Machine Learning with Python-Understanding Logistic Regression

Pre-requisite: Linear Regression
This article discusses the basics of Logistic Regression and its implementation in Python. Logistic regression is basically a supervised classification algorithm. In a classification problem, the target variable(or output), y, can take only discrete values for given set of features(or inputs), X.
We can also say that the target variable is categorical. Based on the number of categories, Logistic regression can be classified as:

binomial: target variable can have only 2 possible types: “0” or “1” which may represent “win” vs “loss”, “pass” vs “fail”, “dead” vs “alive”, etc.
multinomial: target variable can have 3 or more possible types which are not ordered(i.e. types have no quantitative significance) like “disease A” vs “disease B” vs “disease C”.
ordinal: it deals with target variables with ordered categories. For example, a test score can be categorized as:“very poor”, “poor”, “good”, “very good”. Here, each category can be given a score like 0, 1, 2, 3.

First of all, we explore the simplest form of Logistic Regression, i.e Binomial Logistic Regression.

Binomial Logistic Regression

Consider an example dataset which maps the number of hours of study with the result of an exam. The result can take only two values, namely passed(1) or failed(0):

Hours(x)	0.50	0.75	1.00	1.25	1.50	1.75	2.00	2.25	2.50	2.75	3.00	3.25	3.50	3.75	4.00	4.25	4.50	4.75	5.00	5.50
Pass(y)	0	0	0	0	0	0	1	0	1	0	1	0	1	0	1	1	1	1	1	1

So, we have
$y = \left\{\begin{matrix} 0,if fail\\ 1,if pass\\ \end{matrix}\right.$
i.e. y is a categorical target variable which can take only two possible type:“0” or “1”.
In order to generalize our model, we assume that:

The dataset has ‘p’ feature variables and ‘n’ observations.
The feature matrix is represented as:
$\mathbf{X} =\begin{pmatrix} 1 & x_{11} & \cdots & x_{1p} \\ 1 & x_{21} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & x_{n1} & \cdots & x_{np} \end{pmatrix}$
Here, $x_{ij}$ denotes the values of $j^{th}$ feature for $i^{th}$ observation.
Here, we are keeping the convention of letting $x_{i0}$ = 1. (Keep reading, you will understand the logic in a few moments).
The $i^{th}$ observation, $x_i$ , can be represented as:
$x_i = \begin{bmatrix} 1\\ x_{i1}\\ x_{i2}\\ .\\ .\\ x_{ip}\\ \end{bmatrix}$
$h(x_i)$ represents the predicted response for $i^{th}$ observation, i.e. $x_i$ . The formula we use for calculating $h(x_i)$ is called hypothesis.

If you have gone though Linear Regression, you should recall that in Linear Regression, the hypothesis we used for prediction was:
$h(x_i) = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + ..... + \beta_px_{ip}$
where, $\beta_0, \beta_1,…, \beta_p$ are the regression coefficients.
Let regression coefficient matrix/vector, $\beta$ be:
$\beta = \begin{bmatrix} \beta_0\\ \beta_1\\ \beta_2\\ .\\ .\\ \beta_p\\ \end{bmatrix}$
Then, in a more compact form,
$h(x_i) = \beta^Tx_i$

The reason for taking $x_0$ = 1 is pretty clear now.
We needed to do a matrix product, but there was no
actual $x_0$ multiplied to $\beta_0$ in original hypothesis formula. So, we defined $x_0$ = 1.

Now, if we try to apply Linear Regression on above problem, we are likely to get continuous values using the hypothesis we discussed above. Also, it does not make sense for $h(x_i)$ to take values larger that 1 or smaller than 0.
So, some modifications are made to the hypothesis for classification:
$h(x_i) = g(\beta^T x_i) = \frac{1}{1 + e^{-\beta^T x_i}}$
where,
$g(z) = \frac{1}{1 + e^{-z}}$
is called logistic function or the sigmoid function.
Here is a plot showing g(z):

We can infer from above graph that:

g(z) tends towards 1 as $z\rightarrow\infty$
g(z) tends towards 0 as $z\rightarrow-\infty$
g(z) is always bounded between 0 and 1

So, now, we can define conditional probabilities for 2 labels(0 and 1) for $i^{th}$ observation as:
$\newline P(y_i = 1|x_i; \beta) = h(x_i) \newline P(y_i=0|x_i; \beta) = 1 - h(x_i)$
We can write it more compactly as:
$P(y_i|x_i;\beta) = (h(x_i))^{y_i}(1-h(x_i))^{1-y_i}$
Now, we define another term, likelihood of parameters as:
$\newline L(\beta) = \prod_{i=1}^{n}P(y_i|x_i;\beta) \newline or \newline L(\beta) = \prod_{i=1}^{n}(h(x_i))^{y_i}(1-h(x_i))^{1-y_i}$

Likelihood is nothing but the probability of data(training examples), given a model and specific parameter values(here, $\beta$ ). It measures the support provided by the data for each possible value of the $\beta$ . We obtain it by multiplying all $P(y_i|x_i)$ for given $\beta$ .

And for easier calculations, we take log likelihood:
$\newline l(\beta) = log(L(\beta)) \newline or \newline l(\beta) = \sum_{i=1}^{n}y_ilog(h(x_i)) + (1-y_i)log(1-h(x_i))$
The cost function for logistic regression is proportional to inverse of likelihood of parameters. Hence, we can obtain an expression for cost function, J using log likelihood equation as:
$J(\beta) =\sum_{i=1}^{n} - y_ilog(h(x_i)) - (1-y_i)log(1-h(x_i))$
and our aim is to estimate $\beta$ so that cost function is minimized !!

Using Gradient descent algorithm

Firstly, we take partial derivatives of $J(\beta)$ w.r.t each $\beta_j \in \beta$ to derive the stochastic gradient descent rule(we present only the final derived value here):
$\frac{\partial J(\beta)}{\partial \beta_j} = (h(x) - y)x_j$
Here, y and h(x) represent the response vector and predicted response vector(respectively). Also, $x_j$ is the vector representing the observation values for $j^{th}$ feature.
Now, in order to get min $J(\beta)$ ,
$\newline Repeat\{ \newline \beta_j := \beta_j - \alpha\sum_{i=1}^{n}(h(x_i)-y_i)x_{ij} \newline (Simultaneously\hspace{5}update\hspace{5}all\hspace{5}\beta_j) \newline \}$
where $\alpha$ is called learning rate and needs to be set explicitly.
Let us see the python implementation of above technique on a sample dataset (download it from here):

2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.50

import csv 
import numpy as np 
import matplotlib.pyplot as plt 
 
 
def loadCSV(filename): 
    ''' 
    function to load dataset 
    '''
    with open(filename,"r") as csvfile: 
        lines = csv.reader(csvfile) 
        dataset = list(lines) 
        for i in range(len(dataset)): 
            dataset[i] = [float(x) for x in dataset[i]]      
    return np.array(dataset) 
 
 
def normalize(X): 
    ''' 
    function to normalize feature matrix, X 
    '''
    mins = np.min(X, axis = 0) 
    maxs = np.max(X, axis = 0) 
    rng = maxs - mins 
    norm_X = 1 - ((maxs - X)/rng) 
    return norm_X 
 
 
def logistic_func(beta, X): 
    ''' 
    logistic(sigmoid) function 
    '''
    return 1.0/(1 + np.exp(-np.dot(X, beta.T))) 
 
 
def log_gradient(beta, X, y): 
    ''' 
    logistic gradient function 
    '''
    first_calc = logistic_func(beta, X) - y.reshape(X.shape[0], -1) 
    final_calc = np.dot(first_calc.T, X) 
    return final_calc 
 
 
def cost_func(beta, X, y): 
    ''' 
    cost function, J 
    '''
    log_func_v = logistic_func(beta, X) 
    y = np.squeeze(y) 
    step1 = y * np.log(log_func_v) 
    step2 = (1 - y) * np.log(1 - log_func_v) 
    final = -step1 - step2 
    return np.mean(final) 
 
 
def grad_desc(X, y, beta, lr=.01, converge_change=.001): 
    ''' 
    gradient descent function 
    '''
    cost = cost_func(beta, X, y) 
    change_cost = 1
    num_iter = 1
     
    while(change_cost > converge_change): 
        old_cost = cost 
        beta = beta - (lr * log_gradient(beta, X, y)) 
        cost = cost_func(beta, X, y) 
        change_cost = old_cost - cost 
        num_iter += 1
     
    return beta, num_iter  
 
 
def pred_values(beta, X): 
    ''' 
    function to predict labels 
    '''
    pred_prob = logistic_func(beta, X) 
    pred_value = np.where(pred_prob >= .5, 1, 0) 
    return np.squeeze(pred_value) 
 
 
def plot_reg(X, y, beta): 
    ''' 
    function to plot decision boundary 
    '''
    # labelled observations 
    x_0 = X[np.where(y == 0.0)] 
    x_1 = X[np.where(y == 1.0)] 
     
    # plotting points with diff color for diff label 
    plt.scatter([x_0[:, 1]], [x_0[:, 2]], c='b', label='y = 0') 
    plt.scatter([x_1[:, 1]], [x_1[:, 2]], c='r', label='y = 1') 
     
    # plotting decision boundary 
    x1 = np.arange(0, 1, 0.1) 
    x2 = -(beta[0,0] + beta[0,1]*x1)/beta[0,2] 
    plt.plot(x1, x2, c='k', label='reg line') 
 
    plt.xlabel('x1') 
    plt.ylabel('x2') 
    plt.legend() 
    plt.show() 
     
 
     
if __name__ == "__main__": 
    # load the dataset 
    dataset = loadCSV('dataset1.csv') 
     
    # normalizing feature matrix 
    X = normalize(dataset[:, :-1]) 
     
    # stacking columns wth all ones in feature matrix 
    X = np.hstack((np.matrix(np.ones(X.shape[0])).T, X)) 
 
    # response vector 
    y = dataset[:, -1] 
 
    # initial beta values 
    beta = np.matrix(np.zeros(X.shape[1])) 
 
    # beta values after running gradient descent 
    beta, num_iter = grad_desc(X, y, beta) 
 
    # estimated beta values and number of iterations 
    print("Estimated regression coefficients:", beta) 
    print("No. of iterations:", num_iter) 
 
    # predicted labels 
    y_pred = pred_values(beta, X) 
     
    # number of correctly predicted labels 
    print("Correctly predicted labels:", np.sum(y == y_pred)) 
     
    # plotting regression line 
    plot_reg(X, y, beta) 

Estimated regression coefficients: [[  1.70474504  15.04062212 -20.47216021]]
No. of iterations: 2612
Correctly predicted labels: 100

Note: Gradient descent is one of the many way to estimate $\beta$ .
Basically, these are more advanced algorithms which can be easily run in Python once you have defined your cost function and your gradients. These algorithms are:

BFGS(Broyden–Fletcher–Goldfarb–Shanno algorithm)
L-BFGS(Like BFGS but uses limited memory)
Conjugate Gradient

Advantages/disadvantages of using any one of these algorithms over Gradient descent:

Advantages
- Don’t need to pick learning rate
- Often run faster (not always the case)
- Can numerically approximate gradient for you (doesn’t always work out well)
Disadvantages
- More complex
- More of a black box unless you learn the specifics

Multinomial Logistic Regression

In Multinomial Logistic Regression, the output variable can have more than two possible discrete outputs. Consider the Digit Dataset. Here, the output variable is the digit value which can take values out of (0, 12, 3, 4, 5, 6, 7, 8, 9).
Given below is the implementation of Multinomial Logisitc Regression using scikit-learn to make predictions on digit dataset.

from sklearn import datasets, linear_model, metrics 
  
# load the digit dataset 
digits = datasets.load_digits() 
  
# defining feature matrix(X) and response vector(y) 
X = digits.data 
y = digits.target 
 
# splitting X and y into training and testing sets 
from sklearn.model_selection import train_test_split 
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.4, 
                                                    random_state=1) 
  
# create logistic regression object 
reg = linear_model.LogisticRegression() 
  
# train the model using the training sets 
reg.fit(X_train, y_train) 
 
# making predictions on the testing set 
y_pred = reg.predict(X_test) 
  
# comparing actual response values (y_test) with predicted response values (y_pred) 
print("Logistic Regression model accuracy(in %):",  
metrics.accuracy_score(y_test, y_pred)*100) 

Logistic Regression model accuracy(in %): 95.6884561892

At last, here are some points about Logistic regression to ponder upon:

Does NOT assume a linear relationship between the dependent variable and the independent variables, but it does assume linear relationship between the logit of the explanatory variables and the response.
Independent variables can be even the power terms or some other nonlinear transformations of the original independent variables.
The dependent variable does NOT need to be normally distributed, but it typically assumes a distribution from an exponential family (e.g. binomial, Poisson, multinomial, normal,…); binary logistic regression assume binomial distribution of the response.
The homogeneity of variance does NOT need to be satisfied.
Errors need to be independent but NOT normally distributed.
It uses maximum likelihood estimation (MLE) rather than ordinary least squares (OLS) to estimate the parameters, and thus relies on large-sample approximations.

References:

Machine Learning with Python-Understanding Logistic Regression

Machine Learning with Python-Understanding Logistic Regression

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