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doc/src/week38/week38-reveal.html create mode 100644 doc/src/week38/week38-solarized.html create mode 100644 doc/src/week38/week38.html create mode 100644 doc/src/week38/week38.ipynb diff --git a/README.md b/README.md index e69c369eb..1a66beaac 100644 --- a/README.md +++ b/README.md @@ -308,6 +308,7 @@ Recommended prereading: Chapters 1-2 (linear algebra) and chapter 3 (statistics) ### Week 38 September 20-24 - Lab Wednesday: Work on Project 1 - Lecture Thursday: Classification problems and Logistic Regression, from binary cases to several categories + - Video of Lecture at https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h21/forelesningsvideoer/LectureSeptember23.mp4?vrtx=view-as-webpage - Lecture Friday: Logistic Regression and start discussions of gradient optimization - Reading recommendations: diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt index f83d7feb4..bc4fc936d 100644 --- a/doc/BookChapters/chapter3.do.txt +++ b/doc/BookChapters/chapter3.do.txt @@ -1211,6 +1211,967 @@ plt.show() !ec +===== More on Rescaling data ===== + +We end this chapter by adding some words on scaling and how to deal with the intercept for regression cases. + +When you are comparing your own code with for example _Scikit-Learn_'s +library, there are some technicalities to keep in mind. The examples +here demonstrate some of these aspects with potential pitfalls. + +The discussion here focuses on the role of the intercept, how we can +set up the design matrix, what scaling we should use and other topics +which tend confuse us. + +The intercept can be interpreted as the expected value of our +target/output variables when all other predictors are set to zero. +Thus, if we cannot assume that the expected outputs/targets are zero +when all predictors are zero (the columns in the design matrix), it +may be a bad idea to implement a model which penalizes the intercept. +Furthermore, in for example Ridge and Lasso regression, the default solutions +from the library _Scikit-Learn_ (when not shrinking $\beta_0$) for the unknown parameters +$\bm{\beta}$, are derived under the assumption that both $\bm{y}$ and +$\bm{X}$ are zero centered, that is we subtract the mean values. + + +If our predictors represent different scales, then it is important to +standardize the design matrix $\bm{X}$ by subtracting the mean of each +column from the corresponding column and dividing the column with its +standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library, +the results may differ. + +The +"Standadscaler":"https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html" +function in _Scikit-Learn_ does this for us. For the data sets we +have been studying in our various examples, the data are in many cases +already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a +survey of your data, with a critical assessment of them in case you need to scale the data. + +If you need to scale the data, not doing so will give an *unfair* +penalization of the parameters since their magnitude depends on the +scale of their corresponding predictor. + +Suppose as an example that you +you have an input variable given by the heights of different persons. +Human height might be measured in inches or meters or +kilometers. If measured in kilometers, a standard linear regression +model with this predictor would probably give a much bigger +coefficient term, than if measured in millimeters. +This can clearly lead to problems in evaluating the cost/loss functions. + + + +Keep in mind that when you transform your data set before training a model, the same transformation needs to be done +on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as follows + +!bc pycod +""" +#Model training, we compute the mean value of y and X +y_train_mean = np.mean(y_train) +X_train_mean = np.mean(X_train,axis=0) +X_train = X_train - X_train_mean +y_train = y_train - y_train_mean + +# The we fit our model with the training data +trained_model = some_model.fit(X_train,y_train) + + +#Model prediction, we need also to transform our data set used for the prediction. +X_test = X_test - X_train_mean #Use mean from training data +y_pred = trained_model(X_test) +y_pred = y_pred + y_train_mean +""" +!ec + + +Let us try to understand what this may imply mathematically when we +subtract the mean values, also known as *zero centering*. For +simplicity, we will focus on ordinary regression, as done in the above example. + +The cost/loss function for regression is +!bt +\[ +C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,. +\] +!et +Recall also that we use the squared value since this leads to an increase of the penalty for higher differences between predicted and output/target values. + +What we have done is to single out the $\beta_0$ term in the definition of the mean squared error (MSE). +The design matrix +$X$ does in this case not contain any intercept column. +When we take the derivative with respect to $\beta_0$, we want the derivative to obey +!bt +\[ +\frac{\partial C}{\partial \beta_j} = 0, +\] +!et + +for all $j$. For $\beta_0$ we have + +!bt +\[ +\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right). +\] +!et +Multiplying away the constant $2/n$, we obtain +!bt +\[ +\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j. +\] +!et + + +We assume +that every column of $\bm{X}$ is centered, which we can do by subtracting the mean, +!bc pycod +X = X - np.mean(X,axis=0) +!ec + +This means that we need to rewrite $X_{ij}$ as $\tilde{X}_{ij}=X_{ij}-\mu_j$, where +!bt +\[ +\mu_j = \frac{1}{n}\sum_{i=0}^{n-1}X_{ij}. +\] +!et + +Let us special first to the case where we have only two parameters $\beta_0$ and $\beta_1$. +Our result for $\beta_0$ simplifies then to +!bt +\[ +n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1. +\] +!et +Assuming that the matrix elements $X_{i1}$ are centered, what we have is +!bt +\[ +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} \left(X_{i1}-\mu_{1}\right), +\] +!et +where +!bt +\[ +\mu_1=\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}, +\] +!et +and if we define the mean value of the outputs as +!bt +\[ +\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i, +\] +!et +we have +!bt +\[ +\beta_0 = \mu_y - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} (X_{i1}-\mu_{1}), +\] +!et +and it is easy to see that the last sum equals zero! This means that we have +!bt +\[ +\beta_0 = \mu_y, +\] +!et +if the columns of the design matrix are centered. It is straight forward to generalize this results to more values of $\beta$. +We have thus +!bt +\[ +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1} y_i = \overline{\bm{y}}, +\] +!et +the average value of $\bm{y}$. + +Replacing $y_i$ with $y_i - \beta_0 = y_i - \overline{\bm{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise) +!bt +\[ +C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}). +\] +!et + + +If we minimize with respect to $\bm{\beta}$ we have then + +!bt +\[ +\hat{\bm{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}, +\] +!et + +where $\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\bm{y}}$ +and $\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}$. + +For Ridge regression we need to add $\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}$ to the cost function and get then +!bt +\[ +\hat{\bm{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}. +\] +!et + +What does this mean? And why do we insist on all this? Let us look at some examples. + + +This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only. +Note also that we do not split the data into training and test. + +!bc pycod +import numpy as np +import matplotlib.pyplot as plt + +from sklearn.linear_model import LinearRegression + + +np.random.seed(2021) + +def MSE(y_data,y_model): + n = np.size(y_model) + return np.sum((y_data-y_model)**2)/n + + +def fit_beta(X, y): + return np.linalg.pinv(X.T @ X) @ X.T @ y + + +true_beta = [2, 0.5, 3.7] + +x = np.linspace(0, 1, 11) +y = np.sum( + np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0 +) + 0.1 * np.random.normal(size=len(x)) + +degree = 3 +X = np.zeros((len(x), degree)) + +# Include the intercept in the design matrix +for p in range(degree): + X[:, p] = x ** p + +beta = fit_beta(X, y) + +# Intercept is included in the design matrix +skl = LinearRegression(fit_intercept=False).fit(X, y) + +print(f"True beta: {true_beta}") +print(f"Fitted beta: {beta}") +print(f"Sklearn fitted beta: {skl.coef_}") +ypredictOwn = X @ beta +ypredictSKL = skl.predict(X) +print(f"MSE with intercept column") +print(MSE(y,ypredictOwn)) +print(f"MSE with intercept column from SKL") +print(MSE(y,ypredictSKL)) + + +plt.figure() +plt.scatter(x, y, label="Data") +plt.plot(x, X @ beta, label="Fit") +plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)") + + +# Do not include the intercept in the design matrix +X = np.zeros((len(x), degree - 1)) + +for p in range(degree - 1): + X[:, p] = x ** (p + 1) + +# Intercept is not included in the design matrix +skl = LinearRegression(fit_intercept=True).fit(X, y) + +# Use centered values for X and y when computing coefficients +y_offset = np.average(y, axis=0) +X_offset = np.average(X, axis=0) + +beta = fit_beta(X - X_offset, y - y_offset) +intercept = np.mean(y_offset - X_offset @ beta) + +print(f"Manual intercept: {intercept}") +print(f"Fitted beta (wiothout intercept): {beta}") +print(f"Sklearn intercept: {skl.intercept_}") +print(f"Sklearn fitted beta (without intercept): {skl.coef_}") +ypredictOwn = X @ beta +ypredictSKL = skl.predict(X) +print(f"MSE with Manual intercept") +print(MSE(y,ypredictOwn+intercept)) +print(f"MSE with Sklearn intercept") +print(MSE(y,ypredictSKL)) + +plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)") +plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)") +plt.grid() +plt.legend() + +plt.show() + +!ec + +The intercept is the value of our output/target variable +when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). + +Printing the MSE, we see first that both methods give the same MSE, as +they should. However, when we move to for example Ridge regression, +the way we treat the intercept may give a larger or smaller MSE, +meaning that the MSE can be penalized by the value of the +intercept. Not including the intercept in the fit, means that the +regularization term does not include $\beta_0$. For different values +of $\lambda$, this may lead to differeing MSE values. + +To remind the reader, the regularization term, with the intercept in Ridge regression, is given by +!bt +\[ +\lambda \vert\vert \bm{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2, +\] +!et +but when we take out the intercept, this equation becomes +!bt +\[ +\lambda \vert\vert \bm{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2. +\] +!et + +For Lasso regression we have +!bt +\[ +\lambda \vert\vert \bm{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert. +\] +!et + +It means that, when scaling the design matrix and the outputs/targets, +by subtracting the mean values, we have an optimization problem which +is not penalized by the intercept. The MSE value can then be smaller +since it focuses only on the remaining quantities. If we however bring +back the intercept, we will get a MSE which then contains the +intercept. + + +Armed with this wisdom, we attempt first to simply set the intercept equal to _False_ in our implementation of Ridge regression for our well-known vanilla data set. + +!bc pycod +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +from sklearn.model_selection import train_test_split +from sklearn import linear_model + +def MSE(y_data,y_model): + n = np.size(y_model) + return np.sum((y_data-y_model)**2)/n + + +# A seed just to ensure that the random numbers are the same for every run. +# Useful for eventual debugging. +np.random.seed(3155) + +n = 100 +x = np.random.rand(n) +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + +Maxpolydegree = 20 +X = np.zeros((n,Maxpolydegree)) +#We include explicitely the intercept column +for degree in range(Maxpolydegree): + X[:,degree] = x**degree +# We split the data in test and training data +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) + +p = Maxpolydegree +I = np.eye(p,p) +# Decide which values of lambda to use +nlambdas = 6 +MSEOwnRidgePredict = np.zeros(nlambdas) +MSERidgePredict = np.zeros(nlambdas) +lambdas = np.logspace(-4, 2, nlambdas) +for i in range(nlambdas): + lmb = lambdas[i] + OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train + # Note: we include the intercept column and no scaling + RegRidge = linear_model.Ridge(lmb,fit_intercept=False) + RegRidge.fit(X_train,y_train) + # and then make the prediction + ytildeOwnRidge = X_train @ OwnRidgeBeta + ypredictOwnRidge = X_test @ OwnRidgeBeta + ytildeRidge = RegRidge.predict(X_train) + ypredictRidge = RegRidge.predict(X_test) + MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge) + MSERidgePredict[i] = MSE(y_test,ypredictRidge) + print("Beta values for own Ridge implementation") + print(OwnRidgeBeta) + print("Beta values for Scikit-Learn Ridge implementation") + print(RegRidge.coef_) + print("MSE values for own Ridge implementation") + print(MSEOwnRidgePredict[i]) + print("MSE values for Scikit-Learn Ridge implementation") + print(MSERidgePredict[i]) + +# Now plot the results +plt.figure() +plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test') +plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test') + +plt.xlabel('log10(lambda)') +plt.ylabel('MSE') +plt.legend() +plt.show() + +!ec + +The results here agree when we force _Scikit-Learn_'s Ridge function to include the first column in our design matrix. +We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix. +What happens if we do not include the intercept in our fit? +Let us see how we can change this code by zero centering. + +!bc pycod +import numpy as np +import pandas as pd +import matplotlib.pyplot as plt +from sklearn.model_selection import train_test_split +from sklearn import linear_model +from sklearn.preprocessing import StandardScaler + +def MSE(y_data,y_model): + n = np.size(y_model) + return np.sum((y_data-y_model)**2)/n +# A seed just to ensure that the random numbers are the same for every run. +# Useful for eventual debugging. +np.random.seed(315) + +n = 100 +x = np.random.rand(n) +y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + +Maxpolydegree = 20 +X = np.zeros((n,Maxpolydegree-1)) + +for degree in range(1,Maxpolydegree): #No intercept column + X[:,degree-1] = x**(degree) + +# We split the data in test and training data +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) + +#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable +X_train_mean = np.mean(X_train,axis=0) +#Center by removing mean from each feature +X_train_scaled = X_train - X_train_mean +X_test_scaled = X_test - X_train_mean +#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered) +#Remove the intercept from the training data. +y_scaler = np.mean(y_train) +y_train_scaled = y_train - y_scaler + +p = Maxpolydegree-1 +I = np.eye(p,p) +# Decide which values of lambda to use +nlambdas = 6 +MSEOwnRidgePredict = np.zeros(nlambdas) +MSERidgePredict = np.zeros(nlambdas) + +lambdas = np.logspace(-4, 2, nlambdas) +for i in range(nlambdas): + lmb = lambdas[i] + OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled) + intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data + #Add intercept to prediction + ypredictOwnRidge = X_test @ OwnRidgeBeta + intercept_ + #Add intercept to prediction + ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler + RegRidge = linear_model.Ridge(lmb) + RegRidge.fit(X_train,y_train) + ypredictRidge = RegRidge.predict(X_test) + MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge) + MSERidgePredict[i] = MSE(y_test,ypredictRidge) + print("Beta values for own Ridge implementation") + print(OwnRidgeBeta) #Intercept is given by mean of target variable + print("Beta values for Scikit-Learn Ridge implementation") + print(RegRidge.coef_) + print('Intercept from own implementation:') + print(intercept_) + print('Intercept from Scikit-Learn Ridge implementation') + print(RegRidge.intercept_) + print("MSE values for own Ridge implementation") + print(MSEOwnRidgePredict[i]) + print("MSE values for Scikit-Learn Ridge implementation") + print(MSERidgePredict[i]) + + +# Now plot the results +plt.figure() +plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test') +plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test') +plt.xlabel('log10(lambda)') +plt.ylabel('MSE') +plt.legend() +plt.show() +!ec +We see here, when compared to the code which includes explicitely the +intercept column, that our MSE value is actually smaller. This is +because the regularization term does not include the intercept value +$\beta_0$ in the fitting. This applies to Lasso regularization as +well. It means that our optimization is now done only with the +centered matrix and/or vector that enter the fitting procedure. Note +also that the problem with the intercept occurs mainly in these type +of polynomial fitting problem. + +The next example is indeed an example where all these discussions about the role of intercept are not present. + +===== More complicated Example: The Ising model ===== + +The one-dimensional Ising model with nearest neighbor interaction, no +external field and a constant coupling constant $J$ is given by + +!bt +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\end{align} +!et + +where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins +in the system is determined by $L$. For the one-dimensional system +there is no phase transition. + +We will look at a system of $L = 40$ spins with a coupling constant of +$J = 1$. To get enough training data we will generate 10000 states +with their respective energies. + + +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from mpl_toolkits.axes_grid1 import make_axes_locatable +import seaborn as sns +import scipy.linalg as scl +from sklearn.model_selection import train_test_split +import tqdm +sns.set(color_codes=True) +cmap_args=dict(vmin=-1., vmax=1., cmap='seismic') + +L = 40 +n = int(1e4) + +spins = np.random.choice([-1, 1], size=(n, L)) +J = 1.0 + +energies = np.zeros(n) + +for i in range(n): + energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1)) +!ec + +Here we use ordinary least squares +regression to predict the energy for the nearest neighbor +one-dimensional Ising model on a ring, i.e., the endpoints wrap +around. We will use linear regression to fit a value for +the coupling constant to achieve this. + +A more general form for the one-dimensional Ising model is + +!bt +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\end{align} +!et + +Here we allow for interactions beyond the nearest neighbors and a state dependent +coupling constant. This latter expression can be formulated as +a matrix-product +!bt +\begin{align} + \bm{H} = \bm{X} J, +\end{align} +!et + +where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the +elements $-J_{jk}$. This form of writing the energy fits perfectly +with the form utilized in linear regression, that is + +!bt +\begin{align} + \bm{y} = \bm{X}\bm{\beta} + \bm{\epsilon}, +\end{align} +!et + +We split the data in training and test data as discussed in the previous example + +!bc pycod +X = np.zeros((n, L ** 2)) +for i in range(n): + X[i] = np.outer(spins[i], spins[i]).ravel() +y = energies +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2) +!ec + + +In the ordinary least squares method we choose the cost function + +!bt +\begin{align} + C(\bm{X}, \bm{\beta})= \frac{1}{n}\left\{(\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y})\right\}. +\end{align} +!et + +We then find the extremal point of $C$ by taking the derivative with respect to $\bm{\beta}$ as discussed above. +This yields the expression for $\bm{\beta}$ to be + +!bt +\[ + \bm{\beta} = \frac{\bm{X}^T \bm{y}}{\bm{X}^T \bm{X}}, +\] +!et + +which immediately imposes some requirements on $\bm{X}$ as there must exist +an inverse of $\bm{X}^T \bm{X}$. If the expression we are modeling contains an +intercept, i.e., a constant term, we must make sure that the +first column of $\bm{X}$ consists of $1$. We do this here + +!bc pycod +X_train_own = np.concatenate( + (np.ones(len(X_train))[:, np.newaxis], X_train), + axis=1 +) +X_test_own = np.concatenate( + (np.ones(len(X_test))[:, np.newaxis], X_test), + axis=1 +) +!ec + + + +Doing the inversion directly turns out to be a bad idea since the matrix +$\bm{X}^T\bm{X}$ is singular. An alternative approach is to use the _singular +value decomposition_. Using the definition of the Moore-Penrose +pseudoinverse we can write the equation for $\bm{\beta}$ as + +!bt +\[ + \bm{\beta} = \bm{X}^{+}\bm{y}, +\] +!et + +where the pseudoinverse of $\bm{X}$ is given by + +!bt +\[ + \bm{X}^{+} = \frac{\bm{X}^T}{\bm{X}^T\bm{X}}. +\] +!et + +Using singular value decomposition we can decompose the matrix $\bm{X} = \bm{U}\bm{\Sigma} \bm{V}^T$, +where $\bm{U}$ and $\bm{V}$ are orthogonal(unitary) matrices and $\bm{\Sigma}$ contains the singular values (more details below). +where $X^{+} = V\Sigma^{+} U^T$. This reduces the equation for +$\omega$ to +!bt +\begin{align} + \bm{\beta} = \bm{V}\bm{\Sigma}^{+} \bm{U}^T \bm{y}. +\end{align} +!et + +Note that solving this equation by actually doing the pseudoinverse +(which is what we will do) is not a good idea as this operation scales +as $\mathcal{O}(n^3)$, where $n$ is the number of elements in a +general matrix. Instead, doing $QR$-factorization and solving the +linear system as an equation would reduce this down to +$\mathcal{O}(n^2)$ operations. + + +!bc pycod +def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray: + u, s, v = scl.svd(x) + return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y +!ec + +!bc pycod +beta = ols_svd(X_train_own,y_train) +!ec + +When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here + +!bc pycod +J = beta[1:].reshape(L, L) +!ec + +A way of looking at the coefficients in $J$ is to plot the matrices as images. + + +!bc pycod +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J, **cmap_args) +plt.title("OLS", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) +plt.show() +!ec +It is interesting to note that OLS +considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as +valid matrix elements for $J$. +In our discussion below on hyperparameters and Ridge and Lasso regression we will see that +this problem can be removed, partly and only with Lasso regression. + +In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD? + + + + + +Let us now +focus on Ridge and Lasso regression as well. We repeat some of the +basic parts of the Ising model and the setup of the training and test +data. The one-dimensional Ising model with nearest neighbor +interaction, no external field and a constant coupling constant $J$ is +given by + +!bt +\begin{align} + H = -J \sum_{k}^L s_k s_{k + 1}, +\end{align} +!et +where $s_i \in \{-1, 1\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition. + +We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies. + + +!bc pycod +import numpy as np +import matplotlib.pyplot as plt +from mpl_toolkits.axes_grid1 import make_axes_locatable +import seaborn as sns +import scipy.linalg as scl +from sklearn.model_selection import train_test_split +import sklearn.linear_model as skl +import tqdm +sns.set(color_codes=True) +cmap_args=dict(vmin=-1., vmax=1., cmap='seismic') + +L = 40 +n = int(1e4) + +spins = np.random.choice([-1, 1], size=(n, L)) +J = 1.0 + +energies = np.zeros(n) + +for i in range(n): + energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1)) +!ec + +A more general form for the one-dimensional Ising model is + +!bt +\begin{align} + H = - \sum_j^L \sum_k^L s_j s_k J_{jk}. +\end{align} +!et + +Here we allow for interactions beyond the nearest neighbors and a more +adaptive coupling matrix. This latter expression can be formulated as +a matrix-product on the form +!bt +\begin{align} + H = X J, +\end{align} +!et + +where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the +elements $-J_{jk}$. This form of writing the energy fits perfectly +with the form utilized in linear regression, viz. +!bt +\begin{align} + \bm{y} = \bm{X}\bm{\beta} + \bm{\epsilon}. +\end{align} +!et +We organize the data as we did above +!bc pycod +X = np.zeros((n, L ** 2)) +for i in range(n): + X[i] = np.outer(spins[i], spins[i]).ravel() +y = energies +X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96) + +X_train_own = np.concatenate( + (np.ones(len(X_train))[:, np.newaxis], X_train), + axis=1 +) + +X_test_own = np.concatenate( + (np.ones(len(X_test))[:, np.newaxis], X_test), + axis=1 +) +!ec + +We will do all fitting with _Scikit-Learn_, + +!bc pycod +clf = skl.LinearRegression().fit(X_train, y_train) +!ec +When extracting the $J$-matrix we make sure to remove the intercept +!bc pycod +J_sk = clf.coef_.reshape(L, L) +!ec +And then we plot the results +!bc pycod +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_sk, **cmap_args) +plt.title("LinearRegression from Scikit-learn", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) +plt.show() +!ec +The results agree perfectly with our previous discussion where we used our own code. + + +Having explored the ordinary least squares we move on to ridge +regression. In ridge regression we include a _regularizer_. This +involves a new cost function which leads to a new estimate for the +weights $\bm{\beta}$. This results in a penalized regression problem. The +cost function is given by + +!bt +\begin{align} + C(\bm{X}, \bm{\beta}; \lambda) = (\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y}) + \lambda \bm{\beta}^T\bm{\beta}. +\end{align} +!et +!bc pycod +_lambda = 0.1 +clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train) +J_ridge_sk = clf_ridge.coef_.reshape(L, L) +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_ridge_sk, **cmap_args) +plt.title("Ridge from Scikit-learn", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) + +plt.show() +!ec + +In the _Least Absolute Shrinkage and Selection Operator_ (LASSO)-method we get a third cost function. + +!bt +\begin{align} + C(\bm{X}, \bm{\beta}; \lambda) = (\bm{X}\bm{\beta} - \bm{y})^T(\bm{X}\bm{\beta} - \bm{y}) + \lambda \sqrt{\bm{\beta}^T\bm{\beta}}. +\end{align} +!et + +Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from _Scikit-Learn_. + +!bc pycod +clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train) +J_lasso_sk = clf_lasso.coef_.reshape(L, L) +fig = plt.figure(figsize=(20, 14)) +im = plt.imshow(J_lasso_sk, **cmap_args) +plt.title("Lasso from Scikit-learn", fontsize=18) +plt.xticks(fontsize=18) +plt.yticks(fontsize=18) +cb = fig.colorbar(im) +cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18) + +plt.show() +!ec + +It is quite striking how LASSO breaks the symmetry of the coupling +constant as opposed to ridge and OLS. We get a sparse solution with +$J_{j, j + 1} = -1$. + + + + +We see how the different models perform for a different set of values for $\lambda$. + + +!bc pycod +lambdas = np.logspace(-4, 5, 10) + +train_errors = { + "ols_sk": np.zeros(lambdas.size), + "ridge_sk": np.zeros(lambdas.size), + "lasso_sk": np.zeros(lambdas.size) +} + +test_errors = { + "ols_sk": np.zeros(lambdas.size), + "ridge_sk": np.zeros(lambdas.size), + "lasso_sk": np.zeros(lambdas.size) +} + +plot_counter = 1 + +fig = plt.figure(figsize=(32, 54)) + +for i, _lambda in enumerate(tqdm.tqdm(lambdas)): + for key, method in zip( + ["ols_sk", "ridge_sk", "lasso_sk"], + [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)] + ): + method = method.fit(X_train, y_train) + + train_errors[key][i] = method.score(X_train, y_train) + test_errors[key][i] = method.score(X_test, y_test) + + omega = method.coef_.reshape(L, L) + + plt.subplot(10, 5, plot_counter) + plt.imshow(omega, **cmap_args) + plt.title(r"%s, $\lambda = %.4f$" % (key, _lambda)) + plot_counter += 1 + +plt.show() +!ec + +We see that LASSO reaches a good solution for low +values of $\lambda$, but will "wither" when we increase $\lambda$ too +much. Ridge is more stable over a larger range of values for +$\lambda$, but eventually also fades away. + + +To determine which value of $\lambda$ is best we plot the accuracy of +the models when predicting the training and the testing set. We expect +the accuracy of the training set to be quite good, but if the accuracy +of the testing set is much lower this tells us that we might be +subject to an overfit model. The ideal scenario is an accuracy on the +testing set that is close to the accuracy of the training set. + + +!bc pycod +fig = plt.figure(figsize=(20, 14)) + +colors = { + "ols_sk": "r", + "ridge_sk": "y", + "lasso_sk": "c" +} + +for key in train_errors: + plt.semilogx( + lambdas, + train_errors[key], + colors[key], + label="Train {0}".format(key), + linewidth=4.0 + ) + +for key in test_errors: + plt.semilogx( + lambdas, + test_errors[key], + colors[key] + "--", + label="Test {0}".format(key), + linewidth=4.0 + ) +plt.legend(loc="best", fontsize=18) +plt.xlabel(r"$\lambda$", fontsize=18) +plt.ylabel(r"$R^2$", fontsize=18) +plt.tick_params(labelsize=18) +plt.show() +!ec + +From the above figure we can see that LASSO with $\lambda = 10^{-2}$ +achieves a very good accuracy on the test set. This by far surpasses the +other models for all values of $\lambda$. + + + + + ===== Exercises and Projects ===== @@ -1499,6 +2460,7 @@ Here is a simple part of a Python code which reads and plots the data from such files !bc pycod +""" import numpy as np from imageio import imread import matplotlib.pyplot as plt @@ -1514,6 +2476,7 @@ plt.imshow(terrain1, cmap='gray') plt.xlabel('X') plt.ylabel('Y') plt.show() +""" !ec If you should have problems in downloading the digital terrain data, @@ -1537,3 +2500,5 @@ of data presented here (either the terrain data we propose or other data sets). + + diff --git a/doc/LectureNotes/_build/.doctrees/chapter3.doctree b/doc/LectureNotes/_build/.doctrees/chapter3.doctree index b39c4cbe3689f7959148b4c453a34d63dbfc3a1d..5bead3ef4e539cf6121cb036ed435e6c4059712f 100644 GIT binary patch literal 1174808 zcmeFa36LvEdLA}!=X{uRXLo0~oZTgMre~-7b-z9U^flbK+`dorfktuL5c1eUNlg%)tplOMRgOWuvmX~BQ6ylO5sgPyUCaIMV*)*359x^G5 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