Linear models: logististic regression

CSI 4106 - Fall 2024

Marcel Turcotte

Version: Sep 29, 2024 13:24

Preamble

Quote of Day

“The next season of the #Complexity #podcast, ‘The Nature of Intelligence’, explores this question through conversations with cognitive and neuroscientists, animal cognition researchers, and AI experts in six episodes.””

Learning Objectives

• Differentiate between binary classification and multi-class classification paradigms.
• Describe a methodology for converting multi-class classification problems into binary classification tasks.
• Articulate the concept of decision boundary and its significance in classification tasks.
• Implement a logistic regression algorithm, focusing on its application in classification problems.

Classification tasks

Definitions

• Binary classification is a supervised learning task where the objective is to categorize instances (examples) into one of two discrete classes.

• A multi-class classification task is a type of supervised learning problem where the objective is to categorize instances into one of three or more discrete classes.

Binary classification

• Some machine learning algorithms are specifically designed to solve binary classification problems.
  • Logistic regression and support vector machines (SVMs) are such examples.

Later in the presentation, make sure to understand why the logistic regression is specifically designed to solve binary classification problems.

Multi-class classification

• Any multi-class classification problem can be transformed into a binary classification problem.
• One-vs-All (OvA)
  • A separate binary classifier is trained for each class.
  • For each classifier, one class is treated as the positive class, and all other classes are treated as the negative class.
  • The final assignment of a class label is made based on the classifier that outputs the highest confidence score for a given input.

A complete example will be presented at the end of the lecture.

Discussion

To introduce the concept of decision boundaries, let’s reexamine the Iris dataset.

Loading the Iris dataset

from sklearn.datasets import load_iris

# Load the Iris dataset

iris = load_iris()

import pandas as pd

# Create a DataFrame

df = pd.DataFrame(iris.data, columns=iris.feature_names)
df['species'] = iris.target

Pairwise Scatter Plots of Iris Features

import seaborn as sns
import matplotlib.pyplot as plt

# Using string labels to ease visualization

df['species'] = df['species'].map({0: 'setosa', 1: 'versicolor', 2: 'virginica'})

# Display all pairwise scatter plots

sns.pairplot(df, hue='species', markers=["o", "s", "D"])
plt.suptitle("Pairwise Scatter Plots of Iris Features", y=1.02)
plt.show()

Upon examining the scatter plots, it is evident that distinguishing Setosa from the other two varieties (Versicolor and Virginica) is relatively straightforward. In contrast, separating Versicolor from Virginica presents a greater challenge, as their distributions overlap in all the univariate plots displayed on the diagonal.

Here, we define a One-vs-All classification problem to determine whether an example is Setosa or not. It is important to note that this approach will result in an imbalanced dataset, consisting of 50 instances of Setosa and 100 instances of Not Setosa.

Pairwise Scatter Plots of Iris Features

One-vs-All (OvA) on the Iris dataset

import numpy as np

# Transform the target variable into binary classification
# 'setosa' (class 0) vs. 'not setosa' (classes 1 and 2)

y_binary = np.where(iris.target == 0, 0, 1)

# Create a DataFrame for easier plotting with Seaborn

df = pd.DataFrame(iris.data, columns=iris.feature_names)
df['is_setosa'] = y_binary

print(y_binary)

One-vs-All (OvA) on the Iris dataset

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 1 1]

One-vs-All (OvA) on the Iris dataset

# Using string labels for visualization

df['is_setosa'] = df['is_setosa'].map({0: 'setosa', 1: 'not_setosa'})

# Pairwise scatter plots

sns.pairplot(df, hue='is_setosa', markers=["o", "s"])
plt.suptitle('Pairwise Scatter Plots of Iris Attributes \n(Setosa vs. Not Setosa)', y=1.02)
plt.show()

One-vs-All (OvA) on the Iris dataset

Setosa vs not setosa

• Clearly, we have simplified the problem.
• In the majority of the scatter plots, the setosa examples are clustered together.

Decision boundaries

Definition

A decision boundary is a boundary that partitions the underlying feature space into regions corresponding to different class labels.

The term boundary will be clarified over the next slides.

Decision boundary

Consider two attributes, say petal length and sepal width, the decision boundary can be line.

Decision boundary

Consider two attributes, say petal length and sepal width, the decision boundary can be line.

Definition

We say that the data is linearly separable when two classes of data can be perfectly separated by a single linear boundary, such as a line in two-dimensional space or a hyperplane in higher dimensions.

Simple decision boundary

(a) training data, (b) quadratic curve, and (c) linear function.

Attribution: Geurts, P., Irrthum, A. & Wehenkel, L. Supervised learning with decision tree-based methods in computational and systems biology. Mol Biosyst 5 1593–1605 (2009).

The table on the left presents training data for a hypothetical binary classification task in a medical context, where the two attributes, \(X_1\) and \(X_2\), are used to predict the target variable, \(y\), which can take on two values: sick and healthy. You can imagine that \(X_1\) and \(X_2\) are measurements, such as blood pressure and heart rate or cholesterol and glucose levels.

Logistic regression (c) employs a linear decision boundary. In this specific example, the decision boundary is represented by a straight line. Employing logistic regression for this problem results in several classification errors: red dots above the line, which should be classified as ‘sick’, are incorrectly predicted as ‘healthy’. Conversely, green dots below the line, which should be classified as ‘healthy’, are incorrectly predicted as ‘sick’.

Complex decision boundary

Decision trees are capable of generating irregular and non-linear decision boundaries.

Attribution: ibidem.

Make sure to understand the relationships between the eight decision rules delineated in the decision tree and the nine line segments represented in the scatter plot.

Decision boundary

Digression

Observe that separating the data using only the two attributes, \(x1\) and \(x2\), is infeasible. However, by incorporating a third attribute, \(x3\), the data becomes linearly separable, as can be seen on the previous screen.

Decision boundary

• 2 attributes, the linear decision boundary would be a line.
• 2 attributes, the non-linear decision boundary would be a non-linear curve.
• 3 attributes, the linear decision boundary would be a plane.
• 3 attributes, the non-linear decision boundary would be a non-linear surface.
• \(\gt\) 3 attributes, the linear decision boundary would be a hyperplane.
• \(\gt\) 3 attributes, the non-linear decision boundary would be a hypersurface.

In machine learning, it is common for problems to involve hundreds or even thousands of attributes, rendering direct visualization infeasible.

Specifically, a non-linear decision boundary in high-dimensional space is often conceptualized as a non-linear manifold.

Definition (revised)

A decision boundary is a hypersurface that partitions the underlying feature space into regions corresponding to different class labels.

Logistic regression

Logistic (Logit) Regression

• Despite its name, logistic regression serves as a classification algorithm rather than a regression technique.

• The labels in logistic regression are binary values, denoted as \(y_i \in \{0,1\}\), making it a binary classification task.

• The primary objective of logistic regression is to determine the probability that a given instance \(x_i\) belongs to the positive class, i.e., \(y_i = 1\).

The representation of the two classes, negative and positive, by the values 0 and 1, respectively, is not arbitrary. This choice is intrinsically connected to our objective of determining the probability that an instance \(x_i\) belongs to the positive class.

While this learning algorithm may initially seem unremarkable, it is essential to continue engaging with it, as logistic regression will later prove to be crucial in the discussion on artificial neural networks.

Logistic regression

Consider one attributs, say petal length, and the value of the target (0,1).

Fitting a linear regression

\(\ldots\) is not the answer

The resulting line extends infinitely in both directions, but our goal is to constrain values between 0 and 1. Here, 1 indicates a high probability that \(x_i\) belongs to the positive class, while a value near 0 indicates a low probability.

This model nevertheless presents interesting elements. A high function value corresponds to examples of the positive class (Setosa, 1), while a low function value indicates examples of the negative class (Non-Setosa, 0).

Logistic function

In mathematics, the standard logistic function maps a real-valued input from \(\mathbb{R}\) to the open interval \((0,1)\). The function is defined as:

\[ \sigma(t) = \frac{1}{1+e^{-t}} \]

• When the input variable \(t\) is 0, the output of the logistic function is 0.5.
• As \(t\) increases, the output value approaches 1.
• Conversely, as \(t\) becomes more negative, the output value approaches 0.
• The standard logistic function is also commonly referred to as the sigmoid function.

Attribution: Wikipedia

Logistic regression (intuition)

• When the distance to the decision boundary is zero, uncertainty is high, making a probability of 0.5 appropriate.
• As we move away from the decision boundary, confidence increases, warranting higher or lower probabilities accordingly.

Logistic function

An S-shaped curve, such as the standard logistic function (aka sigmoid), is termed a squashing function because it maps a wide input domain to a constrained output range.

\[ \sigma(t) = \frac{1}{1+e^{-t}} \]

Attribution: Wikipedia

Logistic (Logit) Regression

• Analogous to linear regression, logistic regression computes a weighted sum of the input features, expressed as: \[ \theta_0 + \theta_1 x_i^{(1)} + \theta_2 x_i^{(2)} + \ldots + \theta_D x_i^{(D)} \]

• However, using the sigmoid function limits its output to the range \((0,1)\): \[ \sigma(\theta_0 + \theta_1 x_i^{(1)} + \theta_2 x_i^{(2)} + \ldots + \theta_D x_i^{(D)}) \]

Logistic regression

The Logistic Regression model, in its vectorized form, is defined as:

\[ h_\theta(x_i) = \sigma(\theta x_i) = \frac{1}{1+e^{- \theta x_i}} \]

Logistic regression (two attributes)

\[ h_\theta(x_i) = \sigma(\theta x_i) \]

• In logistic regression, the probability of correctly classifying an example increases as its distance from the decision boundary increases.
• This principle holds for both positive and negative classes.
• An example lying on the decision boundary has a 50% probability of belonging to either class.

Logistic regression

• The Logistic Regression model, in its vectorized form, is defined as:

  \[ h_\theta(x_i) = \sigma(\theta x_i) = \frac{1}{1+e^{- \theta x_i}} \]

• Predictions are made as follows:

  • \(y_i = 0\), if \(h_\theta(x_i) < 0.5\)
  • \(y_i = 1\), if \(h_\theta(x_i) \geq 0.5\)

• The values of \(\theta\) are learned using gradient descent.

“StatQuest breaks down complicated Statistics and Machine Learning methods into small, bite-sized pieces that are easy to understand. StatQuest doesn’t dumb down the material, instead, it builds you up so that you are smarter and have a better understanding of Statistics and Machine Learning.” It is often a good place to start to get the intuition to understand statistical and machine learning concepts.

Attribution: StatQuest: Logistic Regression by Josh Starmer.

One-vs-All

One-vs-All classifier (complete)

from sklearn.model_selection import train_test_split
from sklearn.preprocessing import label_binarize

# Load the Iris dataset
iris = load_iris()
X, y = iris.data, iris.target

# Binarize the output
y_bin = label_binarize(y, classes=[0, 1, 2])

# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y_bin, test_size=0.2, random_state=42)

One-vs-All classifier (complete)

# Train a One-vs-All classifier for each class

classifiers = []
for i in range(3):
    clf = LogisticRegression()
    clf.fit(X_train, y_train[:, i])
    classifiers.append(clf)

Each logistic regression finds a hyperplane in a 4-dimensional space that separates the data into two classes.

One-vs-All classifier (complete)

# Predict on a new sample
new_sample = X_test[0].reshape(1, -1)
confidences = [clf.decision_function(new_sample) for clf in classifiers]

# Final assignment
final_class = np.argmax(confidences)

# Printing the result
print(f"Final class assigned: {iris.target_names[final_class]}")
print(f"True class: {iris.target_names[np.argmax(y_test[0])]}")

Final class assigned: versicolor
True class: versicolor

label_binarized

from sklearn.preprocessing import label_binarize

# Original class labels
y_train = np.array([0, 1, 2, 0, 1, 2, 1, 0])

# Binarize the labels
y_train_binarized = label_binarize(y_train, classes=[0, 1, 2])

# Assume y_train_binarized contains the binarized labels
print("Binarized labels:\n", y_train_binarized)

# Convert binarized labels back to the original numerical values
original_labels = [np.argmax(b) for b in y_train_binarized]
print("Original labels:\n", original_labels)

Binarized labels:
 [[1 0 0]
 [0 1 0]
 [0 0 1]
 [1 0 0]
 [0 1 0]
 [0 0 1]
 [0 1 0]
 [1 0 0]]
Original labels:
 [0, 1, 2, 0, 1, 2, 1, 0]

Digits example

UCI ML hand-written digits datasets

Loading the dataset

from sklearn.datasets import load_digits

digits = load_digits()

What is the type of digits.data

type(digits.data)

numpy.ndarray

Developing a logistic regression model for the recognition of handwritten digits.

UCI ML hand-written digits datasets

How many examples (N) and how many attributes (D)?

digits.data.shape

(1797, 64)

Assigning N and D

N, D = digits.data.shape

target has the same number of entries (examples) as data?

digits.target.shape

(1797,)

UCI ML hand-written digits datasets

What are the width and height of those images?

digits.images.shape

(1797, 8, 8)

Assigning width and height

_, width, height = digits.images.shape

UCI ML hand-written digits datasets

Assigning X and y

X = digits.data
y = digits.target

UCI ML hand-written digits datasets

X[0] is a vector of size width * height = D (\(8 \times 8 = 64\)).

X[0]

array([ 0.,  0.,  5., 13.,  9.,  1.,  0.,  0.,  0.,  0., 13., 15., 10.,
       15.,  5.,  0.,  0.,  3., 15.,  2.,  0., 11.,  8.,  0.,  0.,  4.,
       12.,  0.,  0.,  8.,  8.,  0.,  0.,  5.,  8.,  0.,  0.,  9.,  8.,
        0.,  0.,  4., 11.,  0.,  1., 12.,  7.,  0.,  0.,  2., 14.,  5.,
       10., 12.,  0.,  0.,  0.,  0.,  6., 13., 10.,  0.,  0.,  0.])

It corresponds to an \(8 \times 8 = 64\) image.

X[0].reshape(width, height)

array([[ 0.,  0.,  5., 13.,  9.,  1.,  0.,  0.],
       [ 0.,  0., 13., 15., 10., 15.,  5.,  0.],
       [ 0.,  3., 15.,  2.,  0., 11.,  8.,  0.],
       [ 0.,  4., 12.,  0.,  0.,  8.,  8.,  0.],
       [ 0.,  5.,  8.,  0.,  0.,  9.,  8.,  0.],
       [ 0.,  4., 11.,  0.,  1., 12.,  7.,  0.],
       [ 0.,  2., 14.,  5., 10., 12.,  0.,  0.],
       [ 0.,  0.,  6., 13., 10.,  0.,  0.,  0.]])

UCI ML hand-written digits datasets

Plot the first n=5 examples

import matplotlib.pyplot as plt
import numpy as np

plt.figure(figsize=(10,2))
n = 5

for index, (image, label) in enumerate(zip(X[0:n], y[0:n])):
    plt.subplot(1, n, index + 1)
    plt.imshow(np.reshape(image, (width,width)), cmap=plt.cm.gray)
    plt.title(f'y = {label}')

Intensity values close to 0 are represented in black, while high values are represented in white.

Use cmap=plt.cm.binary to invert the images (intensity values close to 0 in white, high values in black).

UCI ML hand-written digits datasets

• In our dataset, each \(x_i\) is an attribute vector of size \(D = 64\).

• This vector is formed by concatenating the rows of an \(8 \times 8\) image.

• The reshape function is employed to convert this 64-dimensional vector back into its original \(8 \times 8\) image format.

UCI ML hand-written digits datasets

• We will train 10 classifiers, each corresponding to a specific digit in a One-vs-All (OvA) approach.

• Each classifier will determine the optimal values of \(\theta_j\) (associated with the pixel features), allowing it to distinguish one digit from all other digits.

UCI ML hand-written digits datasets

Preparing for our machine learning experiment

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.1)

UCI ML hand-written digits datasets

Optimization algorithms generally work best when the attributes have similar ranges.

from sklearn.preprocessing import StandardScaler

scaler = StandardScaler()
X_train = scaler.fit_transform(X_train)
X_test = scaler.transform(X_test)

Discussion: importance of applying fit_transform only to X_train

UCI ML hand-written digits datasets

from sklearn.linear_model import LogisticRegression

clf = LogisticRegression(multi_class='ovr')
clf = clf.fit(X_train, y_train)

Each logistic regression finds a hyperplane in a 64-dimensional space that separates the data into two classes.

Asking the classifier to solve a multiclass task, using one-vs-rest (OvR), aka OvA. Classfiers in sklearn have multi-learning support built-in.

UCI ML hand-written digits datasets

Applying the classifier to our test set

from sklearn.metrics import classification_report

y_pred = clf.predict(X_test)

print(classification_report(y_test, y_pred))

              precision    recall  f1-score   support

           0       1.00      1.00      1.00        13
           1       0.96      1.00      0.98        23
           2       1.00      0.95      0.97        20
           3       1.00      0.95      0.98        21
           4       1.00      0.96      0.98        23
           5       1.00      0.93      0.96        14
           6       1.00      1.00      1.00        13
           7       1.00      1.00      1.00        18
           8       0.89      1.00      0.94        16
           9       0.95      1.00      0.97        19

    accuracy                           0.98       180
   macro avg       0.98      0.98      0.98       180
weighted avg       0.98      0.98      0.98       180

Visualization

How many classes?

clf.classes_

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])

The coefficients and intercepts are in distinct arrays.

(clf.coef_.shape, clf.intercept_.shape)

((10, 64), (10,))

Intercepts are \(\theta_0\), where as coefficents are \(\theta_j, j \in [1,64]\).

Adapted from MNIST digits classification using Logistic regression in Scikit-Learn, visited 2024-09-23.

Visualization

clf.coef_[0].round(2).reshape(width, height)

array([[ 0.  , -0.18, -0.01,  0.21, -0.04, -0.7 , -0.46, -0.08],
       [-0.  , -0.27, -0.07,  0.43,  0.55,  0.88,  0.03, -0.18],
       [-0.04,  0.33,  0.38, -0.1 , -0.95,  0.9 ,  0.09, -0.06],
       [-0.03,  0.26,  0.13, -0.62, -1.74,  0.12,  0.19, -0.01],
       [ 0.  ,  0.37,  0.52, -0.59, -1.74, -0.07,  0.05,  0.  ],
       [-0.08, -0.15,  0.91, -0.97, -0.75,  0.01,  0.28,  0.03],
       [-0.01, -0.22,  0.45,  0.05,  0.29,  0.05, -0.33, -0.48],
       [ 0.01, -0.27, -0.4 ,  0.47, -0.59, -0.1 , -0.28, -0.26]])

Visualization

coef = clf.coef_
plt.imshow(coef[0].reshape(width,height))

Visualization

plt.figure(figsize=(10,5))

for index in range(len(clf.classes_)):
    plt.subplot(2, 5, index + 1)
    plt.title(f'y = {clf.classes_[index]}')
    plt.imshow(clf.coef_[index].reshape(width,width), 
               cmap=plt.cm.RdBu,
               interpolation='bilinear')

Each LogisticRegression learns 64 \(\theta_i\) parameters.

In the images above, red pixels correspond to negative coefficients, while blue pixels correspond to positive coefficients.

For the first classifier, which predicts the digit ‘0’, the model assigns negative weights to high-intensity pixels in the center of the image and positive weights to high-intensity pixels in the oval region surrounding the center.

Attribution: Machine Learning and Logistic Regression, IBM Technology, 2024-07-19.

Prologue

References

Russell, Stuart, and Peter Norvig. 2020. Artificial Intelligence: A Modern Approach. 4th ed. Pearson. http://aima.cs.berkeley.edu/.

Resources

• Logistic Regression 3-class Classifier from sklearn
• Plot the decision surface of decision trees trained on the iris dataset from sklearn
• Decision trees by Jan Kirenz, a Professor at HdM Stuttgart
• CS 320 Apr12-2021 (Part 2) - Decision Boundaries by Tyler Caraza-Harter, an Instructor at UW-Madison

Next lecture

• Cross evaluation and performance measures

3D plot with points below and above the plane

from mpl_toolkits.mplot3d import Axes3D

# Function to generate points
def generate_points_above_below_plane(num_points=100):
    # Define the plane z = ax + by + c
    a, b, c = 1, 1, 0  # Plane coefficients

    # Generate random points
    x1 = np.random.uniform(-10, 10, num_points)
    x2 = np.random.uniform(-10, 10, num_points)

    y1 = np.random.uniform(-10, 10, num_points)
    y2 = np.random.uniform(-10, 10, num_points)

    # Points above the plane
    z_above = a * x1 + b * y1 + c + np.random.normal(20, 2, num_points)

    # Points below the plane
    z_below = a * x2 + b * y2 + c - np.random.normal(20, 2, num_points)

    # Stack the points into arrays
    points_above = np.vstack((x1, y1, z_above)).T
    points_below = np.vstack((x2, y2, z_below)).T

    return points_above, points_below

# Generate points
points_above, points_below = generate_points_above_below_plane()

# Visualization
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Plot points above the plane
ax.scatter(points_above[:, 0], points_above[:, 1], points_above[:, 2], c='r', label='Above the plane')

# Plot points below the plane
ax.scatter(points_below[:, 0], points_below[:, 1], points_below[:, 2], c='b', label='Below the plane')

# Plot the plane itself for reference
xx, yy = np.meshgrid(range(-10, 11), range(-10, 11))
zz = 1 * xx + 1 * yy + 0
ax.plot_surface(xx, yy, zz, alpha=0.2, color='gray')

# Set labels
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')

# Set title and legend
ax.set_title('3D Points Above and Below a Plane')
ax.legend()

# Show plot
plt.show()

Marcel Turcotte

Marcel.Turcotte@uOttawa.ca

School of Electrical Engineering and Computer Science (EECS)

University of Ottawa

Attribution: “An image of a sad Chihuahua amigurumi with a birthday hat.” Generated by DALL-E, via ChatGPT (GPT-4), OpenAI, September 16, 2024.