FYS-STK4155/doc/pub/week38/ipynb/week38.ipynb

Data Analysis and Machine Learning: Logistic Regression

Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

Date: Sep 17, 2020

Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license

Plans for week 38

• Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples

• Friday: Logistic Regression

Thursday September 17

Ridge and LASSO Regression, reminder

The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is our optimization problem is

{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}.

or we can state it as

{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2,

where we have used the definition of a norm-2 vector, that is

\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}.

By minimizing the above equation with respect to the parameters \boldsymbol{\beta} we could then obtain an analytical expression for the parameters \boldsymbol{\beta}. We can add a regularization parameter \lambda by defining a new cost function to be optimized, that is

{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2

which leads to the Ridge regression minimization problem where we require that \vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t, where t is a finite number larger than zero. By defining

C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1,

we have a new optimization equation

{\displaystyle \min_{\boldsymbol{\beta}\in
{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1

which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.

Here we have defined the norm-1 as

\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert.

Various steps in cross-validation

When the repetitive splitting of the data set is done randomly, samples may accidently end up in a fast majority of the splits in either training or test set. Such samples may have an unbalanced influence on either model building or prediction evaluation. To avoid this $k$-fold cross-validation structures the data splitting. The samples are divided into k more or less equally sized exhaustive and mutually exclusive subsets. In turn (at each split) one of these subsets plays the role of the test set while the union of the remaining subsets constitutes the training set. Such a splitting warrants a balanced representation of each sample in both training and test set over the splits. Still the division into the k subsets involves a degree of randomness. This may be fully excluded when choosing k=n. This particular case is referred to as leave-one-out cross-validation (LOOCV).

How to set up the cross-validation for Ridge and/or Lasso

• Define a range of interest for the penalty parameter.

• Divide the data set into training and test set comprising samples \{1, \ldots, n\} \setminus i and \{ i \}, respectively.

• Fit the linear regression model by means of ridge estimation for each \lambda in the grid using the training set, and the corresponding estimate of the error variance \boldsymbol{\sigma}_{-i}^2(\lambda), as

\begin{align*}
\boldsymbol{\beta}_{-i}(\lambda) & =  ( \boldsymbol{X}_{-i, \ast}^{T}
\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1}
\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i}
\end{align*}

• Evaluate the prediction performance of these models on the test set by \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. Or, by the prediction error |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)|, the relative error, the error squared or the R2 score function.

• Repeat the first three steps such that each sample plays the role of the test set once.

• Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as

\begin{align*}
\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}.
\end{align*}

Cross-validation in brief

For the various values of k

1. shuffle the dataset randomly.

2. Split the dataset into k groups.

3. For each unique group:

a. Decide which group to use as set for test data

b. Take the remaining groups as a training data set

c. Fit a model on the training set and evaluate it on the test set

d. Retain the evaluation score and discard the model

5. Summarize the model using the sample of model evaluation scores

Code Example for Cross-validation and $k$-fold Cross-validation

The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial.

In [23]:

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import KFold
from sklearn.linear_model import Ridge
from sklearn.model_selection import cross_val_score
from sklearn.preprocessing import PolynomialFeatures

# A seed just to ensure that the random numbers are the same for every run.
# Useful for eventual debugging.
np.random.seed(3155)

# Generate the data.
nsamples = 100
x = np.random.randn(nsamples)
y = 3*x**2 + np.random.randn(nsamples)

## Cross-validation on Ridge regression using KFold only

# Decide degree on polynomial to fit
poly = PolynomialFeatures(degree = 4)

# Decide which values of lambda to use
nlambdas = 500
lambdas = np.logspace(-3, 5, nlambdas)

# Initialize a KFold instance
k = 5
kfold = KFold(n_splits = k)

# Perform the cross-validation to estimate MSE
scores_KFold = np.zeros((nlambdas, k))

i = 0
for lmb in lambdas:
    ridge = Ridge(alpha = lmb)
    j = 0
    for train_inds, test_inds in kfold.split(x):
        xtrain = x[train_inds]
        ytrain = y[train_inds]

        xtest = x[test_inds]
        ytest = y[test_inds]

        Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
        ridge.fit(Xtrain, ytrain[:, np.newaxis])

        Xtest = poly.fit_transform(xtest[:, np.newaxis])
        ypred = ridge.predict(Xtest)

        scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)

        j += 1
    i += 1


estimated_mse_KFold = np.mean(scores_KFold, axis = 1)

## Cross-validation using cross_val_score from sklearn along with KFold

# kfold is an instance initialized above as:
# kfold = KFold(n_splits = k)

estimated_mse_sklearn = np.zeros(nlambdas)
i = 0
for lmb in lambdas:
    ridge = Ridge(alpha = lmb)

    X = poly.fit_transform(x[:, np.newaxis])
    estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)

    # cross_val_score return an array containing the estimated negative mse for every fold.
    # we have to the the mean of every array in order to get an estimate of the mse of the model
    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)

    i += 1

## Plot and compare the slightly different ways to perform cross-validation

plt.figure()

plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')

plt.xlabel('log10(lambda)')
plt.ylabel('mse')

plt.legend()

plt.show()

Bias-Variance tradeoff with Bootstrap

In [2]:

import matplotlib.pyplot as plt
import numpy as np
from sklearn.linear_model import LinearRegression, Ridge, Lasso
from sklearn.preprocessing import PolynomialFeatures
from sklearn.model_selection import train_test_split
from sklearn.pipeline import make_pipeline
from sklearn.utils import resample

np.random.seed(2018)

n = 40
n_boostraps = 100
maxdegree = 14


# Make data set.
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
error = np.zeros(maxdegree)
bias = np.zeros(maxdegree)
variance = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)

for degree in range(maxdegree):
    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
    y_pred = np.empty((y_test.shape[0], n_boostraps))
    for i in range(n_boostraps):
        x_, y_ = resample(x_train, y_train)
        y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()

    polydegree[degree] = degree
    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
    print('Polynomial degree:', degree)
    print('Error:', error[degree])
    print('Bias^2:', bias[degree])
    print('Var:', variance[degree])
    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))

plt.plot(polydegree, error, label='Error')
plt.plot(polydegree, bias, label='bias')
plt.plot(polydegree, variance, label='Variance')
plt.legend()
plt.show()

Polynomial degree: 0
Error: 0.3214960170351912
Bias^2: 0.3123314713548606
Var: 0.009164545680330616
0.3214960170351912 >= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912
Polynomial degree: 1
Error: 0.08426840630693411
Bias^2: 0.07968918676726029
Var: 0.004579219539673836
0.08426840630693411 >= 0.07968918676726029 + 0.004579219539673836 = 0.08426840630693413
Polynomial degree: 2
Error: 0.10398646080125035
Bias^2: 0.10077114273548984
Var: 0.003215318065760509
0.10398646080125035 >= 0.10077114273548984 + 0.003215318065760509 = 0.10398646080125035
Polynomial degree: 3
Error: 0.06547790180152357
Bias^2: 0.06208238634231953
Var: 0.0033955154592040944
0.06547790180152357 >= 0.06208238634231953 + 0.0033955154592040944 = 0.06547790180152363
Polynomial degree: 4
Error: 0.06844519414009438
Bias^2: 0.06453579006728315
Var: 0.003909404072811231
0.06844519414009438 >= 0.06453579006728315 + 0.003909404072811231 = 0.06844519414009438
Polynomial degree: 5
Error: 0.05227921801205692
Bias^2: 0.04818727730430296
Var: 0.0040919407077539514
0.05227921801205692 >= 0.04818727730430296 + 0.0040919407077539514 = 0.05227921801205691
Polynomial degree: 6
Error: 0.03781367141738885
Bias^2: 0.033657685071527485
Var: 0.004155986345861374
0.03781367141738885 >= 0.033657685071527485 + 0.004155986345861374 = 0.03781367141738886
Polynomial degree: 7
Error: 0.027609773491022314
Bias^2: 0.02299949826036602
Var: 0.004610275230656294
0.027609773491022314 >= 0.02299949826036602 + 0.004610275230656294 = 0.027609773491022314
Polynomial degree: 8
Error: 0.017355848195591845
Bias^2: 0.01033172130665515
Var: 0.007024126888936694
0.017355848195591845 >= 0.01033172130665515 + 0.007024126888936694 = 0.01735584819559184
Polynomial degree: 9
Error: 0.026605727637176654
Bias^2: 0.010018312644139347
Var: 0.016587414993037307
0.026605727637176654 >= 0.010018312644139347 + 0.016587414993037307 = 0.026605727637176654
Polynomial degree: 10
Error: 0.02159270458799264
Bias^2: 0.010516485576652856
Var: 0.011076219011339788
0.02159270458799264 >= 0.010516485576652856 + 0.011076219011339788 = 0.021592704587992645
Polynomial degree: 11
Error: 0.07160048164248561
Bias^2: 0.014436800088969727
Var: 0.05716368155351588
0.07160048164248561 >= 0.014436800088969727 + 0.05716368155351588 = 0.07160048164248561
Polynomial degree: 12
Error: 0.11547777218940905
Bias^2: 0.016285782696075054
Var: 0.099191989493334
0.11547777218940905 >= 0.016285782696075054 + 0.099191989493334 = 0.11547777218940906
Polynomial degree: 13
Error: 0.22842468702288576
Bias^2: 0.01975416527179247
Var: 0.20867052175109335
0.22842468702288576 >= 0.01975416527179247 + 0.20867052175109335 = 0.22842468702288582

Another Example from Scikit-Learn's Repository

In [3]:

"""
============================
Underfitting vs. Overfitting
============================

This example demonstrates the problems of underfitting and overfitting and
how we can use linear regression with polynomial features to approximate
nonlinear functions. The plot shows the function that we want to approximate,
which is a part of the cosine function. In addition, the samples from the
real function and the approximations of different models are displayed. The
models have polynomial features of different degrees. We can see that a
linear function (polynomial with degree 1) is not sufficient to fit the
training samples. This is called **underfitting**. A polynomial of degree 4
approximates the true function almost perfectly. However, for higher degrees
the model will **overfit** the training data, i.e. it learns the noise of the
training data.
We evaluate quantitatively **overfitting** / **underfitting** by using
cross-validation. We calculate the mean squared error (MSE) on the validation
set, the higher, the less likely the model generalizes correctly from the
training data.
"""

print(__doc__)

import numpy as np
import matplotlib.pyplot as plt
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import cross_val_score


def true_fun(X):
    return np.cos(1.5 * np.pi * X)

np.random.seed(0)

n_samples = 30
degrees = [1, 4, 15]

X = np.sort(np.random.rand(n_samples))
y = true_fun(X) + np.random.randn(n_samples) * 0.1

plt.figure(figsize=(14, 5))
for i in range(len(degrees)):
    ax = plt.subplot(1, len(degrees), i + 1)
    plt.setp(ax, xticks=(), yticks=())

    polynomial_features = PolynomialFeatures(degree=degrees[i],
                                             include_bias=False)
    linear_regression = LinearRegression()
    pipeline = Pipeline([("polynomial_features", polynomial_features),
                         ("linear_regression", linear_regression)])
    pipeline.fit(X[:, np.newaxis], y)

    # Evaluate the models using crossvalidation
    scores = cross_val_score(pipeline, X[:, np.newaxis], y,
                             scoring="neg_mean_squared_error", cv=10)

    X_test = np.linspace(0, 1, 100)
    plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
    plt.plot(X_test, true_fun(X_test), label="True function")
    plt.scatter(X, y, edgecolor='b', s=20, label="Samples")
    plt.xlabel("x")
    plt.ylabel("y")
    plt.xlim((0, 1))
    plt.ylim((-2, 2))
    plt.legend(loc="best")
    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
        degrees[i], -scores.mean(), scores.std()))
plt.show()

============================
Underfitting vs. Overfitting
============================

This example demonstrates the problems of underfitting and overfitting and
how we can use linear regression with polynomial features to approximate
nonlinear functions. The plot shows the function that we want to approximate,
which is a part of the cosine function. In addition, the samples from the
real function and the approximations of different models are displayed. The
models have polynomial features of different degrees. We can see that a
linear function (polynomial with degree 1) is not sufficient to fit the
training samples. This is called **underfitting**. A polynomial of degree 4
approximates the true function almost perfectly. However, for higher degrees
the model will **overfit** the training data, i.e. it learns the noise of the
training data.
We evaluate quantitatively **overfitting** / **underfitting** by using
cross-validation. We calculate the mean squared error (MSE) on the validation
set, the higher, the less likely the model generalizes correctly from the
training data.

Cross-validation with Ridge

In [4]:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.model_selection import KFold
from sklearn.linear_model import Ridge
from sklearn.model_selection import cross_val_score
from sklearn.preprocessing import PolynomialFeatures

# A seed just to ensure that the random numbers are the same for every run.
np.random.seed(3155)
# Generate the data.
n = 100
x = np.linspace(-3, 3, n).reshape(-1, 1)
y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
# Decide degree on polynomial to fit
poly = PolynomialFeatures(degree = 10)

# Decide which values of lambda to use
nlambdas = 500
lambdas = np.logspace(-3, 5, nlambdas)
# Initialize a KFold instance
k = 5
kfold = KFold(n_splits = k)
estimated_mse_sklearn = np.zeros(nlambdas)
i = 0
for lmb in lambdas:
    ridge = Ridge(alpha = lmb)
    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
    i += 1
plt.figure()
plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
plt.xlabel('log10(lambda)')
plt.ylabel('MSE')
plt.legend()
plt.show()

The Ising model

The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant J is given by

\begin{equation}
    H = -J \sum_{k}^L s_k s_{k + 1},
\label{_auto1} \tag{1}
\end{equation}

where s_i \in \{-1, 1\} and s_{N + 1} = s_1. The number of spins in the system is determined by L. For the one-dimensional system there is no phase transition.

We will look at a system of L = 40 spins with a coupling constant of J = 1. To get enough training data we will generate 10000 states with their respective energies.

In [5]:

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.

Reformulating the problem to suit regression

A more general form for the one-dimensional Ising model is

\begin{equation}
    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
\label{_auto2} \tag{2}
\end{equation}

Here we allow for interactions beyond the nearest neighbors and a state dependent coupling constant. This latter expression can be formulated as a matrix-product

\begin{equation}
    \boldsymbol{H} = \boldsymbol{X} J,
\label{_auto3} \tag{3}
\end{equation}

where X_{jk} = s_j s_k and J is a matrix which consists of the elements -J_{jk}. This form of writing the energy fits perfectly with the form utilized in linear regression, that is

\begin{equation}
    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon},
\label{_auto4} \tag{4}
\end{equation}

We split the data in training and test data as discussed in the previous example

In [6]:

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)

Linear regression

In the ordinary least squares method we choose the cost function

\begin{equation}
    C(\boldsymbol{X}, \boldsymbol{\beta})= \frac{1}{n}\left\{(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})\right\}.
\label{_auto5} \tag{5}
\end{equation}

We then find the extremal point of C by taking the derivative with respect to \boldsymbol{\beta} as discussed above. This yields the expression for \boldsymbol{\beta} to be

\boldsymbol{\beta} = \frac{\boldsymbol{X}^T \boldsymbol{y}}{\boldsymbol{X}^T \boldsymbol{X}},

which immediately imposes some requirements on \boldsymbol{X} as there must exist an inverse of \boldsymbol{X}^T \boldsymbol{X}. If the expression we are modeling contains an intercept, i.e., a constant term, we must make sure that the first column of \boldsymbol{X} consists of 1. We do this here

In [7]:

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
)

In [8]:

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)

---------------------------------------------------------------------------
LinAlgError                               Traceback (most recent call last)
<ipython-input-8-bc577a44c766> in <module>
      1 def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
      2     return scl.inv(x.T @ x) @ (x.T @ y)
----> 3 beta = ols_inv(X_train_own, y_train)

<ipython-input-8-bc577a44c766> in ols_inv(x, y)
      1 def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:
----> 2     return scl.inv(x.T @ x) @ (x.T @ y)
      3 beta = ols_inv(X_train_own, y_train)

~/anaconda3/lib/python3.6/site-packages/scipy/linalg/basic.py in inv(a, overwrite_a, check_finite)
    977         inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
    978     if info > 0:
--> 979         raise LinAlgError("singular matrix")
    980     if info < 0:
    981         raise ValueError('illegal value in %d-th argument of internal '

LinAlgError: singular matrix

Singular Value decomposition

Doing the inversion directly turns out to be a bad idea since the matrix \boldsymbol{X}^T\boldsymbol{X} is singular. An alternative approach is to use the singular value decomposition. Using the definition of the Moore-Penrose pseudoinverse we can write the equation for \boldsymbol{\beta} as

\boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},

where the pseudoinverse of \boldsymbol{X} is given by

\boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.

Using singular value decomposition we can decompose the matrix \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T, where \boldsymbol{U} and \boldsymbol{V} are orthogonal(unitary) matrices and \boldsymbol{\Sigma} contains the singular values (more details below). where X^{+} = V\Sigma^{+} U^T. This reduces the equation for \omega to

\begin{equation}
    \boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
\label{_auto6} \tag{6}
\end{equation}

Note that solving this equation by actually doing the pseudoinverse (which is what we will do) is not a good idea as this operation scales as \mathcal{O}(n^3), where n is the number of elements in a general matrix. Instead, doing $QR$-factorization and solving the linear system as an equation would reduce this down to \mathcal{O}(n^2) operations.

In [9]:

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

In [10]:

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

In [11]:

J = beta[1:].reshape(L, L)

A way of looking at the coefficients in J is to plot the matrices as images.

In [12]:

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?

The one-dimensional Ising model

Let us bring back the Ising model again, but now with an additional focus on Ridge and Lasso regression as well. We repeat some of the basic parts of the Ising model and the setup of the training and test data. The one-dimensional Ising model with nearest neighbor interaction, no external field and a constant coupling constant J is given by

\begin{equation}
    H = -J \sum_{k}^L s_k s_{k + 1},
\label{_auto7} \tag{7}
\end{equation}

where s_i \in \{-1, 1\} and s_{N + 1} = s_1. The number of spins in the system is determined by L. For the one-dimensional system there is no phase transition.

We will look at a system of L = 40 spins with a coupling constant of J = 1. To get enough training data we will generate 10000 states with their respective energies.

In [13]:

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import make_axes_locatable
import seaborn as sns
import scipy.linalg as scl
from sklearn.model_selection import train_test_split
import sklearn.linear_model as skl
import tqdm
sns.set(color_codes=True)
cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')

L = 40
n = int(1e4)

spins = np.random.choice([-1, 1], size=(n, L))
J = 1.0

energies = np.zeros(n)

for i in range(n):
    energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))

A more general form for the one-dimensional Ising model is

\begin{equation}
    H = - \sum_j^L \sum_k^L s_j s_k J_{jk}.
\label{_auto8} \tag{8}
\end{equation}

Here we allow for interactions beyond the nearest neighbors and a more adaptive coupling matrix. This latter expression can be formulated as a matrix-product on the form

\begin{equation}
    H = X J,
\label{_auto9} \tag{9}
\end{equation}

where X_{jk} = s_j s_k and J is the matrix consisting of the elements -J_{jk}. This form of writing the energy fits perfectly with the form utilized in linear regression, viz.

\begin{equation}
    \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}.
\label{_auto10} \tag{10}
\end{equation}

We organize the data as we did above

In [14]:

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,

In [15]:

clf = skl.LinearRegression().fit(X_train, y_train)

When extracting the $J$-matrix we make sure to remove the intercept

In [16]:

J_sk = clf.coef_.reshape(L, L)

And then we plot the results

In [17]:

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.

Ridge regression

Having explored the ordinary least squares we move on to ridge regression. In ridge regression we include a regularizer. This involves a new cost function which leads to a new estimate for the weights \boldsymbol{\beta}. This results in a penalized regression problem. The cost function is given by

2 2

< < < ! ! M A T H _ B L O C K

In [18]:

_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()

LASSO regression

In the Least Absolute Shrinkage and Selection Operator (LASSO)-method we get a third cost function.

\begin{equation}
    C(\boldsymbol{X}, \boldsymbol{\beta}; \lambda) = (\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y})^T(\boldsymbol{X}\boldsymbol{\beta} - \boldsymbol{y}) + \lambda \sqrt{\boldsymbol{\beta}^T\boldsymbol{\beta}}.
\label{_auto12} \tag{12}
\end{equation}

Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function Lasso from Scikit-Learn.

In [19]:

clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)
J_lasso_sk = clf_lasso.coef_.reshape(L, L)
fig = plt.figure(figsize=(20, 14))
im = plt.imshow(J_lasso_sk, **cmap_args)
plt.title("Lasso from Scikit-learn", fontsize=18)
plt.xticks(fontsize=18)
plt.yticks(fontsize=18)
cb = fig.colorbar(im)
cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)

plt.show()

It is quite striking how LASSO breaks the symmetry of the coupling constant as opposed to ridge and OLS. We get a sparse solution with J_{j, j + 1} = -1.

Performance as function of the regularization parameter

We see how the different models perform for a different set of values for \lambda.

In [20]:

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()

  0%|          | 0/10 [00:00<?, ?it/s]/Users/MortenImac/anaconda3/lib/python3.6/site-packages/sklearn/linear_model/_coordinate_descent.py:529: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations. Duality gap: 3.8501747704274463, tolerance: 1.7070040000000002
  positive)
100%|██████████| 10/10 [00:02<00:00,  4.73it/s]

We see that LASSO reaches a good solution for low values of \lambda, but will "wither" when we increase \lambda too much. Ridge is more stable over a larger range of values for \lambda, but eventually also fades away.

Finding the optimal value of \lambda

To determine which value of \lambda is best we plot the accuracy of the models when predicting the training and the testing set. We expect the accuracy of the training set to be quite good, but if the accuracy of the testing set is much lower this tells us that we might be subject to an overfit model. The ideal scenario is an accuracy on the testing set that is close to the accuracy of the training set.

In [21]:

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.

Friday September 18: Intro to Logistic Regression

Logistic Regression

In linear regression our main interest was centered on learning the coefficients of a functional fit (say a polynomial) in order to be able to predict the response of a continuous variable on some unseen data. The fit to the continuous variable y_i is based on some independent variables \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.

Classification problems

Classification problems, however, are concerned with outcomes taking the form of discrete variables (i.e. categories). We may for example, on the basis of DNA sequencing for a number of patients, like to find out which mutations are important for a certain disease; or based on scans of various patients' brains, figure out if there is a tumor or not; or given a specific physical system, we'd like to identify its state, say whether it is an ordered or disordered system (typical situation in solid state physics); or classify the status of a patient, whether she/he has a stroke or not and many other similar situations.

The most common situation we encounter when we apply logistic regression is that of two possible outcomes, normally denoted as a binary outcome, true or false, positive or negative, success or failure etc.

Optimization and Deep learning

Logistic regression will also serve as our stepping stone towards neural network algorithms and supervised deep learning. For logistic learning, the minimization of the cost function leads to a non-linear equation in the parameters \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.

Basics

We consider the case where the dependent variables, also called the responses or the outcomes, y_i are discrete and only take values from k=0,\dots,K-1 (i.e. K classes).

The goal is to predict the output classes from the design matrix \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}.

Linear classifier

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},
\label{_auto13} \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.

Some selected properties

The main problem with our function is that it takes values on the entire real axis. In the case of logistic regression, however, the labels y_i are discrete variables. A typical example is the credit card data discussed below here, where we can set the state of defaulting the debt to y_i=1 and not to y_i=0 for one the persons in the data set (see the full example below).

One simple way to get a discrete output is to have sign functions that map the output of a linear regressor to values \{0,1\}, f(s_i)=sign(s_i)=1 if s_i\ge 0 and 0 if otherwise. We will encounter this model in our first demonstration of neural networks. Historically it is called the "perceptron" model in the machine learning literature. This model is extremely simple. However, in many cases it is more favorable to use a ``soft" classifier that outputs the probability of a given category. This leads us to the logistic function.

The logistic function

The perceptron 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, it is favorable to have a "soft" classifier that outputs the probability of a given category rather than a single value. For example, given x_i, the classifier outputs the probability of being in a category k. Logistic regression is the most common example of a so-called soft classifier. In logistic regression, the probability that a data point x_i belongs to a category y_i=\{0,1\} is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,

p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}.

Note that 1-p(t)= p(-t).

Examples of likelihood functions used in logistic regression and nueral networks

The following code plots the logistic function, the step function and other functions we will encounter from here and on.

In [22]:

"""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()

Two parameters

We assume now that we have two classes with y_i either 0 or 1. Furthermore we assume also that we have only two parameters \beta in our fitting of the Sigmoid function, that is we define probabilities

\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.

Note that we used

p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}).

Maximum likelihood

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

Warning:

Output truncated. This notebook contains too many cells to display efficiently.