUotQcPDlym9Mvtyyr+N4jZAPhgq96izGtQxAa7U+oDUvwJFX/IfMqrLc2uab9dTrN4jPvRpZKJdx4402YMWNX7Lbbp9Db29/0fM0U4gDwwx/+EA8//LAxI+6YY47BXnvt1aAc32mnnXDXXXdBZ9iA07otpH2F1pzMBv+t1DNiMQmFQhn84zcDAJhEwkJMQURNdrgRFI2aACA6udd39BTdaQeU3voPAIB/+x9QE4SchBrRyE1qSM1qT3JVcyQiSlIAUC43RhitkBTg9OXkkErFIQikBzDI1ppmhXYnyYR5ck2z4n6QIkw7aWqaimw2i2XLlmFoaAh77rkXbr31dscNHbN/OFWIm/3DAeC1117DjTfeaCmIA8Djjz8OoK4cP/fccwEAOidA5wK0aubGv32lIwrxxr/piMUkqKruw4CNgMoESE2jZCEm5aOPgl00SEpnh58Iyo2gzNFTz/6fttxWevtdnLX9Dbg9UU/vBNNuXLHUSNx8jWzMJCVXvaOlSkVtOq7oyh+Tj8ClJ7X2wVZVtbYNTya61C2Dzbte7dWJ/DboOkkmGodjmtPQzokwWZbDRRddgu7uFP74x5cwODjo+jz8KMRfe+01rFy5Eu+//z723ntvfPvb37YUx6lyfPfddwcA6GzAkdOW0PjL85xhBWK3TXGDWSZAUwamNsXCTkxOUdNoUrogYSel6E47APGa5iubwVl5K0HRgEOsGdpVHZqKeY6BLOtGGmeGmaScak/29KtcrEcX7ZCU2VitbhlccKwTtTLOqV0Cca4RiUYaSv/Wqck1RJfHYOedp2Pnnd2P81KIFwoF7Lrrrvj2t7+NqVOn4qKLLsJtt92G8847D4CzclxnOGgB7rCxE2C3rqPkRJXb5EPjL9S3G8klElGIv73T131jnyR9RtHaF2DkH/9ub+EmmKMnc0pnhlP01LVPLRx3+xKk0kA2A1EwpWy2l8hOUubUj69Nx3AiKbeiOFCPSkoFZ1eB1kmq8fk51YkkyVwnctcSBSXAtE+uoUM67f137cy/c4NfF0wvhXg8HrcM1zz99NNxySWXGOTkpBzXWB5akKkYuxmKMIFGH/FIRPS1U+Jkp6L+ut76YY+a2IhkEJITumZYu7wZgUfm9bcajnNK6cwYbYHcgkK+Hj2l0jht+HqsTJHoiTrEtqKFNJNUsejvS1bI14nJzd79OyvJF+jqBaN35Ky3kxRdBzHUd986I8Ck7qx0AKh9/l0QI+FbccFsphD/6KOP8MILL2DevHkACJnxJsW2XTkOBL9bF2iK2CYCTyzrPtqa4SPuZTvLMMS1gDgQFI0PR/TZuxuO5fv7jYsTdJdfQabmU53ebSekd9up1aeFrb/4uaa3RyfXf8WMqAnwtX99Wt+vLNdFwURUptROEBhLbcqMalUziKoZzMQEAJpOLmaUS3V2/M5K3SAqO9qJcmidaGQkhw0bhpDNkhQ8lUpg0qQepNOEuIP2V7cTB01BN24cweDgCKrVKkRRRF9fV9t+4n7JyewfXi6XG/zDI5EIvve97+H999+Hruv4yU9+gkMOOQSAs3IcADSGhVZL7YK5bIY1J13XUSpZ1d66DtcJLNRNwG4kZyGmaBzIjlgIScuNbgR2eredHKOoZpA+sQ0q773f9BgLMbnBFj0BhKDuHiQ7ePSlYxhAktgGcaZ5h88eLfE865jqAUBmuATeZeSPppMoykxMZjhHUqOPcmidiLa1RCISRFFAX19XWwptNzQjDq/JNX5V8353A/0oxK+44gp84xvfgCzL2GuvvXDaacRbyUk5DtTSOjZARf8ESOsCV4gDjT2DkiSAZVmLpxHgPqI8+uzdhJAosiMNj+FETl5RkxuqA95ygMh2nwQAT3KStv2E8w0N8up6M7TSNxUA8OORukKy6KB/MpNUqVR/vZwK53aCygzXJQNuBJUdKgAAIjHR8fZysR51ff8cou+RZbnhfR0NOI5Fd3cag4PDDQrtdt0v7edtBU6qebepwolEFIIgIpttzcwuCIX4hv/8A6ocnDspJ4iYtMOMwM7XDsYkdrOndVQmIAg8stmilZj+8pCVmDoMtqsHkZ2ap3mUmAASPblBmrU7kOry98CF+o4iP/ghAOCkrvruSyzCIBaxRpuSxEKSWAsxAfXCuRk8z4LnWWSGSxZiAgDFYXOCEhNgJSG3v/3vLRUsuHKjr5mEraEejdGdNzreijQt8+jp6UJfXxeSybjv1KtdoShVzWezeQwMDGNkJAtF0RCLRdDf342ennRtN1BHPp/39Rhr1qzGSSd9DcceexQefvgh1+PWrl1rjBsHyC7dmWeeicMPPxwnnngiBgbqO9Uaw0JjueAuEyCtGyNyqpdeSE2KmMTnckXvN9Nn1DRaeBHUqOFTu2MmKKBOUuWyZlwiEQ6RiDX6EUW2gaSGNxZdycNMUGZioigXqwYhOZEVxfylG3HOsgLOWVYY9cw3wL2ORUkik8ljYGAIIyM5aJqGRCKGSZN60N2dQiwWAefqaRRMoZ1MFC5heDiLDRuGDDHo448/ggMP/By+/vWTcNddK4zR4XZQAeZtt92Be+55AL/85aP4978bd5UHBwdx/fXXW/528803Y/bs2XjyySfxta99zWj6BQCV4QO/jDfGNHISRd4YnumUCkT/YvsVcSAm18doI6Vju6y7dH4Jyil6kmbtXr8yiugJAOZu9xokEahUdeMSjzW+VXaCAghJDW8sYnhjvWfRD0G5oRkx2fGNa0ZwzrKCiSja+Xj5IxFFUS1RVbFYBs/z6OlJoa+vG6mUdWhoJwZ1AvXC+pe+9GXcfvsKzJ49G88//xz+9Kc/OB5vFmBGo1EceOBBWL16dcNxS5YsabDtXbt2LY44gqT+c+fOxe9//3vIMqnF6QEXxPUJEDmNWW8dx7GIRCTXqSsNxOSCTkRNZkR22gnlt+qFcnNKZ4af4ngDPL4hpcQkRPMb0DX8Dg6ZCvzmQ+LVk82T+1CCKhTrrx8lqHJZxfBwTVcU4VG1taywrLMGZ91/BxBLuruNFjL1qCqetqbbissPwsnf+Ri6ruOBG7YxnA38+kW1QyJ2TZPT0FDzoNdOgOd57LPPPpg9e2+cfrq7h71dgNnb24d33vmX5Zj77rsPM2bMwB577GH5+4YNG9Bf2xTieR6JRAIjIyPo7++HyvJQA+QTZgIUxDuyAnMaR2aPSQAYY9vYDi9iYo78pmVMOYX06LLRLtURdoLygiVqokh1eUZ+5Z2Jab0sRB1vP2Tqa/jNhzORSpAX04mk8nlngvBDUIMfkeGhxVzRkaDMxGS+Hk/HXYkJqIsMj7+wTt4/vnaqraXEuagdxKTjxlmAAqLRaAdmAVrBst4+8F4CzH/9619Ys2YN7rnnHqxbt87HY5LPgqaz0PTgtEmavplHTvVBBTKEJqOSS/scY/zfrhAXRR48nD+wla+cb7JfUVEsViA+dJ1xeyspnR2RnXYClOa/9KOJnigxAYAglwyCMkdPI93bYdbWw3h7KI2NGRZiTUIwklWNU5mRTErI5eok3oygKDFRuBGUEwqZAqSY5Hib25f9pIvraes9V05BPB51jaqCJAw6NJQWxPP5AiRJRDTafFJLO2jHonfjxkGLAHP16tUYGBjA0UcfDVmWsWHDBpxwwgn46U9/ikmTJmFwcBBTpkyBoijI5/OGa4EGFlqA4lVtbCo+TdExciK9VUQmoKqar2IpdS0wp372UVNmOGmkqscQSwmOY8H/4oa216/1bw32Y2/7XvaAQ4Fhl184h+iptPNsMB4f4FJiEgoSceScon0I9AAA0UNtzLDoSnEGQSWT5C3M5ZTadX8E5YRijtSpYslYQ9RkhiIrUDL1c9rTPS+ceil5vW67OG3RE3Uy/aKTeJ1ba0R0dSUso9grlWpbDcutWvSuXftbXHttvbC9ePFiLF68GADwwQcf4OSTT8ZPf/pTAMCcOXPw2GOPYeHChXjiiScwe/Zs40df0zm04G/nCTbAKKxddIScEokIWJY1BhUAzcN12u6i62hwLXBTl5vJz8miQ9d1SKdchmy20HL6p/VvTf7daltfBNUqdIZpIChz9GSGwooGQX2cT6I3XScoABaS8kNQ69+r7yIxHiTlF5TIuFpjtVtUZceia8nkGE3R8H8vSddm4EVqIszulmpV3nCOauoNy3BorVENovLTf+c3cjILMI844ssNAkw3nHPOObjooovwpS99CclkEsuW1T/Xm2Pk1BERZiTCN4TIdrtdCo5jkUhEUanIhgOjGTzPIRIRkc/XtTrxOCE/2h7jBJLuxZDNFhCPR6D8+Kr6bR4pHSUn43gXgpI/uavxf8EtegKM6KlkSuUANBCUnZxo9MRrVQwwU6BqHD7OJzGQIb8pw1ly/1iEwYfr6q+dVvsJNRMUALz/L+sa3chpZANJ+dJ9jX7qbrUmronjQzOy0mxi0TuW9kGSJBQKxdp0YzEQb3MqM8jl3CNCO6hSXJJEX/13kyb1IJstQXawSW6GIESYb7wzCNmlM6AdCDyLXbfr8z6wg+hI5FStKr7sYOqDCsqurgXmyInOwlNVopFqBpoOplIxUlf4yvkA2iuiBxFB2YnJCfboKV4ZRkHqhsKKgA5wrIoIr6I/DQxkeHSnGAxndRTLOrq7BAyP1Ga1cQyGB+uvz/Ag+VGIpWIommyRdU13JSgAyAwOOxJUKxBEoYGA2FoPoP3vADD/snpk98OLU4b9CVVpx+Mxh2kt3mhHhGltrWk0tjOPQKePMV5QdQ6KHtzjsxOgIN6RyMlpyEE6HbekeX6GG5BzsUgkIigWK449eG7geQ6JRBSlUqXh+Ojv7nK9nz1qMtZhIydz1EThFj1lt9kDguz8i+03egKAYZ1EfIPFJAZyIpJRtSGKAmCQlBNBFR08280ERaMmO9J93S1HTUKbokz74/zou9ZIl3qbtxJVJRIx6LpuTAUeLWhUJYoCOI7FxRdfgr6+XnzqU/tgt912b7oBZLboPeaYE7Bw4RmW259++mnceuut0HUd06ZNw7XXXot0Oo0XX3wR11xzDWRZxtSpU3H99dcjnSa1yFf/k0E1wMhJ5FnsvkPa9/GrVq3C8uXLIcsyTj31VF+2w/Zj7BgzcqITWHSd9CBpmubLfI5liWMBANf6kh20bsAwjGMqCbgTlBs5AVaCciInwJmgstsQvYoTQTkVx1shKAAYypLUzomggDpJeRGUGzFRxNPO6YcTOQVFTGbYSQqo974RohBdo6pkMgZNC46c7Pj1r3+JZ555Gi+//DKSySR+/vNViEYba4gDAxuwaNF8w6J34cLT8YMf3GxY9ObzeXzhC1/Aww8/jMmTJ+OWW25BLpfDkiVLcMghh2D58uXYcccdsWzZMrAsi29961sAgJf/k0W1idVzqxAFBnvukPI+EMD69etx/PHH45FHHoEoijjuuONw4403WmyHFy5ciLPOOqvBTaEZxlBppYPnOUSjku/oByA7eAzDIJMp+ArLYzGpVlsoIpVy30UqHXh6A0E1IyagvfSOEpMb/BTHC0giDvJDoWocOFZFXywHgJKFiKGsDqnWrzucUSFJ5NdhZLiCSExAuSijuy+B4cF8Q3oHkBSvGTRFRW7jSMPfu0xWMRTlIknFeI+GaztUVQXDMq5rWXBZvZds5VIiRnSbGGN2FKhUqmAYFroeoMe2DXPnfhlnnHE63nnnQ7z77n8RiUQcj7Nb9FKFOFWDy7KMyy+/HJMnTwZARo9Tx8snnngCgiBAlmWsX78eu+yyi3FeVWehBCjB4Gop4scff9ywg5pKpYwICABeeOEF7Lfffujq6gLQnu2wE8YwsSQDDgqFsi9iohETfb29iMnsCZXLNc7Is0MQOIhHfdP36im0rbZ1jZoAQO6eYvzfTkyy4H/LfYP0CeMCEIIaVMkXUtU4jJTj4GsWGf3JKnpSQE/t89Kd5tCdJtFMV7eErm4JkZiASExAdx9xQ4ilrJqmoY83QJVb3xUrF0sNl2icnFuRFePiBfMXgGGZhlqY/Quy4LIBC1lR2NtayuWyoZ2Lx6MtNQu3AirPSCbTmDlzlmv9yUkhvn79euN6d3e3MW68XC5jxYoVxnVBEPDmm29izpw5+POf/4wvfelLxv2ICDPYCwCceOKJOOiggyyXe++91/KczMp1gNgOm5+T2Xb40UcfRTabxW233QYvdEwhbkY8HgHDMCgW/aVldv2SKCaaHu+240eL4vb1UBlCPl+CeuDpAJrXoToNe/S0LrkjOKhQbVqTKFdGSSW/yEmxjFw1Ap7VMFySEBF1DOVY9KR0DNU6fLrTHIYz5PXu6ia/UiMAtvpEN9548W10TerGB2++a3kMVZbB2eolblOMIw6iTUpMZnhFUG7aJkpQWpPduQWXDUCtkd9d11inPdujKvrZs0dV7ViwNKy1yRAJ+5rc7mtGLpfDokWLMH36dHzlK18x/r7LLrvghRdewIMPPojzzjsPDz74IABA1ZlAdU5qLXL6yU9+4hg5mTFa22E3dDRyYlkGqRSJfvyqb8lctIjvCIs2ExeLFQcpgg7Y1OVER0OsWswfyFKNpLyQ790OpcTkpsfI3VNc0zm36OmjxM5Yl9wR65L1PJ1jGr+0Ua5ep0uK5P/d0Qq6oxX0JDVERB0RiUFEYlAu64hKLKISC7mqQa5qiMcFxOMCdtrjkxjZMIxEd2PR008EFRQxAcTf28lNQFM1g5hYlnV0x1RNUdnpl3yM0y/52PExGIZxjarasWBxOr9fi96hobp/mF0hDsBQhU+fPt1wHqhUKnj66aeNY4488ki8+eabxnVVY6EEeFE18lpvtdVWmDZtmuViJ6fJkydbXBicbId/8YtfGNfttsNu6FjNSRA4xGIRY3hmLBbxdA2hnex+0jLAWVFuhjlyYhjGKMR7yRD8oJSYjGh+vevtRTGFWHX0Tcoc0xhBmUEjKICQ1HBJQk+Svha1ulNGQypZf6uzOQXpLgmz5+yCzAjRQv1t7SuW86qyjJXLtkVEd36tFNb6JT7/xkYiFaTGL7qTfIDCTFCyi0TATFByxdkx4fRLPjZSyfu+RxwkqEKcorFWxUOSBFNU1ZqxnV8XTC+FuKqqWLhwIQ4//HAsWrTI+DvP81i6dCmmTJmCmTNn4sknn8Ree+1l3K5pgKYFJyVoxUb9M5/5DG699VYMDQ0hGo1izZo1uPLKK43bqe3wvvvui2nTpllsh5uhI7t1ksQjkZAswzNjMQmKorkOO2y2g2eXIVCzOgAoFEruA05SMeTzZTAMmgo9LWt/rdG+giLfu53luhs5bewh1ivNyMm8c/dxvG7VwjLOnwo7QZXUCIqK1bGyULVez5QIIeVLtZ68TP3cqgZks/X34vgDhiEw9etupMTo1vXJnLWoSUnKiZjscCMqxWLx3PjmKrbIzl5Ad6pxrbpjNxQKJZ/OCP52AM2gwxKGHLyx7FizZjXuv/8uQyF+7rlnGwrxdevW4Zvf/Kal2D1z5kxcffXVhpRAVVVMnjwZV1xxBaZMITXOta/LKAVnhImoCHxuN/9R5KpVq3D77bdDlmXMmzcPCxYssKjen3rqKdx6662G7fDSpUshis6OqxQdkxJwnDXMjUYlaJrWkKq5eYibQWUImqbXdE9kpp2XPWwyGUO1KiMSEY1G4ubrJkX18vMPO95uJyfAmaAoOQHeBGUmJmMdDgRlJqesXE8NZc05qnIjqkyeEFV/l4Ztu0YAAAnO+oXyS0xm2Enquyvr192iIDMoUXm5HdiJyXK7pvsqvtNoyi9oVGW3C65UqkbEHolISCRiGB5uPSoPQiH+zGtK4OR00MzxtU3pCDkBjT7ikQj5spgjF6pH8tIv0QiI45haqlhBter9IUzXGlL9jD83OyjwPI/qHx+x3O5ETBRmgjITE4UbQbmRE+BMUMNVd92JnaTchN9R3kroZmJSdQ4i4074otZcl1ZiE5B166/ttXda39dmREWJxSui0l1I0hJxOZQF7IX9H9/0Sde1uMEpqvrww4/x5z//Gfvvvx8ikebCRbsA8+ijj7GQ0xtvvIElS5Ygn89j9uzZWLp0KTKZDE4/vV4TzeVyGB4ext/+Vnc3+M2rauDkdMju49v8O2bkJEkiWJYxoh1aXyoUmivEARIBKYoKUeR9EQ1Adgh5nkexWPIc6ElJMpcrQVUVQ/kr/+lR45hm5ATUCaoVcvpI2BYi41JbsZFTM2Iyw60+ZSYlvbYTE+MaBYnNyMk4xoGkSmx9R9VOUJlqArfdn2m4DyUqt2jHTFJOx5hJyu0clKTcdhwp2iEqgERV//rXGzj//P/Fxo0bscMOO+KUU+bj858/uOFYJwHm5ZdfjX32qW+ezJ07F1dddRX23HNPXHLJJZg5cyZOOOEE43ZN03DKKafgmGOOMVwxAWD1yxpaMC31REwEvrDn+LawjOGj62AYsx4JvgvfLMtAELiGHTa3Y4k+Soeqqp5FymhUgiQJyGYLUFW5NiaojEwmh/Ju3kU7M5yICSDFcTs+ErYFAFR157zebPbll5g0nQUDveES4arQdca4AO0TEwBU2QiqbF1kaCYmAJbaFQCkxTwWfb0xohBEAQzDuKrJWZ4Fy7OuxMMwLBFXNvkMEVGn1rSHUFVVHL/4Pzh+8X9cj3GDoijYfvudsHr1U3jwwZ/hwAMPhuCyQ+lk0bt27TPG7R9++CHK5TL23HNPAMBXv/rVBgvfhx9+GNFo1EJMAKDqpI4Y2KUDlsatYsySStLAyyKViqFclg1L1Wagjb4AUCxWPLdq7fUrqq9ygrmons0WoGlaw/k1TUdx14OhbfD+0L7Xs5eh4g4Kms4iIzfXeJmPdYLENUZmoyEmM6psBCU9BhGN7yUlKBpFUYK66+ESysVqA+FQgjKnfTRyMssR7PfTahocutNn1+SYIyZKUGYysx9vJqgHfuA8ft4JHMdh111nYNo09/s4CTD/8Y/Xjet2MWN/f79FzKiqKpYvX47ly5c3nFvTGIxSqmU7X3DnahdjFjnxPAdB4Gr6JW9i4nnOKGgriurZ8S2K/vVRNLqi7gZOxGQ599Y7e64XICpuN5ijJxo1UbhFT4OVtGvB24zRElO7KOlE11TV3XddzFFUWszj9KOj+MpRWyGaiCCaaGzxEETB0cmAghd48AIPTVUNYjKDaqY4jnNN5agC3a12RdFKJOVH5+R0u9n4z0vM+Nxzz2G77baz7ORRKGrwl/FGx8jJ/DrHYhJ4nq95O3s/azvRNHPDBEhqRmpG1hl4TvejpFcuyygWS97EJApIpfxFL14oiqkGYqKwE9RgpZ4GNSOoiipA1jjjQuFETE4oqDFUdBE5bXQ7Rl4ENVhOY7BMyFbkdCw4hmxW2ElKlVWosmqQkJOIU5EVsBwH1mUMFG2ZYVgWjIt3DyU2lmWMix2tRE5+dE5OAsy+vnqkZBczDgwMWMSMTz/9NL74xS86nlvTmMAv442ORk7mfrdCoezL7yYWkxCJCDaiaVR6k/MT/RIRbhYc6lfW+1HSy+VKqFQqtW1g909UNBpBPB5FJpMHO8n9gzogTjX+3yx6AoCS4t7sSAnKTEwUTgRVURsjLkpSZVVsuOhgUFBjlosZOS1puWRV91oXjZqs628kqMFKNwYrVk+oVKSKjcUovjqXpDiKrEKQBFdtlJmk7GkdJSlKVE71KUpSDMu6RlwALCT12MoZjop0N7Csd+Q0e/Y+eOmlv2J4mKjT1679Lfbdd3/j9qlTp0KSJLz00ktkDY89hgMOOMC4/eWXX8bs2c6+YFrANScfpeCOo2PkxPMcUqlYbWprGV6zyKgQk2EYZLPWQrlTBMSyrGH6RRwxG89pvh+NrkhRXXFUlJuRSMQgigJGRnJGXcKJoMzEROFGUB/LUxz/boYTMVGYCcqJmChoQ7Af8KzLLletcJ5VU01Jyg4zQZlJSWBt/VkRktp/dW4fkl11ooskIsbFDEo6zdphdE1vMlST1J9oEb0ZHl2xa80quAu9vV1IJGKuRW6KVi16Tz31BBxyyGGYMWMmFixYgL///e8AgGXLluHaa6/F4YcfjlKphJNPPtm4//vvv2+ILu3YHNO6jkkJurtjKJUqRvRjts21g+NYxONRVKvOCm67RooWvr30TvR+fM1viAg5m6dxDMMglUrU2lyc1b7mArkTOVGYC+R2YrLrjYzzFVOQeG8Nl1udyY2Y3NI8L3KyI8VlHaMmO7JV5x5CewSYLZP3qKoyeOb3WeRGGkWM+WFnTy5zlKS67OjRH5ZmUgJz7emBH2wPQIeukx83nucRiUgWq17aLGz+Ae3tTaNcVlAue793dgQhwvzJs0DO2x7NN5IR4MQ5wZ2vHXQscspmves/AHEUpI6VXq0lQL0xOJ8vewoxGYYc77fwzXEsurqSkGW5qdc0jaCaEZMXnNK7gSKJUCqK+690URZQlAWUFQ5lZXQiOTdiaoasmkKm2rwGN1hKoaq6uGO6RFAip2OvT3UhEhMRiYmQK7JxkWKSow+536Zi3SNKppHUz364E1iWA8NwYFkWHMdC0xQUCgUMDQ1jYGAYlUoVkYiIvr5u9PSkEY9HMTIyglwu56u3Ttd1/PCHN+OEE47GSSd9Da+++rLrsfl8HnPnzsUHH3xg/O2FF17AEUccgUMPPRQ33XST8fdQSjAKOE1RiUSI2NFLWKnrpG2llcZgajqmKAqKxTJ0vTkxCQKPZDKOQqHkT+YwaQdgpPlPFTWJc0vnSopkRFCUmCgqCu8rgqIEFeHVlqMmN7hFTQBQVEi6lakmkBadIxqKqspB5BwaglnVsYY2uUvFvvv24d9v13cUc8P1HwlKUJXagFVz5MTViMoeQdF2FycZgRk/+2Fdo0Y+p/Q1YMAweo14dFSrZVQqZeg68VeKRER8//vX4/nnn8fuu++B/ff/LI455gTXrvu1a5/Bf//7Dn7845/jgw/exwUXnIM5c55qOP6VV17BkiVL8O677xp/K5fLuOSSS3D//fdjq622wllnnYU//elP2G+//aAoOkY5ds8Ccq7xLYqPyW6dExKJKHieOFZ6CSt1nVij+BVu0sI3kRQwnsXKSISY1mezBV/E1AreKXn3cdmJicIeQRVl9zpTscojWxYtF6D1dK4VOEVQgyXrc3GLoMyg0RNACGrH7aPYdvsu9PbHkewmFzOoONMJnMAbROXUh0dlBGZR5oO3um92MAwDhmFrli3WqEpVZeTzBVx22VLccccd2HXX3fDcc2uxfr37JJ4//vF5HHTQoWBZFp/4xLaYMmUrSxsKxUMPPYTLLrvMslv36quvYtttt8U222wDnudxxBFH4NlnnwVAdEmBFsQngM5pzDv76IReWVY9G3cBkmpFo2LN/9k7qaY2KsTFQDMsWxmGQbVKusrNDcDxeBSCQArfXkVyOyZ3RbDeI3oCgLIiIMI7k0Sz3TugTlBqG5M1smURPOdMaAz0mtWvFX6iJjPajaDs0VMqUjUIFQCm9PNYB6BYFJHZWIAUJbflTcM+xWj9+KqpsUw1iufkubs1CzMsgwdu2b7p2hvu4xBVpVJJ7L///pgxYw/PsVBOQkynsePUx8kMJ8fJ559/HgARDGsBFrHJV2F8I6cxJSdd15FMxgyPJy8IAo9YTKqNmvJ+oeLxKBgGyGbzRn2pWCyhWCyB49ja0EYiPZBlBSzLQtd1ZDK5tkdgNyOoDaUu4/9uBDVYJMXlCO/+ycpVyJcsJrr0oLloUnjO+TkxtZ3TwWJjIbY32pxonEAJyh41eYES1PtDdb/0dEzF5C4Vf/83ISigXnzPbCwgQZu5bROJKVGVHIY3uJFUq8RkB8MwEEUByWQChULF17w6ZyGmvwSmmUhTUXT4MGTwjSBTxHYxZuQkSWJN7+TdiAuQVEuSSD2KYRhj580JNBpTFBX5vHN9SVU1lEoVlEoVcBxrCCsZhkMyGa/NH6t6poxOcCIoMzFR2AmKEhO5jWtKUABJ3QArSQUpllM0FusLhGAmx1szyvMqkjtFT++PNO7ofThIoqmeLmDDoAqWYSCKnGEzDDiTlFzrCuBrWinFoUuAktT9y1qzTHGDKPIGMZVK7j+2d9zxI/zhD78HABQKeU8nTDc4OU729pIBE5oebCo2EXROY0JOtJCtaZqvLz+d6EsN5vgm02Q5jkMiETFcDb0K30QhnkC5XDbSSlEUalFVCqqqGUTViq+03xSPwkxMFE4ERaMmM4pV3jWKonCLmvzCTlJOKZ0ZgwVyezriXrMzE5QTMQFAIgbka8FPTxeHoREVPT0RDA2VIYochgcKpFY5UhuBXtM1ybD5hLmQVFDERDZQEigWmxMTAMyfvxDz5y8EAPz2t0/j17/+JQ4++DB8/PFHeP/995qOIDdjjz32wDvvvIP//ve/mDZtGn71q1/hpJNOAgAoitaBgvj4WqZ0jJzM1rh0Kz/p4Dtthv34+rkad/oAGFM1CoUyZFmuEV/zVpREIoZ8vmhJK81TWwWBt7SsUGMxPx7olKCcoiaKZvUncnudoJyIiYJGUV7Rlh2MhxjWDkpSScnfRkGmLDYlqLcHExB42ygsXoes1N/fZgTV3R/H8EAByS5CbpSkIjFCjuWi9QfCTFLBEROHVIoQU7HY2k7ogQcehH/84zWccsrxAICLLroUkUgE69evx5lnnonHH3/c9b6SJOG6667DN7/5TVQqFcyZMwdz5hAxkqbqcBG+t4Ug61ftomMiTEFgkUpFLQ6XiQS57tRfRyeoODkW0Km/WVM9IRoVIQhCbfdO9SxmE4V4BNls3nXaR+OaOIiiAEkSwLKsQWJe9bK/f+wVZUiICM3XG+HVpuQEABXZStjpaK0Q3CRqciMnRXOvexSq5Bd0StK5aZhGTZa1uBDUB8PRBnKiMBMUUCcoABgaIe/Z0BAhn+GBes2JEpQZZpK6+2pvZb5f8DyHdDqJUqmKQiGYnd0gRJjfe6gCl/mxbaErAVxwTPPNmk6jY5ET7aezCjGbR0DFYtmlHmXtkaNWKMTqRPUsZlPT+kwm21JNSVVVlEoqSqUyWJaBKIo1O9Y4ZLlOVPbH333rCl79yPmNHSyQv5dltilBrc+SGltMcl6vnZiAuhWvrgN9SYdt9BajJjvW5aKuBNWwliYRlKwwrgTlBqcICoBjFAWQSKpcLE94YgoKqqJDUYIrFKnK+O7UAR0kJ1lWYW9zclKJe01QMd/PnPaR0eberSjJZBy6rmNkZHReS5qmo1yuoFyu1HZpeIiiiHg8WqtT1SPEdDqBfaNV/NnmuEGJicKNoDLF+gtXrDANBOVETBT05RjMWaMuJ7Ki8BM1UazLkZ01SlJOUROFnaA+GK7vyjkRVLP0zgxKUACMNA8Akl1xg6DuvnISqlW5rQ0OJ/A8i1QqiXJ54hETAKiaDjVAWXeQ3lDtYox1TtYIKJEgH9ZcruAh2tRrPW8x34VvlmWRTidQqRBrlCCh63qtx8pcpxLR1RWp6alIRDVrK9UzxbMTlJmYKIqVmnulSxTlBxuydbKanBr9l8tvFEUJykxMzWAnKIpT55TB8zyy2Tx+9JvGNOiG/yHPj2G6IQg8eJ5HLEYm+tD3w49djxPI7m4SlUoV+fzEIyaA7EYHSk7qZhw5NXMJaFWISaX9xWLZ9GsYXCvKaCHLimF8XygUazWyOBiGwX5xGX/6t9YQNZnhleJRFCsMuCaSGL9SLZoyAqMjqnW5KHi2/S+EV3p3+IxhACQt5zge2SzpXzvrYLvMoU68uq5b6oI8z0EURSQScbBs/YdDlmVfrxfHsUink6hWlQlLTAAhp0DTus2ZnJyg6zo4jkUk4l+ISfvvANSIqfmXWJJIqpXLFTxHQQWFSERCNBpBJlMvtheLZbAsC0kS8NnpAh57qfk5yjLbNF2joDPoElH/H8Rmmc0HQ4Q0p3Q1vhf2lM6OkTxhyr5U84jknx9GkXDZqHUiKEpKACUm1iCmVkEMDokQl2WJEJfWDRVFMcjK6XPFcYxBTLlc6zbGYwlN0ZsOLG39fOM73AAYY3LiOBaCQCao+Amx63qnAmIxCV1dqaaF6FgsAkkSkcnkWtIojQa0/SWTaWx/IcMSiPDz8zsx+O1b7grqXLH+YRAF529hqVInLztJtSlwN7BuhPwAOJGUFwaznCdB5YvwRVAH7zxi/D2RiIFlWWQywWxDaZpmqhuSxl2ib4tA03RLVMXzLNLpFGRZQS5Xxni3cniBSAmCi5y0CWBLMGbkRKx6OV+5v7nwnc0WoOvEW4m2C5DG3pjpl6+KeDwGlmUwMtJ+K0qrSCZJquCn/UXXdRy4Ywa/+3ejmZyZmACgKjMNBGUmJjMoScUjrT9np9oOJalkrDm506iJwo2gPt5Yj76aERRgJyby2mazAe6Pm6DrcEj/BIgih0MPPQif/OQn8X/+zwHYb7//g09+cnRtLmMBRVWhBBg5KROAnDoeu9UdLtlafan5LxApPsZQrSooFkuWwjcpRFeRyxUwNDSCcrkCQeDR3Z2GIPCoVmVfVsBBPKd0mhRlM5l8S2R44I6N89ucUJUZVGtpnhsx1Y8FhnOMcTGj3c2q9UMs1g+19vEYzLavKDYTEyX9ThGTExRFRbFYRrlcxcqVKzF79mw88cQTOPvsM1tuCB8PaIoW+GW80dHIqT46XEapVIUg8E0HFdBG32KxYupzc/92qapWG5xZhqIoxo4ZCdHJ1r5fwWUrzymVSqBabX8X0BxB2aMmO6o+6lB2UILqTrq/dk5RkxPWD7GY3GP9oNqjJjPMEZQ5aqJwip7MNaZkMm40b481WJb86CQSM7D11tvhtNO+AVVVW/ISHy9omh5oKWMiDDjooAgTSCajFitdNxEmUG/0Jf7h3opvp1YU4lde36FJpYg4r5UWlGagfXmlUhnl8ugKpAfumMEvX+32PK5QqhNMRGp87ZrtKQzUvvO9Xa2tzdYB4khQzeBVg3JL75JJ8n45WTl3GgwDpFJJqKqOTKYEGuE38ySfSFBkFYqPhnr/5wvsVG2jo1KCjM3Wgvbb2UEL35lMAbquegrn6O5YNpt3rF+Zd2iIVYqIRCJqtKBUKtWWd/IEQUAy2diXNxocuftwU4IyExMAlCu6haD8LmPjSP3/rRIVBU3xJHdzCAMbBlVsGAS63Wc1GAR14mc1VKsxMmeuiW97J8EwQDqdgq7rtRap8Y8aWoWmuU+Vae98gZ2qbXQ0rWtUhFu/bLQepWn1wrdX/SYeJ5MwnHbHnECsUsq1FhTW2J3hOA6yrNSIqrnmJRIREYtFXclwNDhydxLe2EnKTkwU5Qr5u1MUZYab5oUSldsoPnvUZEel2pygNgz6f30OnzGMkRGSJrMsA47jjJS5XfuaVkGIKVnz9So2NdubyFBlLdDISW2jnBA0xlznRCMn2uhbqcgolyu+W1EAIJPJtrV1bt1KZho0L2SqhlWiQOUJ7ThltgKvKMqO4QxZSzzWWA/xEuNVqjoGhsgx/T2t1VMyWfIFmNTnne4MZ1R0p5sfl0iQXdd8vmjb3k8Z6u5O1A6BeipHovxNl5gAQNXUQF8jdXOuOTmBRlKtFr5pEVqWZRQKwbSi0J2/SqVqfCmogFNVVVQqMgSBA8tyYyZPOHL3YUiSiHt+39xaplSur6VQdCcpJ1Sq1ucxMKT5JihKTACJkOwE5RQ1uRHU12YPGyO48rUGusbtfWpfEwfAGBFVEOJaSkwMw2BkZNMmJgDQA95h0zf33Tp7Wkcjp1hM8l34JsM5E7Vt3s6odO1fCqqjYlkGqqoiEpFaNp9rB9FoBJGIiBP3z+Inf2zN8paSlCS2/iUbGGqN4CicCMoJdoI6Zm9CTLSB2w2KotSm55htliPgOL6pGNcLxPc7ZSKmlu4+IaEoquO04/bPN/5kPaaRU6xmCOa38D0erSgsyyAWi6JaraJQKIHneUhS3XyuUiG/3kHXnojSnNbSdBy/7wgA4IE/d1mOM0dNTsjmybpSCYfx5dXm9x3cSMi5r7fRR8ocNZlBCcqr1mQmqFQqCUVRUSi4E5MdZptlqxi37gpRrco+fkB0JJMkYiKpnO8lTGg0G7Pe3vnGXz4xJuRkLnwDpGep6vFFGY9WFFqQNUsF6K93oVACx3GQJMFo6g0qzUgk4uA4xlHQefy+IwZBeRFTpVp/nZqRlBNKpfoHe3Cj7EhQbvBbBB/OqLjgq7rxerYLc0oOmN1LkyCz5Wgbiv19IZNSOI7FyEhxQuxIBQVN0wKtOWkOcwXHGh0nJ/Oo8VKpAlHkEI1GkEjEXJ0liUKYHdNWFOpk0EwqoKoqikXV0tRLp7n4dcm0I5VK1HaK3EWHNIq661n3vXkzMZmRzatIJTjPqMkOcxTlFjWZMTJcRVe3t85gtMTkBFlWahq3kuFeWn9fFAwMDGDDhvWYOXMmOI7DyEhhsyImAFCrClSPCditnW8zj5wEgQwfMBe+y2UV5XLV5iwZgyyTPrlIRIKmachkRmcO1wpo+tiKVMDc1NuKSyYF8adKQFXVpnUXM06fk2lKUG7I5lWUSyrSaedoyBw12TG4UYYgNP+gjgxXjX+bEdS5RwS3oeGGRvdSAc888xtce+21mDZtGvbf/7OYN+84TJ06raPrGGsEv1s3/ibiHY6cmJoDgdKwI2d3loxEyPBLAKhWyTDMdoqdrSKI9LHRJdO5OZnW2OotMFUUvYRFNpw+h/TmmUnKLWqiKNfIJ5OhTp3+U7ZCnvwa+4mKgOYEFbTpnxfo+3L88SfggAPm4Mkn1+D555/D//t/L2525NSX5gPdYetLj/m83QZ0bMABAESjnKFVaQazOVy1KhtfbkEQXPVHQYB4BXHIZltr3m0F9LmIogBV1SDLCiRJDKQFBiAk1Yycyk2ionRaaBo1AXVyorATD42a7LAfd+ZBrc3BCwqJRByiyGNkpBiIpciaNatx3313QpZlHHPMCTj66GMst7/55j/xve9dA1mWMXnyZFx66ZVIJpsPMAhiwMHmiI4mljxPSKenJ41EIgZR5GHXM0UiEpLJOLLZPCqVqqPzgCgK6O5OIZVKIBKRfE3/bQaicUn4tjsZDapVGfl8EUNDGVQqFUSjEhiGPO9YLNJ0Jp8f0EiqHdBIyg12YgLcyajd4zoJ+pkLipgGBjZg5crbcNttd+Ceex7AL3/5KN55523LMbfcsgxnnHEW7r33AWyzzbZ44IH7R/24Wyo6Sk65XAUbN+aQz1fAMCySyQR6erqRSMRRrZZx9913olwuYWQk51rrqVblGlFlUC5XwPMcurpSSKeTiEalljvGaec59YoaK9AibTabx9BQBvl8AQBRvXd3pw0pQTv4xiE5fOOQxii3WdQEAOWygmy2imy2NSIx15j8HDceUVM8HoMoCshkgiEmAHjxxb9gr71mI5VKIxqN4sADD8Latc9YjtE0DcUi+VxVKmVI0viOV9qU0fHEUtcZVCoKKhUFDKNDFHkMDq7H4sXfhCzLOO2008DzHGRZ9VTp2odfShK1SNEM/VGzuhE1qjdP+x0LOBXcnZqT4/F6c3I7O3+UoJY7DACwo1y2RkWUoFIpko45RU1mtBJBERGrt8VyUIjHY7U6YiFQX+3BwQH09vYZ13t7+/CPf7xuOebss8/Deef9D37wg+8jEolixYp7Anv8LQ1jul9IiErFZZctRTKZxsqV9yCRSCKRiKOnpxvJZBySJDT1fKKQZcVIlwqFEliWQSqVRFdXymjsNUMQeKTTSRSLxTElpmiUpG+ZjHt0SJuTR0ZyRhQZjUro6elq6TWh+MYhOVz41faIwG8Ulc2UjUszfOd4tRbt1t+b0aayzRCLRRGJiMhkioESEwDH9N9cYqhUyrjuuitxyy234fHHn8JXvjIPV111WaBr2JIwLiX5pUuvQSxGitHEOL4MUeQhSTzi8RgSCQbVqoJqtVIrhDc/n1nnYvdyqlRIIT0alcZUaQ7Y/cX9fVGcmpNJ5OXenGwH1U6dd2QRN/2ysU/PHjXZse7j2hy4pHNKYiekbKaMVLpxBNaF88qQZUCWrZNQSAM3Y6i6g3pPYrEoolGpRkzBR2n9/ZPwyit/M65v3DiIvr5+4/rbb/8HkiRhxoyZAIAvf/lo3HHHjwJfx5aCcVFaJZNJW2TDoFpVazWqAjKZIjRNQzweQ09PV60QLvqKHojdagnDw1lkswXwPGe0zYiiYIyZ6jTIlGG+JWKyg24OZLMFDA9bNwecam7UPtjsi3TekUWcd6T/NhEzWpk44hVBAfb3hrg8xGJR9PSkkUzGIYqtRYhmRKMRg5hkuTPp4+zZ++Cll/6K4eFhlMtlrF37W+y77/7G7VOnboMNG9bjvffeBQA899yzmD59RkfWsiWgo1KCICAILCSJr6U2TE2sWa3phprfl44hz2bztSiEuG2yLGvUqDoRSdHIpZPGabTmJoqCIdcQRcGIIN1w7UPN9UrFgnOdi0ZRXiREI6gL5/nXb5kjRJ7na9qwqu+JvdFoBLFYBNlsCdVqZ8WDa9asxv333wVZVnDEEV/GiSeegvPPX4z58xdi+vQZ+OMfn8ePfvRDADq6unrw7W9/B1tvPbXpOUMpgTMmPDmZIQhsLf0TwHFszSyu4vghpo26uVy+IS2krSeiKBqtJ+24Y9rRjuo7CIgij0SCpLF0GnGz5mQ3gnIjJopkUvIVIaXSkZbIyQyzpxPVhlERq9NmB6npRceEmDqFkJycsUmRkxk8z0KSuBpRcUZE9frr/8B7772LuXOP8EUQtPWEnqfdnTJakG9H9T0a0LHrpRKpU9HmZFEUwbJM0wjRTlJe5FTMlRGNe2+NX3VGa8+hGWhTr3mwarlcRT6fr0kwYps0MQEhOblhkyUnM3iegSTx+Mtf/ozzzjsXRx11FM4//4Lar63/eg8dKS6KJL2QZdmXjS+VKASl+vYL+rjFovPYdacI0Yl4r31I9EVMZriRVJDEZAcl3mefXYtvfetb+PSnP43PfvYAzJlzCPr6+rxPMEERkpMzNgtyAoCRkRF8+cuH4YQTTsI555yLaJQM8aS9bZVKa2ZxjXUQuRaFWHfKqBleoeBMEJ0CfVy/AxdoE6yZeM3NyZfe3bwSbScnwJmgOklOFILA4403XseTTz6FZ59di1mz9sB3v3tl5x+4QwjJyRmbDTkBwHvv/Ref+MS2xnWOY4xiOiEqtTbPrlWiAkRRtPT70UbeRCI25hIF2ouYyxWNbfpWYG5Orj8fkv59587G452IyQxKUmNBTNRPK58vo1xWjB+K0QxT9eqXe++9d3HDDdcgl8uht7cXl19+DVKp1pxKmyEkJ2eMv2lLgDATEwCoqo5iUcbwcBEbN+ZQLlchCAK6u9Po7qaCQO+XQNfR0O8nSaQnUNM0cBw36n4/v6gTU6EtYgKcJyfTtqAfnGsdy+JFTABQKoxNKktmFdaJCSCkNBpi8uqX03Ud3/72t3DSSafi3nsfwE477YIf//ie0T6VED4w/r4IYwRNA0olBaWSApYlv8DULI4ONPBrv8txLFiWwchIFizL1c6TMuxiK5XOtGrQQaJBj6iytwXdeHYEoiji7Bv998R1OmoSRepcUfEUkbYCc78cAKNfbrvttgdAXAai0Sj22+8zAICTTz4NudzYTyPeErHFkJMZhKhklEoyWBaGPCEalYxR5mSHq5EAYrForaE0Z4yAphEM2Vlqrd/PLySJzM7LZPIdGZNEYVbbX7uAQyTSjbOuGW56n+8v7mxzKyGmBAqFCkqlYEfRevXLffjh++jp6cVVV12Gt956E9tvvyPOO++CQNcQwhmbVVrXDjSNtHNkMiVs3JhHsVipeYkn0dOTNoZ45nI5vP32W5YhBHaQL3Wx5jrg3e/nF9RehRjijd2WOa1N/eB/467HdJqYSBqbQLEYPDEB3v1yqqrib397CfPmHYt7730QW289FbfeelPg6wjRiC2enMzQdUpUZWzcmEehUAHLsiiVCvjmNxfhhz/8IYrFki//J+qVPTxct0dJpeK1WlfUd/NrvS2js0M97aC1rWyWFPu/v1hyJKJ6c7I4qtqP8xroWLAKisXgiQkg/XJDQxuN6/Z+uZ6eXkyb9gmjDeXggw/DG2+83nCeEMEjJCcXUKLauDGH4447HrFYHN///o1IpRLo6elyNc9zgr3fD9CRSHj7OMViUUiSUJs2PHYzjOrElIeiWOs7ZpL6/mIJw8MZo3Wmuztt6oMcHVERYkqiVKp2jJgA7365WbN2x8jIMN56618AgOef/z122WV6x9YToo7NSkrQCaiqit/8ZjU+97mDEIlEDE8qSeIN1TKtLVWrVQD+v5TUx4n2+5nbaOLxel/gWE2gAUZfdKdaKlHk294g4HkO6TQhpkKh89oxr365119/DTfffANKpTImTZqESy+9At3dPYE9figlcEYg5OSlE3nrrTdx/fVXI5/PY889P4Xzz794zNwBOgsrUTEMDL2QH6sXM8xqbp7noGk6CgV/AsugEPRuIN0gkKR6c3KlIjetm/E8i1QqhUqlinx+/K1+xwIhOTlj1GmdH1/lK664FOeeewEefPAR6LqOVaseG+3DThCYrV5yyGZL0HUYVi/JZMK3URwdNaVpZPevVCrXDOfqdiKdBCWmTCY4mYJ1g6BYa4wmdbd4PNrwA0XbcbYkYgrhjlGTk5ev8rp1H6NSqWDmzFkAgC9+8Qj87ndPj/ZhJyCsnlTZbKnBk4oUjd3PQJ0UstkCyuUKMpk8hoezxjw/a/E5uJWbialTu4Fk/Hi97qbrOhIJ4uX097+/gl/+8jGUy0VUq0pITCEABEBOTjqRDRs2+L59c0W1qiKfrxrmeaqqIR6PWszzqE+crusNJnEUdcO5vK34bCa89plKksQaMY2dTIFMTq7bEq9fvx53330XPve5z+G4447Dq6++PCbrCDGxMWpy8tKJeN2+JUCWNQtRKYqGaDSK7u4urFv3EY4++ij8+c9/8rR4MbedWJ0x6S6Z1BJR1YWd7Q8UHS0YRscxx3wNa9b8Brfffjf23nvfcVlHiImHUZOTl07E6/YtDbKsoVCoYmiogD/96SWcdtqp2HPPT2GfffYxWe96E4yum8dm1fvjqIUvme/n/vZSYspmx4+YiEg1BVkmUeZuu83CmWcuwu677zmq865ZsxonnfQ1HHvsUXj44Ydcj3vhhT/ga187clSPFaJzGDU5eelEpkzZCqIoGqH66tW/NvqUtnS88cYb+MIXvoRLLlmKQoFICGhtiRIVx/mLhMzDO4vFkjHxhJwnYiGqSGT8I6b6/EAV2WwJrUgwmsHPBg0ADA1txP/9vzePqUwjRGsY9X5+f/8kLFiwCIsXn2XoRGbMmGnRiXz3u1fhhhuuQrFYxE477YJ5847zdW4vicJzz63FnXeugK7r2HrrrXHxxZcFamXRaRxxxFHG/xVFh6LIKBRkk9WLhHg81rInFe2PA+pOkl1dSWiaXnNRYMdccW6GebBpJhMcMQHejbwU1113FU47bUHN7zvEREQgYqNDD/0CDj30C5a/LVv2A+P/O+20M1auvK+lc9JfwDvvvB+CIGLhwtOx116zjQ9ZoZDHsmXX4Y477kN//yTcccePcNddK3DuueeP/gmNM6jVS7FoJioS7bTqSWVu5KUz3XQdSKcTxpipIB0OvMCyQCqVhKrqgRMT4G/w5c9//iB22WU6dtttVqCPHSJYTNj2FS+JgqIo+N//vQj9/ZMAADvssCPWr183XsvtGMyeVENDeZTLclueVNGoVGuFyWJ4OFPbFaTj0KnuqHPDLgFi2pdKpaDrOrLZIoImJsB7A+btt/+NZ5/9LU45ZQyc8UKMChNWpu31C5hOd+GAAz4HgExa/fGP78W8eceO9TLHFKqqW6xe/HpSRaOR2hTceo+edRw6ZzhMMkzwwy4BQkzpdBK6riOTKXqOnm8XXoMvf/e7ZzA4OIj580+GosgYHBzAokXzcdttd3RkPSHax4SNnPxKEPL5PM4//xzsuONOOPzwuWOxtAkB6kk1MlLCxo05lEoyeJ6MXKcNxQyj4+GHf45isdB0uGddd0SHXerGsMtEIubamOwXJGJKQtfRUWICvDdozjjjLDz44CO4556f4nvfuwV9ff0hMU1QTFhy8iNBGBwcxP/8z3zsuOPOuOiiS8d6iRMGlKjMnlSapmHp0stx9913oVQqg+P8vdWqqqFUKiOTIQJJRVEtRNVqGw0lJoZhkMmUOkpMgHWD5tRTT8AhhxxmbND885//6OhjhwgWE9aVYGBgAxYtmo8VK+5FNBrFwoWn48ILLzHm0KuqigULTsEBB3wOp546f9zWOVGxYsVtWLXqMSxfvgK77TYdgsBD1/W2B4jS+X50pLufsVkMoyOVSoFhGIyMFFtqhN6SEDb+OmPCkhPQ3Mpi/fr1WLLkQuyww07G8dOn77pFR1BmfPjhBxAEAZMmTQaAmtWLUHNQqBMVrS+1Upy2j82yj5ki0JFKJcGyLDKZoufo+C0ZITk5Y0KTU6fgpZ+ieOGFP+Cmm27Az3/+yzFeYWcRpCeVfczU+++/j0cffRRz5hyAWbNmIZMphcTkgZCcnDFhd+s6BS/9FMXmrCDWdQaViopKRQVQNogqmYwDiLfkSUX7/ehA0Ww2hz/84TksX34bJk+egnnzjsPxx5/U8ecUYvPDhC2Idwpe+ikKqiDe/FG3ehkcNHtSxU0WLf48qQAdO++8E372s4fwyCO/wrHHngBNG73A06tX7rnn1uLUU0/AKaccj4sv/l9ks/5HWoWYuNjiIqdQQdwMhKiqVRVABaLIQRQ5xOMxJBIMZFmpqcqrDhGVjmQyAZ7nkckUMWnSFBxzzAmjXtGW3CmwpWOLi5xCBbF/+PGkohFVIpGojc0qQlWDS4XDToEtF1scOXnpp8wK4gsuOMdQEG/psHtSqSrxpOrp6UJ3dwqiyGNkJFhiArzNCp06Bej1EJs2tjhyChXEowclqqGhAkZGiHleNlsKnJiAsFNgS8YWR06dVhB7FW/fe+9dnH32mTjllOPxrW+dvckXbwkxlSHLndELhJ0CWy62SJ1Tp0BV7ebi7eWXX20Ub3VdxwknHI1zzjkf++33GSxffit0XceiRYvHeeUTF1tCp0Coc3LGFrdb10l4GZ29+eY/EY1GDSfQk08+DblcftzWuynAy8xw/fr1eOutN6FpGtau/S2AsFNgc0FITgHCS6bw4Yfvo6enF1dddRneeutNbL/9jjjvvAvGY6mbFJqZGU6fPgPPPffX8VhWiA5ji6s5dRJexVtVVfG3v72EefOOxb33Poitt56KW2+9aSyXGCLEJoOQnAKEV/G2p6cX06Z9AtOnzwAAHHzwYXjjjdcbzhMiRIiQnAKFl0xh1qzdMTIyjLfe+hcA4Pnnf49ddpk+XssNEWJCI9ytCxjNbF6mT5+B119/DTfffANKpTImTZqESy+9At3dPeO97BDjiHC3zhkhOW0i8LJ5efPNf+J737sGsixj8uTJuPTSK5FMhh/6TQEhOTkjTOs2AfgZFHnLLctwxhln4d57H8A222yLBx64f5xW6w0voepbb72J+fNPxnHHfRXXXXclFCW4QQshNh2E5LQJwI/Ni6ZpKBYLAEiPmSRJ47FUT/gh2iuuuBTnnnsBHnzwEei6jlWrHhufxYYYV4TktAnAq/kVAM4++zxcd91V+PKXD8Nf//pnHHXU0WO9TF/wItp16z5GpVLBzJnEruaLXzwCv/vd0+O13BDjiJCcNgF46acqlTKuu+5K3HLLbXj88afwla/Mw1VXXTaWS/QNL6L1Q8Qhtgw0VYiHhbqJge23/wRefPFF4/0olbL4xCemGtdfffUdxONRzJlDZAtnnHEK7rzz9gn5/sViIhhGNdaWTEYQi4nG9XQ6CkHgjOvFYtxyPcSWgzBy2gTwmc98Bn/84x8xNDSEUqmENWvW4IADDjBu33bbbbFu3Tq8/Tap3TzzzDOYNWtiunhOnjwZg4ODxvUNGzZg0qRJrrcPDAxYbg+x5SAkp00AkydPxnnnnYeTTz4ZRx11FObOnYvdd98dCxYswN///nek02lce+21OPfcc3HEEUfg4YcfxjXXXDPey3aEF9FOnToVkiThpZdeAgA89thjlttDbDloqnMKEaITWLVqFW6//XbIsox58+ZhwYIFWLBgARYvXoxZs2bhn//8J5YsWYJCoYAZM2bg2muvhSiK473sEGOMkJxChAgxIRGmdSFChJiQCMkpRIgQExIhOYUIEWJCIiSnECFCTEiE5BQiRIgJiZCcQoQIMSERklOIECEmJP4/uij4BsdHgT0AAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {
"filenames": {
- "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_78_0.png"
- },
- "needs_background": "light"
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter3_184_0.png"
+ }
},
"output_type": "display_data"
}
@@ -2428,7 +4527,7 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 33,
"metadata": {
"collapsed": false,
"editable": true
@@ -2441,7 +4540,7 @@
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
- "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mscipy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmisc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimread\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+ "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mscipy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmisc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mimread\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mNameError\u001b[0m: name 'scipy' is not defined"
]
}
@@ -2467,6 +4566,7 @@
},
"outputs": [],
"source": [
+ "\"\"\"\n",
"import numpy as np\n",
"from imageio import imread\n",
"import matplotlib.pyplot as plt\n",
@@ -2481,7 +4581,8 @@
"plt.imshow(terrain1, cmap='gray')\n",
"plt.xlabel('X')\n",
"plt.ylabel('Y')\n",
- "plt.show()"
+ "plt.show()\n",
+ "\"\"\""
]
},
{
diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.py b/doc/LectureNotes/_build/jupyter_execute/chapter3.py
index c827ef0fe..1b4dd4f8f 100644
--- a/doc/LectureNotes/_build/jupyter_execute/chapter3.py
+++ b/doc/LectureNotes/_build/jupyter_execute/chapter3.py
@@ -1167,6 +1167,969 @@ plt.ylabel('MSE')
plt.legend()
plt.show()
+## More on Rescaling data
+
+We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.
+
+When you are comparing your own code with for example **Scikit-Learn**'s
+library, there are some technicalities to keep in mind. The examples
+here demonstrate some of these aspects with potential pitfalls.
+
+The discussion here focuses on the role of the intercept, how we can
+set up the design matrix, what scaling we should use and other topics
+which tend confuse us.
+
+The intercept can be interpreted as the expected value of our
+target/output variables when all other predictors are set to zero.
+Thus, if we cannot assume that the expected outputs/targets are zero
+when all predictors are zero (the columns in the design matrix), it
+may be a bad idea to implement a model which penalizes the intercept.
+Furthermore, in for example Ridge and Lasso regression, the default solutions
+from the library **Scikit-Learn** (when not shrinking $\beta_0$) for the unknown parameters
+$\boldsymbol{\beta}$, are derived under the assumption that both $\boldsymbol{y}$ and
+$\boldsymbol{X}$ are zero centered, that is we subtract the mean values.
+
+
+If our predictors represent different scales, then it is important to
+standardize the design matrix $\boldsymbol{X}$ by subtracting the mean of each
+column from the corresponding column and dividing the column with its
+standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
+the results may differ.
+
+The
+[Standadscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)
+function in **Scikit-Learn** does this for us. For the data sets we
+have been studying in our various examples, the data are in many cases
+already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
+survey of your data, with a critical assessment of them in case you need to scale the data.
+
+If you need to scale the data, not doing so will give an *unfair*
+penalization of the parameters since their magnitude depends on the
+scale of their corresponding predictor.
+
+Suppose as an example that you
+you have an input variable given by the heights of different persons.
+Human height might be measured in inches or meters or
+kilometers. If measured in kilometers, a standard linear regression
+model with this predictor would probably give a much bigger
+coefficient term, than if measured in millimeters.
+This can clearly lead to problems in evaluating the cost/loss functions.
+
+
+
+Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
+on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows
+
+"""
+#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+"""
+
+Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as *zero centering*. For
+simplicity, we will focus on ordinary regression, as done in the above example.
+
+The cost/loss function for regression is
+
+$$
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+$$
+
+Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.
+
+What we have done is to single out the $\beta_0$ term in the definition of the mean squared error (MSE).
+The design matrix
+$X$ does in this case not contain any intercept column.
+When we take the derivative with respect to $\beta_0$, we want the derivative to obey
+
+$$
+\frac{\partial C}{\partial \beta_j} = 0,
+$$
+
+for all $j$. For $\beta_0$ we have
+
+$$
+\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
+$$
+
+Multiplying away the constant $2/n$, we obtain
+
+$$
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
+$$
+
+We assume
+that every column of $\boldsymbol{X}$ is centered, which we can do by subtracting the mean,
+
+X = X - np.mean(X,axis=0)
+
+This means that we need to rewrite $X_{ij}$ as $\tilde{X}_{ij}=X_{ij}-\mu_j$, where
+
+$$
+\mu_j = \frac{1}{n}\sum_{i=0}^{n-1}X_{ij}.
+$$
+
+Let us special first to the case where we have only two parameters $\beta_0$ and $\beta_1$.
+Our result for $\beta_0$ simplifies then to
+
+$$
+n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
+$$
+
+Assuming that the matrix elements $X_{i1}$ are centered, what we have is
+
+$$
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} \left(X_{i1}-\mu_{1}\right),
+$$
+
+where
+
+$$
+\mu_1=\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1},
+$$
+
+and if we define the mean value of the outputs as
+
+$$
+\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
+$$
+
+we have
+
+$$
+\beta_0 = \mu_y - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}-\mu_{1}),
+$$
+
+and it is easy to see that the last sum equals zero! This means that we have
+
+$$
+\beta_0 = \mu_y,
+$$
+
+if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of $\beta$.
+We have thus
+
+$$
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\boldsymbol{y}},
+$$
+
+the average value of $\boldsymbol{y}$.
+
+Replacing $y_i$ with $y_i - \beta_0 = y_i - \overline{\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)
+
+$$
+C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
+$$
+
+If we minimize with respect to $\boldsymbol{\beta}$ we have then
+
+$$
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
+$$
+
+where $\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}$
+and $\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}$.
+
+For Ridge regression we need to add $\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}$ to the cost function and get then
+
+$$
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
+$$
+
+What does this mean? And why do we insist on all this? Let us look at some examples.
+
+
+This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.
+Note also that we do not split the data into training and test.
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+from sklearn.linear_model import LinearRegression
+
+
+np.random.seed(2021)
+
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+def fit_beta(X, y):
+ return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+
+true_beta = [2, 0.5, 3.7]
+
+x = np.linspace(0, 1, 11)
+y = np.sum(
+ np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0
+) + 0.1 * np.random.normal(size=len(x))
+
+degree = 3
+X = np.zeros((len(x), degree))
+
+# Include the intercept in the design matrix
+for p in range(degree):
+ X[:, p] = x ** p
+
+beta = fit_beta(X, y)
+
+# Intercept is included in the design matrix
+skl = LinearRegression(fit_intercept=False).fit(X, y)
+
+print(f"True beta: {true_beta}")
+print(f"Fitted beta: {beta}")
+print(f"Sklearn fitted beta: {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column")
+print(MSE(y,ypredictOwn))
+print(f"MSE with intercept column from SKL")
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x, y, label="Data")
+plt.plot(x, X @ beta, label="Fit")
+plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
+
+
+# Do not include the intercept in the design matrix
+X = np.zeros((len(x), degree - 1))
+
+for p in range(degree - 1):
+ X[:, p] = x ** (p + 1)
+
+# Intercept is not included in the design matrix
+skl = LinearRegression(fit_intercept=True).fit(X, y)
+
+# Use centered values for X and y when computing coefficients
+y_offset = np.average(y, axis=0)
+X_offset = np.average(X, axis=0)
+
+beta = fit_beta(X - X_offset, y - y_offset)
+intercept = np.mean(y_offset - X_offset @ beta)
+
+print(f"Manual intercept: {intercept}")
+print(f"Fitted beta (wiothout intercept): {beta}")
+print(f"Sklearn intercept: {skl.intercept_}")
+print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept")
+print(MSE(y,ypredictOwn+intercept))
+print(f"MSE with Sklearn intercept")
+print(MSE(y,ypredictSKL))
+
+plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
+plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
+plt.grid()
+plt.legend()
+
+plt.show()
+
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case).
+
+Printing the MSE, we see first that both methods give the same MSE, as
+they should. However, when we move to for example Ridge regression,
+the way we treat the intercept may give a larger or smaller MSE,
+meaning that the MSE can be penalized by the value of the
+intercept. Not including the intercept in the fit, means that the
+regularization term does not include $\beta_0$. For different values
+of $\lambda$, this may lead to differeing MSE values.
+
+To remind the reader, the regularization term, with the intercept in Ridge regression, is given by
+
+$$
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
+$$
+
+but when we take out the intercept, this equation becomes
+
+$$
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
+$$
+
+For Lasso regression we have
+
+$$
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
+$$
+
+It means that, when scaling the design matrix and the outputs/targets,
+by subtracting the mean values, we have an optimization problem which
+is not penalized by the intercept. The MSE value can then be smaller
+since it focuses only on the remaining quantities. If we however bring
+back the intercept, we will get a MSE which then contains the
+intercept.
+
+
+Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known vanilla data set.
+
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree))
+#We include explicitely the intercept column
+for degree in range(Maxpolydegree):
+ X[:,degree] = x**degree
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+p = Maxpolydegree
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 2, nlambdas)
+for i in range(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+ # Note: we include the intercept column and no scaling
+ RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+ RegRidge.fit(X_train,y_train)
+ # and then make the prediction
+ ytildeOwnRidge = X_train @ OwnRidgeBeta
+ ypredictOwnRidge = X_test @ OwnRidgeBeta
+ ytildeRidge = RegRidge.predict(X_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta)
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.
+We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
+What happens if we do not include the intercept in our fit?
+Let us see how we can change this code by zero centering.
+
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+from sklearn.preprocessing import StandardScaler
+
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(315)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree-1))
+
+for degree in range(1,Maxpolydegree): #No intercept column
+ X[:,degree-1] = x**(degree)
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean = np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
+X_test_scaled = X_test - X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler = np.mean(y_train)
+y_train_scaled = y_train - y_scaler
+
+p = Maxpolydegree-1
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 2, nlambdas)
+for i in range(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
+ intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
+ RegRidge = linear_model.Ridge(lmb)
+ RegRidge.fit(X_train,y_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta) #Intercept is given by mean of target variable
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print('Intercept from own implementation:')
+ print(intercept_)
+ print('Intercept from Scikit-Learn Ridge implementation')
+ print(RegRidge.intercept_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+We see here, when compared to the code which includes explicitely the
+intercept column, that our MSE value is actually smaller. This is
+because the regularization term does not include the intercept value
+$\beta_0$ in the fitting. This applies to Lasso regularization as
+well. It means that our optimization is now done only with the
+centered matrix and/or vector that enter the fitting procedure. Note
+also that the problem with the intercept occurs mainly in these type
+of polynomial fitting problem.
+
+The next example is indeed an example where all these discussions about the role of intercept are not present.
+
+## More complicated Example: The Ising model
+
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant $J$ is given by
+
+
+
+
+$$
+\begin{equation}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\label{_auto1} \tag{1}
+\end{equation}
+$$
+
+where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins
+in the system is determined by $L$. For the one-dimensional system
+there is no phase transition.
+
+We will look at a system of $L = 40$ spins with a coupling constant of
+$J = 1$. To get enough training data we will generate 10000 states
+with their respective energies.
+
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+ energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+A more general form for the one-dimensional Ising model is
+
+
+
+
+$$
+\begin{equation}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\label{_auto2} \tag{2}
+\end{equation}
+$$
+
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+
+
+
+
+$$
+\begin{equation}
+ \boldsymbol{H} = \boldsymbol{X} J,
+\label{_auto3} \tag{3}
+\end{equation}
+$$
+
+where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the
+elements $-J_{jk}$. This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+
+
+
+$$
+\begin{equation}
+ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\label{_auto4} \tag{4}
+\end{equation}
+$$
+
+We split the data in training and test data as discussed in the previous example
+
+X = np.zeros((n, L ** 2))
+for i in range(n):
+ X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+In the ordinary least squares method we choose the cost function
+
+
+
+
+$$
+\begin{equation}
+ C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\label{_auto5} \tag{5}
+\end{equation}
+$$
+
+We then find the extremal point of $C$ by taking the derivative with respect to $\boldsymbol{\beta}$ as discussed above.
+This yields the expression for $\boldsymbol{\beta}$ to be
+
+$$
+\boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
+
+which immediately imposes some requirements on $\boldsymbol{X}$ as there must exist
+an inverse of $\boldsymbol{X}^T \boldsymbol{X}$. If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of $\boldsymbol{X}$ consists of $1$. We do this here
+
+X_train_own = np.concatenate(
+ (np.ones(len(X_train))[:, np.newaxis], X_train),
+ axis=1
+)
+X_test_own = np.concatenate(
+ (np.ones(len(X_test))[:, np.newaxis], X_test),
+ axis=1
+)
+
+Doing the inversion directly turns out to be a bad idea since the matrix
+$\boldsymbol{X}^T\boldsymbol{X}$ is singular. An alternative approach is to use the **singular
+value decomposition**. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for $\boldsymbol{\beta}$ as
+
+$$
+\boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+where the pseudoinverse of $\boldsymbol{X}$ is given by
+
+$$
+\boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+Using singular value decomposition we can decompose the matrix $\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T$,
+where $\boldsymbol{U}$ and $\boldsymbol{V}$ are orthogonal(unitary) matrices and $\boldsymbol{\Sigma}$ contains the singular values (more details below).
+where $X^{+} = V\Sigma^{+} U^T$. This reduces the equation for
+$\omega$ to
+
+
+
+
+$$
+\begin{equation}
+ \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\label{_auto6} \tag{6}
+\end{equation}
+$$
+
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as $\mathcal{O}(n^3)$, where $n$ is the number of elements in a
+general matrix. Instead, doing $QR$-factorization and solving the
+linear system as an equation would reduce this down to
+$\mathcal{O}(n^2)$ operations.
+
+def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+ u, s, v = scl.svd(x)
+ return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
+beta = ols_svd(X_train_own,y_train)
+
+When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here
+
+J = beta[1:].reshape(L, L)
+
+A way of looking at the coefficients in $J$ is to plot the matrices as images.
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J, **cmap_args)
+plt.title("OLS", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
+It is interesting to note that OLS
+considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as
+valid matrix elements for $J$.
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+
+
+
+
+Let us now
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data. The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant $J$ is
+given by
+
+
+
+
+$$
+\begin{equation}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\label{_auto7} \tag{7}
+\end{equation}
+$$
+
+where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.
+
+We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies.
+
+import numpy as np
+import matplotlib.pyplot as plt
+from mpl_toolkits.axes_grid1 import make_axes_locatable
+import seaborn as sns
+import scipy.linalg as scl
+from sklearn.model_selection import train_test_split
+import sklearn.linear_model as skl
+import tqdm
+sns.set(color_codes=True)
+cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')
+
+L = 40
+n = int(1e4)
+
+spins = np.random.choice([-1, 1], size=(n, L))
+J = 1.0
+
+energies = np.zeros(n)
+
+for i in range(n):
+ energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+
+A more general form for the one-dimensional Ising model is
+
+
+
+
+$$
+\begin{equation}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\label{_auto8} \tag{8}
+\end{equation}
+$$
+
+Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+
+
+
+
+$$
+\begin{equation}
+ H = X J,
+\label{_auto9} \tag{9}
+\end{equation}
+$$
+
+where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the
+elements $-J_{jk}$. This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
+
+
+
+
+$$
+\begin{equation}
+ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
+\label{_auto10} \tag{10}
+\end{equation}
+$$
+
+We organize the data as we did above
+
+X = np.zeros((n, L ** 2))
+for i in range(n):
+ X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)
+
+X_train_own = np.concatenate(
+ (np.ones(len(X_train))[:, np.newaxis], X_train),
+ axis=1
+)
+
+X_test_own = np.concatenate(
+ (np.ones(len(X_test))[:, np.newaxis], X_test),
+ axis=1
+)
+
+We will do all fitting with **Scikit-Learn**,
+
+clf = skl.LinearRegression().fit(X_train, y_train)
+
+When extracting the $J$-matrix we make sure to remove the intercept
+
+J_sk = clf.coef_.reshape(L, L)
+
+And then we plot the results
+
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_sk, **cmap_args)
+plt.title("LinearRegression from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+plt.show()
+
+The results agree perfectly with our previous discussion where we used our own code.
+
+
+Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a **regularizer**. This
+involves a new cost function which leads to a new estimate for the
+weights $\boldsymbol{\beta}$. This results in a penalized regression problem. The
+cost function is given by
+
+6
+0
+
+<
+<
+<
+!
+!
+M
+A
+T
+H
+_
+B
+L
+O
+C
+K
+
+_lambda = 0.1
+clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)
+J_ridge_sk = clf_ridge.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_ridge_sk, **cmap_args)
+plt.title("Ridge from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
+In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function.
+
+
+
+
+$$
+\begin{equation}
+ C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
+\label{_auto12} \tag{12}
+\end{equation}
+$$
+
+Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**.
+
+clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
+J_lasso_sk = clf_lasso.coef_.reshape(L, L)
+fig = plt.figure(figsize=(20, 14))
+im = plt.imshow(J_lasso_sk, **cmap_args)
+plt.title("Lasso from Scikit-learn", fontsize=18)
+plt.xticks(fontsize=18)
+plt.yticks(fontsize=18)
+cb = fig.colorbar(im)
+cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
+
+plt.show()
+
+It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+$J_{j, j + 1} = -1$.
+
+
+
+
+We see how the different models perform for a different set of values for $\lambda$.
+
+lambdas = np.logspace(-4, 5, 10)
+
+train_errors = {
+ "ols_sk": np.zeros(lambdas.size),
+ "ridge_sk": np.zeros(lambdas.size),
+ "lasso_sk": np.zeros(lambdas.size)
+}
+
+test_errors = {
+ "ols_sk": np.zeros(lambdas.size),
+ "ridge_sk": np.zeros(lambdas.size),
+ "lasso_sk": np.zeros(lambdas.size)
+}
+
+plot_counter = 1
+
+fig = plt.figure(figsize=(32, 54))
+
+for i, _lambda in enumerate(tqdm.tqdm(lambdas)):
+ for key, method in zip(
+ ["ols_sk", "ridge_sk", "lasso_sk"],
+ [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]
+ ):
+ method = method.fit(X_train, y_train)
+
+ train_errors[key][i] = method.score(X_train, y_train)
+ test_errors[key][i] = method.score(X_test, y_test)
+
+ omega = method.coef_.reshape(L, L)
+
+ plt.subplot(10, 5, plot_counter)
+ plt.imshow(omega, **cmap_args)
+ plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda))
+ plot_counter += 1
+
+plt.show()
+
+We see that LASSO reaches a good solution for low
+values of $\lambda$, but will "wither" when we increase $\lambda$ too
+much. Ridge is more stable over a larger range of values for
+$\lambda$, but eventually also fades away.
+
+
+To determine which value of $\lambda$ is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
+
+fig = plt.figure(figsize=(20, 14))
+
+colors = {
+ "ols_sk": "r",
+ "ridge_sk": "y",
+ "lasso_sk": "c"
+}
+
+for key in train_errors:
+ plt.semilogx(
+ lambdas,
+ train_errors[key],
+ colors[key],
+ label="Train {0}".format(key),
+ linewidth=4.0
+ )
+
+for key in test_errors:
+ plt.semilogx(
+ lambdas,
+ test_errors[key],
+ colors[key] + "--",
+ label="Test {0}".format(key),
+ linewidth=4.0
+ )
+plt.legend(loc="best", fontsize=18)
+plt.xlabel(r"$\lambda$", fontsize=18)
+plt.ylabel(r"$R^2$", fontsize=18)
+plt.tick_params(labelsize=18)
+plt.show()
+
+From the above figure we can see that LASSO with $\lambda = 10^{-2}$
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of $\lambda$.
+
+
+
+
+
+
## Exercises and Projects
@@ -1439,6 +2402,7 @@ scipy.misc.imread
Here is a simple part of a Python code which reads and plots the data
from such files
+"""
import numpy as np
from imageio import imread
import matplotlib.pyplot as plt
@@ -1454,6 +2418,7 @@ plt.imshow(terrain1, cmap='gray')
plt.xlabel('X')
plt.ylabel('Y')
plt.show()
+"""
If you should have problems in downloading the digital terrain data,
we provide two examples under the data folder of project 1. One is
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index 413c54051..18cd038f8 100644
--- a/doc/LectureNotes/chapter3.ipynb
+++ b/doc/LectureNotes/chapter3.ipynb
@@ -1598,6 +1598,1604 @@
"cell_type": "markdown",
"metadata": {},
"source": [
+ "## More on Rescaling data\n",
+ "\n",
+ "We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases.\n",
+ "\n",
+ "When you are comparing your own code with for example **Scikit-Learn**'s\n",
+ "library, there are some technicalities to keep in mind. The examples\n",
+ "here demonstrate some of these aspects with potential pitfalls.\n",
+ "\n",
+ "The discussion here focuses on the role of the intercept, how we can\n",
+ "set up the design matrix, what scaling we should use and other topics\n",
+ "which tend confuse us.\n",
+ "\n",
+ "The intercept can be interpreted as the expected value of our\n",
+ "target/output variables when all other predictors are set to zero.\n",
+ "Thus, if we cannot assume that the expected outputs/targets are zero\n",
+ "when all predictors are zero (the columns in the design matrix), it\n",
+ "may be a bad idea to implement a model which penalizes the intercept.\n",
+ "Furthermore, in for example Ridge and Lasso regression, the default solutions\n",
+ "from the library **Scikit-Learn** (when not shrinking $\\beta_0$) for the unknown parameters\n",
+ "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n",
+ "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n",
+ "\n",
+ "\n",
+ "If our predictors represent different scales, then it is important to\n",
+ "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n",
+ "column from the corresponding column and dividing the column with its\n",
+ "standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,\n",
+ "the results may differ. \n",
+ "\n",
+ "The\n",
+ "[Standadscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)\n",
+ "function in **Scikit-Learn** does this for us. For the data sets we\n",
+ "have been studying in our various examples, the data are in many cases\n",
+ "already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a\n",
+ "survey of your data, with a critical assessment of them in case you need to scale the data.\n",
+ "\n",
+ "If you need to scale the data, not doing so will give an *unfair*\n",
+ "penalization of the parameters since their magnitude depends on the\n",
+ "scale of their corresponding predictor.\n",
+ "\n",
+ "Suppose as an example that you \n",
+ "you have an input variable given by the heights of different persons.\n",
+ "Human height might be measured in inches or meters or\n",
+ "kilometers. If measured in kilometers, a standard linear regression\n",
+ "model with this predictor would probably give a much bigger\n",
+ "coefficient term, than if measured in millimeters.\n",
+ "This can clearly lead to problems in evaluating the cost/loss functions.\n",
+ "\n",
+ "\n",
+ "\n",
+ "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n",
+ "on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\"\"\"\n",
+ "#Model training, we compute the mean value of y and X\n",
+ "y_train_mean = np.mean(y_train)\n",
+ "X_train_mean = np.mean(X_train,axis=0)\n",
+ "X_train = X_train - X_train_mean\n",
+ "y_train = y_train - y_train_mean\n",
+ "\n",
+ "# The we fit our model with the training data\n",
+ "trained_model = some_model.fit(X_train,y_train)\n",
+ "\n",
+ "\n",
+ "#Model prediction, we need also to transform our data set used for the prediction.\n",
+ "X_test = X_test - X_train_mean #Use mean from training data\n",
+ "y_pred = trained_model(X_test)\n",
+ "y_pred = y_pred + y_train_mean\n",
+ "\"\"\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let us try to understand what this may imply mathematically when we\n",
+ "subtract the mean values, also known as *zero centering*. For\n",
+ "simplicity, we will focus on ordinary regression, as done in the above example.\n",
+ "\n",
+ "The cost/loss function for regression is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.\n",
+ "\n",
+ "What we have done is to single out the $\\beta_0$ term in the definition of the mean squared error (MSE).\n",
+ "The design matrix\n",
+ "$X$ does in this case not contain any intercept column.\n",
+ "When we take the derivative with respect to $\\beta_0$, we want the derivative to obey"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "for all $j$. For $\\beta_0$ we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Multiplying away the constant $2/n$, we obtain"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We assume \n",
+ "that every column of $\\boldsymbol{X}$ is centered, which we can do by subtracting the mean,"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = X - np.mean(X,axis=0)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This means that we need to rewrite $X_{ij}$ as $\\tilde{X}_{ij}=X_{ij}-\\mu_j$, where"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_j = \\frac{1}{n}\\sum_{i=0}^{n-1}X_{ij}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let us special first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n",
+ "Our result for $\\beta_0$ simplifies then to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Assuming that the matrix elements $X_{i1}$ are centered, what we have is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} \\left(X_{i1}-\\mu_{1}\\right),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_1=\\frac{1}{n}\\sum_{i=0}^{n-1} (X_{i1},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "and if we define the mean value of the outputs as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\mu_y - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} (X_{i1}-\\mu_{1}),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "and it is easy to see that the last sum equals zero! This means that we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\mu_y,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of $\\beta$.\n",
+ "We have thus"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1} y_i = \\overline{\\boldsymbol{y}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "the average value of $\\boldsymbol{y}$.\n",
+ "\n",
+ "Replacing $y_i$ with $y_i - \\beta_0 = y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
+ "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n",
+ "\n",
+ "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "What does this mean? And why do we insist on all this? Let us look at some examples.\n",
+ "\n",
+ "\n",
+ "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n",
+ "Note also that we do not split the data into training and test."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "from sklearn.linear_model import LinearRegression\n",
+ "\n",
+ "\n",
+ "np.random.seed(2021)\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "\n",
+ "\n",
+ "def fit_beta(X, y):\n",
+ " return np.linalg.pinv(X.T @ X) @ X.T @ y\n",
+ "\n",
+ "\n",
+ "true_beta = [2, 0.5, 3.7]\n",
+ "\n",
+ "x = np.linspace(0, 1, 11)\n",
+ "y = np.sum(\n",
+ " np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0\n",
+ ") + 0.1 * np.random.normal(size=len(x))\n",
+ "\n",
+ "degree = 3\n",
+ "X = np.zeros((len(x), degree))\n",
+ "\n",
+ "# Include the intercept in the design matrix\n",
+ "for p in range(degree):\n",
+ " X[:, p] = x ** p\n",
+ "\n",
+ "beta = fit_beta(X, y)\n",
+ "\n",
+ "# Intercept is included in the design matrix\n",
+ "skl = LinearRegression(fit_intercept=False).fit(X, y)\n",
+ "\n",
+ "print(f\"True beta: {true_beta}\")\n",
+ "print(f\"Fitted beta: {beta}\")\n",
+ "print(f\"Sklearn fitted beta: {skl.coef_}\")\n",
+ "ypredictOwn = X @ beta\n",
+ "ypredictSKL = skl.predict(X)\n",
+ "print(f\"MSE with intercept column\")\n",
+ "print(MSE(y,ypredictOwn))\n",
+ "print(f\"MSE with intercept column from SKL\")\n",
+ "print(MSE(y,ypredictSKL))\n",
+ "\n",
+ "\n",
+ "plt.figure()\n",
+ "plt.scatter(x, y, label=\"Data\")\n",
+ "plt.plot(x, X @ beta, label=\"Fit\")\n",
+ "plt.plot(x, skl.predict(X), label=\"Sklearn (fit_intercept=False)\")\n",
+ "\n",
+ "\n",
+ "# Do not include the intercept in the design matrix\n",
+ "X = np.zeros((len(x), degree - 1))\n",
+ "\n",
+ "for p in range(degree - 1):\n",
+ " X[:, p] = x ** (p + 1)\n",
+ "\n",
+ "# Intercept is not included in the design matrix\n",
+ "skl = LinearRegression(fit_intercept=True).fit(X, y)\n",
+ "\n",
+ "# Use centered values for X and y when computing coefficients\n",
+ "y_offset = np.average(y, axis=0)\n",
+ "X_offset = np.average(X, axis=0)\n",
+ "\n",
+ "beta = fit_beta(X - X_offset, y - y_offset)\n",
+ "intercept = np.mean(y_offset - X_offset @ beta)\n",
+ "\n",
+ "print(f\"Manual intercept: {intercept}\")\n",
+ "print(f\"Fitted beta (wiothout intercept): {beta}\")\n",
+ "print(f\"Sklearn intercept: {skl.intercept_}\")\n",
+ "print(f\"Sklearn fitted beta (without intercept): {skl.coef_}\")\n",
+ "ypredictOwn = X @ beta\n",
+ "ypredictSKL = skl.predict(X)\n",
+ "print(f\"MSE with Manual intercept\")\n",
+ "print(MSE(y,ypredictOwn+intercept))\n",
+ "print(f\"MSE with Sklearn intercept\")\n",
+ "print(MSE(y,ypredictSKL))\n",
+ "\n",
+ "plt.plot(x, X @ beta + intercept, \"--\", label=\"Fit (manual intercept)\")\n",
+ "plt.plot(x, skl.predict(X), \"--\", label=\"Sklearn (fit_intercept=True)\")\n",
+ "plt.grid()\n",
+ "plt.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The intercept is the value of our output/target variable\n",
+ "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n",
+ "\n",
+ "Printing the MSE, we see first that both methods give the same MSE, as\n",
+ "they should. However, when we move to for example Ridge regression,\n",
+ "the way we treat the intercept may give a larger or smaller MSE,\n",
+ "meaning that the MSE can be penalized by the value of the\n",
+ "intercept. Not including the intercept in the fit, means that the\n",
+ "regularization term does not include $\\beta_0$. For different values\n",
+ "of $\\lambda$, this may lead to differeing MSE values. \n",
+ "\n",
+ "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "but when we take out the intercept, this equation becomes"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For Lasso regression we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It means that, when scaling the design matrix and the outputs/targets,\n",
+ "by subtracting the mean values, we have an optimization problem which\n",
+ "is not penalized by the intercept. The MSE value can then be smaller\n",
+ "since it focuses only on the remaining quantities. If we however bring\n",
+ "back the intercept, we will get a MSE which then contains the\n",
+ "intercept.\n",
+ "\n",
+ "\n",
+ "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known vanilla data set."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn import linear_model\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "\n",
+ "\n",
+ "# A seed just to ensure that the random numbers are the same for every run.\n",
+ "# Useful for eventual debugging.\n",
+ "np.random.seed(3155)\n",
+ "\n",
+ "n = 100\n",
+ "x = np.random.rand(n)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+ "\n",
+ "Maxpolydegree = 20\n",
+ "X = np.zeros((n,Maxpolydegree))\n",
+ "#We include explicitely the intercept column\n",
+ "for degree in range(Maxpolydegree):\n",
+ " X[:,degree] = x**degree\n",
+ "# We split the data in test and training data\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+ "\n",
+ "p = Maxpolydegree\n",
+ "I = np.eye(p,p)\n",
+ "# Decide which values of lambda to use\n",
+ "nlambdas = 6\n",
+ "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+ "MSERidgePredict = np.zeros(nlambdas)\n",
+ "lambdas = np.logspace(-4, 2, nlambdas)\n",
+ "for i in range(nlambdas):\n",
+ " lmb = lambdas[i]\n",
+ " OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n",
+ " # Note: we include the intercept column and no scaling\n",
+ " RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n",
+ " RegRidge.fit(X_train,y_train)\n",
+ " # and then make the prediction\n",
+ " ytildeOwnRidge = X_train @ OwnRidgeBeta\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta\n",
+ " ytildeRidge = RegRidge.predict(X_train)\n",
+ " ypredictRidge = RegRidge.predict(X_test)\n",
+ " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+ " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+ " print(\"Beta values for own Ridge implementation\")\n",
+ " print(OwnRidgeBeta)\n",
+ " print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+ " print(RegRidge.coef_)\n",
+ " print(\"MSE values for own Ridge implementation\")\n",
+ " print(MSEOwnRidgePredict[i])\n",
+ " print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+ " print(MSERidgePredict[i])\n",
+ "\n",
+ "# Now plot the results\n",
+ "plt.figure()\n",
+ "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')\n",
+ "plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n",
+ "\n",
+ "plt.xlabel('log10(lambda)')\n",
+ "plt.ylabel('MSE')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n",
+ "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n",
+ "What happens if we do not include the intercept in our fit?\n",
+ "Let us see how we can change this code by zero centering."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn import linear_model\n",
+ "from sklearn.preprocessing import StandardScaler\n",
+ "\n",
+ "def MSE(y_data,y_model):\n",
+ " n = np.size(y_model)\n",
+ " return np.sum((y_data-y_model)**2)/n\n",
+ "# A seed just to ensure that the random numbers are the same for every run.\n",
+ "# Useful for eventual debugging.\n",
+ "np.random.seed(315)\n",
+ "\n",
+ "n = 100\n",
+ "x = np.random.rand(n)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
+ "\n",
+ "Maxpolydegree = 20\n",
+ "X = np.zeros((n,Maxpolydegree-1))\n",
+ "\n",
+ "for degree in range(1,Maxpolydegree): #No intercept column\n",
+ " X[:,degree-1] = x**(degree)\n",
+ "\n",
+ "# We split the data in test and training data\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n",
+ "\n",
+ "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n",
+ "X_train_mean = np.mean(X_train,axis=0)\n",
+ "#Center by removing mean from each feature\n",
+ "X_train_scaled = X_train - X_train_mean \n",
+ "X_test_scaled = X_test - X_train_mean\n",
+ "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)\n",
+ "#Remove the intercept from the training data.\n",
+ "y_scaler = np.mean(y_train) \n",
+ "y_train_scaled = y_train - y_scaler \n",
+ "\n",
+ "p = Maxpolydegree-1\n",
+ "I = np.eye(p,p)\n",
+ "# Decide which values of lambda to use\n",
+ "nlambdas = 6\n",
+ "MSEOwnRidgePredict = np.zeros(nlambdas)\n",
+ "MSERidgePredict = np.zeros(nlambdas)\n",
+ "\n",
+ "lambdas = np.logspace(-4, 2, nlambdas)\n",
+ "for i in range(nlambdas):\n",
+ " lmb = lambdas[i]\n",
+ " OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)\n",
+ " intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data\n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ \n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n",
+ " RegRidge = linear_model.Ridge(lmb)\n",
+ " RegRidge.fit(X_train,y_train)\n",
+ " ypredictRidge = RegRidge.predict(X_test)\n",
+ " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
+ " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
+ " print(\"Beta values for own Ridge implementation\")\n",
+ " print(OwnRidgeBeta) #Intercept is given by mean of target variable\n",
+ " print(\"Beta values for Scikit-Learn Ridge implementation\")\n",
+ " print(RegRidge.coef_)\n",
+ " print('Intercept from own implementation:')\n",
+ " print(intercept_)\n",
+ " print('Intercept from Scikit-Learn Ridge implementation')\n",
+ " print(RegRidge.intercept_)\n",
+ " print(\"MSE values for own Ridge implementation\")\n",
+ " print(MSEOwnRidgePredict[i])\n",
+ " print(\"MSE values for Scikit-Learn Ridge implementation\")\n",
+ " print(MSERidgePredict[i])\n",
+ "\n",
+ "\n",
+ "# Now plot the results\n",
+ "plt.figure()\n",
+ "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')\n",
+ "plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')\n",
+ "plt.xlabel('log10(lambda)')\n",
+ "plt.ylabel('MSE')\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see here, when compared to the code which includes explicitely the\n",
+ "intercept column, that our MSE value is actually smaller. This is\n",
+ "because the regularization term does not include the intercept value\n",
+ "$\\beta_0$ in the fitting. This applies to Lasso regularization as\n",
+ "well. It means that our optimization is now done only with the\n",
+ "centered matrix and/or vector that enter the fitting procedure. Note\n",
+ "also that the problem with the intercept occurs mainly in these type\n",
+ "of polynomial fitting problem.\n",
+ "\n",
+ "The next example is indeed an example where all these discussions about the role of intercept are not present.\n",
+ "\n",
+ "## More complicated Example: The Ising model\n",
+ "\n",
+ "The one-dimensional Ising model with nearest neighbor interaction, no\n",
+ "external field and a constant coupling constant $J$ is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+ "\\label{_auto1} \\tag{1}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n",
+ "in the system is determined by $L$. For the one-dimensional system\n",
+ "there is no phase transition.\n",
+ "\n",
+ "We will look at a system of $L = 40$ spins with a coupling constant of\n",
+ "$J = 1$. To get enough training data we will generate 10000 states\n",
+ "with their respective energies."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+ "import seaborn as sns\n",
+ "import scipy.linalg as scl\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "import tqdm\n",
+ "sns.set(color_codes=True)\n",
+ "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+ "\n",
+ "L = 40\n",
+ "n = int(1e4)\n",
+ "\n",
+ "spins = np.random.choice([-1, 1], size=(n, L))\n",
+ "J = 1.0\n",
+ "\n",
+ "energies = np.zeros(n)\n",
+ "\n",
+ "for i in range(n):\n",
+ " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we use ordinary least squares\n",
+ "regression to predict the energy for the nearest neighbor\n",
+ "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n",
+ "around. We will use linear regression to fit a value for\n",
+ "the coupling constant to achieve this.\n",
+ "\n",
+ "A more general form for the one-dimensional Ising model is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+ "\\label{_auto2} \\tag{2}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we allow for interactions beyond the nearest neighbors and a state dependent\n",
+ "coupling constant. This latter expression can be formulated as\n",
+ "a matrix-product"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{H} = \\boldsymbol{X} J,\n",
+ "\\label{_auto3} \\tag{3}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n",
+ "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+ "with the form utilized in linear regression, that is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n",
+ "\\label{_auto4} \\tag{4}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We split the data in training and test data as discussed in the previous example"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = np.zeros((n, L ** 2))\n",
+ "for i in range(n):\n",
+ " X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+ "y = energies\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In the ordinary least squares method we choose the cost function"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n",
+ "\\label{_auto5} \\tag{5}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n",
+ "This yields the expression for $\\boldsymbol{\\beta}$ to be"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n",
+ "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n",
+ "intercept, i.e., a constant term, we must make sure that the\n",
+ "first column of $\\boldsymbol{X}$ consists of $1$. We do this here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X_train_own = np.concatenate(\n",
+ " (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+ " axis=1\n",
+ ")\n",
+ "X_test_own = np.concatenate(\n",
+ " (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+ " axis=1\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Doing the inversion directly turns out to be a bad idea since the matrix\n",
+ "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n",
+ "value decomposition**. Using the definition of the Moore-Penrose\n",
+ "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where the pseudoinverse of $\\boldsymbol{X}$ is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Using singular value decomposition we can decompose the matrix $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n",
+ "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n",
+ "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n",
+ "$\\omega$ to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n",
+ "\\label{_auto6} \\tag{6}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Note that solving this equation by actually doing the pseudoinverse\n",
+ "(which is what we will do) is not a good idea as this operation scales\n",
+ "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n",
+ "general matrix. Instead, doing $QR$-factorization and solving the\n",
+ "linear system as an equation would reduce this down to\n",
+ "$\\mathcal{O}(n^2)$ operations."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
+ " u, s, v = scl.svd(x)\n",
+ " return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "beta = ols_svd(X_train_own,y_train)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "J = beta[1:].reshape(L, L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A way of looking at the coefficients in $J$ is to plot the matrices as images."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J, **cmap_args)\n",
+ "plt.title(\"OLS\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It is interesting to note that OLS\n",
+ "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n",
+ "valid matrix elements for $J$.\n",
+ "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n",
+ "this problem can be removed, partly and only with Lasso regression. \n",
+ "\n",
+ "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "Let us now \n",
+ "focus on Ridge and Lasso regression as well. We repeat some of the\n",
+ "basic parts of the Ising model and the setup of the training and test\n",
+ "data. The one-dimensional Ising model with nearest neighbor\n",
+ "interaction, no external field and a constant coupling constant $J$ is\n",
+ "given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = -J \\sum_{k}^L s_k s_{k + 1},\n",
+ "\\label{_auto7} \\tag{7}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n",
+ "\n",
+ "We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n",
+ "import seaborn as sns\n",
+ "import scipy.linalg as scl\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "import sklearn.linear_model as skl\n",
+ "import tqdm\n",
+ "sns.set(color_codes=True)\n",
+ "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n",
+ "\n",
+ "L = 40\n",
+ "n = int(1e4)\n",
+ "\n",
+ "spins = np.random.choice([-1, 1], size=(n, L))\n",
+ "J = 1.0\n",
+ "\n",
+ "energies = np.zeros(n)\n",
+ "\n",
+ "for i in range(n):\n",
+ " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A more general form for the one-dimensional Ising model is"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n",
+ "\\label{_auto8} \\tag{8}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we allow for interactions beyond the nearest neighbors and a more\n",
+ "adaptive coupling matrix. This latter expression can be formulated as\n",
+ "a matrix-product on the form"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " H = X J,\n",
+ "\\label{_auto9} \\tag{9}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n",
+ "elements $-J_{jk}$. This form of writing the energy fits perfectly\n",
+ "with the form utilized in linear regression, viz."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon}.\n",
+ "\\label{_auto10} \\tag{10}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We organize the data as we did above"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "X = np.zeros((n, L ** 2))\n",
+ "for i in range(n):\n",
+ " X[i] = np.outer(spins[i], spins[i]).ravel()\n",
+ "y = energies\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)\n",
+ "\n",
+ "X_train_own = np.concatenate(\n",
+ " (np.ones(len(X_train))[:, np.newaxis], X_train),\n",
+ " axis=1\n",
+ ")\n",
+ "\n",
+ "X_test_own = np.concatenate(\n",
+ " (np.ones(len(X_test))[:, np.newaxis], X_test),\n",
+ " axis=1\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We will do all fitting with **Scikit-Learn**,"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "clf = skl.LinearRegression().fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When extracting the $J$-matrix we make sure to remove the intercept"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "J_sk = clf.coef_.reshape(L, L)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "And then we plot the results"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_sk, **cmap_args)\n",
+ "plt.title(\"LinearRegression from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The results agree perfectly with our previous discussion where we used our own code.\n",
+ "\n",
+ "\n",
+ "Having explored the ordinary least squares we move on to ridge\n",
+ "regression. In ridge regression we include a **regularizer**. This\n",
+ "involves a new cost function which leads to a new estimate for the\n",
+ "weights $\\boldsymbol{\\beta}$. This results in a penalized regression problem. The\n",
+ "cost function is given by"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "6\n",
+ "0\n",
+ " \n",
+ "<\n",
+ "<\n",
+ "<\n",
+ "!\n",
+ "!\n",
+ "M\n",
+ "A\n",
+ "T\n",
+ "H\n",
+ "_\n",
+ "B\n",
+ "L\n",
+ "O\n",
+ "C\n",
+ "K"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "_lambda = 0.1\n",
+ "clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)\n",
+ "J_ridge_sk = clf_ridge.coef_.reshape(L, L)\n",
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_ridge_sk, **cmap_args)\n",
+ "plt.title(\"Ridge from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{equation}\n",
+ " C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\sqrt{\\boldsymbol{\\beta}^T\\boldsymbol{\\beta}}.\n",
+ "\\label{_auto12} \\tag{12}\n",
+ "\\end{equation}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n",
+ "J_lasso_sk = clf_lasso.coef_.reshape(L, L)\n",
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "im = plt.imshow(J_lasso_sk, **cmap_args)\n",
+ "plt.title(\"Lasso from Scikit-learn\", fontsize=18)\n",
+ "plt.xticks(fontsize=18)\n",
+ "plt.yticks(fontsize=18)\n",
+ "cb = fig.colorbar(im)\n",
+ "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "It is quite striking how LASSO breaks the symmetry of the coupling\n",
+ "constant as opposed to ridge and OLS. We get a sparse solution with\n",
+ "$J_{j, j + 1} = -1$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "We see how the different models perform for a different set of values for $\\lambda$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "lambdas = np.logspace(-4, 5, 10)\n",
+ "\n",
+ "train_errors = {\n",
+ " \"ols_sk\": np.zeros(lambdas.size),\n",
+ " \"ridge_sk\": np.zeros(lambdas.size),\n",
+ " \"lasso_sk\": np.zeros(lambdas.size)\n",
+ "}\n",
+ "\n",
+ "test_errors = {\n",
+ " \"ols_sk\": np.zeros(lambdas.size),\n",
+ " \"ridge_sk\": np.zeros(lambdas.size),\n",
+ " \"lasso_sk\": np.zeros(lambdas.size)\n",
+ "}\n",
+ "\n",
+ "plot_counter = 1\n",
+ "\n",
+ "fig = plt.figure(figsize=(32, 54))\n",
+ "\n",
+ "for i, _lambda in enumerate(tqdm.tqdm(lambdas)):\n",
+ " for key, method in zip(\n",
+ " [\"ols_sk\", \"ridge_sk\", \"lasso_sk\"],\n",
+ " [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]\n",
+ " ):\n",
+ " method = method.fit(X_train, y_train)\n",
+ "\n",
+ " train_errors[key][i] = method.score(X_train, y_train)\n",
+ " test_errors[key][i] = method.score(X_test, y_test)\n",
+ "\n",
+ " omega = method.coef_.reshape(L, L)\n",
+ "\n",
+ " plt.subplot(10, 5, plot_counter)\n",
+ " plt.imshow(omega, **cmap_args)\n",
+ " plt.title(r\"%s, $\\lambda = %.4f$\" % (key, _lambda))\n",
+ " plot_counter += 1\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see that LASSO reaches a good solution for low\n",
+ "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n",
+ "much. Ridge is more stable over a larger range of values for\n",
+ "$\\lambda$, but eventually also fades away.\n",
+ "\n",
+ "\n",
+ "To determine which value of $\\lambda$ is best we plot the accuracy of\n",
+ "the models when predicting the training and the testing set. We expect\n",
+ "the accuracy of the training set to be quite good, but if the accuracy\n",
+ "of the testing set is much lower this tells us that we might be\n",
+ "subject to an overfit model. The ideal scenario is an accuracy on the\n",
+ "testing set that is close to the accuracy of the training set."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "fig = plt.figure(figsize=(20, 14))\n",
+ "\n",
+ "colors = {\n",
+ " \"ols_sk\": \"r\",\n",
+ " \"ridge_sk\": \"y\",\n",
+ " \"lasso_sk\": \"c\"\n",
+ "}\n",
+ "\n",
+ "for key in train_errors:\n",
+ " plt.semilogx(\n",
+ " lambdas,\n",
+ " train_errors[key],\n",
+ " colors[key],\n",
+ " label=\"Train {0}\".format(key),\n",
+ " linewidth=4.0\n",
+ " )\n",
+ "\n",
+ "for key in test_errors:\n",
+ " plt.semilogx(\n",
+ " lambdas,\n",
+ " test_errors[key],\n",
+ " colors[key] + \"--\",\n",
+ " label=\"Test {0}\".format(key),\n",
+ " linewidth=4.0\n",
+ " )\n",
+ "plt.legend(loc=\"best\", fontsize=18)\n",
+ "plt.xlabel(r\"$\\lambda$\", fontsize=18)\n",
+ "plt.ylabel(r\"$R^2$\", fontsize=18)\n",
+ "plt.tick_params(labelsize=18)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n",
+ "achieves a very good accuracy on the test set. This by far surpasses the\n",
+ "other models for all values of $\\lambda$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
"## Exercises and Projects\n",
"\n",
"\n",
@@ -1980,6 +3578,7 @@
},
"outputs": [],
"source": [
+ "\"\"\"\n",
"import numpy as np\n",
"from imageio import imread\n",
"import matplotlib.pyplot as plt\n",
@@ -1994,7 +3593,8 @@
"plt.imshow(terrain1, cmap='gray')\n",
"plt.xlabel('X')\n",
"plt.ylabel('Y')\n",
- "plt.show()"
+ "plt.show()\n",
+ "\"\"\""
]
},
{
diff --git a/doc/LectureNotes/schedule.md b/doc/LectureNotes/schedule.md
index a33d7bf3e..8fb9c6ab3 100644
--- a/doc/LectureNotes/schedule.md
+++ b/doc/LectureNotes/schedule.md
@@ -78,6 +78,7 @@ For the reading assignments we use the following abbreviations:
### Week 38 September 20-24
- Lab Wednesday: Work on Project 1
- Lecture Thursday: Classification problems and Logistic Regression, from binary cases to several categories
+ - Video of Lecture at https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureSeptember23.mp4?vrtx=view-as-webpage
- Lecture Friday: Logistic Regression and gradient optimization
- Reading recommendations:
diff --git a/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz b/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz
index 51d029e51..4a81a6d5e 100644
Binary files a/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz and b/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz differ
diff --git a/doc/pub/week38/ipynb/week38.ipynb b/doc/pub/week38/ipynb/week38.ipynb
index d836998bd..2d7f8b9a9 100644
--- a/doc/pub/week38/ipynb/week38.ipynb
+++ b/doc/pub/week38/ipynb/week38.ipynb
@@ -236,10 +236,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
@@ -400,10 +397,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"#Model training, we compute the mean value of y and X\n",
@@ -511,10 +505,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"X = X - np.mean(X,axis=0)"
@@ -722,10 +713,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
@@ -889,10 +877,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
@@ -977,10 +962,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
@@ -1118,10 +1100,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
@@ -1234,10 +1213,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"X = np.zeros((n, L ** 2))\n",
@@ -1301,10 +1277,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"X_train_own = np.concatenate(\n",
@@ -1320,10 +1293,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
@@ -1408,10 +1378,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n",
@@ -1422,10 +1389,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"beta = ols_svd(X_train_own,y_train)"
@@ -1441,10 +1405,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"J = beta[1:].reshape(L, L)"
@@ -1460,10 +1421,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"fig = plt.figure(figsize=(20, 14))\n",
@@ -1529,10 +1487,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
@@ -1638,10 +1593,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"X = np.zeros((n, L ** 2))\n",
@@ -1671,10 +1623,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"clf = skl.LinearRegression().fit(X_train, y_train)"
@@ -1690,10 +1639,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"J_sk = clf.coef_.reshape(L, L)"
@@ -1709,10 +1655,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"fig = plt.figure(figsize=(20, 14))\n",
@@ -1767,10 +1710,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"_lambda = 0.1\n",
@@ -1821,10 +1761,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n",
@@ -1858,10 +1795,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"lambdas = np.logspace(-4, 5, 10)\n",
@@ -1924,10 +1858,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"fig = plt.figure(figsize=(20, 14))\n",
@@ -2118,10 +2049,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"# Common imports\n",
@@ -2192,10 +2120,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])\n",
@@ -2279,10 +2204,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"\"\"\"The sigmoid function (or the logistic curve) is a\n",
@@ -2701,10 +2623,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
@@ -2747,10 +2666,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
@@ -2813,10 +2729,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)"
@@ -2832,10 +2745,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"correlation_matrix = cancerpd.corr().round(1)"
@@ -2858,10 +2768,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
@@ -3678,10 +3585,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
@@ -3716,10 +3620,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"pt.axis(\"equal\")\n",
@@ -3737,10 +3638,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"x = guesses[-1]\n",
@@ -3757,10 +3655,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"def f1d(alpha):\n",
@@ -3782,10 +3677,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"pt.axis(\"equal\")\n",
@@ -4161,10 +4053,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"x = 2*np.random.rand(m,1)\n",
@@ -4331,10 +4220,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"\n",
@@ -4395,10 +4281,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"# Importing various packages\n",
@@ -4482,10 +4365,7 @@
{
"cell_type": "code",
"execution_count": null,
- "metadata": {
- "collapsed": false,
- "editable": true
- },
+ "metadata": {},
"outputs": [],
"source": [
"from random import random, seed\n",
@@ -4553,7 +4433,25 @@
]
}
],
- "metadata": {},
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.8.5"
+ }
+ },
"nbformat": 4,
"nbformat_minor": 4
}
diff --git a/doc/src/week38/._week38-bs000.html b/doc/src/week38/._week38-bs000.html
new file mode 100644
index 000000000..d5042016a
--- /dev/null
+++ b/doc/src/week38/._week38-bs000.html
@@ -0,0 +1,489 @@
+
+
+
+
+
+
+
+
+Data Analysis and Machine Learning: Logistic Regression
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
+our optimization problem is
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.
+$$
+
+or we can state it as
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,
+$$
+
+where we have used the definition of a norm-2 vector, that is
+$$
+\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.
+$$
+
+
+By minimizing the above equation with respect to the parameters
+\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the
+parameters \( \boldsymbol{\beta} \). We can add a regularization parameter \( \lambda \) by
+defining a new cost function to be optimized, that is
+
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2
+$$
+
+
+which leads to the Ridge regression minimization problem where we
+require that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is
+a finite number larger than zero. By defining
+
+$$
+C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,
+$$
+
+
+we have a new optimization equation
+$$
+{\displaystyle \min_{\boldsymbol{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1
+$$
+
+which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.
+
+
+Here we have defined the norm-1 as
+$$
+\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.
+$$
+
+
+When the repetitive splitting of the data set is done randomly,
+samples may accidently end up in a fast majority of the splits in
+either training or test set. Such samples may have an unbalanced
+influence on either model building or prediction evaluation. To avoid
+this \( k \)-fold cross-validation structures the data splitting. The
+samples are divided into \( k \) more or less equally sized exhaustive and
+mutually exclusive subsets. In turn (at each split) one of these
+subsets plays the role of the test set while the union of the
+remaining subsets constitutes the training set. Such a splitting
+warrants a balanced representation of each sample in both training and
+test set over the splits. Still the division into the \( k \) subsets
+involves a degree of randomness. This may be fully excluded when
+choosing \( k=n \). This particular case is referred to as leave-one-out
+cross-validation (LOOCV).
+
+
How to set up the cross-validation for Ridge and/or Lasso
+
+
+
Define a range of interest for the penalty parameter.
+
Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
+
Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as
Evaluate the prediction performance of these models on the test set by \( C[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
+
Repeat the first three steps such that each sample plays the role of the test set once.
+
Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data.
Code Example for Cross-validation and \( k \)-fold Cross-validation
+
+
+The code here uses Ridge regression with cross-validation (CV) resampling and \( k \)-fold CV in order to fit a specific polynomial.
+
+
+
+
importnumpyasnp
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimport KFold
+fromsklearn.linear_modelimport Ridge
+fromsklearn.model_selectionimport cross_val_score
+fromsklearn.preprocessingimport PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+# Generate the data.
+nsamples =100
+x = np.random.randn(nsamples)
+y =3*x**2+ np.random.randn(nsamples)
+
+## Cross-validation on Ridge regression using KFold only
+
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree =6)
+
+# Decide which values of lambda to use
+nlambdas =500
+lambdas = np.logspace(-3, 5, nlambdas)
+
+# Initialize a KFold instance
+k =5
+kfold = KFold(n_splits = k)
+
+# Perform the cross-validation to estimate MSE
+scores_KFold = np.zeros((nlambdas, k))
+
+i =0
+for lmb in lambdas:
+ ridge = Ridge(alpha = lmb)
+ j =0
+ for train_inds, test_inds in kfold.split(x):
+ xtrain = x[train_inds]
+ ytrain = y[train_inds]
+
+ xtest = x[test_inds]
+ ytest = y[test_inds]
+
+ Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
+ ridge.fit(Xtrain, ytrain[:, np.newaxis])
+
+ Xtest = poly.fit_transform(xtest[:, np.newaxis])
+ ypred = ridge.predict(Xtest)
+
+ scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
+
+ j +=1
+ i +=1
+
+
+estimated_mse_KFold = np.mean(scores_KFold, axis =1)
+
+## Cross-validation using cross_val_score from sklearn along with KFold
+
+# kfold is an instance initialized above as:
+# kfold = KFold(n_splits = k)
+
+estimated_mse_sklearn = np.zeros(nlambdas)
+i =0
+for lmb in lambdas:
+ ridge = Ridge(alpha = lmb)
+
+ X = poly.fit_transform(x[:, np.newaxis])
+ estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
+
+ # cross_val_score return an array containing the estimated negative mse for every fold.
+ # we have to the the mean of every array in order to get an estimate of the mse of the model
+ estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+
+ i +=1
+
+## Plot and compare the slightly different ways to perform cross-validation
+
+plt.figure()
+
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label ='cross_val_score')
+plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label ='KFold')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('mse')
+
+plt.legend()
+
+plt.show()
+
+When you are comparing your own code with for example Scikit-Learn's
+library, there are some technicalities to keep in mind. The examples
+here demonstrate some of these aspects with potential pitfalls.
+
+
+The discussion here focuses on the role of the intercept, how we can
+set up the design matrix, what scaling we should use and other topics
+which tend confuse us.
+
+
+The intercept can be interpreted as the expected value of our
+target/output variables when all other predictors are set to zero.
+Thus, if we cannot assume that the expected outputs/targets are zero
+when all predictors are zero (the columns in the design matrix), it
+may be a bad idea to implement a model which penalizes the intercept.
+Furthermore, in for example Ridge and Lasso regression, the default solutions
+from the library Scikit-Learn (when not shrinking \( \beta_0 \)) for the unknown parameters
+\( \boldsymbol{\beta} \), are derived under the assumption that both \( \boldsymbol{y} \) and
+\( \boldsymbol{X} \) are zero centered, that is we subtract the mean values.
+
+
+If our predictors represent different scales, then it is important to
+standardize the design matrix \( \boldsymbol{X} \) by subtracting the mean of each
+column from the corresponding column and dividing the column with its
+standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
+the results may differ.
+
+
+The
+Standadscaler
+function in Scikit-Learn does this for us. For the data sets we
+have been studying in our various examples, the data are in many cases
+already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
+survey of your data, with a critical assessment of them in case you need to scale the data.
+
+
+If you need to scale the data, not doing so will give an unfair
+penalization of the parameters since their magnitude depends on the
+scale of their corresponding predictor.
+
+
+Suppose as an example that you
+you have an input variable given by the heights of different persons.
+Human height might be measured in inches or meters or
+kilometers. If measured in kilometers, a standard linear regression
+model with this predictor would probably give a much bigger
+coefficient term, than if measured in millimeters.
+This can clearly lead to problems in evaluating the cost/loss functions.
+
+
+Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
+on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows
+
+
+
+
+
#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+
What does centering (subtracting the mean values) mean mathematically?
+
+
+Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as zero centering. For
+simplicity, we will focus on ordinary regression, as done in the above example.
+
+
+The cost/loss function for regression is
+$$
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
+$$
+
+Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.
+
+
+What we have done is to single out the \( \beta_0 \) term in the definition of the mean squared error (MSE).
+The design matrix
+\( X \) does in this case not contain any intercept column.
+When we take the derivative with respect to \( \beta_0 \), we want the derivative to obey
+$$
+\frac{\partial C}{\partial \beta_j} = 0,
+$$
+
+
+for all \( j \). For \( \beta_0 \) we have
+
+$$
+\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
+$$
+
+Multiplying away the constant \( 2/n \), we obtain
+$$
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
+$$
+
+
+We assume
+that every column of \( \boldsymbol{X} \) is centered, which we can do by subtracting the mean,
+
+
+
+
X = X - np.mean(X,axis=0)
+
+
+This means that we need to rewrite \( X_{ij} \) as \( \tilde{X}_{ij}=X_{ij}-\mu_j \), where
+$$
+\mu_j = \frac{1}{n}\sum_{i=0}^{n-1}X_{ij}.
+$$
+
+
+Let us special first to the case where we have only two parameters \( \beta_0 \) and \( \beta_1 \).
+Our result for \( \beta_0 \) simplifies then to
+$$
+n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
+$$
+
+Assuming that the matrix elements \( X_{i1} \) are centered, what we have is
+$$
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} \left(X_{i1}-\mu_{1}\right),
+$$
+
+where
+$$
+\mu_1=\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1},
+$$
+
+and if we define the mean value of the outputs as
+$$
+\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
+$$
+
+we have
+$$
+\beta_0 = \mu_y - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}-\mu_{1}),
+$$
+
+and it is easy to see that the last sum equals zero! This means that we have
+$$
+\beta_0 = \mu_y,
+$$
+
+if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of \( \beta \).
+We have thus
+$$
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\boldsymbol{y}},
+$$
+
+the average value of \( \boldsymbol{y} \).
+
+
+Replacing \( y_i \) with \( y_i - \beta_0 = y_i - \overline{\boldsymbol{y}} \) and centering also our design matrix results in a cost function (in vector-matrix disguise)
+$$
+C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
+$$
+
+
+If we minimize with respect to \( \boldsymbol{\beta} \) we have then
+
+$$
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
+$$
+
+
+For Ridge regression we need to add \( \lambda \boldsymbol{\beta}^T\boldsymbol{\beta} \) to the cost function and get then
+$$
+\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
+$$
+
+
+What does this mean? And why do we insist on all this? Let us look at some examples.
+
+
+This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (code example thanks to Øyvind Sigmundson Schøyen). Here our scaling of the data is done by subtracting the mean values only.
+Note also that we do not split the data into training and test.
+
+
+
+
+
importnumpyasnp
+importmatplotlib.pyplotasplt
+
+fromsklearn.linear_modelimport LinearRegression
+
+
+np.random.seed(2021)
+
+defMSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+deffit_beta(X, y):
+ return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+
+true_beta = [2, 0.5, 3.7]
+
+x = np.linspace(0, 1, 11)
+y = np.sum(
+ np.asarray([x ** p * b for p, b inenumerate(true_beta)]), axis=0
+) +0.1* np.random.normal(size=len(x))
+
+degree =3
+X = np.zeros((len(x), degree))
+
+# Include the intercept in the design matrix
+for p inrange(degree):
+ X[:, p] = x ** p
+
+beta = fit_beta(X, y)
+
+# Intercept is included in the design matrix
+skl = LinearRegression(fit_intercept=False).fit(X, y)
+
+print(f"True beta: {true_beta}")
+print(f"Fitted beta: {beta}")
+print(f"Sklearn fitted beta: {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column")
+print(MSE(y,ypredictOwn))
+print(f"MSE with intercept column from SKL")
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x, y, label="Data")
+plt.plot(x, X @ beta, label="Fit")
+plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
+
+
+# Do not include the intercept in the design matrix
+X = np.zeros((len(x), degree -1))
+
+for p inrange(degree -1):
+ X[:, p] = x ** (p +1)
+
+# Intercept is not included in the design matrix
+skl = LinearRegression(fit_intercept=True).fit(X, y)
+
+# Use centered values for X and y when computing coefficients
+y_offset = np.average(y, axis=0)
+X_offset = np.average(X, axis=0)
+
+beta = fit_beta(X - X_offset, y - y_offset)
+intercept = np.mean(y_offset - X_offset @ beta)
+
+print(f"Manual intercept: {intercept}")
+print(f"Fitted beta (wiothout intercept): {beta}")
+print(f"Sklearn intercept: {skl.intercept_}")
+print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept")
+print(MSE(y,ypredictOwn+intercept))
+print(f"MSE with Sklearn intercept")
+print(MSE(y,ypredictSKL))
+
+plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
+plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
+plt.grid()
+plt.legend()
+
+plt.show()
+
+
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case).
+
+
+Printing the MSE, we see first that both methods give the same MSE, as
+they should. However, when we move to for example Ridge regression,
+the way we treat the intercept may give a larger or smaller MSE,
+meaning that the MSE can be penalized by the value of the
+intercept. Not including the intercept in the fit, means that the
+regularization term does not include \( \beta_0 \). For different values
+of \( \lambda \), this may lead to differeing MSE values.
+
+
+To remind the reader, the regularization term, with the intercept in Ridge regression is given by
+$$
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
+$$
+
+but when we take out the intercept, this equation becomes
+$$
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
+$$
+
+
+For Lasso regression we have
+$$
+\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
+$$
+
+
+It means that, when scaling the design matrix and the outputs/targets, by subtracting the mean values, we have an optimization problem which is not penalized by the intercept. The MSE value can then be smaller since it focuses only on the remaining quantities. If we however bring back the intercept, we will get a MSE which then contains the intercept.
+
+
+Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known vanilla data set.
+
+
+
+
+
importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimport train_test_split
+fromsklearnimport linear_model
+
+defMSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+n =100
+x = np.random.rand(n)
+y = np.exp(-x**2) +1.5* np.exp(-(x-2)**2)
+
+Maxpolydegree =20
+X = np.zeros((n,Maxpolydegree))
+#We include explicitely the intercept column
+for degree inrange(Maxpolydegree):
+ X[:,degree] = x**degree
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+p = Maxpolydegree
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas =6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 2, nlambdas)
+for i inrange(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+ # Note: we include the intercept column and no scaling
+ RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+ RegRidge.fit(X_train,y_train)
+ # and then make the prediction
+ ytildeOwnRidge = X_train @ OwnRidgeBeta
+ ypredictOwnRidge = X_test @ OwnRidgeBeta
+ ytildeRidge = RegRidge.predict(X_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta)
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label ='MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label ='MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix.
+We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
+What happens if we do not include the intercept in our fit?
+Let us see how we can change this code by zero centering (thanks to Stian Bilek for inpouts here).
+
+
importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimport train_test_split
+fromsklearnimport linear_model
+fromsklearn.preprocessingimport StandardScaler
+
+defMSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(315)
+
+n =100
+x = np.random.rand(n)
+y = np.exp(-x**2) +1.5* np.exp(-(x-2)**2)
+
+Maxpolydegree =20
+X = np.zeros((n,Maxpolydegree-1))
+
+for degree inrange(1,Maxpolydegree): #No intercept column
+ X[:,degree-1] = x**(degree)
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean = np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
+X_test_scaled = X_test - X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler = np.mean(y_train)
+y_train_scaled = y_train - y_scaler
+
+p = Maxpolydegree-1
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas =6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 2, nlambdas)
+for i inrange(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
+ intercept_ = y_scaler - X_train_mean@OwnRidgeBeta#The intercept can be shifted so the model can predict on uncentered data
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
+ RegRidge = linear_model.Ridge(lmb)
+ RegRidge.fit(X_train,y_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta) #Intercept is given by mean of target variable
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print('Intercept from own implementation:')
+ print(intercept_)
+ print('Intercept from Scikit-Learn Ridge implementation')
+ print(RegRidge.intercept_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label ='MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label ='MSE SL Ridge Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+We see here, when compared to the code which includes explicitely the
+intercept column, that our MSE value is actually smaller. This is
+because the regularization term does not include the intercept value
+\( \beta_0 \) in the fitting. This applies to Lasso regularization as
+well. It means that our optimization is now done only with the
+centered matrix and/or vector that enter the fitting procedure. Note
+also that the problem with the intercept occurs mainly in these type
+of polynomial fitting problem.
+
+
+The next example is indeed an example where all these discussions about the role of intercept are not present.
+
+
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
+
+$$
+\begin{align}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{1}
+\end{align}
+$$
+
+
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
+in the system is determined by \( L \). For the one-dimensional system
+there is no phase transition.
+
+
+We will look at a system of \( L = 40 \) spins with a coupling constant of
+\( J = 1 \). To get enough training data we will generate 10000 states
+with their respective energies.
+
+
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+
+A more general form for the one-dimensional Ising model is
+
+$$
+\begin{align}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{2}
+\end{align}
+$$
+
+
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+$$
+\begin{align}
+ \boldsymbol{H} = \boldsymbol{X} J,
+\tag{3}
+\end{align}
+$$
+
+
+where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+$$
+\begin{align}
+ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\tag{4}
+\end{align}
+$$
+
+
+We split the data in training and test data as discussed in the previous example
+
+
+
+
+
X = np.zeros((n, L **2))
+for i inrange(n):
+ X[i] = np.outer(spins[i], spins[i]).ravel()
+y = energies
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+In the ordinary least squares method we choose the cost function
+
+$$
+\begin{align}
+ C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
+\tag{5}
+\end{align}
+$$
+
+
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
+$$
+ \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},
+$$
+
+
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+
+
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the singular
+value decomposition. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+
+$$
+ \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
+$$
+
+
+where the pseudoinverse of \( \boldsymbol{X} \) is given by
+
+$$
+ \boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
+$$
+
+
+Using singular value decomposition we can decompose the matrix \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
+$$
+\begin{align}
+ \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\tag{6}
+\end{align}
+$$
+
+
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
+
+
+
+
+
defols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+ u, s, v = scl.svd(x)
+ return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
+
+
+
+
beta = ols_svd(X_train_own,y_train)
+
+
+When extracting the \( J \)-matrix we need to make sure that we remove the intercept, as is done here
+
+
+
+
+
J = beta[1:].reshape(L, L)
+
+
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+
+Let us bring back the Ising model again, but now with an additional
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data. The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant \( J \) is
+given by
+
+$$
+\begin{align}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{7}
+\end{align}
+$$
+
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
+
+
+We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
+
+
+A more general form for the one-dimensional Ising model is
+
+$$
+\begin{align}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{8}
+\end{align}
+$$
+
+
+Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+$$
+\begin{align}
+ H = X J,
+\tag{9}
+\end{align}
+$$
+
+
+where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
+$$
+\begin{align}
+ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
+\tag{10}
+\end{align}
+$$
+
+We organize the data as we did above
+
+Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a regularizer. This
+involves a new cost function which leads to a new estimate for the
+weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
+cost function is given by
+
+$$
+\begin{align}
+ C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
+\tag{11}
+\end{align}
+$$
+
+
+In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
+
+$$
+\begin{align}
+ C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
+\tag{12}
+\end{align}
+$$
+
+
+Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-Learn.
+
+
+It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+\( J_{j, j + 1} = -1 \).
+
+
+We see that LASSO reaches a good solution for low
+values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
+much. Ridge is more stable over a larger range of values for
+\( \lambda \), but eventually also fades away.
+
+
+To determine which value of \( \lambda \) is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
+
+
+From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of \( \lambda \).
+
+
+In linear regression our main interest was centered on learning the
+coefficients of a functional fit (say a polynomial) in order to be
+able to predict the response of a continuous variable on some unseen
+data. The fit to the continuous variable \( y_i \) is based on some
+independent variables \( \boldsymbol{x}_i \). Linear regression resulted in
+analytical expressions for standard ordinary Least Squares or Ridge
+regression (in terms of matrices to invert) for several quantities,
+ranging from the variance and thereby the confidence intervals of the
+parameters \( \boldsymbol{\beta} \) to the mean squared error. If we can invert
+the product of the design matrices, linear regression gives then a
+simple recipe for fitting our data.
+
+
+Classification problems, however, are concerned with outcomes taking
+the form of discrete variables (i.e. categories). We may for example,
+on the basis of DNA sequencing for a number of patients, like to find
+out which mutations are important for a certain disease; or based on
+scans of various patients' brains, figure out if there is a tumor or
+not; or given a specific physical system, we'd like to identify its
+state, say whether it is an ordered or disordered system (typical
+situation in solid state physics); or classify the status of a
+patient, whether she/he has a stroke or not and many other similar
+situations.
+
+
+The most common situation we encounter when we apply logistic
+regression is that of two possible outcomes, normally denoted as a
+binary outcome, true or false, positive or negative, success or
+failure etc.
+
+
+Logistic regression will also serve as our stepping stone towards
+neural network algorithms and supervised deep learning. For logistic
+learning, the minimization of the cost function leads to a non-linear
+equation in the parameters \( \boldsymbol{\beta} \). The optimization of the
+problem calls therefore for minimization algorithms. This forms the
+bottle neck of all machine learning algorithms, namely how to find
+reliable minima of a multi-variable function. This leads us to the
+family of gradient descent methods. The latter are the working horses
+of basically all modern machine learning algorithms.
+
+
+We note also that many of the topics discussed here on logistic
+regression are also commonly used in modern supervised Deep Learning
+models, as we will see later.
+
+
+We consider the case where the dependent variables, also called the
+responses or the outcomes, \( y_i \) are discrete and only take values
+from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
+
+
+The goal is to predict the
+output classes from the design matrix \( \boldsymbol{X}\in\mathbb{R}^{n\times p} \)
+made of \( n \) samples, each of which carries \( p \) features or predictors. The
+primary goal is to identify the classes to which new unseen samples
+belong.
+
+
+Let us specialize to the case of two classes only, with outputs
+\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
+credit card user that could default or not on her/his credit card
+debt. That is
+
+$$
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
+$$
+
+
+Before moving to the logistic model, let us try to use our linear
+regression model to classify these two outcomes. We could for example
+fit a linear model to the default case if \( y_i > 0.5 \) and the no
+default case \( y_i \leq 0.5 \).
+
+
+We would then have our
+weighted linear combination, namely
+$$
+\begin{equation}
+\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{\beta} + \boldsymbol{\epsilon},
+\tag{13}
+\end{equation}
+$$
+
+where \( \boldsymbol{y} \) is a vector representing the possible outcomes, \( \boldsymbol{X} \) is our
+\( n\times p \) design matrix and \( \boldsymbol{\beta} \) represents our estimators/predictors.
+
+
+The main problem with our function is that it takes values on the
+entire real axis. In the case of logistic regression, however, the
+labels \( y_i \) are discrete variables. A typical example is the credit
+card data discussed below here, where we can set the state of
+defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons
+in the data set (see the full example below).
+
+
+One simple way to get a discrete output is to have sign
+functions that map the output of a linear regressor to values \( \{0,1\} \),
+\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise.
+We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
+literature. This model is extremely simple. However, in many cases it is more
+favorable to use a ``soft" classifier that outputs
+the probability of a given category. This leads us to the logistic function.
+
+
+The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
+
+
+
+
+
# Common imports
+importos
+importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.linear_modelimport LinearRegression, Ridge, Lasso
+fromsklearn.model_selectionimport train_test_split
+fromsklearn.utilsimport resample
+fromsklearn.metricsimport mean_squared_error
+fromIPython.displayimport display
+frompylabimport plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] ='serif'
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR ="Results"
+FIGURE_ID ="Results/FigureFiles"
+DATA_ID ="DataFiles/"
+
+ifnot os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+ifnot os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+ifnot os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+defimage_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+defdata_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+defsave_fig(fig_id):
+ plt.savefig(image_path(fig_id) +".png", format='png')
+
+infile =open(data_path("chddata.csv"),'r')
+
+# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
+chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
+chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
+output = chd['CHD']
+age = chd['Age']
+agegroup = chd['Agegroup']
+numberID = chd['ID']
+display(chd)
+
+plt.scatter(age, output, marker='o')
+plt.axis([18,70.0,-0.1, 1.2])
+plt.xlabel(r'Age')
+plt.ylabel(r'CHD')
+plt.title(r'Age distribution and Coronary heart disease')
+plt.show()
+
+What we could attempt however is to plot the mean value for each group.
+
+
+
+
+
agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
+group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
+plt.plot(group, agegroupmean, "r-")
+plt.axis([0,9,0, 1.0])
+plt.xlabel(r'Age group')
+plt.ylabel(r'CHD mean values')
+plt.title(r'Mean values for each age group')
+plt.show()
+
+
+We are now trying to find a function \( f(y\vert x) \), that is a function which gives us an expected value for the output \( y \) with a given input \( x \).
+In standard linear regression with a linear dependence on \( x \), we would write this in terms of our model
+$$
+f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
+$$
+
+
+This expression implies however that \( f(y_i\vert x_i) \) could take any
+value from minus infinity to plus infinity. If we however let
+\( f(y\vert y) \) be represented by the mean value, the above example
+shows us that we can constrain the function to take values between
+zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
+at our last curve we see also that it has an S-shaped form. This leads
+us to a very popular model for the function \( f \), namely the so-called
+Sigmoid function or logistic model. We will consider this function as
+representing the probability for finding a value of \( y_i \) with a given
+\( x_i \).
+
+
+Another widely studied model, is the so-called
+perceptron model, which is an example of a "hard classification" model. We
+will encounter this model when we discuss neural networks as
+well. Each datapoint is deterministically assigned to a category (i.e
+\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
+classifier that outputs the probability of a given category rather
+than a single value. For example, given \( x_i \), the classifier
+outputs the probability of being in a category \( k \). Logistic regression
+is the most common example of a so-called soft classifier. In logistic
+regression, the probability that a data point \( x_i \)
+belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
+$$
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
+$$
+
+Note that \( 1-p(t)= p(-t) \).
+
+
+We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities
+$$
+\begin{align*}
+p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
+p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
+\end{align*}
+$$
+
+where \( \boldsymbol{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
+
+
+Note that we used
+$$
+p(y_i=0\vert x_i, \boldsymbol{\beta}) = 1-p(y_i=1\vert x_i, \boldsymbol{\beta}).
+$$
+
+
+In order to define the total likelihood for all possible outcomes from a
+dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
+\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
+We aim thus at maximizing
+the probability of seeing the observed data. We can then approximate the
+likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is
+$$
+\begin{align*}
+P(\mathcal{D}|\boldsymbol{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\boldsymbol{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]^{1-y_i}\nonumber \\
+\end{align*}
+$$
+
+from which we obtain the log-likelihood and our cost/loss function
+$$
+\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\boldsymbol{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\boldsymbol{\beta}))\right]\right).
+$$
+
+
+Reordering the logarithms, we can rewrite the cost/loss function as
+$$
+\mathcal{C}(\boldsymbol{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+$$
+
+
+The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \).
+Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
+$$
+\mathcal{C}(\boldsymbol{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+$$
+
+This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression,
+in practice we often supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression.
+
+
+Let us now define a vector \( \boldsymbol{y} \) with \( n \) elements \( y_i \), an
+\( n\times p \) matrix \( \boldsymbol{X} \) which contains the \( x_i \) values and a
+vector \( \boldsymbol{p} \) of fitted probabilities \( p(y_i\vert x_i,\boldsymbol{\beta}) \). We can rewrite in a more compact form the first
+derivative of cost function as
+
+$$
+\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
+$$
+
+
+If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements
+\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as
+
+$$
+\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
+$$
+
+
+Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
+$$
+\log{ \frac{p(\boldsymbol{\beta}\boldsymbol{x})}{1-p(\boldsymbol{\beta}\boldsymbol{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
+$$
+
+Here we defined \( \boldsymbol{x}=[1,x_1,x_2,\dots,x_p] \) and \( \boldsymbol{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to
+$$
+p(\boldsymbol{\beta}\boldsymbol{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
+$$
+
+
+Till now we have mainly focused on two classes, the so-called binary
+system. Suppose we wish to extend to \( K \) classes. Let us for the sake
+of simplicity assume we have only two predictors. We have then following model
+
+$$
+\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
+$$
+
+and
+$$
+\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1,
+$$
+
+and so on till the class \( C=K-1 \) class
+$$
+\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
+$$
+
+
+and the model is specified in term of \( K-1 \) so-called log-odds or
+logit transformations.
+
+
+In our discussion of neural networks we will encounter the above again
+in terms of a slightly modified function, the so-called Softmax function.
+
+
+The softmax function is used in various multiclass classification
+methods, such as multinomial logistic regression (also known as
+softmax regression), multiclass linear discriminant analysis, naive
+Bayes classifiers, and artificial neural networks. Specifically, in
+multinomial logistic regression and linear discriminant analysis, the
+input to the function is the result of \( K \) distinct linear functions,
+and the predicted probability for the \( k \)-th class given a sample
+vector \( \boldsymbol{x} \) and a weighting vector \( \boldsymbol{\beta} \) is (with two
+predictors):
+
+$$
+p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
+$$
+
+It is easy to extend to more predictors. The final class is
+$$
+p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
+$$
+
+
+and they sum to one. Our earlier discussions were all specialized to
+the case with two classes only. It is easy to see from the above that
+what we derived earlier is compatible with these equations.
+
+
+To find the optimal parameters we would typically use a gradient
+descent method. Newton's method and gradient descent methods are
+discussed in the material on optimization
+methods.
+
+
+In the above example we note two things. In the first plot we display
+the overlap of benign and malignant tumors as functions of the various
+features in the Wisconsing breast cancer data set. We see that for
+some of the features we can distinguish clearly the benign and
+malignant cases while for other features we cannot. This can point to
+us which features may be of greater interest when we wish to classify
+a benign or not benign tumour.
+
+
+In the second figure we have computed the so-called correlation
+matrix, which in our case with thirty features becomes a \( 30\times 30 \)
+matrix.
+
+
+We constructed this matrix using pandas via the statements
+
+Diagonalizing this matrix we can in turn say something about which
+features are of relevance and which are not. This leads us to
+the classical Principal Component Analysis (PCA) theorem with
+applications. This will be discussed later this semester (week 43).
+
+
+Almost every problem in machine learning and data science starts with
+a dataset \( X \), a model \( g(\beta) \), which is a function of the
+parameters \( \beta \) and a cost function \( C(X, g(\beta)) \) that allows
+us to judge how well the model \( g(\beta) \) explains the observations
+\( X \). The model is fit by finding the values of \( \beta \) that minimize
+the cost function. Ideally we would be able to solve for \( \beta \)
+analytically, however this is not possible in general and we must use
+some approximative/numerical method to compute the minimum.
+
+
+In our discussion on Logistic Regression we studied the
+case of
+two classes, with \( y_i \) either
+\( 0 \) or \( 1 \). Furthermore we assumed also that we have only two
+parameters \( \beta \) in our fitting, that is we
+defined probabilities
+
+$$
+\begin{align*}
+p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
+p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}),
+\end{align*}
+$$
+
+where \( \boldsymbol{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
+
+
+Our compact equations used a definition of a vector \( \boldsymbol{y} \) with \( n \)
+elements \( y_i \), an \( n\times p \) matrix \( \boldsymbol{X} \) which contains the
+\( x_i \) values and a vector \( \boldsymbol{p} \) of fitted probabilities
+\( p(y_i\vert x_i,\boldsymbol{\beta}) \). We rewrote in a more compact form
+the first derivative of the cost function as
+
+$$
+\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right).
+$$
+
+
+If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements
+\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as
+
+$$
+\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}.
+$$
+
+This defines what is called the Hessian matrix.
+
+
+If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
+
+
+Our iterative scheme is then given by
+
+$$
+\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T}\right)^{-1}_{\boldsymbol{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right)_{\boldsymbol{\beta}^{\mathrm{old}}},
+$$
+
+or in matrix form as
+
+$$
+\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X} \right)^{-1}\times \left(-\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{p}) \right)_{\boldsymbol{\beta}^{\mathrm{old}}}.
+$$
+
+The right-hand side is computed with the old values of \( \beta \).
+
+
+If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.
+
+
+Let us quickly remind ourselves how we derive the above method.
+
+
+Perhaps the most celebrated of all one-dimensional root-finding
+routines is Newton's method, also called the Newton-Raphson
+method. This method requires the evaluation of both the
+function \( f \) and its derivative \( f' \) at arbitrary points.
+If you can only calculate the derivative
+numerically and/or your function is not of the smooth type, we
+normally discourage the use of this method.
+
+
+The Newton-Raphson formula consists geometrically of extending the
+tangent line at a current point until it crosses zero, then setting
+the next guess to the abscissa of that zero-crossing. The mathematics
+behind this method is rather simple. Employing a Taylor expansion for
+\( x \) sufficiently close to the solution \( s \), we have
+
+$$
+ f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots.
+ \tag{14}
+$$
+
+
+For small enough values of the function and for well-behaved
+functions, the terms beyond linear are unimportant, hence we obtain
+
+$$
+ f(x)+(s-x)f'(x)\approx 0,
+$$
+
+yielding
+$$
+ s\approx x-\frac{f(x)}{f'(x)}.
+$$
+
+
+Having in mind an iterative procedure, it is natural to start iterating with
+$$
+ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
+$$
+
+
+The above is Newton-Raphson's method. It has a simple geometric
+interpretation, namely \( x_{n+1} \) is the point where the tangent from
+\( (x_n,f(x_n)) \) crosses the \( x \)-axis. Close to the solution,
+Newton-Raphson converges fast to the desired result. However, if we
+are far from a root, where the higher-order terms in the series are
+important, the Newton-Raphson formula can give grossly inaccurate
+results. For instance, the initial guess for the root might be so far
+from the true root as to let the search interval include a local
+maximum or minimum of the function. If an iteration places a trial
+guess near such a local extremum, so that the first derivative nearly
+vanishes, then Newton-Raphson may fail totally
+
+
+Newton's method can be generalized to systems of several non-linear equations
+and variables. Consider the case with two equations
+$$
+ \begin{array}{cc} f_1(x_1,x_2) &=0\\
+ f_2(x_1,x_2) &=0,\end{array}
+$$
+
+which we Taylor expand to obtain
+
+$$
+ \begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1
+ \partial f_1/\partial x_1+h_2
+ \partial f_1/\partial x_2+\dots\\
+ 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1
+ \partial f_2/\partial x_1+h_2
+ \partial f_2/\partial x_2+\dots
+ \end{array}.
+$$
+
+Defining the Jacobian matrix \( {\bf \boldsymbol{J}} \) we have
+$$
+ {\bf \boldsymbol{J}}=\left( \begin{array}{cc}
+ \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\
+ \partial f_2/\partial x_1 &\partial f_2/\partial x_2
+ \end{array} \right),
+$$
+
+we can rephrase Newton's method as
+$$
+\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)=
+\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+
+\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right),
+$$
+
+where we have defined
+$$
+ \left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)=
+ -{\bf \boldsymbol{J}}^{-1}
+ \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right).
+$$
+
+We need thus to compute the inverse of the Jacobian matrix and it
+is to understand that difficulties may
+arise in case \( {\bf \boldsymbol{J}} \) is nearly singular.
+
+
+It is rather straightforward to extend the above scheme to systems of
+more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.
+
+
+The basic idea of gradient descent is
+that a function \( F(\mathbf{x}) \),
+\( \mathbf{x} \equiv (x_1,\cdots,x_n) \), decreases fastest if one goes from \( \bf {x} \) in the
+direction of the negative gradient \( -\nabla F(\mathbf{x}) \).
+
+
+It can be shown that if
+$$
+\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k),
+$$
+
+with \( \gamma_k > 0 \).
+
+
+For \( \gamma_k \) small enough, then \( F(\mathbf{x}_{k+1}) \leq
+F(\mathbf{x}_k) \). This means that for a sufficiently small \( \gamma_k \)
+we are always moving towards smaller function values, i.e a minimum.
+
+
+The previous observation is the basis of the method of steepest
+descent, which is also referred to as just gradient descent (GD). One
+starts with an initial guess \( \mathbf{x}_0 \) for a minimum of \( F \) and
+computes new approximations according to
+
+$$
+\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0.
+$$
+
+
+The parameter \( \gamma_k \) is often referred to as the step length or
+the learning rate within the context of Machine Learning.
+
+
+Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
+minimum of the function \( F \). In general we do not know if we are in a
+global or local minimum. In the special case when \( F \) is a convex
+function, all local minima are also global minima, so in this case
+gradient descent can converge to the global solution. The advantage of
+this scheme is that it is conceptually simple and straightforward to
+implement. However the method in this form has some severe
+limitations:
+
+
+In machine learing we are often faced with non-convex high dimensional
+cost functions with many local minima. Since GD is deterministic we
+will get stuck in a local minimum, if the method converges, unless we
+have a very good intial guess. This also implies that the scheme is
+sensitive to the chosen initial condition.
+
+
+Note that the gradient is a function of \( \mathbf{x} =
+(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.
+
+
+The gradient descent method
+is sensitive to the choice of learning rate \( \gamma_k \). This is due
+to the fact that we are only guaranteed that \( F(\mathbf{x}_{k+1}) \leq
+F(\mathbf{x}_k) \) for sufficiently small \( \gamma_k \). The problem is to
+determine an optimal learning rate. If the learning rate is chosen too
+small the method will take a long time to converge and if it is too
+large we can experience erratic behavior.
+
+
+Many of these shortcomings can be alleviated by introducing
+randomness. One such method is that of Stochastic Gradient Descent
+(SGD), see below.
+
+
+Ideally we want our cost/loss function to be convex(concave).
+
+
+First we give the definition of a convex set: A set \( C \) in
+\( \mathbb{R}^n \) is said to be convex if, for all \( x \) and \( y \) in \( C \) and
+all \( t \in (0,1) \) , the point \( (1 − t)x + ty \) also belongs to
+C. Geometrically this means that every point on the line segment
+connecting \( x \) and \( y \) is in \( C \) as discussed below.
+
+
+The convex subsets of \( \mathbb{R} \) are the intervals of
+\( \mathbb{R} \). Examples of convex sets of \( \mathbb{R}^2 \) are the
+regular polygons (triangles, rectangles, pentagons, etc...).
+
+
+Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
+
+
+In the following we state first and second-order conditions which
+ensures convexity of a function \( f \). We write \( D_f \) to denote the
+domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more
+details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.
+
+
+
+
+
+Suppose \( f \) is differentiable (i.e \( \nabla f(x) \) is well defined for
+all \( x \) in the domain of \( f \)). Then \( f \) is convex if and only if \( D_f \)
+is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds
+for all \( x,y \in D_f \). This condition means that for a convex function
+the first order Taylor expansion (right hand side above) at any point
+a global under estimator of the function. To convince yourself you can
+make a drawing of \( f(x) = x^2+1 \) and draw the tangent line to \( f(x) \) and
+note that it is always below the graph.
+
+
+
+
+
+
+
+
+Assume that \( f \) is twice
+differentiable, i.e the Hessian matrix exists at each point in
+\( D_f \). Then \( f \) is convex if and only if \( D_f \) is a convex set and its
+Hessian is positive semi-definite for all \( x\in D_f \). For a
+single-variable function this reduces to \( f''(x) \geq 0 \). Geometrically this means that \( f \) has nonnegative curvature
+everywhere.
+
+
+
+
+
+This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
+
+
+The next result is of great importance to us and the reason why we are
+going on about convex functions. In machine learning we frequently
+have to minimize a loss/cost function in order to find the best
+parameters for the model we are considering.
+
+
+Ideally we want the
+global minimum (for high-dimensional models it is hard to know
+if we have local or global minimum). However, if the cost/loss function
+is convex the following result provides invaluable information:
+
+
+
+
+
+Consider the problem of finding \( x \in \mathbb{R}^n \) such that \( f(x) \)
+is minimal, where \( f \) is convex and differentiable. Then, any point
+\( x^* \) that satisfies \( \nabla f(x^*) = 0 \) is a global minimum.
+
+
+
+
+
+This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
+
+
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
+
Using the second order condition show that the following functions are convex on the specified domain.
+
+
+
\( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
+
\( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
+
+
+
Let \( f(x) = x^2 \) and \( g(x) = e^x \). Show that \( f(g(x)) \) and \( g(f(x)) \) is convex for \( x \in \mathbb{R} \). Also show that if \( f(x) \) is any convex function than \( h(x) = e^{f(x)} \) is convex.
+
A norm is any function that satisfy the following properties
+
+
+
\( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).
+
\( f(x+y) \leq f(x) + f(y) \)
+
\( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)
+
+
+
+
+Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
+
+
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent (different from the above steepest descent discussion), which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type
+$$
+\begin{equation*}
+\boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}.
+\end{equation*}
+$$
+
+
+In the iterative process we end up with a problem like
+
+$$
+\begin{equation*}
+ \boldsymbol{r}= \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x},
+\end{equation*}
+$$
+
+where \( \boldsymbol{r} \) is the so-called residual or error in the iterative process.
+
+
+When we have found the exact solution, \( \boldsymbol{r}=0 \).
+
+
+The residual is zero when we reach the minimum of the quadratic equation
+$$
+\begin{equation*}
+ P(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T\boldsymbol{b},
+\end{equation*}
+$$
+
+
+with the constraint that the matrix \( \boldsymbol{A} \) is positive definite and
+symmetric. This defines also the Hessian and we want it to be positive definite.
+
+
+We denote the initial guess for \( \boldsymbol{x} \) as \( \boldsymbol{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\boldsymbol{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+
+One can show that the solution \( \boldsymbol{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \boldsymbol{r}_1 \) (see below for definition)
+to be the gradient of \( f \) at \( \boldsymbol{x}=\boldsymbol{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
+\end{equation*}
+$$
+
+and
+\( \boldsymbol{x}_0=0 \) it is equal \( -\boldsymbol{b} \).
+
+
+In the CG method we define so-called conjugate directions and two vectors
+\( \boldsymbol{s} \) and \( \boldsymbol{t} \)
+are said to be
+conjugate if
+$$
+\begin{equation*}
+\boldsymbol{s}^T\boldsymbol{A}\boldsymbol{t}= 0.
+\end{equation*}
+$$
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \boldsymbol{x}_i \) obeying the above criterion, namely
+$$
+\begin{equation*}
+\boldsymbol{x}_i^T\boldsymbol{A}\boldsymbol{x}_j= 0.
+\end{equation*}
+$$
+
+Two vectors are conjugate if they are orthogonal with respect to
+this inner product. Being conjugate is a symmetric relation: if \( \boldsymbol{s} \) is conjugate to \( \boldsymbol{t} \), then \( \boldsymbol{t} \) is conjugate to \( \boldsymbol{s} \).
+
+An example is given by the eigenvectors of the matrix
+$$
+\begin{equation*}
+\boldsymbol{v}_i^T\boldsymbol{A}\boldsymbol{v}_j= \lambda\boldsymbol{v}_i^T\boldsymbol{v}_j,
+\end{equation*}
+$$
+
+which is zero unless \( i=j \).
+
+Assume now that we have a symmetric positive-definite matrix \( \boldsymbol{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+$$
+\begin{equation*}
+\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+\alpha_i\boldsymbol{p}_{i}.
+\end{equation*}
+$$
+
+We assume that \( \boldsymbol{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
+Then the \( \boldsymbol{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
+$ \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}$ in this basis, namely
+
+$$
+\begin{equation*}
+ \boldsymbol{x} = \sum^{n}_{i=1} \alpha_i \boldsymbol{p}_i.
+\end{equation*}
+$$
+
+If we choose the conjugate vectors \( \boldsymbol{p}_k \) carefully,
+then we may not need all of them to obtain a good approximation to the solution
+\( \boldsymbol{x} \).
+We want to regard the conjugate gradient method as an iterative method.
+This will us to solve systems where \( n \) is so large that the direct
+method would take too much time.
+
+
+We denote the initial guess for \( \boldsymbol{x} \) as \( \boldsymbol{x}_0 \).
+We can assume without loss of generality that
+$$
+\begin{equation*}
+\boldsymbol{x}_0=0,
+\end{equation*}
+$$
+
+or consider the system
+$$
+\begin{equation*}
+\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0,
+\end{equation*}
+$$
+
+instead.
+
+One can show that the solution \( \boldsymbol{x} \) is also the unique minimizer of the quadratic form
+$$
+\begin{equation*}
+ f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n.
+\end{equation*}
+$$
+
+This suggests taking the first basis vector \( \boldsymbol{p}_1 \)
+to be the gradient of \( f \) at \( \boldsymbol{x}=\boldsymbol{x}_0 \),
+which equals
+$$
+\begin{equation*}
+\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b},
+\end{equation*}
+$$
+
+and
+\( \boldsymbol{x}_0=0 \) it is equal \( -\boldsymbol{b} \).
+The other vectors in the basis will be conjugate to the gradient,
+hence the name conjugate gradient method.
+
+Let \( \boldsymbol{r}_k \) be the residual at the \( k \)-th step:
+$$
+\begin{equation*}
+\boldsymbol{r}_k=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k.
+\end{equation*}
+$$
+
+Note that \( \boldsymbol{r}_k \) is the negative gradient of \( f \) at
+\( \boldsymbol{x}=\boldsymbol{x}_k \),
+so the gradient descent method would be to move in the direction \( \boldsymbol{r}_k \).
+Here, we insist that the directions \( \boldsymbol{p}_k \) are conjugate to each other,
+so we take the direction closest to the gradient \( \boldsymbol{r}_k \)
+under the conjugacy constraint.
+This gives the following expression
+$$
+\begin{equation*}
+\boldsymbol{p}_{k+1}=\boldsymbol{r}_k-\frac{\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{r}_k}{\boldsymbol{p}_k^T\boldsymbol{A}\boldsymbol{p}_k} \boldsymbol{p}_k.
+\end{equation*}
+$$
+
Revisiting some of our first Linear Regression Encounters
+
+
+We will use linear regression as a case study for the gradient descent
+methods. Linear regression is a great test case for the gradient
+descent methods discussed in the lectures since it has several
+desirable properties such as:
+
+
+
An analytical solution (recall homework set 1).
+
The gradient can be computed analytically.
+
The cost function is convex which guarantees that gradient descent converges for small enough learning rates
+
+
+We revisit an example similar to what we had in the first homework set. We had a function of the type
+
+
+
+
+
x =2*np.random.rand(m,1)
+y =4+3*x+np.random.randn(m,1)
+
+
+with \( x_i \in [0,1] \) is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \).
+The linear regression model is given by
+$$
+h_\beta(x) = \boldsymbol{y} = \beta_0 + \beta_1 x,
+$$
+
+such that
+$$
+\boldsymbol{y}_i = \beta_0 + \beta_1 x_i.
+$$
+
+
+Computing \( \partial C(\beta) / \partial \beta_0 \) and \( \partial C(\beta) / \partial \beta_1 \) we can show that the gradient can be written as
+$$
+\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
+\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
+\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}),
+$$
+
+where \( X \) is the design matrix defined above.
+
+
+The Hessian matrix of \( C(\beta) \) is given by
+$$
+\boldsymbol{H} \equiv \begin{bmatrix}
+\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\
+\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\
+\end{bmatrix} = \frac{2}{n}X^T X.
+$$
+
+This result implies that \( C(\beta) \) is a convex function since the matrix \( X^T X \) always is positive semi-definite.
+
+
+We can now write a program that minimizes \( C(\beta) \) using the gradient descent method with a constant learning rate \( \gamma \) according to
+$$
+\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots
+$$
+
+
+We can use the expression we computed for the gradient and let use a
+\( \beta_0 \) be chosen randomly and let \( \gamma = 0.001 \). Stop iterating
+when \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \). Note that the code below does not include the latter stop criterion.
+
+
+And finally we can compare our solution for \( \beta \) with the analytic result given by
+\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
+
+
+We have also discussed Ridge regression where the loss function contains a regularized term given by the \( L_2 \) norm of \( \beta \),
+$$
+C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0.
+$$
+
+
+In order to minimize \( C_{\text{ridge}}(\beta) \) using GD we only have adjust the gradient as follows
+$$
+\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
+\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
+\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
+$$
+
+
+We can easily extend our program to minimize \( C_{\text{ridge}}(\beta) \) using gradient descent and compare with the analytical solution given by
+$$
+\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
+$$
+
+
Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
+
GD is sensitive to initial conditions. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
+
Gradients are computationally expensive to calculate for large datasets. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over all \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.
+
GD is very sensitive to choices of learning rates. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would adaptively choose the learning rates to match the landscape.
+
GD treats all directions in parameter space uniformly. Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.
+
GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points. This leads us to our next topic, Stochastic Gradient Methods.
+
+
+
+
+
+
diff --git a/doc/src/week38/README.txt b/doc/src/week38/README.txt
new file mode 100644
index 000000000..589bb9764
--- /dev/null
+++ b/doc/src/week38/README.txt
@@ -0,0 +1,2 @@
+This IPython notebook week38.ipynb does not require any additional
+programs.
diff --git a/doc/src/week38/ipynb-week38-src.tar.gz b/doc/src/week38/ipynb-week38-src.tar.gz
new file mode 100644
index 000000000..4a81a6d5e
Binary files /dev/null and b/doc/src/week38/ipynb-week38-src.tar.gz differ
diff --git a/doc/src/week38/reveal.js/.gitignore b/doc/src/week38/reveal.js/.gitignore
new file mode 100644
index 000000000..a5df3133d
--- /dev/null
+++ b/doc/src/week38/reveal.js/.gitignore
@@ -0,0 +1,8 @@
+.DS_Store
+.svn
+log/*.log
+tmp/**
+node_modules/
+.sass-cache
+css/reveal.min.css
+js/reveal.min.js
diff --git a/doc/src/week38/reveal.js/.travis.yml b/doc/src/week38/reveal.js/.travis.yml
new file mode 100644
index 000000000..165d9ae9f
--- /dev/null
+++ b/doc/src/week38/reveal.js/.travis.yml
@@ -0,0 +1,5 @@
+language: node_js
+node_js:
+ - 0.10
+before_script:
+ - npm install -g grunt-cli
\ No newline at end of file
diff --git a/doc/src/week38/reveal.js/CONTRIBUTING.md b/doc/src/week38/reveal.js/CONTRIBUTING.md
new file mode 100644
index 000000000..c2091e88f
--- /dev/null
+++ b/doc/src/week38/reveal.js/CONTRIBUTING.md
@@ -0,0 +1,23 @@
+## Contributing
+
+Please keep the [issue tracker](http://github.com/hakimel/reveal.js/issues) limited to **bug reports**, **feature requests** and **pull requests**.
+
+
+### Personal Support
+If you have personal support or setup questions the best place to ask those are [StackOverflow](http://stackoverflow.com/questions/tagged/reveal.js).
+
+
+### Bug Reports
+When reporting a bug make sure to include information about which browser and operating system you are on as well as the necessary steps to reproduce the issue. If possible please include a link to a sample presentation where the bug can be tested.
+
+
+### Pull Requests
+- Should follow the coding style of the file you work in, most importantly:
+ - Tabs to indent
+ - Single-quoted strings
+- Should be made towards the **dev branch**
+- Should be submitted from a feature/topic branch (not your master)
+
+
+### Plugins
+Please do not submit plugins as pull requests. They should be maintained in their own separate repository. More information here: https://github.com/hakimel/reveal.js/wiki/Plugin-Guidelines
diff --git a/doc/src/week38/reveal.js/Gruntfile.js b/doc/src/week38/reveal.js/Gruntfile.js
new file mode 100644
index 000000000..b257e8f32
--- /dev/null
+++ b/doc/src/week38/reveal.js/Gruntfile.js
@@ -0,0 +1,140 @@
+/* global module:false */
+module.exports = function(grunt) {
+ var port = grunt.option('port') || 8000;
+ // Project configuration
+ grunt.initConfig({
+ pkg: grunt.file.readJSON('package.json'),
+ meta: {
+ banner:
+ '/*!\n' +
+ ' * reveal.js <%= pkg.version %> (<%= grunt.template.today("yyyy-mm-dd, HH:MM") %>)\n' +
+ ' * http://lab.hakim.se/reveal-js\n' +
+ ' * MIT licensed\n' +
+ ' *\n' +
+ ' * Copyright (C) 2014 Hakim El Hattab, http://hakim.se\n' +
+ ' */'
+ },
+
+ qunit: {
+ files: [ 'test/*.html' ]
+ },
+
+ uglify: {
+ options: {
+ banner: '<%= meta.banner %>\n'
+ },
+ build: {
+ src: 'js/reveal.js',
+ dest: 'js/reveal.min.js'
+ }
+ },
+
+ cssmin: {
+ compress: {
+ files: {
+ 'css/reveal.min.css': [ 'css/reveal.css' ]
+ }
+ }
+ },
+
+ sass: {
+ main: {
+ files: {
+ 'css/theme/darkgray.css': 'css/theme/source/darkgray.scss',
+ 'css/theme/beigesmall.css': 'css/theme/source/beigesmall.scss',
+ 'css/theme/cbc.css': 'css/theme/source/cbc.scss',
+ 'css/theme/default.css': 'css/theme/source/default.scss',
+ 'css/theme/beige.css': 'css/theme/source/beige.scss',
+ 'css/theme/night.css': 'css/theme/source/night.scss',
+ 'css/theme/serif.css': 'css/theme/source/serif.scss',
+ 'css/theme/simple.css': 'css/theme/source/simple.scss',
+ 'css/theme/sky.css': 'css/theme/source/sky.scss',
+ 'css/theme/moon.css': 'css/theme/source/moon.scss',
+ 'css/theme/solarized.css': 'css/theme/source/solarized.scss',
+ 'css/theme/blood.css': 'css/theme/source/blood.scss'
+ }
+ }
+ },
+
+ jshint: {
+ options: {
+ curly: false,
+ eqeqeq: true,
+ immed: true,
+ latedef: true,
+ newcap: true,
+ noarg: true,
+ sub: true,
+ undef: true,
+ eqnull: true,
+ browser: true,
+ expr: true,
+ globals: {
+ head: false,
+ module: false,
+ console: false,
+ unescape: false
+ }
+ },
+ files: [ 'Gruntfile.js', 'js/reveal.js' ]
+ },
+
+ connect: {
+ server: {
+ options: {
+ port: port,
+ base: '.'
+ }
+ }
+ },
+
+ zip: {
+ 'reveal-js-presentation.zip': [
+ 'index.html',
+ 'css/**',
+ 'js/**',
+ 'lib/**',
+ 'images/**',
+ 'plugin/**'
+ ]
+ },
+
+ watch: {
+ main: {
+ files: [ 'Gruntfile.js', 'js/reveal.js', 'css/reveal.css' ],
+ tasks: 'default'
+ },
+ theme: {
+ files: [ 'css/theme/source/*.scss', 'css/theme/template/*.scss' ],
+ tasks: 'themes'
+ }
+ }
+
+ });
+
+ // Dependencies
+ grunt.loadNpmTasks( 'grunt-contrib-qunit' );
+ grunt.loadNpmTasks( 'grunt-contrib-jshint' );
+ grunt.loadNpmTasks( 'grunt-contrib-cssmin' );
+ grunt.loadNpmTasks( 'grunt-contrib-uglify' );
+ grunt.loadNpmTasks( 'grunt-contrib-watch' );
+ grunt.loadNpmTasks( 'grunt-contrib-sass' );
+ grunt.loadNpmTasks( 'grunt-contrib-connect' );
+ grunt.loadNpmTasks( 'grunt-zip' );
+
+ // Default task
+ grunt.registerTask( 'default', [ 'jshint', 'cssmin', 'uglify', 'qunit' ] );
+
+ // Theme task
+ grunt.registerTask( 'themes', [ 'sass' ] );
+
+ // Package presentation to archive
+ grunt.registerTask( 'package', [ 'default', 'zip' ] );
+
+ // Serve presentation locally
+ grunt.registerTask( 'serve', [ 'connect', 'watch' ] );
+
+ // Run tests
+ grunt.registerTask( 'test', [ 'jshint', 'qunit' ] );
+
+};
diff --git a/doc/src/week38/reveal.js/LICENSE b/doc/src/week38/reveal.js/LICENSE
new file mode 100644
index 000000000..09623076f
--- /dev/null
+++ b/doc/src/week38/reveal.js/LICENSE
@@ -0,0 +1,19 @@
+Copyright (C) 2015 Hakim El Hattab, http://hakim.se
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in
+all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
+THE SOFTWARE.
\ No newline at end of file
diff --git a/doc/src/week38/reveal.js/README.md b/doc/src/week38/reveal.js/README.md
new file mode 100644
index 000000000..573b19597
--- /dev/null
+++ b/doc/src/week38/reveal.js/README.md
@@ -0,0 +1,1052 @@
+# reveal.js [](https://travis-ci.org/hakimel/reveal.js)
+
+A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/).
+
+reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere.
+
+
+#### More reading:
+- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer.
+- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history.
+- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own!
+- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks.
+- [Plugins](https://github.com/hakimel/reveal.js/wiki/Plugins,-Tools-and-Hardware): A list of plugins that can be used to extend reveal.js.
+
+## Online Editor
+
+Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com).
+
+
+## Instructions
+
+### Markup
+
+Markup hierarchy needs to be ``
`` where the ```` represents one slide and can be repeated indefinitely. If you place multiple ````'s inside of another ```` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and it will be included in the horizontal sequence. For example:
+
+```html
+
+```
+
+### Markdown
+
+It's possible to write your slides using Markdown. To enable Markdown, add the ```data-markdown``` attribute to your `````` elements and wrap the contents in a ```
+
+```
+
+#### External Markdown
+
+You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file. The ```data-charset``` attribute is optional and specifies which charset to use when loading the external file.
+
+When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup).
+
+```html
+
+
+```
+
+#### Element Attributes
+
+Special syntax (in html comment) is available for adding attributes to Markdown elements. This is useful for fragments, amongst other things.
+
+```html
+
+
+
+```
+
+#### Slide Attributes
+
+Special syntax (in html comment) is available for adding attributes to the slide `` elements generated by your Markdown.
+
+```html
+
+
+
+```
+
+
+### Configuration
+
+At the end of your page you need to initialize reveal by running the following code. Note that all config values are optional and will default as specified below.
+
+```javascript
+Reveal.initialize({
+
+ // Display controls in the bottom right corner
+ controls: true,
+
+ // Display a presentation progress bar
+ progress: true,
+
+ // Display the page number of the current slide
+ slideNumber: false,
+
+ // Push each slide change to the browser history
+ history: false,
+
+ // Enable keyboard shortcuts for navigation
+ keyboard: true,
+
+ // Enable the slide overview mode
+ overview: true,
+
+ // Vertical centering of slides
+ center: true,
+
+ // Enables touch navigation on devices with touch input
+ touch: true,
+
+ // Loop the presentation
+ loop: false,
+
+ // Change the presentation direction to be RTL
+ rtl: false,
+
+ // Turns fragments on and off globally
+ fragments: true,
+
+ // Flags if the presentation is running in an embedded mode,
+ // i.e. contained within a limited portion of the screen
+ embedded: false,
+
+ // Flags if we should show a help overlay when the questionmark
+ // key is pressed
+ help: true,
+
+ // Number of milliseconds between automatically proceeding to the
+ // next slide, disabled when set to 0, this value can be overwritten
+ // by using a data-autoslide attribute on your slides
+ autoSlide: 0,
+
+ // Stop auto-sliding after user input
+ autoSlideStoppable: true,
+
+ // Enable slide navigation via mouse wheel
+ mouseWheel: false,
+
+ // Hides the address bar on mobile devices
+ hideAddressBar: true,
+
+ // Opens links in an iframe preview overlay
+ previewLinks: false,
+
+ // Transition style
+ transition: 'default', // none/fade/slide/convex/concave/zoom
+
+ // Transition speed
+ transitionSpeed: 'default', // default/fast/slow
+
+ // Transition style for full page slide backgrounds
+ backgroundTransition: 'default', // none/fade/slide/convex/concave/zoom
+
+ // Number of slides away from the current that are visible
+ viewDistance: 3,
+
+ // Parallax background image
+ parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'"
+
+ // Parallax background size
+ parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px"
+
+ // Amount to move parallax background (horizontal and vertical) on slide change
+ // Number, e.g. 100
+ parallaxBackgroundHorizontal: '',
+ parallaxBackgroundVertical: ''
+
+});
+```
+
+
+The configuration can be updated after initialization using the ```configure``` method:
+
+```javascript
+// Turn autoSlide off
+Reveal.configure({ autoSlide: 0 });
+
+// Start auto-sliding every 5s
+Reveal.configure({ autoSlide: 5000 });
+```
+
+
+### Dependencies
+
+Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example:
+
+```javascript
+Reveal.initialize({
+ dependencies: [
+ // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/
+ { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } },
+
+ // Interpret Markdown in elements
+ { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
+ { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } },
+
+ // Syntax highlight for elements
+ { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } },
+
+ // Zoom in and out with Alt+click
+ { src: 'plugin/zoom-js/zoom.js', async: true },
+
+ // Speaker notes
+ { src: 'plugin/notes/notes.js', async: true },
+
+ // Remote control your reveal.js presentation using a touch device
+ { src: 'plugin/remotes/remotes.js', async: true },
+
+ // MathJax
+ { src: 'plugin/math/math.js', async: true }
+ ]
+});
+```
+
+You can add your own extensions using the same syntax. The following properties are available for each dependency object:
+- **src**: Path to the script to load
+- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false
+- **callback**: [optional] Function to execute when the script has loaded
+- **condition**: [optional] Function which must return true for the script to be loaded
+
+
+### Ready Event
+
+A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`.
+
+```javascript
+Reveal.addEventListener( 'ready', function( event ) {
+ // event.currentSlide, event.indexh, event.indexv
+} );
+```
+
+
+### Presentation Size
+
+All presentations have a normal size, that is the resolution at which they are authored. The framework will automatically scale presentations uniformly based on this size to ensure that everything fits on any given display or viewport.
+
+See below for a list of configuration options related to sizing, including default values:
+
+```javascript
+Reveal.initialize({
+
+ ...
+
+ // The "normal" size of the presentation, aspect ratio will be preserved
+ // when the presentation is scaled to fit different resolutions. Can be
+ // specified using percentage units.
+ width: 960,
+ height: 700,
+
+ // Factor of the display size that should remain empty around the content
+ margin: 0.1,
+
+ // Bounds for smallest/largest possible scale to apply to content
+ minScale: 0.2,
+ maxScale: 1.5
+
+});
+```
+
+
+### Auto-sliding
+
+Presentations can be configured to progress through slides automatically, without any user input. To enable this you will need to tell the framework how many milliseconds it should wait between slides:
+
+```javascript
+// Slide every five seconds
+Reveal.configure({
+ autoSlide: 5000
+});
+```
+When this is turned on a control element will appear that enables users to pause and resume auto-sliding. Alternatively, sliding can be paused or resumed by pressing »a« on the keyboard. Sliding is paused automatically as soon as the user starts navigating. You can disable these controls by specifying ```autoSlideStoppable: false``` in your reveal.js config.
+
+You can also override the slide duration for individual slides and fragments by using the ```data-autoslide``` attribute:
+
+```html
+
+
After 2 seconds the first fragment will be shown.
+
After 10 seconds the next fragment will be shown.
+
Now, the fragment is displayed for 2 seconds before the next slide is shown.
+
+```
+
+Whenever the auto-slide mode is resumed or paused the ```autoslideresumed``` and ```autoslidepaused``` events are fired.
+
+
+### Keyboard Bindings
+
+If you're unhappy with any of the default keyboard bindings you can override them using the ```keyboard``` config option:
+
+```javascript
+Reveal.configure({
+ keyboard: {
+ 13: 'next', // go to the next slide when the ENTER key is pressed
+ 27: function() {}, // do something custom when ESC is pressed
+ 32: null // don't do anything when SPACE is pressed (i.e. disable a reveal.js default binding)
+ }
+});
+```
+
+### Lazy Loading
+
+When working on presentation with a lot of media or iframe content it's important to load lazily. Lazy loading means that reveal.js will only load content for the few slides nearest to the current slide. The number of slides that are preloaded is determined by the `viewDistance` configuration option.
+
+To enable lazy loading all you need to do is change your "src" attributes to "data-src" as shown below. This is supported for image, video, audio and iframe elements. Lazy loaded iframes will also unload when the containing slide is no longer visible.
+
+```html
+
+
+
+
+
+```
+
+
+### API
+
+The ``Reveal`` object exposes a JavaScript API for controlling navigation and reading state:
+
+```javascript
+// Navigation
+Reveal.slide( indexh, indexv, indexf );
+Reveal.left();
+Reveal.right();
+Reveal.up();
+Reveal.down();
+Reveal.prev();
+Reveal.next();
+Reveal.prevFragment();
+Reveal.nextFragment();
+
+// Toggle presentation states, optionally pass true/false to force on/off
+Reveal.toggleOverview();
+Reveal.togglePause();
+Reveal.toggleAutoSlide();
+
+// Change a config value at runtime
+Reveal.configure({ controls: true });
+
+// Returns the present configuration options
+Reveal.getConfig();
+
+// Fetch the current scale of the presentation
+Reveal.getScale();
+
+// Retrieves the previous and current slide elements
+Reveal.getPreviousSlide();
+Reveal.getCurrentSlide();
+
+Reveal.getIndices(); // { h: 0, v: 0 } }
+Reveal.getProgress(); // 0-1
+Reveal.getTotalSlides();
+
+// State checks
+Reveal.isFirstSlide();
+Reveal.isLastSlide();
+Reveal.isOverview();
+Reveal.isPaused();
+Reveal.isAutoSliding();
+```
+
+### Slide Changed Event
+
+A 'slidechanged' event is fired each time the slide is changed (regardless of state). The event object holds the index values of the current slide as well as a reference to the previous and current slide HTML nodes.
+
+Some libraries, like MathJax (see [#226](https://github.com/hakimel/reveal.js/issues/226#issuecomment-10261609)), get confused by the transforms and display states of slides. Often times, this can be fixed by calling their update or render function from this callback.
+
+```javascript
+Reveal.addEventListener( 'slidechanged', function( event ) {
+ // event.previousSlide, event.currentSlide, event.indexh, event.indexv
+} );
+```
+
+### Presentation State
+
+The presentation's current state can be fetched by using the `getState` method. A state object contains all of the information required to put the presentation back as it was when `getState` was first called. Sort of like a snapshot. It's a simple object that can easily be stringified and persisted or sent over the wire.
+
+```javascript
+Reveal.slide( 1 );
+// we're on slide 1
+
+var state = Reveal.getState();
+
+Reveal.slide( 3 );
+// we're on slide 3
+
+Reveal.setState( state );
+// we're back on slide 1
+```
+
+### Slide States
+
+If you set ``data-state="somestate"`` on a slide ````, "somestate" will be applied as a class on the document element when that slide is opened. This allows you to apply broad style changes to the page based on the active slide.
+
+Furthermore you can also listen to these changes in state via JavaScript:
+
+```javascript
+Reveal.addEventListener( 'somestate', function() {
+ // TODO: Sprinkle magic
+}, false );
+```
+
+### Slide Backgrounds
+
+Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your `````` elements. Four different types of backgrounds are supported: color, image, video and iframe. Below are a few examples.
+
+```html
+
+
All CSS color formats are supported, like rgba() or hsl().
+
+
+
This slide will have a full-size background image.
+
+
+
This background image will be sized to 100px and repeated.
+
+
+
Video. Multiple sources can be defined using a comma separated list. Video will loop when the data-background-video-loop attribute is provided.
+
+
+
Embeds a web page as a background. Note that the page won't be interactive.
+
+```
+
+Backgrounds transition using a fade animation by default. This can be changed to a linear sliding transition by passing ```backgroundTransition: 'slide'``` to the ```Reveal.initialize()``` call. Alternatively you can set ```data-background-transition``` on any section with a background to override that specific transition.
+
+
+### Parallax Background
+
+If you want to use a parallax scrolling background, set the first two config properties below when initializing reveal.js (the other two are optional).
+
+```javascript
+Reveal.initialize({
+
+ // Parallax background image
+ parallaxBackgroundImage: '', // e.g. "https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg"
+
+ // Parallax background size
+ parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" - currently only pixels are supported (don't use % or auto)
+
+ // Amount of pixels to move the parallax background per slide step,
+ // a value of 0 disables movement along the given axis
+ // These are optional, if they aren't specified they'll be calculated automatically
+ parallaxBackgroundHorizontal: 200,
+ parallaxBackgroundVertical: 50
+
+});
+```
+
+Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://lab.hakim.se/reveal-js/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px).
+
+
+
+### Slide Transitions
+The global presentation transition is set using the ```transition``` config value. You can override the global transition for a specific slide by using the ```data-transition``` attribute:
+
+```html
+
+
This slide will override the presentation transition and zoom!
+
+
+
+
Choose from three transition speeds: default, fast or slow!
Friday: Logistic Regression and Optimization methods
+
+
+
+
+
+
Thursday September 23
+
+
+
+
+
Ridge and LASSO Regression, reminder
+
+
+The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is
+our optimization problem is
+
+By minimizing the above equation with respect to the parameters
+\( \boldsymbol{\beta} \) we could then obtain an analytical expression for the
+parameters \( \boldsymbol{\beta} \). We can add a regularization parameter \( \lambda \) by
+defining a new cost function to be optimized, that is
+
+
+which leads to the Ridge regression minimization problem where we
+require that \( \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t \), where \( t \) is
+a finite number larger than zero. By defining
+
+
+When the repetitive splitting of the data set is done randomly,
+samples may accidently end up in a fast majority of the splits in
+either training or test set. Such samples may have an unbalanced
+influence on either model building or prediction evaluation. To avoid
+this \( k \)-fold cross-validation structures the data splitting. The
+samples are divided into \( k \) more or less equally sized exhaustive and
+mutually exclusive subsets. In turn (at each split) one of these
+subsets plays the role of the test set while the union of the
+remaining subsets constitutes the training set. Such a splitting
+warrants a balanced representation of each sample in both training and
+test set over the splits. Still the division into the \( k \) subsets
+involves a degree of randomness. This may be fully excluded when
+choosing \( k=n \). This particular case is referred to as leave-one-out
+cross-validation (LOOCV).
+
+
+
+
+
How to set up the cross-validation for Ridge and/or Lasso
+
+
+
Define a range of interest for the penalty parameter.
+
Divide the data set into training and test set comprising samples \( \{1, \ldots, n\} \setminus i \) and \( \{ i \} \), respectively.
+
Fit the linear regression model by means of ridge estimation for each \( \lambda \) in the grid using the training set, and the corresponding estimate of the error variance \( \boldsymbol{\sigma}_{-i}^2(\lambda) \), as
Evaluate the prediction performance of these models on the test set by \( C[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.
+
Repeat the first three steps such that each sample plays the role of the test set once.
+
Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data.
+
+
+
+
+
+
Cross-validation in brief
+
+
+For the various values of \( k \)
+
+
+
shuffle the dataset randomly.
+
Split the dataset into \( k \) groups.
+
For each unique group:
+
+
+
Decide which group to use as set for test data
+
Take the remaining groups as a training data set
+
Fit a model on the training set and evaluate it on the test set
+
Retain the evaluation score and discard the model
+
+
Summarize the model using the sample of model evaluation scores
+
+
+
+
+
+
Code Example for Cross-validation and \( k \)-fold Cross-validation
+
+
+The code here uses Ridge regression with cross-validation (CV) resampling and \( k \)-fold CV in order to fit a specific polynomial.
+
+
+
+
importnumpyasnp
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimport KFold
+fromsklearn.linear_modelimport Ridge
+fromsklearn.model_selectionimport cross_val_score
+fromsklearn.preprocessingimport PolynomialFeatures
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+# Generate the data.
+nsamples = 100
+x = np.random.randn(nsamples)
+y = 3*x**2 + np.random.randn(nsamples)
+
+## Cross-validation on Ridge regression using KFold only
+
+# Decide degree on polynomial to fit
+poly = PolynomialFeatures(degree = 6)
+
+# Decide which values of lambda to use
+nlambdas = 500
+lambdas = np.logspace(-3, 5, nlambdas)
+
+# Initialize a KFold instance
+k = 5
+kfold = KFold(n_splits = k)
+
+# Perform the cross-validation to estimate MSE
+scores_KFold = np.zeros((nlambdas, k))
+
+i = 0
+for lmb in lambdas:
+ ridge = Ridge(alpha = lmb)
+ j = 0
+ for train_inds, test_inds in kfold.split(x):
+ xtrain = x[train_inds]
+ ytrain = y[train_inds]
+
+ xtest = x[test_inds]
+ ytest = y[test_inds]
+
+ Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
+ ridge.fit(Xtrain, ytrain[:, np.newaxis])
+
+ Xtest = poly.fit_transform(xtest[:, np.newaxis])
+ ypred = ridge.predict(Xtest)
+
+ scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
+
+ j += 1
+ i += 1
+
+
+estimated_mse_KFold = np.mean(scores_KFold, axis = 1)
+
+## Cross-validation using cross_val_score from sklearn along with KFold
+
+# kfold is an instance initialized above as:
+# kfold = KFold(n_splits = k)
+
+estimated_mse_sklearn = np.zeros(nlambdas)
+i = 0
+for lmb in lambdas:
+ ridge = Ridge(alpha = lmb)
+
+ X = poly.fit_transform(x[:, np.newaxis])
+ estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
+
+ # cross_val_score return an array containing the estimated negative mse for every fold.
+ # we have to the the mean of every array in order to get an estimate of the mse of the model
+ estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
+
+ i += 1
+
+## Plot and compare the slightly different ways to perform cross-validation
+
+plt.figure()
+
+plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
+plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('mse')
+
+plt.legend()
+
+plt.show()
+
+
+
+
+
+
To think about, first part
+
+
+When you are comparing your own code with for example Scikit-Learn's
+library, there are some technicalities to keep in mind. The examples
+here demonstrate some of these aspects with potential pitfalls.
+
+
+The discussion here focuses on the role of the intercept, how we can
+set up the design matrix, what scaling we should use and other topics
+which tend confuse us.
+
+
+The intercept can be interpreted as the expected value of our
+target/output variables when all other predictors are set to zero.
+Thus, if we cannot assume that the expected outputs/targets are zero
+when all predictors are zero (the columns in the design matrix), it
+may be a bad idea to implement a model which penalizes the intercept.
+Furthermore, in for example Ridge and Lasso regression, the default solutions
+from the library Scikit-Learn (when not shrinking \( \beta_0 \)) for the unknown parameters
+\( \boldsymbol{\beta} \), are derived under the assumption that both \( \boldsymbol{y} \) and
+\( \boldsymbol{X} \) are zero centered, that is we subtract the mean values.
+
+
+
+
+
More thinking
+
+
+If our predictors represent different scales, then it is important to
+standardize the design matrix \( \boldsymbol{X} \) by subtracting the mean of each
+column from the corresponding column and dividing the column with its
+standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
+the results may differ.
+
+
+The
+Standadscaler
+function in Scikit-Learn does this for us. For the data sets we
+have been studying in our various examples, the data are in many cases
+already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
+survey of your data, with a critical assessment of them in case you need to scale the data.
+
+
+If you need to scale the data, not doing so will give an unfair
+penalization of the parameters since their magnitude depends on the
+scale of their corresponding predictor.
+
+
+Suppose as an example that you
+you have an input variable given by the heights of different persons.
+Human height might be measured in inches or meters or
+kilometers. If measured in kilometers, a standard linear regression
+model with this predictor would probably give a much bigger
+coefficient term, than if measured in millimeters.
+This can clearly lead to problems in evaluating the cost/loss functions.
+
+
+
+
+
Still thinking
+
+
+Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
+on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows
+
+
+
+
+
#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+
+
+
+
+
+
What does centering (subtracting the mean values) mean mathematically?
+
+
+Let us try to understand what this may imply mathematically when we
+subtract the mean values, also known as zero centering. For
+simplicity, we will focus on ordinary regression, as done in the above example.
+
+
+
+Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values.
+
+
+What we have done is to single out the \( \beta_0 \) term in the definition of the mean squared error (MSE).
+The design matrix
+\( X \) does in this case not contain any intercept column.
+When we take the derivative with respect to \( \beta_0 \), we want the derivative to obey
+
+Let us special first to the case where we have only two parameters \( \beta_0 \) and \( \beta_1 \).
+Our result for \( \beta_0 \) simplifies then to
+
+Replacing \( y_i \) with \( y_i - \beta_0 = y_i - \overline{\boldsymbol{y}} \) and centering also our design matrix results in a cost function (in vector-matrix disguise)
+
+What does this mean? And why do we insist on all this? Let us look at some examples.
+
+
+
+
+
Linear Regression code, Intercept handling first
+
+
+This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (code example thanks to Øyvind Sigmundson Schøyen). Here our scaling of the data is done by subtracting the mean values only.
+Note also that we do not split the data into training and test.
+
+
+
+
+
importnumpyasnp
+importmatplotlib.pyplotasplt
+
+fromsklearn.linear_modelimport LinearRegression
+
+
+np.random.seed(2021)
+
+defMSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+deffit_beta(X, y):
+ return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+
+true_beta = [2, 0.5, 3.7]
+
+x = np.linspace(0, 1, 11)
+y = np.sum(
+ np.asarray([x ** p * b for p, b inenumerate(true_beta)]), axis=0
+) + 0.1 * np.random.normal(size=len(x))
+
+degree = 3
+X = np.zeros((len(x), degree))
+
+# Include the intercept in the design matrix
+for p inrange(degree):
+ X[:, p] = x ** p
+
+beta = fit_beta(X, y)
+
+# Intercept is included in the design matrix
+skl = LinearRegression(fit_intercept=False).fit(X, y)
+
+print(f"True beta: {true_beta}")
+print(f"Fitted beta: {beta}")
+print(f"Sklearn fitted beta: {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column")
+print(MSE(y,ypredictOwn))
+print(f"MSE with intercept column from SKL")
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x, y, label="Data")
+plt.plot(x, X @ beta, label="Fit")
+plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
+
+
+# Do not include the intercept in the design matrix
+X = np.zeros((len(x), degree - 1))
+
+for p inrange(degree - 1):
+ X[:, p] = x ** (p + 1)
+
+# Intercept is not included in the design matrix
+skl = LinearRegression(fit_intercept=True).fit(X, y)
+
+# Use centered values for X and y when computing coefficients
+y_offset = np.average(y, axis=0)
+X_offset = np.average(X, axis=0)
+
+beta = fit_beta(X - X_offset, y - y_offset)
+intercept = np.mean(y_offset - X_offset @ beta)
+
+print(f"Manual intercept: {intercept}")
+print(f"Fitted beta (wiothout intercept): {beta}")
+print(f"Sklearn intercept: {skl.intercept_}")
+print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept")
+print(MSE(y,ypredictOwn+intercept))
+print(f"MSE with Sklearn intercept")
+print(MSE(y,ypredictSKL))
+
+plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
+plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
+plt.grid()
+plt.legend()
+
+plt.show()
+
+
+The intercept is the value of our output/target variable
+when all our features are zero and our function crosses the \( y \)-axis (for a one-dimensional case).
+
+
+Printing the MSE, we see first that both methods give the same MSE, as
+they should. However, when we move to for example Ridge regression,
+the way we treat the intercept may give a larger or smaller MSE,
+meaning that the MSE can be penalized by the value of the
+intercept. Not including the intercept in the fit, means that the
+regularization term does not include \( \beta_0 \). For different values
+of \( \lambda \), this may lead to differeing MSE values.
+
+
+To remind the reader, the regularization term, with the intercept in Ridge regression is given by
+
+It means that, when scaling the design matrix and the outputs/targets, by subtracting the mean values, we have an optimization problem which is not penalized by the intercept. The MSE value can then be smaller since it focuses only on the remaining quantities. If we however bring back the intercept, we will get a MSE which then contains the intercept.
+
+
+
+
+
Code Examples
+
+
+Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known vanilla data set.
+
+
+
+
+
importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimport train_test_split
+fromsklearnimport linear_model
+
+defMSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree))
+#We include explicitely the intercept column
+for degree inrange(Maxpolydegree):
+ X[:,degree] = x**degree
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+p = Maxpolydegree
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 2, nlambdas)
+for i inrange(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+ # Note: we include the intercept column and no scaling
+ RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+ RegRidge.fit(X_train,y_train)
+ # and then make the prediction
+ ytildeOwnRidge = X_train @ OwnRidgeBeta
+ ypredictOwnRidge = X_test @ OwnRidgeBeta
+ ytildeRidge = RegRidge.predict(X_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta)
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix.
+We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
+What happens if we do not include the intercept in our fit?
+Let us see how we can change this code by zero centering (thanks to Stian Bilek for inpouts here).
+
+
+
+
+
Taking out the mean
+
+
+
+
importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.model_selectionimport train_test_split
+fromsklearnimport linear_model
+fromsklearn.preprocessingimport StandardScaler
+
+defMSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(315)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree-1))
+
+for degree inrange(1,Maxpolydegree): #No intercept column
+ X[:,degree-1] = x**(degree)
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean = np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
+X_test_scaled = X_test - X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler = np.mean(y_train)
+y_train_scaled = y_train - y_scaler
+
+p = Maxpolydegree-1
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 2, nlambdas)
+for i inrange(nlambdas):
+ lmb = lambdas[i]
+ OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
+ intercept_ = y_scaler - X_train_mean@OwnRidgeBeta#The intercept can be shifted so the model can predict on uncentered data
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
+ RegRidge = linear_model.Ridge(lmb)
+ RegRidge.fit(X_train,y_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+ MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+ print("Beta values for own Ridge implementation")
+ print(OwnRidgeBeta) #Intercept is given by mean of target variable
+ print("Beta values for Scikit-Learn Ridge implementation")
+ print(RegRidge.coef_)
+ print('Intercept from own implementation:')
+ print(intercept_)
+ print('Intercept from Scikit-Learn Ridge implementation')
+ print(RegRidge.intercept_)
+ print("MSE values for own Ridge implementation")
+ print(MSEOwnRidgePredict[i])
+ print("MSE values for Scikit-Learn Ridge implementation")
+ print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
+
+We see here, when compared to the code which includes explicitely the
+intercept column, that our MSE value is actually smaller. This is
+because the regularization term does not include the intercept value
+\( \beta_0 \) in the fitting. This applies to Lasso regularization as
+well. It means that our optimization is now done only with the
+centered matrix and/or vector that enter the fitting procedure. Note
+also that the problem with the intercept occurs mainly in these type
+of polynomial fitting problem.
+
+
+The next example is indeed an example where all these discussions about the role of intercept are not present.
+
+
+
+
+
More complicated Example: The Ising model
+
+
+The one-dimensional Ising model with nearest neighbor interaction, no
+external field and a constant coupling constant \( J \) is given by
+
+
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins
+in the system is determined by \( L \). For the one-dimensional system
+there is no phase transition.
+
+
+We will look at a system of \( L = 40 \) spins with a coupling constant of
+\( J = 1 \). To get enough training data we will generate 10000 states
+with their respective energies.
+
+
+Here we use ordinary least squares
+regression to predict the energy for the nearest neighbor
+one-dimensional Ising model on a ring, i.e., the endpoints wrap
+around. We will use linear regression to fit a value for
+the coupling constant to achieve this.
+
+
+
+
+
Reformulating the problem to suit regression
+
+
+A more general form for the one-dimensional Ising model is
+
+
+Here we allow for interactions beyond the nearest neighbors and a state dependent
+coupling constant. This latter expression can be formulated as
+a matrix-product
+
+where \( X_{jk} = s_j s_k \) and \( J \) is a matrix which consists of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, that is
+
+
+We then find the extremal point of \( C \) by taking the derivative with respect to \( \boldsymbol{\beta} \) as discussed above.
+This yields the expression for \( \boldsymbol{\beta} \) to be
+
+
+which immediately imposes some requirements on \( \boldsymbol{X} \) as there must exist
+an inverse of \( \boldsymbol{X}^T \boldsymbol{X} \). If the expression we are modeling contains an
+intercept, i.e., a constant term, we must make sure that the
+first column of \( \boldsymbol{X} \) consists of \( 1 \). We do this here
+
+
+Doing the inversion directly turns out to be a bad idea since the matrix
+\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the singular
+value decomposition. Using the definition of the Moore-Penrose
+pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
+
+
+Using singular value decomposition we can decompose the matrix \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
+where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
+where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
+\( \omega \) to
+
+Note that solving this equation by actually doing the pseudoinverse
+(which is what we will do) is not a good idea as this operation scales
+as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
+general matrix. Instead, doing \( QR \)-factorization and solving the
+linear system as an equation would reduce this down to
+\( \mathcal{O}(n^2) \) operations.
+
+
+
+
+
defols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+ u, s, v = scl.svd(x)
+ return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y
+
+
+
+
+
beta = ols_svd(X_train_own,y_train)
+
+
+When extracting the \( J \)-matrix we need to make sure that we remove the intercept, as is done here
+
+
+
+
+
J = beta[1:].reshape(L, L)
+
+
+A way of looking at the coefficients in \( J \) is to plot the matrices as images.
+
+
+It is interesting to note that OLS
+considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
+valid matrix elements for \( J \).
+In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
+this problem can be removed, partly and only with Lasso regression.
+
+
+In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
+
+
+
+
+
The one-dimensional Ising model
+
+
+Let us bring back the Ising model again, but now with an additional
+focus on Ridge and Lasso regression as well. We repeat some of the
+basic parts of the Ising model and the setup of the training and test
+data. The one-dimensional Ising model with nearest neighbor
+interaction, no external field and a constant coupling constant \( J \) is
+given by
+
+
+
+where \( s_i \in \{-1, 1\} \) and \( s_{N + 1} = s_1 \). The number of spins in the system is determined by \( L \). For the one-dimensional system there is no phase transition.
+
+
+We will look at a system of \( L = 40 \) spins with a coupling constant of \( J = 1 \). To get enough training data we will generate 10000 states with their respective energies.
+
+
+Here we allow for interactions beyond the nearest neighbors and a more
+adaptive coupling matrix. This latter expression can be formulated as
+a matrix-product on the form
+
+$$
+\begin{align}
+ H = X J,
+\tag{9}
+\end{align}
+$$
+
+
+
+where \( X_{jk} = s_j s_k \) and \( J \) is the matrix consisting of the
+elements \( -J_{jk} \). This form of writing the energy fits perfectly
+with the form utilized in linear regression, viz.
+
+The results perfectly with our previous discussion where we used our own code.
+
+
+
+
+
Ridge regression
+
+
+Having explored the ordinary least squares we move on to ridge
+regression. In ridge regression we include a regularizer. This
+involves a new cost function which leads to a new estimate for the
+weights \( \boldsymbol{\beta} \). This results in a penalized regression problem. The
+cost function is given by
+
+
+Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from Scikit-Learn.
+
+
+It is quite striking how LASSO breaks the symmetry of the coupling
+constant as opposed to ridge and OLS. We get a sparse solution with
+\( J_{j, j + 1} = -1 \).
+
+
+
+
+
Performance as function of the regularization parameter
+
+
+We see how the different models perform for a different set of values for \( \lambda \).
+
+
+We see that LASSO reaches a good solution for low
+values of \( \lambda \), but will "wither" when we increase \( \lambda \) too
+much. Ridge is more stable over a larger range of values for
+\( \lambda \), but eventually also fades away.
+
+
+
+
+
Finding the optimal value of \( \lambda \)
+
+
+To determine which value of \( \lambda \) is best we plot the accuracy of
+the models when predicting the training and the testing set. We expect
+the accuracy of the training set to be quite good, but if the accuracy
+of the testing set is much lower this tells us that we might be
+subject to an overfit model. The ideal scenario is an accuracy on the
+testing set that is close to the accuracy of the training set.
+
+
+From the above figure we can see that LASSO with \( \lambda = 10^{-2} \)
+achieves a very good accuracy on the test set. This by far surpasses the
+other models for all values of \( \lambda \).
+
+
+
+
+
Logistic Regression
+
+
+In linear regression our main interest was centered on learning the
+coefficients of a functional fit (say a polynomial) in order to be
+able to predict the response of a continuous variable on some unseen
+data. The fit to the continuous variable \( y_i \) is based on some
+independent variables \( \boldsymbol{x}_i \). Linear regression resulted in
+analytical expressions for standard ordinary Least Squares or Ridge
+regression (in terms of matrices to invert) for several quantities,
+ranging from the variance and thereby the confidence intervals of the
+parameters \( \boldsymbol{\beta} \) to the mean squared error. If we can invert
+the product of the design matrices, linear regression gives then a
+simple recipe for fitting our data.
+
+
+
+
+
Classification problems
+
+
+Classification problems, however, are concerned with outcomes taking
+the form of discrete variables (i.e. categories). We may for example,
+on the basis of DNA sequencing for a number of patients, like to find
+out which mutations are important for a certain disease; or based on
+scans of various patients' brains, figure out if there is a tumor or
+not; or given a specific physical system, we'd like to identify its
+state, say whether it is an ordered or disordered system (typical
+situation in solid state physics); or classify the status of a
+patient, whether she/he has a stroke or not and many other similar
+situations.
+
+
+The most common situation we encounter when we apply logistic
+regression is that of two possible outcomes, normally denoted as a
+binary outcome, true or false, positive or negative, success or
+failure etc.
+
+
+
+
+
Optimization and Deep learning
+
+
+Logistic regression will also serve as our stepping stone towards
+neural network algorithms and supervised deep learning. For logistic
+learning, the minimization of the cost function leads to a non-linear
+equation in the parameters \( \boldsymbol{\beta} \). The optimization of the
+problem calls therefore for minimization algorithms. This forms the
+bottle neck of all machine learning algorithms, namely how to find
+reliable minima of a multi-variable function. This leads us to the
+family of gradient descent methods. The latter are the working horses
+of basically all modern machine learning algorithms.
+
+
+We note also that many of the topics discussed here on logistic
+regression are also commonly used in modern supervised Deep Learning
+models, as we will see later.
+
+
+
+
+
Basics
+
+
+We consider the case where the dependent variables, also called the
+responses or the outcomes, \( y_i \) are discrete and only take values
+from \( k=0,\dots,K-1 \) (i.e. \( K \) classes).
+
+
+The goal is to predict the
+output classes from the design matrix \( \boldsymbol{X}\in\mathbb{R}^{n\times p} \)
+made of \( n \) samples, each of which carries \( p \) features or predictors. The
+primary goal is to identify the classes to which new unseen samples
+belong.
+
+
+Let us specialize to the case of two classes only, with outputs
+\( y_i=0 \) and \( y_i=1 \). Our outcomes could represent the status of a
+credit card user that could default or not on her/his credit card
+debt. That is
+
+
+Before moving to the logistic model, let us try to use our linear
+regression model to classify these two outcomes. We could for example
+fit a linear model to the default case if \( y_i > 0.5 \) and the no
+default case \( y_i \leq 0.5 \).
+
+
+We would then have our
+weighted linear combination, namely
+
+
+where \( \boldsymbol{y} \) is a vector representing the possible outcomes, \( \boldsymbol{X} \) is our
+\( n\times p \) design matrix and \( \boldsymbol{\beta} \) represents our estimators/predictors.
+
+
+
+
+
Some selected properties
+
+
+The main problem with our function is that it takes values on the
+entire real axis. In the case of logistic regression, however, the
+labels \( y_i \) are discrete variables. A typical example is the credit
+card data discussed below here, where we can set the state of
+defaulting the debt to \( y_i=1 \) and not to \( y_i=0 \) for one the persons
+in the data set (see the full example below).
+
+
+One simple way to get a discrete output is to have sign
+functions that map the output of a linear regressor to values \( \{0,1\} \),
+\( f(s_i)=sign(s_i)=1 \) if \( s_i\ge 0 \) and 0 if otherwise.
+We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning
+literature. This model is extremely simple. However, in many cases it is more
+favorable to use a ``soft" classifier that outputs
+the probability of a given category. This leads us to the logistic function.
+
+
+
+
+
Simple example
+
+
+The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.
+
+
+
+
+
# Common imports
+importos
+importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.linear_modelimport LinearRegression, Ridge, Lasso
+fromsklearn.model_selectionimport train_test_split
+fromsklearn.utilsimport resample
+fromsklearn.metricsimport mean_squared_error
+fromIPython.displayimport display
+frompylabimport plt, mpl
+plt.style.use('seaborn')
+mpl.rcParams['font.family'] = 'serif'
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+ifnot os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+ifnot os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+ifnot os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+defimage_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+defdata_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+defsave_fig(fig_id):
+ plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("chddata.csv"),'r')
+
+# Read the chd data as csv file and organize the data into arrays with age group, age, and chd
+chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
+chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
+output = chd['CHD']
+age = chd['Age']
+agegroup = chd['Agegroup']
+numberID = chd['ID']
+display(chd)
+
+plt.scatter(age, output, marker='o')
+plt.axis([18,70.0,-0.1, 1.2])
+plt.xlabel(r'Age')
+plt.ylabel(r'CHD')
+plt.title(r'Age distribution and Coronary heart disease')
+plt.show()
+
+
+
+
+
+
Plotting the mean value for each group
+
+
+What we could attempt however is to plot the mean value for each group.
+
+
+
+
+
agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
+group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
+plt.plot(group, agegroupmean, "r-")
+plt.axis([0,9,0, 1.0])
+plt.xlabel(r'Age group')
+plt.ylabel(r'CHD mean values')
+plt.title(r'Mean values for each age group')
+plt.show()
+
+
+We are now trying to find a function \( f(y\vert x) \), that is a function which gives us an expected value for the output \( y \) with a given input \( x \).
+In standard linear regression with a linear dependence on \( x \), we would write this in terms of our model
+
+$$
+f(y_i\vert x_i)=\beta_0+\beta_1 x_i.
+$$
+
+
+
+This expression implies however that \( f(y_i\vert x_i) \) could take any
+value from minus infinity to plus infinity. If we however let
+\( f(y\vert y) \) be represented by the mean value, the above example
+shows us that we can constrain the function to take values between
+zero and one, that is we have \( 0 \le f(y_i\vert x_i) \le 1 \). Looking
+at our last curve we see also that it has an S-shaped form. This leads
+us to a very popular model for the function \( f \), namely the so-called
+Sigmoid function or logistic model. We will consider this function as
+representing the probability for finding a value of \( y_i \) with a given
+\( x_i \).
+
+
+
+
+
The logistic function
+
+
+Another widely studied model, is the so-called
+perceptron model, which is an example of a "hard classification" model. We
+will encounter this model when we discuss neural networks as
+well. Each datapoint is deterministically assigned to a category (i.e
+\( y_i=0 \) or \( y_i=1 \)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft"
+classifier that outputs the probability of a given category rather
+than a single value. For example, given \( x_i \), the classifier
+outputs the probability of being in a category \( k \). Logistic regression
+is the most common example of a so-called soft classifier. In logistic
+regression, the probability that a data point \( x_i \)
+belongs to a category \( y_i=\{0,1\} \) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,
+
Examples of likelihood functions used in logistic regression and nueral networks
+
+
+The following code plots the logistic function, the step function and other functions we will encounter from here and on.
+
+
+
+
+
"""The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
+
+importnumpy
+importmatplotlib.pyplotasplt
+importmathasmt
+
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0if z >= 0.0else0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""tanh Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.tanh(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('tanh function')
+
+plt.show()
+
+
+
+
+
+
Two parameters
+
+
+We assume now that we have two classes with \( y_i \) either \( 0 \) or \( 1 \). Furthermore we assume also that we have only two parameters \( \beta \) in our fitting of the Sigmoid function, that is we define probabilities
+
+In order to define the total likelihood for all possible outcomes from a
+dataset \( \mathcal{D}=\{(y_i,x_i)\} \), with the binary labels
+\( y_i\in\{0,1\} \) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle.
+We aim thus at maximizing
+the probability of seeing the observed data. We can then approximate the
+likelihood in terms of the product of the individual probabilities of a specific outcome \( y_i \), that is
+
+The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \( \beta \).
+Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that
+
+
+This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression,
+in practice we often supplement the cross-entropy with additional regularization terms, usually \( L_1 \) and \( L_2 \) regularization as we did for Ridge and Lasso regression.
+
+
+
+
+
Minimizing the cross entropy
+
+
+The cross entropy is a convex function of the weights \( \boldsymbol{\beta} \) and,
+therefore, any local minimizer is a global minimizer.
+
+
+Minimizing this
+cost function with respect to the two parameters \( \beta_0 \) and \( \beta_1 \) we obtain
+
+
+Let us now define a vector \( \boldsymbol{y} \) with \( n \) elements \( y_i \), an
+\( n\times p \) matrix \( \boldsymbol{X} \) which contains the \( x_i \) values and a
+vector \( \boldsymbol{p} \) of fitted probabilities \( p(y_i\vert x_i,\boldsymbol{\beta}) \). We can rewrite in a more compact form the first
+derivative of cost function as
+
+
+If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements
+\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as
+
+
+Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
+
+Till now we have mainly focused on two classes, the so-called binary
+system. Suppose we wish to extend to \( K \) classes. Let us for the sake
+of simplicity assume we have only two predictors. We have then following model
+
+
+and the model is specified in term of \( K-1 \) so-called log-odds or
+logit transformations.
+
+
+
+
+
More classes
+
+
+In our discussion of neural networks we will encounter the above again
+in terms of a slightly modified function, the so-called Softmax function.
+
+
+The softmax function is used in various multiclass classification
+methods, such as multinomial logistic regression (also known as
+softmax regression), multiclass linear discriminant analysis, naive
+Bayes classifiers, and artificial neural networks. Specifically, in
+multinomial logistic regression and linear discriminant analysis, the
+input to the function is the result of \( K \) distinct linear functions,
+and the predicted probability for the \( k \)-th class given a sample
+vector \( \boldsymbol{x} \) and a weighting vector \( \boldsymbol{\beta} \) is (with two
+predictors):
+
+
+and they sum to one. Our earlier discussions were all specialized to
+the case with two classes only. It is easy to see from the above that
+what we derived earlier is compatible with these equations.
+
+
+To find the optimal parameters we would typically use a gradient
+descent method. Newton's method and gradient descent methods are
+discussed in the material on optimization
+methods.
+
+
+
+
+
Friday September 24
+
+
+
+
+
Wisconsin Cancer Data
+
+
+We show here how we can use a simple regression case on the breast
+cancer data using Logistic regression as our algorithm for
+classification.
+
+
+
+
+
importmatplotlib.pyplotasplt
+importnumpyasnp
+fromsklearn.model_selectionimport train_test_split
+fromsklearn.datasetsimport load_breast_cancer
+fromsklearn.linear_modelimport LogisticRegression
+
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+# Logistic Regression
+logreg = LogisticRegression(solver='lbfgs')
+logreg.fit(X_train, y_train)
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+#now scale the data
+fromsklearn.preprocessingimport StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled, y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+
+
+
+
+
+
Using the correlation matrix
+
+
+In addition to the above scores, we could also study the covariance (and the correlation matrix).
+We use Pandas to compute the correlation matrix.
+
+
+
+
importmatplotlib.pyplotasplt
+importnumpyasnp
+fromsklearn.model_selectionimport train_test_split
+fromsklearn.datasetsimport load_breast_cancer
+fromsklearn.linear_modelimport LogisticRegression
+cancer = load_breast_cancer()
+importpandasaspd
+# Making a data frame
+cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
+
+fig, axes = plt.subplots(15,2,figsize=(10,20))
+malignant = cancer.data[cancer.target == 0]
+benign = cancer.data[cancer.target == 1]
+ax = axes.ravel()
+
+for i inrange(30):
+ _, bins = np.histogram(cancer.data[:,i], bins =50)
+ ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)
+ ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)
+ ax[i].set_title(cancer.feature_names[i])
+ ax[i].set_yticks(())
+ax[0].set_xlabel("Feature magnitude")
+ax[0].set_ylabel("Frequency")
+ax[0].legend(["Malignant", "Benign"], loc ="best")
+fig.tight_layout()
+plt.show()
+
+importseabornassns
+correlation_matrix = cancerpd.corr().round(1)
+# use the heatmap function from seaborn to plot the correlation matrix
+# annot = True to print the values inside the square
+plt.figure(figsize=(15,8))
+sns.heatmap(data=correlation_matrix, annot=True)
+plt.show()
+
+
+
+
+
+
Discussing the correlation data
+
+
+In the above example we note two things. In the first plot we display
+the overlap of benign and malignant tumors as functions of the various
+features in the Wisconsing breast cancer data set. We see that for
+some of the features we can distinguish clearly the benign and
+malignant cases while for other features we cannot. This can point to
+us which features may be of greater interest when we wish to classify
+a benign or not benign tumour.
+
+
+In the second figure we have computed the so-called correlation
+matrix, which in our case with thirty features becomes a \( 30\times 30 \)
+matrix.
+
+
+We constructed this matrix using pandas via the statements
+
+Diagonalizing this matrix we can in turn say something about which
+features are of relevance and which are not. This leads us to
+the classical Principal Component Analysis (PCA) theorem with
+applications. This will be discussed later this semester (week 43).
+
+
+
+
+
Other measures in classification studies: Cancer Data again
+Almost every problem in machine learning and data science starts with
+a dataset \( X \), a model \( g(\beta) \), which is a function of the
+parameters \( \beta \) and a cost function \( C(X, g(\beta)) \) that allows
+us to judge how well the model \( g(\beta) \) explains the observations
+\( X \). The model is fit by finding the values of \( \beta \) that minimize
+the cost function. Ideally we would be able to solve for \( \beta \)
+analytically, however this is not possible in general and we must use
+some approximative/numerical method to compute the minimum.
+
+
+
+
+
Revisiting our Logistic Regression case
+
+
+In our discussion on Logistic Regression we studied the
+case of
+two classes, with \( y_i \) either
+\( 0 \) or \( 1 \). Furthermore we assumed also that we have only two
+parameters \( \beta \) in our fitting, that is we
+defined probabilities
+
+
+
+where \( \boldsymbol{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
+
+
+
+
+
The equations to solve
+
+
+Our compact equations used a definition of a vector \( \boldsymbol{y} \) with \( n \)
+elements \( y_i \), an \( n\times p \) matrix \( \boldsymbol{X} \) which contains the
+\( x_i \) values and a vector \( \boldsymbol{p} \) of fitted probabilities
+\( p(y_i\vert x_i,\boldsymbol{\beta}) \). We rewrote in a more compact form
+the first derivative of the cost function as
+
+
+If we in addition define a diagonal matrix \( \boldsymbol{W} \) with elements
+\( p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta}) \), we can obtain a compact expression of the second derivative as
+
+
+
+This defines what is called the Hessian matrix.
+
+
+
+
+
Solving using Newton-Raphson's method
+
+
+If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.
+
+
+
+The right-hand side is computed with the old values of \( \beta \).
+
+
+If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.
+
+
+
+
+
Brief reminder on Newton-Raphson's method
+
+
+Let us quickly remind ourselves how we derive the above method.
+
+
+Perhaps the most celebrated of all one-dimensional root-finding
+routines is Newton's method, also called the Newton-Raphson
+method. This method requires the evaluation of both the
+function \( f \) and its derivative \( f' \) at arbitrary points.
+If you can only calculate the derivative
+numerically and/or your function is not of the smooth type, we
+normally discourage the use of this method.
+
+
+
+
+
The equations
+
+
+The Newton-Raphson formula consists geometrically of extending the
+tangent line at a current point until it crosses zero, then setting
+the next guess to the abscissa of that zero-crossing. The mathematics
+behind this method is rather simple. Employing a Taylor expansion for
+\( x \) sufficiently close to the solution \( s \), we have
+
+
+For small enough values of the function and for well-behaved
+functions, the terms beyond linear are unimportant, hence we obtain
+
+
+$$
+ f(x)+(s-x)f'(x)\approx 0,
+$$
+
+
+yielding
+
+$$
+ s\approx x-\frac{f(x)}{f'(x)}.
+$$
+
+
+
+Having in mind an iterative procedure, it is natural to start iterating with
+
+$$
+ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.
+$$
+
+
+
+
+
+
Simple geometric interpretation
+
+
+The above is Newton-Raphson's method. It has a simple geometric
+interpretation, namely \( x_{n+1} \) is the point where the tangent from
+\( (x_n,f(x_n)) \) crosses the \( x \)-axis. Close to the solution,
+Newton-Raphson converges fast to the desired result. However, if we
+are far from a root, where the higher-order terms in the series are
+important, the Newton-Raphson formula can give grossly inaccurate
+results. For instance, the initial guess for the root might be so far
+from the true root as to let the search interval include a local
+maximum or minimum of the function. If an iteration places a trial
+guess near such a local extremum, so that the first derivative nearly
+vanishes, then Newton-Raphson may fail totally
+
+
+
+
+
Extending to more than one variable
+
+
+Newton's method can be generalized to systems of several non-linear equations
+and variables. Consider the case with two equations
+
+
+We need thus to compute the inverse of the Jacobian matrix and it
+is to understand that difficulties may
+arise in case \( {\bf \boldsymbol{J}} \) is nearly singular.
+
+
+It is rather straightforward to extend the above scheme to systems of
+more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.
+
+
+
+
+
Steepest descent
+
+
+The basic idea of gradient descent is
+that a function \( F(\mathbf{x}) \),
+\( \mathbf{x} \equiv (x_1,\cdots,x_n) \), decreases fastest if one goes from \( \bf {x} \) in the
+direction of the negative gradient \( -\nabla F(\mathbf{x}) \).
+
+
+For \( \gamma_k \) small enough, then \( F(\mathbf{x}_{k+1}) \leq
+F(\mathbf{x}_k) \). This means that for a sufficiently small \( \gamma_k \)
+we are always moving towards smaller function values, i.e a minimum.
+
+
+
+
+
More on Steepest descent
+
+
+The previous observation is the basis of the method of steepest
+descent, which is also referred to as just gradient descent (GD). One
+starts with an initial guess \( \mathbf{x}_0 \) for a minimum of \( F \) and
+computes new approximations according to
+
+
+The parameter \( \gamma_k \) is often referred to as the step length or
+the learning rate within the context of Machine Learning.
+
+
+
+
+
The ideal
+
+
+Ideally the sequence \( \{\mathbf{x}_k \}_{k=0} \) converges to a global
+minimum of the function \( F \). In general we do not know if we are in a
+global or local minimum. In the special case when \( F \) is a convex
+function, all local minima are also global minima, so in this case
+gradient descent can converge to the global solution. The advantage of
+this scheme is that it is conceptually simple and straightforward to
+implement. However the method in this form has some severe
+limitations:
+
+
+In machine learing we are often faced with non-convex high dimensional
+cost functions with many local minima. Since GD is deterministic we
+will get stuck in a local minimum, if the method converges, unless we
+have a very good intial guess. This also implies that the scheme is
+sensitive to the chosen initial condition.
+
+
+Note that the gradient is a function of \( \mathbf{x} =
+(x_1,\cdots,x_n) \) which makes it expensive to compute numerically.
+
+
+
+
+
The sensitiveness of the gradient descent
+
+
+The gradient descent method
+is sensitive to the choice of learning rate \( \gamma_k \). This is due
+to the fact that we are only guaranteed that \( F(\mathbf{x}_{k+1}) \leq
+F(\mathbf{x}_k) \) for sufficiently small \( \gamma_k \). The problem is to
+determine an optimal learning rate. If the learning rate is chosen too
+small the method will take a long time to converge and if it is too
+large we can experience erratic behavior.
+
+
+Many of these shortcomings can be alleviated by introducing
+randomness. One such method is that of Stochastic Gradient Descent
+(SGD), see below.
+
+
+
+
+
Convex functions
+
+
+Ideally we want our cost/loss function to be convex(concave).
+
+
+First we give the definition of a convex set: A set \( C \) in
+\( \mathbb{R}^n \) is said to be convex if, for all \( x \) and \( y \) in \( C \) and
+all \( t \in (0,1) \) , the point \( (1 − t)x + ty \) also belongs to
+C. Geometrically this means that every point on the line segment
+connecting \( x \) and \( y \) is in \( C \) as discussed below.
+
+
+The convex subsets of \( \mathbb{R} \) are the intervals of
+\( \mathbb{R} \). Examples of convex sets of \( \mathbb{R}^2 \) are the
+regular polygons (triangles, rectangles, pentagons, etc...).
+
+
+
+
+
Convex function
+
+
+Convex function: Let \( X \subset \mathbb{R}^n \) be a convex set. Assume that the function \( f: X \rightarrow \mathbb{R} \) is continuous, then \( f \) is said to be convex if
for all \( x_1, x_2 \in X \) and for all \( t \in [0,1] \). If \( \leq \) is replaced with a strict inequaltiy in the definition, we demand \( x_1 \neq x_2 \) and \( t\in(0,1) \) then \( f \) is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \( f(x_1) \) and \( f(x_2) \), the value of the function on the interval \( [x_1,x_2] \) is always below the line as illustrated below.
+
+
+
+
+
Conditions on convex functions
+
+
+In the following we state first and second-order conditions which
+ensures convexity of a function \( f \). We write \( D_f \) to denote the
+domain of \( f \), i.e the subset of \( R^n \) where \( f \) is defined. For more
+details and proofs we refer to: S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press.
+
+
+
+First order condition
+
+Suppose \( f \) is differentiable (i.e \( \nabla f(x) \) is well defined for
+all \( x \) in the domain of \( f \)). Then \( f \) is convex if and only if \( D_f \)
+is a convex set and
+$$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$
+
holds
+for all \( x,y \in D_f \). This condition means that for a convex function
+the first order Taylor expansion (right hand side above) at any point
+a global under estimator of the function. To convince yourself you can
+make a drawing of \( f(x) = x^2+1 \) and draw the tangent line to \( f(x) \) and
+note that it is always below the graph.
+
+
+
+
+Second order condition
+
+Assume that \( f \) is twice
+differentiable, i.e the Hessian matrix exists at each point in
+\( D_f \). Then \( f \) is convex if and only if \( D_f \) is a convex set and its
+Hessian is positive semi-definite for all \( x\in D_f \). For a
+single-variable function this reduces to \( f''(x) \geq 0 \). Geometrically this means that \( f \) has nonnegative curvature
+everywhere.
+
+
+
+This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.
+
+
+
+
+
More on convex functions
+
+
+The next result is of great importance to us and the reason why we are
+going on about convex functions. In machine learning we frequently
+have to minimize a loss/cost function in order to find the best
+parameters for the model we are considering.
+
+
+Ideally we want the
+global minimum (for high-dimensional models it is hard to know
+if we have local or global minimum). However, if the cost/loss function
+is convex the following result provides invaluable information:
+
+
+
+Any minimum is global for convex functions
+
+Consider the problem of finding \( x \in \mathbb{R}^n \) such that \( f(x) \)
+is minimal, where \( f \) is convex and differentiable. Then, any point
+\( x^* \) that satisfies \( \nabla f(x^*) = 0 \) is a global minimum.
+
+
+
+This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.
+
+
+
+
+
Some simple problems
+
+
+
Show that \( f(x)=x^2 \) is convex for \( x \in \mathbb{R} \) using the definition of convexity. Hint: If you re-write the definition, \( f \) is convex if the following holds for all \( x,y \in D_f \) and any \( \lambda \in [0,1] \) $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$.
+
Using the second order condition show that the following functions are convex on the specified domain.
+
+
+
\( f(x) = e^x \) is convex for \( x \in \mathbb{R} \).
+
\( g(x) = -\ln(x) \) is convex for \( x \in (0,\infty) \).
+
+
Let \( f(x) = x^2 \) and \( g(x) = e^x \). Show that \( f(g(x)) \) and \( g(f(x)) \) is convex for \( x \in \mathbb{R} \). Also show that if \( f(x) \) is any convex function than \( h(x) = e^{f(x)} \) is convex.
+
A norm is any function that satisfy the following properties
+
+
+
\( f(\alpha x) = |\alpha| f(x) \) for all \( \alpha \in \mathbb{R} \).
+
\( f(x+y) \leq f(x) + f(y) \)
+
\( f(x) \leq 0 \) for all \( x \in \mathbb{R}^n \) with equality if and only if \( x = 0 \)
+
+
+
+
+
+Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).
+
+Before we proceed, we would like to discuss the approach called the
+standard Steepest descent (different from the above steepest descent discussion), which again leads to us having to be able
+to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).
+
+
+The success of the CG method
+for finding solutions of non-linear problems is based on the theory
+of conjugate gradients for linear systems of equations. It belongs to
+the class of iterative methods for solving problems from linear
+algebra of the type
+
+with the constraint that the matrix \( \boldsymbol{A} \) is positive definite and
+symmetric. This defines also the Hessian and we want it to be positive definite.
+
+
+
+
+
Steepest descent method
+
+
+We denote the initial guess for \( \boldsymbol{x} \) as \( \boldsymbol{x}_0 \).
+We can assume without loss of generality that
+
+
+This suggests taking the first basis vector \( \boldsymbol{r}_1 \) (see below for definition)
+to be the gradient of \( f \) at \( \boldsymbol{x}=\boldsymbol{x}_0 \),
+which equals
+
+In the CG method we define so-called conjugate directions and two vectors
+\( \boldsymbol{s} \) and \( \boldsymbol{t} \)
+are said to be
+conjugate if
+
+
+The philosophy of the CG method is to perform searches in various conjugate directions
+of our vectors \( \boldsymbol{x}_i \) obeying the above criterion, namely
+
+
+Two vectors are conjugate if they are orthogonal with respect to
+this inner product. Being conjugate is a symmetric relation: if \( \boldsymbol{s} \) is conjugate to \( \boldsymbol{t} \), then \( \boldsymbol{t} \) is conjugate to \( \boldsymbol{s} \).
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+An example is given by the eigenvectors of the matrix
+
+Assume now that we have a symmetric positive-definite matrix \( \boldsymbol{A} \) of size
+\( n\times n \). At each iteration \( i+1 \) we obtain the conjugate direction of a vector
+
+
+We assume that \( \boldsymbol{p}_{i} \) is a sequence of \( n \) mutually conjugate directions.
+Then the \( \boldsymbol{p}_{i} \) form a basis of \( R^n \) and we can expand the solution
+$ \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}$ in this basis, namely
+
+
+If we choose the conjugate vectors \( \boldsymbol{p}_k \) carefully,
+then we may not need all of them to obtain a good approximation to the solution
+\( \boldsymbol{x} \).
+We want to regard the conjugate gradient method as an iterative method.
+This will us to solve systems where \( n \) is so large that the direct
+method would take too much time.
+
+
+We denote the initial guess for \( \boldsymbol{x} \) as \( \boldsymbol{x}_0 \).
+We can assume without loss of generality that
+
+
+This suggests taking the first basis vector \( \boldsymbol{p}_1 \)
+to be the gradient of \( f \) at \( \boldsymbol{x}=\boldsymbol{x}_0 \),
+which equals
+
+
+and
+\( \boldsymbol{x}_0=0 \) it is equal \( -\boldsymbol{b} \).
+The other vectors in the basis will be conjugate to the gradient,
+hence the name conjugate gradient method.
+
+
+
+
+
+
Conjugate gradient method
+
+
+
+Let \( \boldsymbol{r}_k \) be the residual at the \( k \)-th step:
+
+
+Note that \( \boldsymbol{r}_k \) is the negative gradient of \( f \) at
+\( \boldsymbol{x}=\boldsymbol{x}_k \),
+so the gradient descent method would be to move in the direction \( \boldsymbol{r}_k \).
+Here, we insist that the directions \( \boldsymbol{p}_k \) are conjugate to each other,
+so we take the direction closest to the gradient \( \boldsymbol{r}_k \)
+under the conjugacy constraint.
+This gives the following expression
+
Revisiting some of our first Linear Regression Encounters
+
+
+We will use linear regression as a case study for the gradient descent
+methods. Linear regression is a great test case for the gradient
+descent methods discussed in the lectures since it has several
+desirable properties such as:
+
+
+
An analytical solution (recall homework set 1).
+
The gradient can be computed analytically.
+
The cost function is convex which guarantees that gradient descent converges for small enough learning rates
+
+
+
+We revisit an example similar to what we had in the first homework set. We had a function of the type
+
+
+
+
+
x = 2*np.random.rand(m,1)
+y = 4+3*x+np.random.randn(m,1)
+
+
+with \( x_i \in [0,1] \) is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution \( \cal {N}(0,1) \).
+The linear regression model is given by
+
+It is convenient to write \( \mathbf{\boldsymbol{y}} = X\beta \) where \( X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by (we keep the intercept here)
+
+We can use the expression we computed for the gradient and let use a
+\( \beta_0 \) be chosen randomly and let \( \gamma = 0.001 \). Stop iterating
+when \( ||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8} \). Note that the code below does not include the latter stop criterion.
+
+
+And finally we can compare our solution for \( \beta \) with the analytic result given by
+\( \beta= (X^TX)^{-1} X^T \mathbf{y} \).
+
Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.
+
GD is sensitive to initial conditions. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.
+
Gradients are computationally expensive to calculate for large datasets. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \( E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2 \); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over all \( n \) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm.
+
GD is very sensitive to choices of learning rates. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would adaptively choose the learning rates to match the landscape.
+
GD treats all directions in parameter space uniformly. Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.
+
GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points. This leads us to our next topic, Stochastic Gradient Methods.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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+