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. +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 @@ -463,7 +470,7 @@ This can clearly lead to problems in evaluating the cost/loss functions.
-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) +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.
@@ -514,7 +519,7 @@ plt.show()
-The cost/loss function for Ridge regression is: +Let us try to understand what this may imply mathematically when we subtract the mean values, also known as zero centering. To catch many birds with just one stone, we will focus on Ridge regression. +
+The cost/loss function for Ridge regression is $$ -C(\beta_0, \beta_1, ... , \beta_P) = \sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip}\beta_p)^2 + \lambda \sum_{p=1}^P \beta_p^2. +C(\beta_0, \beta_1, ... , \beta_{p-1}) = \sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2 + \lambda \sum_{j=1}^{p-1} \beta_i^2. $$
-Notice that the intercept is left out of the \( L_2 \) regularization term. The design matrix +Note that the intercept term $\beta_0$is left out of the \( L_2 \) regularization term. The design matrix \( X \) does in this case not contain any intercept column. We want $$ -\frac{\partial L}{\partial \beta_j} = 0, +\frac{\partial C}{\partial \beta_j} = 0, $$
-for all \( j \), so lets start with \( \beta_0 \). This means that we have +for all \( j \), so let us start with \( \beta_0 \). This means that we have $$ -\frac{\partial L}{\partial \beta_0} = -2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p). -$$ - -
-We want to solve -$$ --2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p) = 0, +\frac{\partial C}{\partial \beta_0} = -2\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right), $$
which gives $$ -\sum_{i=1}^{n} \beta_0 = \sum_{i=1}^{n}y_i - \sum_{i=1}^{n} \sum_{p=1}^P X_{ip} \beta_p, +\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j. $$
-or -$ n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip}$. - -
-If we assume that every column of \( X \) is centered, whic we can do by subtracting the mean, +If 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)
-the sum $ \sum_{i=1}^{n} X_{ip} $ - -
+the sum \( \sum_{i=0}^{n-1} X_{ij} \) can be rewritten as $$ -\sum_{i=1}^{n} (X_{ip} - \frac{1}{n}\sum_{i=1}^{n} X_{ip}) = \sum_{i=1}^{n} X_{ip} - \sum_{i=1}^{n} \frac{1}{n} \sum_{i=1}^{n}X_{ip}, +\sum_{i=0}^{n-1} \left(X_{ij} - \frac{1}{n}\sum_{i=0}^{n-1} X_{ij}) = \sum_{i=0}^{n-1} X_{ij} - \sum_{i=0}^{n-1} \frac{1}{n} \sum_{i=0}^{n-1}X_{ij}, $$ resulting in $$ -\sum_{i=1}^{n} X_{ip} - n \frac{1}{n} \sum_{i=1}^{n}X_{ip} = 0. +\sum_{i=0}^{n-1} X_{ij} - n \frac{1}{n} \sum_{i=0}^{n-1}X_{ij} = 0. $$
Finally we have $$ -n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip}, +n\beta_0 = \sum_{i=0}^{n-1} y_i - \sum_{j=1}^{p-1}\beta_j \sum_{i=0}^{n-1} X_{ij}, $$ or $$ -\beta_0 = \frac{1}{n}\sum_{i=1}^{n} y_i = y_{average}. +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\boldsymbol{y}}, $$ +the average value of \( \boldsymbol{y] \). +
-Replacing \( y_i \) with \( y_i - \beta_0 = y_i - y_{average} \) in the loss function will give us (in vector-matrix disguise) +Replacing \( y_i \) with \( y_i - \beta_0 = y_i - \overline{\boldsymbol{y}} \) in the cost function will give us (in vector-matrix disguise) $$ C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}, $$ @@ -493,201 +489,9 @@ which has the solution
\( \beta = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}} \). -where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - y_{average} \) +where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \) and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \). -
- - -
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 R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-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))
-X[:,0] = 1.0
-
-for polydegree in range(1, Maxpolydegree):
- for degree in range(polydegree):
- 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)
-
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X_train @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y_train,ytildeOLS))
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-
-p = len(OLSbeta)
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 4
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSEOwnRidgeTrain = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-MSERidgeTrain = np.zeros(nlambdas)
-
-lambdas = np.logspace(-4, 4, 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
- # include lasso using Scikit-Learn
- # Note: we include the intercept
- 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)
- MSEOwnRidgeTrain[i] = MSE(y_train,ytildeOwnRidge)
- MSERidgePredict[i] = MSE(y_test,ypredictRidge)
- MSERidgeTrain[i] = MSE(y_train,ytildeRidge)
- print("Beta values for own Ridge implementation")
- print(OwnRidgeBeta)
- print("Beta values for Scikit-Learn Ridge implementation")
- print(RegRidge.coef_)
-# Now plot the results
-plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'b', label = 'MSE Ridge train')
-plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE Ridge train')
-plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-- - -
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 R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-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 = 5
-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)
-X_train_scaled = X_train - X_train_mean #Center by removing mean from each feature
-X_test_scaled = X_test - X_train_mean
-
-y_scaler = np.mean(y_train) #The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)
-y_train_scaled = y_train - y_scaler #Remove the intercept from the training data.
-
-
-p = Maxpolydegree-1
-I = np.eye(p,p)
-# Decide which values of lambda to use
-nlambdas = 4
-MSEOwnRidgePredict = np.zeros(nlambdas)
-MSERidgePredict = np.zeros(nlambdas)
-
-lambdas = np.logspace(-4, 1, 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
-
- ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ #Add intercept to prediction
- #EQUIVALENT PREDICTION:
- ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler #Add intercept to prediction
- print("Values for own Ridge prediction")
- print(ypredictOwnRidge)
-
-
-
- RegRidge = linear_model.Ridge(lmb)
- RegRidge.fit(X_train,y_train)
- ypredictRidge = RegRidge.predict(X_test)
- print("Values for SL Ridge prediction")
- print(ypredictRidge)
-
-
- 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_)
-
-
-
-# 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()
-
@@ -714,7 +518,7 @@ plt.show()
-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. +Armed with this wisdom, we attempt first simply set the intercept eqault to False in our implementation of Ridge regression for a vanilla data set.
import numpy as np
+import pandas as pd
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')
+from sklearn import linear_model
-L = 40
-n = int(1e4)
+def MSE(y_data,y_model):
+ n = np.size(y_model)
+ return np.sum((y_data-y_model)**2)/n
-spins = np.random.choice([-1, 1], size=(n, L))
-J = 1.0
-energies = np.zeros(n)
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
-for i in range(n):
- energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))
+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 intercpt column
+X[:,0] = 1.0
+
+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 = 4
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 4, 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
+ # include lasso using Scikit-Learn
+ # 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_)
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
-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. +The results here agree when we force Scikit-Learn's Ridge function to include the first column in our design matrix. +The problem however is that can easily lead to a larger mean-squared error! + +
+Let us see how we can change this code by zero centering.
@@ -489,7 +522,7 @@ 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 in range(n):
- X[i] = np.outer(spins[i], spins[i]).ravel()
-y = energies
+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 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 = 4
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 1, 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_
+ #EQUIVALENT PREDICTION:
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
+ print("Values for own Ridge prediction")
+ print(ypredictOwnRidge)
+ RegRidge = linear_model.Ridge(lmb)
+ RegRidge.fit(X_train,y_train)
+ ypredictRidge = RegRidge.predict(X_test)
+ print("Values for SL Ridge prediction")
+ print(ypredictRidge)
+ 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_)
+
+# 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()
@@ -482,7 +530,7 @@ X_train, X_test, y_train, y_test = train_tes
-In the ordinary least squares method we choose the cost function +The one-dimensional Ising model with nearest neighbor interaction, no +external field and a constant coupling constant \( J \) is given by $$ \begin{align} - C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}. -\tag{5} + H = -J \sum_{k}^L s_k s_{k + 1}, +\tag{1} \end{align} $$
-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 - -$$ - \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}}, -$$ +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.
-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 +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.
-
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
-)
+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.
-
-
def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
- return scl.inv(x.T @ x) @ (x.T @ y)
-beta = ols_inv(X_train_own, y_train)
-
@@ -480,7 +494,7 @@ beta = ols_inv(X_train_own, y_train)
-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 +A more general form for the one-dimensional Ising model is -$$ - \boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y}, -$$ - -
-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 $$ \begin{align} - \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}. -\tag{6} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\tag{2} \end{align} $$
-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. +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
-
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
+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)
-
-
-
-
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?
-
@@ -519,7 +487,7 @@ In this case our matrix inversion was actually possible. The obvious question no
-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 +In the ordinary least squares method we choose the cost function $$ \begin{align} - H = -J \sum_{k}^L s_k s_{k + 1}, -\tag{7} + C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}. +\tag{5} \end{align} $$ -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 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 + +$$ + \boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}}, +$$
-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. +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
-
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 - -$$ -\begin{align} - H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. -\tag{8} -\end{align} -$$ - -
-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 -
- - -
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(
+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)
+def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
+ return scl.inv(x.T @ x) @ (x.T @ y)
+beta = ols_inv(X_train_own, 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 perfectly with our previous discussion where we used our own code.
-
@@ -566,7 +485,7 @@ The results perfectly with our previous discussion where we used our own code.
26
27
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs018.html b/doc/pub/week38/html/._week38-bs018.html
index 6fb23cd20..8cd02d15c 100644
--- a/doc/pub/week38/html/._week38-bs018.html
+++ b/doc/pub/week38/html/._week38-bs018.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,38 +414,90 @@ MathJax.Hub.Config({
-Ridge regression
+Singular Value decomposition
-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
+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
$$
\begin{align}
- C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
-\tag{11}
+ \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
+\tag{6}
\end{align}
$$
+
+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.
+
-
_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)
+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?
+
@@ -467,7 +524,7 @@ plt.show()
27
28
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs019.html b/doc/pub/week38/html/._week38-bs019.html
index 77d4edb1b..697fe8fac 100644
--- a/doc/pub/week38/html/._week38-bs019.html
+++ b/doc/pub/week38/html/._week38-bs019.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,40 +414,136 @@ MathJax.Hub.Config({
-LASSO regression
+The one-dimensional Ising model
-In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.
+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
$$
\begin{align}
- C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
-\tag{12}
+ H = -J \sum_{k}^L s_k s_{k + 1},
+\tag{7}
\end{align}
$$
+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.
+
-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.
+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.
-
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)
+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
+
+$$
+\begin{align}
+ H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
+\tag{8}
+\end{align}
+$$
+
+
+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
+
+
+
+
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()
-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
-\( J_{j, j + 1} = -1 \).
+The results perfectly with our previous discussion where we used our own code.
@@ -470,7 +571,7 @@ constant as opposed to ridge and OLS. We get a sparse solution with
28
29
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs020.html b/doc/pub/week38/html/._week38-bs020.html
index 589a2cb5c..e3cf5f3ed 100644
--- a/doc/pub/week38/html/._week38-bs020.html
+++ b/doc/pub/week38/html/._week38-bs020.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,57 +414,38 @@ MathJax.Hub.Config({
-Performance as function of the regularization parameter
+Ridge regression
-We see how the different models perform for a different set of values for \( \lambda \).
+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
+
+$$
+\begin{align}
+ C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta}.
+\tag{11}
+\end{align}
+$$
-
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
+_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()
-
-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.
-
@@ -486,7 +472,7 @@ much. Ridge is more stable over a larger range of values for
29
30
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs021.html b/doc/pub/week38/html/._week38-bs021.html
index 0a0dc4868..92c18e7cd 100644
--- a/doc/pub/week38/html/._week38-bs021.html
+++ b/doc/pub/week38/html/._week38-bs021.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,54 +414,40 @@ MathJax.Hub.Config({
-Finding the optimal value of \( \lambda \)
+LASSO regression
-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.
+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.
-
fig = plt.figure(figsize=(20, 14))
+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)
-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 \( \lambda \).
+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
+\( J_{j, j + 1} = -1 \).
@@ -484,7 +475,7 @@ other models for all values of \( \lambda \).
30
31
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs022.html b/doc/pub/week38/html/._week38-bs022.html
index 6817af07e..d7a5cdefd 100644
--- a/doc/pub/week38/html/._week38-bs022.html
+++ b/doc/pub/week38/html/._week38-bs022.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -407,22 +412,58 @@ MathJax.Hub.Config({
-
+
-Logistic Regression
+Performance as function of the regularization parameter
-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 \( \hat{x}_i \). 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 \( \hat{\beta} \) 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.
+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.
@@ -450,7 +491,7 @@ simple recipe for fitting our data.
31
32
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs023.html b/doc/pub/week38/html/._week38-bs023.html
index 06ae8ca43..d2cc08158 100644
--- a/doc/pub/week38/html/._week38-bs023.html
+++ b/doc/pub/week38/html/._week38-bs023.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -407,27 +412,56 @@ MathJax.Hub.Config({
-
+
-Classification problems
+Finding the optimal value of \( \lambda \)
-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.
+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.
-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.
+
+
+
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 \( \lambda \).
@@ -455,7 +489,7 @@ failure etc.
32
33
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs024.html b/doc/pub/week38/html/._week38-bs024.html
index 3f503bc03..82957f115 100644
--- a/doc/pub/week38/html/._week38-bs024.html
+++ b/doc/pub/week38/html/._week38-bs024.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -407,25 +412,22 @@ MathJax.Hub.Config({
-
+
-Optimization and Deep learning
+Logistic Regression
-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 \( \hat{\beta} \). 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.
+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 \( \hat{x}_i \). 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 \( \hat{\beta} \) 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.
@@ -453,7 +455,7 @@ models, as we will see later.
33
34
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs025.html b/doc/pub/week38/html/._week38-bs025.html
index 330b937fd..c46487a47 100644
--- a/doc/pub/week38/html/._week38-bs025.html
+++ b/doc/pub/week38/html/._week38-bs025.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,29 +414,25 @@ MathJax.Hub.Config({
-Basics
+Classification problems
-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).
+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 goal is to predict the
-output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \)
-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
-
-$$
-y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
-$$
+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.
@@ -459,7 +460,7 @@ $$
34
35
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs026.html b/doc/pub/week38/html/._week38-bs026.html
index 3fe0da66a..cd4ddf0cd 100644
--- a/doc/pub/week38/html/._week38-bs026.html
+++ b/doc/pub/week38/html/._week38-bs026.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,26 +414,23 @@ MathJax.Hub.Config({
-Linear classifier
+Optimization and Deep learning
-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 \( y_i \leq 0.5 \).
+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 \( \hat{\beta} \). 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 would then have our
-weighted linear combination, namely
-$$
-\begin{equation}
-\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
-\tag{13}
-\end{equation}
-$$
-
-where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our
-\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.
+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.
@@ -456,7 +458,7 @@ where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \
35
36
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs027.html b/doc/pub/week38/html/._week38-bs027.html
index 05986f353..41accb3ca 100644
--- a/doc/pub/week38/html/._week38-bs027.html
+++ b/doc/pub/week38/html/._week38-bs027.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -407,26 +412,31 @@ MathJax.Hub.Config({
-
+
-Some selected properties
+Basics
-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).
+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).
-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 goal is to predict the
+output classes from the design matrix \( \hat{X}\in\mathbb{R}^{n\times p} \)
+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
+
+$$
+y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}.
+$$
@@ -454,7 +464,7 @@ the probability of a given category. This leads us to the logistic function.
36
37
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs028.html b/doc/pub/week38/html/._week38-bs028.html
index c0a0bb6f3..92fd9cd9e 100644
--- a/doc/pub/week38/html/._week38-bs028.html
+++ b/doc/pub/week38/html/._week38-bs028.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,69 +414,27 @@ MathJax.Hub.Config({
-Simple example
+Linear classifier
-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.
+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
+$$
+\begin{equation}
+\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon},
+\tag{13}
+\end{equation}
+$$
-
-
# Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.model_selection import train_test_split
-from sklearn.utils import resample
-from sklearn.metrics import mean_squared_error
-from IPython.display import display
-from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
+where \( \hat{y} \) is a vector representing the possible outcomes, \( \hat{X} \) is our
+\( n\times p \) design matrix and \( \hat{\beta} \) represents our estimators/predictors.
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
- os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
- os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
- os.makedirs(DATA_ID)
-
-def image_path(fig_id):
- return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
- return os.path.join(DATA_ID, dat_id)
-
-def save_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()
-
@@ -498,7 +461,7 @@ plt.show()
37
38
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs029.html b/doc/pub/week38/html/._week38-bs029.html
index e1844c7a6..3a4c82f7b 100644
--- a/doc/pub/week38/html/._week38-bs029.html
+++ b/doc/pub/week38/html/._week38-bs029.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,41 +414,24 @@ MathJax.Hub.Config({
-Plotting the mean value for each group
+Some selected properties
-What we could attempt however is to plot the mean value for each group.
+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).
-
-
-
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
-\( x_i \).
+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.
@@ -471,7 +459,7 @@ representing the probability for finding a value of \( y_i \) with a given
38
39
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs030.html b/doc/pub/week38/html/._week38-bs030.html
index 915bf852a..0af859c11 100644
--- a/doc/pub/week38/html/._week38-bs030.html
+++ b/doc/pub/week38/html/._week38-bs030.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,26 +414,69 @@ MathJax.Hub.Config({
-The logistic function
+Simple example
-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,
-$$
-p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
-$$
+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.
-Note that \( 1-p(t)= p(-t) \).
+
+
+
# Common imports
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.linear_model import LinearRegression, Ridge, Lasso
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
+from sklearn.metrics import mean_squared_error
+from IPython.display import display
+from pylab import 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/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+def save_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()
+
@@ -455,7 +503,7 @@ Note that \( 1-p(t)= p(-t) \).
39
40
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs031.html b/doc/pub/week38/html/._week38-bs031.html
index fadd68ad2..ab192b242 100644
--- a/doc/pub/week38/html/._week38-bs031.html
+++ b/doc/pub/week38/html/._week38-bs031.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,69 +414,42 @@ MathJax.Hub.Config({
-Examples of likelihood functions used in logistic regression and nueral networks
+Plotting the mean value for each group
-The following code plots the logistic function, the step function and other functions we will encounter from here and on.
+What we could attempt however is to plot the mean value for each group.
-
"""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."""
-
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
-
-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: 1.0 if z >= 0.0 else 0.0)
-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')
-
+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
+\( x_i \).
+
@@ -498,7 +476,7 @@ plt.show()
40
41
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs032.html b/doc/pub/week38/html/._week38-bs032.html
index c9b19bb0a..dc81f68e5 100644
--- a/doc/pub/week38/html/._week38-bs032.html
+++ b/doc/pub/week38/html/._week38-bs032.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,24 +414,25 @@ MathJax.Hub.Config({
-Two parameters
+The logistic function
-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
+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,
$$
-\begin{align*}
-p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
-p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
-\end{align*}
+p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.
$$
-where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
-
-
-Note that we used
-$$
-p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
-$$
+Note that \( 1-p(t)= p(-t) \).
@@ -454,7 +460,7 @@ $$
41
42
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs033.html b/doc/pub/week38/html/._week38-bs033.html
index 31966ad47..36ee8c2b0 100644
--- a/doc/pub/week38/html/._week38-bs033.html
+++ b/doc/pub/week38/html/._week38-bs033.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -407,28 +412,71 @@ MathJax.Hub.Config({
-
+
-Maximum likelihood
+Examples of likelihood functions used in logistic regression and nueral networks
-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
-$$
-\begin{align*}
-P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\
-\end{align*}
-$$
+The following code plots the logistic function, the step function and other functions we will encounter from here and on.
-from which we obtain the log-likelihood and our cost/loss function
-$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
-$$
+
+
+
"""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."""
+
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+
+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: 1.0 if z >= 0.0 else 0.0)
+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()
+
@@ -455,7 +503,7 @@ $$
42
43
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs034.html b/doc/pub/week38/html/._week38-bs034.html
index bc8a0f0d0..c3fc78420 100644
--- a/doc/pub/week38/html/._week38-bs034.html
+++ b/doc/pub/week38/html/._week38-bs034.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,24 +414,25 @@ MathJax.Hub.Config({
-The cost function rewritten
+Two parameters
-Reordering the logarithms, we can rewrite the cost/loss function as
+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
$$
-\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+\begin{align*}
+p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\
+p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}),
+\end{align*}
$$
+where \( \hat{\beta} \) are the weights we wish to extract from data, in our case \( \beta_0 \) and \( \beta_1 \).
+
-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
+Note that we used
$$
-\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).
$$
-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.
-
@@ -453,7 +459,7 @@ in practice we often supplement the cross-entropy with additional regularization
43
44
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs035.html b/doc/pub/week38/html/._week38-bs035.html
index 3a3a516b3..becd83eea 100644
--- a/doc/pub/week38/html/._week38-bs035.html
+++ b/doc/pub/week38/html/._week38-bs035.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -407,25 +412,26 @@ MathJax.Hub.Config({
-
+
-Minimizing the cross entropy
+Maximum likelihood
-The cross entropy is a convex function of the weights \( \hat{\beta} \) 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
-
+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
$$
-\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
+\begin{align*}
+P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\
+\end{align*}
$$
-and
+from which we obtain the log-likelihood and our cost/loss function
$$
-\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right).
$$
@@ -454,7 +460,7 @@ $$
44
45
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs036.html b/doc/pub/week38/html/._week38-bs036.html
index f88ecef63..744d676b1 100644
--- a/doc/pub/week38/html/._week38-bs036.html
+++ b/doc/pub/week38/html/._week38-bs036.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,25 +414,23 @@ MathJax.Hub.Config({
-A more compact expression
+The cost function rewritten
-Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
-\( n\times p \) matrix \( \hat{X} \) which contains the \( x_i \) values and a
-vector \( \hat{p} \) of fitted probabilities \( p(y_i\vert x_i,\hat{\beta}) \). We can rewrite in a more compact form the first
-derivative of cost function as
-
+Reordering the logarithms, we can rewrite the cost/loss function as
$$
-\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).
+\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
$$
-If we in addition define a diagonal matrix \( \hat{W} \) with elements
-\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as
+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
+$$
+\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right).
+$$
-$$
-\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.
-$$
+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.
@@ -455,7 +458,7 @@ $$
45
46
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs037.html b/doc/pub/week38/html/._week38-bs037.html
index 57164ae35..6448312d3 100644
--- a/doc/pub/week38/html/._week38-bs037.html
+++ b/doc/pub/week38/html/._week38-bs037.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,17 +414,23 @@ MathJax.Hub.Config({
-Extending to more predictors
+Minimizing the cross entropy
-Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
+The cross entropy is a convex function of the weights \( \hat{\beta} \) 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
+
$$
-\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right),
$$
-Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to
+and
$$
-p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right).
$$
@@ -448,7 +459,7 @@ $$
46
47
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs038.html b/doc/pub/week38/html/._week38-bs038.html
index fbd63feff..6b2a6fb1b 100644
--- a/doc/pub/week38/html/._week38-bs038.html
+++ b/doc/pub/week38/html/._week38-bs038.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,30 +414,25 @@ MathJax.Hub.Config({
-Including more classes
+A more compact expression
-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
+Let us now define a vector \( \hat{y} \) with \( n \) elements \( y_i \), an
+\( n\times p \) matrix \( \hat{X} \) which contains the \( x_i \) values and a
+vector \( \hat{p} \) of fitted probabilities \( p(y_i\vert x_i,\hat{\beta}) \). We can rewrite in a more compact form the first
+derivative of cost function as
$$
-\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1,
-$$
-
-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
-$$
-\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1,
+\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right).
$$
-and the model is specified in term of \( K-1 \) so-called log-odds or
-logit transformations.
+If we in addition define a diagonal matrix \( \hat{W} \) with elements
+\( p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta}) \), we can obtain a compact expression of the second derivative as
+
+$$
+\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}.
+$$
@@ -460,7 +460,7 @@ and the model is specified in term of \( K-1 \) so-called log-odds or
47
48
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs039.html b/doc/pub/week38/html/._week38-bs039.html
index c704af8da..2956657ba 100644
--- a/doc/pub/week38/html/._week38-bs039.html
+++ b/doc/pub/week38/html/._week38-bs039.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,43 +414,19 @@ MathJax.Hub.Config({
-More classes
+Extending to more predictors
-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 \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two
-predictors):
-
+Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \( p \) predictors
$$
-p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
+\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p.
$$
-It is easy to extend to more predictors. The final class is
+Here we defined \( \hat{x}=[1,x_1,x_2,\dots,x_p] \) and \( \hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p] \) leading to
$$
-p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}},
+p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}.
$$
-
-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.
-
@@ -472,7 +453,7 @@ methods.
48
49
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs040.html b/doc/pub/week38/html/._week38-bs040.html
index 5e76b4d0c..7d996ad52 100644
--- a/doc/pub/week38/html/._week38-bs040.html
+++ b/doc/pub/week38/html/._week38-bs040.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,7 +414,30 @@ MathJax.Hub.Config({
-Friday September 24
+Including more classes
+
+
+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.
@@ -437,7 +465,7 @@ MathJax.Hub.Config({
49
50
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs041.html b/doc/pub/week38/html/._week38-bs041.html
index 560f7bb10..011541c40 100644
--- a/doc/pub/week38/html/._week38-bs041.html
+++ b/doc/pub/week38/html/._week38-bs041.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,42 +414,43 @@ MathJax.Hub.Config({
-Wisconsin Cancer Data
+More classes
-We show here how we can use a simple regression case on the breast
-cancer data using Logistic regression as our algorithm for
-classification.
+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 \( \hat{x} \) and a weighting vector \( \hat{\beta} \) is (with two
+predictors):
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
+$$
+p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}.
+$$
-# Load the data
-cancer = load_breast_cancer()
+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.
-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
-from sklearn.preprocessing import 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)))
-
@@ -471,7 +477,7 @@ logreg.fit(X_train_scaled, y_train)
50
51
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/._week38-bs042.html b/doc/pub/week38/html/._week38-bs042.html
index a27a31827..2ca873953 100644
--- a/doc/pub/week38/html/._week38-bs042.html
+++ b/doc/pub/week38/html/._week38-bs042.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -409,49 +414,8 @@ MathJax.Hub.Config({
-Using the correlation matrix
+Friday September 24
-
-In addition to the above scores, we could also study the covariance (and the correlation matrix).
-We use Pandas to compute the correlation matrix.
-
-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-cancer = load_breast_cancer()
-import pandas as pd
-# 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 in range(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()
-
-import seaborn as sns
-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()
-
@@ -478,7 +442,7 @@ plt.show()
51
52
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/week38-bs.html b/doc/pub/week38/html/week38-bs.html
index a6e6e80b8..0d1b2580c 100644
--- a/doc/pub/week38/html/week38-bs.html
+++ b/doc/pub/week38/html/week38-bs.html
@@ -74,10 +74,13 @@ Automatically generated HTML file from DocOnce source
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -319,81 +322,83 @@ MathJax.Hub.Config({
More thinking
Still thinking
Linear Regression code, Intercept handling first
- What does centering mean mathematically?
- More complicated Example: The Ising model
- Reformulating the problem to suit regression
- Linear regression
- Singular Value decomposition
- The one-dimensional Ising model
- Ridge regression
- LASSO regression
- Performance as function of the regularization parameter
- Finding the optimal value of \( \lambda \)
- Logistic Regression
- Classification problems
- Optimization and Deep learning
- Basics
- Linear classifier
- Some selected properties
- Simple example
- Plotting the mean value for each group
- The logistic function
- Examples of likelihood functions used in logistic regression and nueral networks
- Two parameters
- Maximum likelihood
- The cost function rewritten
- Minimizing the cross entropy
- A more compact expression
- Extending to more predictors
- Including more classes
- More classes
- Friday September 24
- Wisconsin Cancer Data
- Using the correlation matrix
- Discussing the correlation data
- Other measures in classification studies: Cancer Data again
- Optimization, the central part of any Machine Learning algortithm
- Revisiting our Logistic Regression case
- The equations to solve
- Solving using Newton-Raphson's method
- Brief reminder on Newton-Raphson's method
- The equations
- Simple geometric interpretation
- Extending to more than one variable
- Steepest descent
- More on Steepest descent
- The ideal
- The sensitiveness of the gradient descent
- Convex functions
- Convex function
- Conditions on convex functions
- More on convex functions
- Some simple problems
- Friday September 25
- Standard steepest descent
- Gradient method
- Steepest descent method
- Steepest descent method
- Final expressions
- Steepest descent example
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method and iterations
- Conjugate gradient method
- Conjugate gradient method
- Conjugate gradient method
- Revisiting some of our first Linear Regression Encounters
- Gradient descent example
- The derivative of the cost/loss function
- The Hessian matrix
- Simple program
- Gradient Descent Example
- And a corresponding example using scikit-learn
- Gradient descent and Ridge
- Program example for gradient descent with Ridge Regression
- Using gradient descent methods, limitations
+ What does centering (subtracting the mean values) mean mathematically?
+ Code Examples
+ Taking out the mean
+ More complicated Example: The Ising model
+ Reformulating the problem to suit regression
+ Linear regression
+ Singular Value decomposition
+ The one-dimensional Ising model
+ Ridge regression
+ LASSO regression
+ Performance as function of the regularization parameter
+ Finding the optimal value of \( \lambda \)
+ Logistic Regression
+ Classification problems
+ Optimization and Deep learning
+ Basics
+ Linear classifier
+ Some selected properties
+ Simple example
+ Plotting the mean value for each group
+ The logistic function
+ Examples of likelihood functions used in logistic regression and nueral networks
+ Two parameters
+ Maximum likelihood
+ The cost function rewritten
+ Minimizing the cross entropy
+ A more compact expression
+ Extending to more predictors
+ Including more classes
+ More classes
+ Friday September 24
+ Wisconsin Cancer Data
+ Using the correlation matrix
+ Discussing the correlation data
+ Other measures in classification studies: Cancer Data again
+ Optimization, the central part of any Machine Learning algortithm
+ Revisiting our Logistic Regression case
+ The equations to solve
+ Solving using Newton-Raphson's method
+ Brief reminder on Newton-Raphson's method
+ The equations
+ Simple geometric interpretation
+ Extending to more than one variable
+ Steepest descent
+ More on Steepest descent
+ The ideal
+ The sensitiveness of the gradient descent
+ Convex functions
+ Convex function
+ Conditions on convex functions
+ More on convex functions
+ Some simple problems
+ Friday September 25
+ Standard steepest descent
+ Gradient method
+ Steepest descent method
+ Steepest descent method
+ Final expressions
+ Steepest descent example
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method and iterations
+ Conjugate gradient method
+ Conjugate gradient method
+ Conjugate gradient method
+ Revisiting some of our first Linear Regression Encounters
+ Gradient descent example
+ The derivative of the cost/loss function
+ The Hessian matrix
+ Simple program
+ Gradient Descent Example
+ And a corresponding example using scikit-learn
+ Gradient descent and Ridge
+ Program example for gradient descent with Ridge Regression
+ Using gradient descent methods, limitations
@@ -452,7 +457,7 @@ MathJax.Hub.Config({
9
10
...
- 87
+ 89
»
diff --git a/doc/pub/week38/html/week38-reveal.html b/doc/pub/week38/html/week38-reveal.html
index 2cf1ea365..c2f081444 100644
--- a/doc/pub/week38/html/week38-reveal.html
+++ b/doc/pub/week38/html/week38-reveal.html
@@ -458,14 +458,16 @@ $$\beta_0$$
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.
+standard deviation. Most machine learning libraries do this as a deafult. This means that if you compare your code with the results from a given library,
+the results may differ. Tracing back the differences may often lead to an increased confusion.
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.
+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
@@ -515,7 +517,7 @@ y_pred = y_pred + y_train_mean
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)
+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.
@@ -595,42 +597,35 @@ plt.show()
-What does centering mean mathematically?
-Here is a mathematical explanation of the zero centering:
+What does centering (subtracting the mean values) mean mathematically?
-The cost/loss function for Ridge regression is:
+Let us try to understand what this may imply mathematically when we subtract the mean values, also known as zero centering. To catch many birds with just one stone, we will focus on Ridge regression.
+
+The cost/loss function for Ridge regression is
$$
-C(\beta_0, \beta_1, ... , \beta_P) = \sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip}\beta_p)^2 + \lambda \sum_{p=1}^P \beta_p^2.
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2 + \lambda \sum_{j=1}^{p-1} \beta_i^2.
$$
-Notice that the intercept is left out of the \( L_2 \) regularization term. The design matrix
+Note that the intercept term $\beta_0$is left out of the \( L_2 \) regularization term. The design matrix
\( X \) does in this case not contain any intercept column. We want
$$
-\frac{\partial L}{\partial \beta_j} = 0,
+\frac{\partial C}{\partial \beta_j} = 0,
$$
-for all \( j \), so lets start with \( \beta_0 \). This means that we have
+for all \( j \), so let us start with \( \beta_0 \). This means that we have
$$
-\frac{\partial L}{\partial \beta_0} = -2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p).
-$$
-
-
-
-We want to solve
-
-$$
--2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p) = 0,
+\frac{\partial C}{\partial \beta_0} = -2\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right),
$$
@@ -638,36 +633,30 @@ $$
which gives
$$
-\sum_{i=1}^{n} \beta_0 = \sum_{i=1}^{n}y_i - \sum_{i=1}^{n} \sum_{p=1}^P X_{ip} \beta_p,
+\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.
$$
-or
-$ n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip}$.
-
-
-If we assume that every column of \( X \) is centered, whic we can do by subtracting the mean,
+If 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)
-the sum $ \sum_{i=1}^{n} X_{ip} $
-
-
+the sum \( \sum_{i=0}^{n-1} X_{ij} \)
can be rewritten as
$$
-\sum_{i=1}^{n} (X_{ip} - \frac{1}{n}\sum_{i=1}^{n} X_{ip}) = \sum_{i=1}^{n} X_{ip} - \sum_{i=1}^{n} \frac{1}{n} \sum_{i=1}^{n}X_{ip},
+\sum_{i=0}^{n-1} \left(X_{ij} - \frac{1}{n}\sum_{i=0}^{n-1} X_{ij}) = \sum_{i=0}^{n-1} X_{ij} - \sum_{i=0}^{n-1} \frac{1}{n} \sum_{i=0}^{n-1}X_{ij},
$$
resulting in
$$
-\sum_{i=1}^{n} X_{ip} - n \frac{1}{n} \sum_{i=1}^{n}X_{ip} = 0.
+\sum_{i=0}^{n-1} X_{ij} - n \frac{1}{n} \sum_{i=0}^{n-1}X_{ij} = 0.
$$
@@ -675,19 +664,21 @@ $$
Finally we have
$$
-n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip},
+n\beta_0 = \sum_{i=0}^{n-1} y_i - \sum_{j=1}^{p-1}\beta_j \sum_{i=0}^{n-1} X_{ij},
$$
or
$$
-\beta_0 = \frac{1}{n}\sum_{i=1}^{n} y_i = y_{average}.
+\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 \( y_i - \beta_0 = y_i - y_{average} \) in the loss function will give us (in vector-matrix disguise)
+Replacing \( y_i \) with \( y_i - \beta_0 = y_i - \overline{\boldsymbol{y}} \) in the cost function will give us (in vector-matrix disguise)
$$
C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta},
@@ -699,8 +690,16 @@ which has the solution
\( \beta = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}} \).
-where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - y_{average} \)
+where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \)
and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \).
+
+
+
+
+Code Examples
+
+
+Armed with this wisdom, we attempt first simply set the intercept eqault to False in our implementation of Ridge regression for a vanilla data set.
@@ -711,8 +710,6 @@ and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \).
from sklearn.model_selection import train_test_split
from sklearn import linear_model
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
def MSE(y_data,y_model):
n = np.size(y_model)
return np.sum((y_data-y_model)**2)/n
@@ -728,42 +725,29 @@ y = np.exp(-x**2) + 20
X = np.zeros((n,Maxpolydegree))
+We include explicitely the intercpt column
X[:,0] = 1.0
-for polydegree in range(1, Maxpolydegree):
- for degree in range(polydegree):
- X[:,degree] = x**degree
+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)
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X_train @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y_train,ytildeOLS))
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-
-p = len(OLSbeta)
+p = Maxpolydegree
I = np.eye(p,p)
# Decide which values of lambda to use
nlambdas = 4
MSEOwnRidgePredict = np.zeros(nlambdas)
-MSEOwnRidgeTrain = np.zeros(nlambdas)
MSERidgePredict = np.zeros(nlambdas)
-MSERidgeTrain = np.zeros(nlambdas)
lambdas = np.logspace(-4, 4, 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
# include lasso using Scikit-Learn
- # Note: we include the intercept
+ # 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
@@ -772,18 +756,14 @@ lambdas = np.logspace(-4, print("Beta values for own Ridge implementation")
print(OwnRidgeBeta)
print("Beta values for Scikit-Learn Ridge implementation")
print(RegRidge.coef_)
# Now plot the results
plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'b', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
plt.xlabel('log10(lambda)')
@@ -792,6 +772,17 @@ 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.
+The problem however is that can easily lead to a larger mean-squared error!
+
+
+Let us see how we can change this code by zero centering.
+
+
+
+
+Taking out the mean
+
import numpy as np
@@ -801,13 +792,9 @@ plt.show()
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
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)
@@ -816,15 +803,12 @@ n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-Maxpolydegree = 5
+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)
@@ -834,11 +818,13 @@ X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=#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)
-X_train_scaled = X_train - X_train_mean #Center by removing mean from each feature
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
X_test_scaled = X_test - X_train_mean
-
-y_scaler = np.mean(y_train) #The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)
-y_train_scaled = y_train - y_scaler #Remove the intercept from the training data.
+#The model intercept (called y_scaler) is given by the mean of 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
@@ -853,25 +839,20 @@ lambdas = np.logspace(-4, @OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
-
- ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
#EQUIVALENT PREDICTION:
- ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
print("Values for own Ridge prediction")
print(ypredictOwnRidge)
-
-
-
RegRidge = linear_model.Ridge(lmb)
RegRidge.fit(X_train,y_train)
ypredictRidge = RegRidge.predict(X_test)
print("Values for SL Ridge prediction")
print(ypredictRidge)
-
-
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")
@@ -881,14 +862,10 @@ lambdas = np.logspace(-4, print('Intercept from Scikit-Learn Ridge implementation')
print(RegRidge.intercept_)
-
-
# 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()
diff --git a/doc/pub/week38/html/week38-solarized.html b/doc/pub/week38/html/week38-solarized.html
index c89b349fd..748ba6002 100644
--- a/doc/pub/week38/html/week38-solarized.html
+++ b/doc/pub/week38/html/week38-solarized.html
@@ -94,10 +94,13 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -616,14 +619,16 @@ Furthermore, in for example Ridge and Lasso regression, the solutions
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.
+standard deviation. Most machine learning libraries do this as a deafult. This means that if you compare your code with the results from a given library,
+the results may differ. Tracing back the differences may often lead to an increased confusion.
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.
+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
@@ -672,7 +677,7 @@ y_pred = y_pred + y_train_mean
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)
+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.
@@ -751,81 +756,72 @@ plt.show()
-
What does centering mean mathematically?
-Here is a mathematical explanation of the zero centering:
+What does centering (subtracting the mean values) mean mathematically?
-The cost/loss function for Ridge regression is:
+Let us try to understand what this may imply mathematically when we subtract the mean values, also known as zero centering. To catch many birds with just one stone, we will focus on Ridge regression.
+
+The cost/loss function for Ridge regression is
$$
-C(\beta_0, \beta_1, ... , \beta_P) = \sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip}\beta_p)^2 + \lambda \sum_{p=1}^P \beta_p^2.
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2 + \lambda \sum_{j=1}^{p-1} \beta_i^2.
$$
-Notice that the intercept is left out of the \( L_2 \) regularization term. The design matrix
+Note that the intercept term $\beta_0$is left out of the \( L_2 \) regularization term. The design matrix
\( X \) does in this case not contain any intercept column. We want
$$
-\frac{\partial L}{\partial \beta_j} = 0,
+\frac{\partial C}{\partial \beta_j} = 0,
$$
-for all \( j \), so lets start with \( \beta_0 \). This means that we have
+for all \( j \), so let us start with \( \beta_0 \). This means that we have
$$
-\frac{\partial L}{\partial \beta_0} = -2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p).
-$$
-
-
-We want to solve
-$$
--2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p) = 0,
+\frac{\partial C}{\partial \beta_0} = -2\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right),
$$
which gives
$$
-\sum_{i=1}^{n} \beta_0 = \sum_{i=1}^{n}y_i - \sum_{i=1}^{n} \sum_{p=1}^P X_{ip} \beta_p,
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
$$
-or
-$ n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip}$.
-
-
-If we assume that every column of \( X \) is centered, whic we can do by subtracting the mean,
+If 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)
-the sum $ \sum_{i=1}^{n} X_{ip} $
-
-
+the sum \( \sum_{i=0}^{n-1} X_{ij} \)
can be rewritten as
$$
-\sum_{i=1}^{n} (X_{ip} - \frac{1}{n}\sum_{i=1}^{n} X_{ip}) = \sum_{i=1}^{n} X_{ip} - \sum_{i=1}^{n} \frac{1}{n} \sum_{i=1}^{n}X_{ip},
+\sum_{i=0}^{n-1} \left(X_{ij} - \frac{1}{n}\sum_{i=0}^{n-1} X_{ij}) = \sum_{i=0}^{n-1} X_{ij} - \sum_{i=0}^{n-1} \frac{1}{n} \sum_{i=0}^{n-1}X_{ij},
$$
resulting in
$$
-\sum_{i=1}^{n} X_{ip} - n \frac{1}{n} \sum_{i=1}^{n}X_{ip} = 0.
+\sum_{i=0}^{n-1} X_{ij} - n \frac{1}{n} \sum_{i=0}^{n-1}X_{ij} = 0.
$$
Finally we have
$$
-n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip},
+n\beta_0 = \sum_{i=0}^{n-1} y_i - \sum_{j=1}^{p-1}\beta_j \sum_{i=0}^{n-1} X_{ij},
$$
or
$$
-\beta_0 = \frac{1}{n}\sum_{i=1}^{n} y_i = y_{average}.
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\boldsymbol{y}},
$$
+the average value of \( \boldsymbol{y] \).
+
-Replacing \( y_i \) with \( y_i - \beta_0 = y_i - y_{average} \) in the loss function will give us (in vector-matrix disguise)
+Replacing \( y_i \) with \( y_i - \beta_0 = y_i - \overline{\boldsymbol{y}} \) in the cost function will give us (in vector-matrix disguise)
$$
C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta},
$$
@@ -835,9 +831,17 @@ which has the solution
\( \beta = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}} \).
-where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - y_{average} \)
+where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \)
and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \).
+
+
+
+
Code Examples
+
+
+Armed with this wisdom, we attempt first simply set the intercept eqault to False in our implementation of Ridge regression for a vanilla data set.
+
@@ -847,8 +851,6 @@ and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \).
from sklearn.model_selection import train_test_split
from sklearn import linear_model
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
def MSE(y_data,y_model):
n = np.size(y_model)
return np.sum((y_data-y_model)**2)/n
@@ -864,42 +866,29 @@ y = np.exp(-x**2) + 20
X = np.zeros((n,Maxpolydegree))
+We include explicitely the intercpt column
X[:,0] = 1.0
-for polydegree in range(1, Maxpolydegree):
- for degree in range(polydegree):
- X[:,degree] = x**degree
+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)
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X_train @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y_train,ytildeOLS))
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-
-p = len(OLSbeta)
+p = Maxpolydegree
I = np.eye(p,p)
# Decide which values of lambda to use
nlambdas = 4
MSEOwnRidgePredict = np.zeros(nlambdas)
-MSEOwnRidgeTrain = np.zeros(nlambdas)
MSERidgePredict = np.zeros(nlambdas)
-MSERidgeTrain = np.zeros(nlambdas)
lambdas = np.logspace(-4, 4, 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
# include lasso using Scikit-Learn
- # Note: we include the intercept
+ # 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
@@ -908,18 +897,14 @@ lambdas = np.logspace(-4, print("Beta values for own Ridge implementation")
print(OwnRidgeBeta)
print("Beta values for Scikit-Learn Ridge implementation")
print(RegRidge.coef_)
# Now plot the results
plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'b', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
plt.xlabel('log10(lambda)')
@@ -928,6 +913,17 @@ 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.
+The problem however is that can easily lead to a larger mean-squared error!
+
+
+Let us see how we can change this code by zero centering.
+
+
+
+
+
Taking out the mean
+
import numpy as np
@@ -937,13 +933,9 @@ plt.show()
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
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)
@@ -952,15 +944,12 @@ n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-Maxpolydegree = 5
+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)
@@ -970,11 +959,13 @@ X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=#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)
-X_train_scaled = X_train - X_train_mean #Center by removing mean from each feature
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
X_test_scaled = X_test - X_train_mean
-
-y_scaler = np.mean(y_train) #The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)
-y_train_scaled = y_train - y_scaler #Remove the intercept from the training data.
+#The model intercept (called y_scaler) is given by the mean of 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
@@ -989,25 +980,20 @@ lambdas = np.logspace(-4, @OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
-
- ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
#EQUIVALENT PREDICTION:
- ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
print("Values for own Ridge prediction")
print(ypredictOwnRidge)
-
-
-
RegRidge = linear_model.Ridge(lmb)
RegRidge.fit(X_train,y_train)
ypredictRidge = RegRidge.predict(X_test)
print("Values for SL Ridge prediction")
print(ypredictRidge)
-
-
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")
@@ -1017,14 +1003,10 @@ lambdas = np.logspace(-4, print('Intercept from Scikit-Learn Ridge implementation')
print(RegRidge.intercept_)
-
-
# 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()
diff --git a/doc/pub/week38/html/week38.html b/doc/pub/week38/html/week38.html
index d98d373ee..d229b1d1c 100644
--- a/doc/pub/week38/html/week38.html
+++ b/doc/pub/week38/html/week38.html
@@ -99,10 +99,13 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'linear-regression-code-intercept-handling-first'),
- ('What does centering mean mathematically?',
+ ('What does centering (subtracting the mean values) mean '
+ 'mathematically?',
2,
None,
- 'what-does-centering-mean-mathematically'),
+ 'what-does-centering-subtracting-the-mean-values-mean-mathematically'),
+ ('Code Examples', 2, None, 'code-examples'),
+ ('Taking out the mean', 2, None, 'taking-out-the-mean'),
('More complicated Example: The Ising model',
2,
None,
@@ -621,14 +624,16 @@ Furthermore, in for example Ridge and Lasso regression, the solutions
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.
+standard deviation. Most machine learning libraries do this as a deafult. This means that if you compare your code with the results from a given library,
+the results may differ. Tracing back the differences may often lead to an increased confusion.
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.
+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
@@ -677,7 +682,7 @@ y_pred = y_pred 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)
+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.
@@ -756,81 +761,72 @@ plt.show()
-
What does centering mean mathematically?
-Here is a mathematical explanation of the zero centering:
+What does centering (subtracting the mean values) mean mathematically?
-The cost/loss function for Ridge regression is:
+Let us try to understand what this may imply mathematically when we subtract the mean values, also known as zero centering. To catch many birds with just one stone, we will focus on Ridge regression.
+
+The cost/loss function for Ridge regression is
$$
-C(\beta_0, \beta_1, ... , \beta_P) = \sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip}\beta_p)^2 + \lambda \sum_{p=1}^P \beta_p^2.
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2 + \lambda \sum_{j=1}^{p-1} \beta_i^2.
$$
-Notice that the intercept is left out of the \( L_2 \) regularization term. The design matrix
+Note that the intercept term $\beta_0$is left out of the \( L_2 \) regularization term. The design matrix
\( X \) does in this case not contain any intercept column. We want
$$
-\frac{\partial L}{\partial \beta_j} = 0,
+\frac{\partial C}{\partial \beta_j} = 0,
$$
-for all \( j \), so lets start with \( \beta_0 \). This means that we have
+for all \( j \), so let us start with \( \beta_0 \). This means that we have
$$
-\frac{\partial L}{\partial \beta_0} = -2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p).
-$$
-
-
-We want to solve
-$$
--2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p) = 0,
+\frac{\partial C}{\partial \beta_0} = -2\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right),
$$
which gives
$$
-\sum_{i=1}^{n} \beta_0 = \sum_{i=1}^{n}y_i - \sum_{i=1}^{n} \sum_{p=1}^P X_{ip} \beta_p,
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
$$
-or
-$ n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip}$.
-
-
-If we assume that every column of \( X \) is centered, whic we can do by subtracting the mean,
+If 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)
-the sum $ \sum_{i=1}^{n} X_{ip} $
-
-
+the sum \( \sum_{i=0}^{n-1} X_{ij} \)
can be rewritten as
$$
-\sum_{i=1}^{n} (X_{ip} - \frac{1}{n}\sum_{i=1}^{n} X_{ip}) = \sum_{i=1}^{n} X_{ip} - \sum_{i=1}^{n} \frac{1}{n} \sum_{i=1}^{n}X_{ip},
+\sum_{i=0}^{n-1} \left(X_{ij} - \frac{1}{n}\sum_{i=0}^{n-1} X_{ij}) = \sum_{i=0}^{n-1} X_{ij} - \sum_{i=0}^{n-1} \frac{1}{n} \sum_{i=0}^{n-1}X_{ij},
$$
resulting in
$$
-\sum_{i=1}^{n} X_{ip} - n \frac{1}{n} \sum_{i=1}^{n}X_{ip} = 0.
+\sum_{i=0}^{n-1} X_{ij} - n \frac{1}{n} \sum_{i=0}^{n-1}X_{ij} = 0.
$$
Finally we have
$$
-n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip},
+n\beta_0 = \sum_{i=0}^{n-1} y_i - \sum_{j=1}^{p-1}\beta_j \sum_{i=0}^{n-1} X_{ij},
$$
or
$$
-\beta_0 = \frac{1}{n}\sum_{i=1}^{n} y_i = y_{average}.
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\boldsymbol{y}},
$$
+the average value of \( \boldsymbol{y] \).
+
-Replacing \( y_i \) with \( y_i - \beta_0 = y_i - y_{average} \) in the loss function will give us (in vector-matrix disguise)
+Replacing \( y_i \) with \( y_i - \beta_0 = y_i - \overline{\boldsymbol{y}} \) in the cost function will give us (in vector-matrix disguise)
$$
C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta},
$$
@@ -840,9 +836,17 @@ which has the solution
\( \beta = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}} \).
-where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - y_{average} \)
+where \( \boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}} \)
and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \).
+
+
+
+
Code Examples
+
+
+Armed with this wisdom, we attempt first simply set the intercept eqault to False in our implementation of Ridge regression for a vanilla data set.
+
@@ -852,8 +856,6 @@ and \( \tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj} \).
from sklearn.model_selection import train_test_split
from sklearn import linear_model
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
def MSE(y_data,y_model):
n = np.size(y_model)
return np.sum((y_data-y_model)**2)/n
@@ -869,42 +871,29 @@ y = np.e
Maxpolydegree = 20
X = np.zeros((n,Maxpolydegree))
+We include explicitely the intercpt column
X[:,0] = 1.0
-for polydegree in range(1, Maxpolydegree):
- for degree in range(polydegree):
- X[:,degree] = x**degree
+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)
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X_train @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y_train,ytildeOLS))
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-
-p = len(OLSbeta)
+p = Maxpolydegree
I = np.eye(p,p)
# Decide which values of lambda to use
nlambdas = 4
MSEOwnRidgePredict = np.zeros(nlambdas)
-MSEOwnRidgeTrain = np.zeros(nlambdas)
MSERidgePredict = np.zeros(nlambdas)
-MSERidgeTrain = np.zeros(nlambdas)
lambdas = np.logspace(-4, 4, 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
# include lasso using Scikit-Learn
- # Note: we include the intercept
+ # 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
@@ -913,18 +902,14 @@ lambdas = np.= RegRidge.predict(X_train)
ypredictRidge = RegRidge.predict(X_test)
MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
- MSEOwnRidgeTrain[i] = MSE(y_train,ytildeOwnRidge)
MSERidgePredict[i] = MSE(y_test,ypredictRidge)
- MSERidgeTrain[i] = MSE(y_train,ytildeRidge)
print("Beta values for own Ridge implementation")
print(OwnRidgeBeta)
print("Beta values for Scikit-Learn Ridge implementation")
print(RegRidge.coef_)
# Now plot the results
plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'b', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
plt.xlabel('log10(lambda)')
@@ -933,6 +918,17 @@ 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.
+The problem however is that can easily lead to a larger mean-squared error!
+
+
+Let us see how we can change this code by zero centering.
+
+
+
+
+
Taking out the mean
+
import numpy as np
@@ -942,13 +938,9 @@ plt.show()
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
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)
@@ -957,15 +949,12 @@ n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-Maxpolydegree = 5
+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)
@@ -975,11 +964,13 @@ X_train, X_test, y_train, y_test = train_tes
#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)
-X_train_scaled = X_train - X_train_mean #Center by removing mean from each feature
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
X_test_scaled = X_test - X_train_mean
-
-y_scaler = np.mean(y_train) #The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)
-y_train_scaled = y_train - y_scaler #Remove the intercept from the training data.
+#The model intercept (called y_scaler) is given by the mean of 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
@@ -994,25 +985,20 @@ lambdas = np.= 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
-
- ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
#EQUIVALENT PREDICTION:
- ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
print("Values for own Ridge prediction")
print(ypredictOwnRidge)
-
-
-
RegRidge = linear_model.Ridge(lmb)
RegRidge.fit(X_train,y_train)
ypredictRidge = RegRidge.predict(X_test)
print("Values for SL Ridge prediction")
print(ypredictRidge)
-
-
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")
@@ -1022,14 +1008,10 @@ lambdas = np.print('Intercept from Scikit-Learn Ridge implementation')
print(RegRidge.intercept_)
-
-
# 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()
diff --git a/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz b/doc/pub/week38/ipynb/ipynb-week38-src.tar.gz
index 534605b92..ebfa58c4b 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 29c201f3d..573e0db46 100644
--- a/doc/pub/week38/ipynb/week38.ipynb
+++ b/doc/pub/week38/ipynb/week38.ipynb
@@ -382,13 +382,15 @@
"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.\n",
+ "standard deviation. Most machine learning libraries do this as a deafult. This means that if you compare your code with the results from a given library,\n",
+ "the results may differ. Tracing back the differences may often lead to an increased confusion.\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.\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",
@@ -440,7 +442,7 @@
"source": [
"## Linear Regression code, Intercept handling first\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*)"
+ "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."
]
},
{
@@ -528,13 +530,12 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "## What does centering mean mathematically?\n",
- "Here is a mathematical explanation of the zero centering:\n",
+ "## What does centering (subtracting the mean values) mean mathematically?\n",
"\n",
"\n",
+ "Let us try to understand what this may imply mathematically when we subtract the mean values, also known as *zero centering*. To catch many birds with just one stone, we will focus on Ridge regression.\n",
"\n",
- "\n",
- "The cost/loss function for Ridge regression is:"
+ "The cost/loss function for Ridge regression is"
]
},
{
@@ -542,7 +543,7 @@
"metadata": {},
"source": [
"$$\n",
- "C(\\beta_0, \\beta_1, ... , \\beta_P) = \\sum_{i=1}^{n} (y_i - \\beta_0 - \\sum_{p=1}^P X_{ip}\\beta_p)^2 + \\lambda \\sum_{p=1}^P \\beta_p^2.\n",
+ "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2 + \\lambda \\sum_{j=1}^{p-1} \\beta_i^2.\n",
"$$"
]
},
@@ -550,7 +551,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Notice that the intercept is left out of the $L_2$ regularization term. The design matrix\n",
+ "Note that the intercept term $\\beta_0$is left out of the $L_2$ regularization term. The design matrix\n",
"$X$ does in this case not contain any intercept column. We want"
]
},
@@ -559,7 +560,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\frac{\\partial L}{\\partial \\beta_j} = 0,\n",
+ "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n",
"$$"
]
},
@@ -567,7 +568,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "for all $j$, so lets start with $\\beta_0$. This means that we have"
+ "for all $j$, so let us start with $\\beta_0$. This means that we have"
]
},
{
@@ -575,23 +576,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\frac{\\partial L}{\\partial \\beta_0} = -2\\sum_{i=1}^{n} (y_i - \\beta_0 - \\sum_{p=1}^P X_{ip} \\beta_p).\n",
- "$$"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "We want to solve"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "$$\n",
- "-2\\sum_{i=1}^{n} (y_i - \\beta_0 - \\sum_{p=1}^P X_{ip} \\beta_p) = 0,\n",
+ "\\frac{\\partial C}{\\partial \\beta_0} = -2\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right),\n",
"$$"
]
},
@@ -607,7 +592,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\sum_{i=1}^{n} \\beta_0 = \\sum_{i=1}^{n}y_i - \\sum_{i=1}^{n} \\sum_{p=1}^P X_{ip} \\beta_p,\n",
+ "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n",
"$$"
]
},
@@ -615,13 +600,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "or\n",
- "$ n\\beta_0 = \\sum_{i=1}^{n} y_i - \\sum_{p=1}^P\\beta_p \\sum_{i=1}^{n} X_{ip}$.\n",
- "\n",
- "\n",
- "\n",
- "\n",
- "If we assume that every column of $X$ is centered, whic we can do by subtracting the mean,"
+ "If we assume that every column of $\\boldsymbol{X}$ is centered, which we can do by subtracting the mean,"
]
},
{
@@ -640,8 +619,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "the sum $ \\sum_{i=1}^{n} X_{ip} $\n",
- "\n",
+ "the sum $\\sum_{i=0}^{n-1} X_{ij}$\n",
"can be rewritten as"
]
},
@@ -650,7 +628,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\sum_{i=1}^{n} (X_{ip} - \\frac{1}{n}\\sum_{i=1}^{n} X_{ip}) = \\sum_{i=1}^{n} X_{ip} - \\sum_{i=1}^{n} \\frac{1}{n} \\sum_{i=1}^{n}X_{ip},\n",
+ "\\sum_{i=0}^{n-1} \\left(X_{ij} - \\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij}) = \\sum_{i=0}^{n-1} X_{ij} - \\sum_{i=0}^{n-1} \\frac{1}{n} \\sum_{i=0}^{n-1}X_{ij},\n",
"$$"
]
},
@@ -666,7 +644,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\sum_{i=1}^{n} X_{ip} - n \\frac{1}{n} \\sum_{i=1}^{n}X_{ip} = 0.\n",
+ "\\sum_{i=0}^{n-1} X_{ij} - n \\frac{1}{n} \\sum_{i=0}^{n-1}X_{ij} = 0.\n",
"$$"
]
},
@@ -682,7 +660,7 @@
"metadata": {},
"source": [
"$$\n",
- "n\\beta_0 = \\sum_{i=1}^{n} y_i - \\sum_{p=1}^P\\beta_p \\sum_{i=1}^{n} X_{ip},\n",
+ "n\\beta_0 = \\sum_{i=0}^{n-1} y_i - \\sum_{j=1}^{p-1}\\beta_j \\sum_{i=0}^{n-1} X_{ij},\n",
"$$"
]
},
@@ -698,7 +676,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\beta_0 = \\frac{1}{n}\\sum_{i=1}^{n} y_i = y_{average}.\n",
+ "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1} y_i = \\overline{\\boldsymbol{y}},\n",
"$$"
]
},
@@ -706,7 +684,9 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Replacing $y_i$ with $y_i - \\beta_0 = y_i - y_{average}$ in the loss function will give us (in vector-matrix disguise)"
+ "the average value of $\\boldsymbol{y]$.\n",
+ "\n",
+ "Replacing $y_i$ with $y_i - \\beta_0 = y_i - \\overline{\\boldsymbol{y}}$ in the cost function will give us (in vector-matrix disguise)"
]
},
{
@@ -725,8 +705,12 @@
"which has the solution\n",
"\n",
"$\\beta = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}$.\n",
- "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - y_{average}$\n",
- "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=1}^{n-1}X_{kj}$."
+ "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n",
+ "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=1}^{n-1}X_{kj}$.\n",
+ "\n",
+ "## Code Examples\n",
+ "\n",
+ "Armed with this wisdom, we attempt first simply set the intercept eqault to **False** in our implementation of Ridge regression for a vanilla data set."
]
},
{
@@ -744,8 +728,6 @@
"from sklearn.model_selection import train_test_split\n",
"from sklearn import linear_model\n",
"\n",
- "def R2(y_data, y_model):\n",
- " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n",
"def MSE(y_data,y_model):\n",
" n = np.size(y_model)\n",
" return np.sum((y_data-y_model)**2)/n\n",
@@ -761,42 +743,29 @@
"\n",
"Maxpolydegree = 20\n",
"X = np.zeros((n,Maxpolydegree))\n",
+ "We include explicitely the intercpt column\n",
"X[:,0] = 1.0\n",
"\n",
- "for polydegree in range(1, Maxpolydegree):\n",
- " for degree in range(polydegree):\n",
- " X[:,degree] = x**degree\n",
+ "for degree in range(Maxpolydegree):\n",
+ " X[:,degree] = x**degree\n",
"\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",
- "# matrix inversion to find beta\n",
- "OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train\n",
- "print(OLSbeta)\n",
- "# and then make the prediction\n",
- "ytildeOLS = X_train @ OLSbeta\n",
- "print(\"Training MSE for OLS\")\n",
- "print(MSE(y_train,ytildeOLS))\n",
- "ypredictOLS = X_test @ OLSbeta\n",
- "print(\"Test MSE OLS\")\n",
- "print(MSE(y_test,ypredictOLS))\n",
- "\n",
- "p = len(OLSbeta)\n",
+ "p = Maxpolydegree\n",
"I = np.eye(p,p)\n",
"# Decide which values of lambda to use\n",
"nlambdas = 4\n",
"MSEOwnRidgePredict = np.zeros(nlambdas)\n",
- "MSEOwnRidgeTrain = np.zeros(nlambdas)\n",
"MSERidgePredict = np.zeros(nlambdas)\n",
- "MSERidgeTrain = np.zeros(nlambdas)\n",
"\n",
"lambdas = np.logspace(-4, 4, 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",
" # include lasso using Scikit-Learn\n",
- " # Note: we include the intercept\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",
@@ -805,18 +774,14 @@
" ytildeRidge = RegRidge.predict(X_train)\n",
" ypredictRidge = RegRidge.predict(X_test)\n",
" MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
- " MSEOwnRidgeTrain[i] = MSE(y_train,ytildeOwnRidge)\n",
" MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
- " MSERidgeTrain[i] = MSE(y_train,ytildeRidge)\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",
"# Now plot the results\n",
"plt.figure()\n",
- "plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'b', label = 'MSE Ridge train')\n",
"plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')\n",
- "plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE Ridge train')\n",
"plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n",
"\n",
"plt.xlabel('log10(lambda)')\n",
@@ -825,6 +790,18 @@
"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",
+ "The problem however is that can easily lead to a larger mean-squared error!\n",
+ "\n",
+ "Let us see how we can change this code by zero centering.\n",
+ "\n",
+ "## Taking out the mean"
+ ]
+ },
{
"cell_type": "code",
"execution_count": null,
@@ -841,13 +818,9 @@
"from sklearn import linear_model\n",
"from sklearn.preprocessing import StandardScaler\n",
"\n",
- "def R2(y_data, y_model):\n",
- " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\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(315)\n",
@@ -856,15 +829,12 @@
"x = np.random.rand(n)\n",
"y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n",
"\n",
- "Maxpolydegree = 5\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",
- "\n",
- "\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",
@@ -874,11 +844,13 @@
"\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",
- "X_train_scaled = X_train - X_train_mean #Center by removing mean from each feature\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",
- "\n",
- "y_scaler = np.mean(y_train) #The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)\n",
- "y_train_scaled = y_train - y_scaler #Remove the intercept from the training data.\n",
+ "#The model intercept (called y_scaler) is given by the mean of 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",
"\n",
"p = Maxpolydegree-1\n",
@@ -893,25 +865,20 @@
" 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",
- " \n",
- " ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ #Add intercept to prediction\n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ \n",
" #EQUIVALENT PREDICTION:\n",
- " ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler #Add intercept to prediction\n",
+ " #Add intercept to prediction\n",
+ " ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n",
" print(\"Values for own Ridge prediction\")\n",
" print(ypredictOwnRidge)\n",
- "\n",
- " \n",
- "\n",
" RegRidge = linear_model.Ridge(lmb)\n",
" RegRidge.fit(X_train,y_train)\n",
" ypredictRidge = RegRidge.predict(X_test)\n",
" print(\"Values for SL Ridge prediction\")\n",
" print(ypredictRidge)\n",
- "\n",
- "\n",
" MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n",
" MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n",
- "\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",
@@ -921,14 +888,10 @@
" print('Intercept from Scikit-Learn Ridge implementation')\n",
" print(RegRidge.intercept_)\n",
"\n",
- "\n",
- "\n",
"# Now plot the results\n",
- "\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",
- "\n",
"plt.xlabel('log10(lambda)')\n",
"plt.ylabel('MSE')\n",
"plt.legend()\n",
@@ -1603,7 +1566,7 @@
"metadata": {},
"source": [
"3\n",
- "2\n",
+ "1\n",
" \n",
"<\n",
"<\n",
diff --git a/doc/src/week38/week38.do.txt b/doc/src/week38/week38.do.txt
index 34f5c0c87..4f0c87ecf 100644
--- a/doc/src/week38/week38.do.txt
+++ b/doc/src/week38/week38.do.txt
@@ -271,13 +271,15 @@ $\bm{X}$ 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.
+standard deviation. Most machine learning libraries do this as a deafult. This means that if you compare your code with the results from a given library,
+the results may differ. Tracing back the differences may often lead to an increased confusion.
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.
+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
@@ -318,7 +320,7 @@ y_pred = y_pred + y_train_mean
!split
===== 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*)
+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.
!bc pycod
import numpy as np
@@ -395,78 +397,57 @@ plt.show()
!ec
!split
-===== What does centering mean mathematically? =====
-Here is a mathematical explanation of the zero centering:
+===== 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*. To catch many birds with just one stone, we will focus on Ridge regression.
-
-The cost/loss function for Ridge regression is:
-
-
+The cost/loss function for Ridge regression is
!bt
\[
-C(\beta_0, \beta_1, ... , \beta_P) = \sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip}\beta_p)^2 + \lambda \sum_{p=1}^P \beta_p^2.
+C(\beta_0, \beta_1, ... , \beta_{p-1}) = \sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2 + \lambda \sum_{j=1}^{p-1} \beta_i^2.
\]
!et
-
-
-
-Notice that the intercept is left out of the $L_2$ regularization term. The design matrix
+Note that the intercept term $\beta_0$is left out of the $L_2$ regularization term. The design matrix
$X$ does in this case not contain any intercept column. We want
!bt
\[
-\frac{\partial L}{\partial \beta_j} = 0,
+\frac{\partial C}{\partial \beta_j} = 0,
\]
!et
-for all $j$, so lets start with $\beta_0$. This means that we have
+for all $j$, so let us start with $\beta_0$. This means that we have
!bt
\[
-\frac{\partial L}{\partial \beta_0} = -2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p).
+\frac{\partial C}{\partial \beta_0} = -2\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right),
\]
!et
-We want to solve
-!bt
-\[
--2\sum_{i=1}^{n} (y_i - \beta_0 - \sum_{p=1}^P X_{ip} \beta_p) = 0,
-\]
-!et
-
-
which gives
!bt
\[
-\sum_{i=1}^{n} \beta_0 = \sum_{i=1}^{n}y_i - \sum_{i=1}^{n} \sum_{p=1}^P X_{ip} \beta_p,
+\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
\]
!et
-or
-$ n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip}$.
-
-
-
-
-If we assume that every column of $X$ is centered, whic we can do by subtracting the mean,
+If 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
-the sum $ \sum_{i=1}^{n} X_{ip} $
-
+the sum $\sum_{i=0}^{n-1} X_{ij}$
can be rewritten as
!bt
\[
-\sum_{i=1}^{n} (X_{ip} - \frac{1}{n}\sum_{i=1}^{n} X_{ip}) = \sum_{i=1}^{n} X_{ip} - \sum_{i=1}^{n} \frac{1}{n} \sum_{i=1}^{n}X_{ip},
+\sum_{i=0}^{n-1} \left(X_{ij} - \frac{1}{n}\sum_{i=0}^{n-1} X_{ij}) = \sum_{i=0}^{n-1} X_{ij} - \sum_{i=0}^{n-1} \frac{1}{n} \sum_{i=0}^{n-1}X_{ij},
\]
!et
resulting in
!bt
\[
-\sum_{i=1}^{n} X_{ip} - n \frac{1}{n} \sum_{i=1}^{n}X_{ip} = 0.
+\sum_{i=0}^{n-1} X_{ij} - n \frac{1}{n} \sum_{i=0}^{n-1}X_{ij} = 0.
\]
!et
@@ -474,17 +455,18 @@ resulting in
Finally we have
!bt
\[
-n\beta_0 = \sum_{i=1}^{n} y_i - \sum_{p=1}^P\beta_p \sum_{i=1}^{n} X_{ip},
+n\beta_0 = \sum_{i=0}^{n-1} y_i - \sum_{j=1}^{p-1}\beta_j \sum_{i=0}^{n-1} X_{ij},
\]
!et
or
!bt
\[
-\beta_0 = \frac{1}{n}\sum_{i=1}^{n} y_i = y_{average}.
+\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\bm{y}},
\]
!et
+the average value of $\bm{y]$.
-Replacing $y_i$ with $y_i - \beta_0 = y_i - y_{average}$ in the loss function will give us (in vector-matrix disguise)
+Replacing $y_i$ with $y_i - \beta_0 = y_i - \overline{\bm{y}}$ in the cost function will give us (in vector-matrix disguise)
!bt
\[
C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}) + \lambda \boldsymbol{\beta}^T\boldsymbol{\beta},
@@ -494,11 +476,13 @@ C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T
which has the solution
$\beta = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}$.
-where $\boldsymbol{\tilde{y}} = \boldsymbol{y} - y_{average}$
+where $\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\bm{y}}$
and $\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=1}^{n-1}X_{kj}$.
+!split
+===== Code Examples =====
-
+Armed with this wisdom, we attempt first simply set the intercept eqault to _False_ in our implementation of Ridge regression for a vanilla data set.
!bc pycod
import numpy as np
@@ -507,8 +491,6 @@ import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn import linear_model
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
def MSE(y_data,y_model):
n = np.size(y_model)
return np.sum((y_data-y_model)**2)/n
@@ -524,42 +506,29 @@ y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
Maxpolydegree = 20
X = np.zeros((n,Maxpolydegree))
+We include explicitely the intercpt column
X[:,0] = 1.0
-for polydegree in range(1, Maxpolydegree):
- for degree in range(polydegree):
- X[:,degree] = x**degree
+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)
-# matrix inversion to find beta
-OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
-print(OLSbeta)
-# and then make the prediction
-ytildeOLS = X_train @ OLSbeta
-print("Training MSE for OLS")
-print(MSE(y_train,ytildeOLS))
-ypredictOLS = X_test @ OLSbeta
-print("Test MSE OLS")
-print(MSE(y_test,ypredictOLS))
-
-p = len(OLSbeta)
+p = Maxpolydegree
I = np.eye(p,p)
# Decide which values of lambda to use
nlambdas = 4
MSEOwnRidgePredict = np.zeros(nlambdas)
-MSEOwnRidgeTrain = np.zeros(nlambdas)
MSERidgePredict = np.zeros(nlambdas)
-MSERidgeTrain = np.zeros(nlambdas)
lambdas = np.logspace(-4, 4, 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
# include lasso using Scikit-Learn
- # Note: we include the intercept
+ # 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
@@ -568,18 +537,14 @@ for i in range(nlambdas):
ytildeRidge = RegRidge.predict(X_train)
ypredictRidge = RegRidge.predict(X_test)
MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
- MSEOwnRidgeTrain[i] = MSE(y_train,ytildeOwnRidge)
MSERidgePredict[i] = MSE(y_test,ypredictRidge)
- MSERidgeTrain[i] = MSE(y_train,ytildeRidge)
print("Beta values for own Ridge implementation")
print(OwnRidgeBeta)
print("Beta values for Scikit-Learn Ridge implementation")
print(RegRidge.coef_)
# Now plot the results
plt.figure()
-plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'b', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE Ridge Test')
-plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE Ridge train')
plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
plt.xlabel('log10(lambda)')
@@ -589,8 +554,13 @@ plt.show()
!ec
+The results here agree when we force _Scikit-Learn_'s Ridge function to include the first column in our design matrix.
+The problem however is that can easily lead to a larger mean-squared error!
+Let us see how we can change this code by zero centering.
+!split
+===== Taking out the mean =====
!bc pycod
import numpy as np
import pandas as pd
@@ -599,13 +569,9 @@ from sklearn.model_selection import train_test_split
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler
-def R2(y_data, y_model):
- return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
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)
@@ -614,15 +580,12 @@ n = 100
x = np.random.rand(n)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
-Maxpolydegree = 5
+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)
@@ -632,11 +595,13 @@ 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)
-X_train_scaled = X_train - X_train_mean #Center by removing mean from each feature
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean
X_test_scaled = X_test - X_train_mean
-
-y_scaler = np.mean(y_train) #The model intercept (called y_scaler) is given by the mean of target variable (IF X is centered)
-y_train_scaled = y_train - y_scaler #Remove the intercept from the training data.
+#The model intercept (called y_scaler) is given by the mean of 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
@@ -651,25 +616,20 @@ 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
-
- ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_
#EQUIVALENT PREDICTION:
- ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler #Add intercept to prediction
+ #Add intercept to prediction
+ ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler
print("Values for own Ridge prediction")
print(ypredictOwnRidge)
-
-
-
RegRidge = linear_model.Ridge(lmb)
RegRidge.fit(X_train,y_train)
ypredictRidge = RegRidge.predict(X_test)
print("Values for SL Ridge prediction")
print(ypredictRidge)
-
-
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")
@@ -679,20 +639,15 @@ for i in range(nlambdas):
print('Intercept from Scikit-Learn Ridge implementation')
print(RegRidge.intercept_)
-
-
# 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