Logistic Regression

1. Logistic Model

Consider a model with features
$x_1, x_2, x_3 ... x_n$
. Let the binary output be denoted by
$y$
, that can take the values 0 or 1. Let
$p$
be the probability of
$y = 1$
, we can denote it as
$p = P(y=1)$
. The mathematical relationship between these variables can be denoted as:
$ln(\frac{p}{1-p}) = \theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3\\$
Here the term
$\frac{p}{1−p}$
is known as the odds and denotes the likelihood of the event taking place. Thus
$ln(\frac{p}{1−p})$
is known as the log odds and is simply used to map the probability that lies between 0 and 1 to a range between (−∞, +∞). The terms
$\theta_1,\theta_2,\theta_3,...$
are parameters (or weights) that we will estimate during training.

It is actually Sigmoid!

$ln(\frac{p}{1-p}) = \theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3\\ \frac{p}{1-p} = e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}\\ e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3} - p(e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}) = p\\ p + p(e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}) = e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}\\ p(1 + e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}) = e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}\\ p = \frac {e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}}{1 + e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}}\\ p = \frac {\frac{e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}}{e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}}}{\frac{1 + e^{b_0+b_1x_1+b_2x_2+b_3x_3}}{e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}}}\\ p = \frac {1}{1+\frac{1}{e^{\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3}}}\\ p = \frac {1}{1+e^{-(\theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3)}}\\ S(x)=\frac{1}{1+e^{-x}}$
Now we will be using the above equation to make our predictions. Before that we will train our model to obtain the values of our parameters
$\theta_1,\theta_2,\theta_3,...$
that result in least error.

2. Define the Loss Function

A L2 Loss function such as Least Squared Error will do the job.
$L = \sum_{i=1}^n (y_{true}-y_{predicted})^2$

3. Utilize the Gradient Descent Algorithm

You might know that the partial derivative of a function at its minimum value is equal to 0. So gradient descent basically uses this concept to estimate the parameters or weights of our model by minimizing the loss function.
1.
Initialize the weights,
$\theta_0=0$
and
$\theta_1=0$
.
2.
Calculate the partial derivative with respect to
$\theta_0$
and
$\theta_1$
$d_{\theta_0} = -2 \sum_{i=1}^n(y_i - \bar{y_i}) \times \bar{y_i} \times (1 - \bar{y_i})\\ d_{\theta_1} = -2 \sum_{i=1}^n(y_i - \bar{y_i}) \times \bar{y_i} \times (1 - \bar{y_i}) \times x_i$
3.
Update the weights - values of
$b_0$
and
$b_1$
$\theta_0 = \theta_0 - l \times d_{\theta_0} \\ \theta_1 = \theta_1 - l \times d_{\theta_1}$

Python Implementation

1
# Importing libraries
2
import numpy as np
3
import pandas as pd
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from sklearn.model_selection import train_test_split
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from math import exp
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# Preparing the dataset
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data = pd.DataFrame({'feature' : [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], 'label' : [0,0,0,0,0,0,0,1,1,1,1,1,1,1,1]})
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# Divide the data to training set and test set
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X_train, X_test, y_train, y_test = train_test_split(data['feature'], data['label'], test_size=0.30)
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## Logistic Regression Model
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# Helper function to normalize data
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def normalize(X):
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return X - X.mean()
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# Method to make predictions
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def predict(X, theta0, theta1):
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# Here the predict function is: 1/(1+e^(-x))
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return np.array([1 / (1 + exp(-(theta0 + theta1*x))) for x in X])
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# Method to train the model
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def logistic_regression(X, Y):
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# Normalizing the data
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X = normalize(X)
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# Initializing variables
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theta0 = 0
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theta1 = 0
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learning_rate = 0.001
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epochs = 300
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# Training iteration
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for epoch in range(epochs):
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y_pred = predict(X, theta0, theta1)
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## Here the loss function is: sum(y-y_pred)^2 a.k.a least squared error (LSE)
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# Derivative of loss w.r.t. theta0
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theta0_d = -2 * sum((Y - y_pred) * y_pred * (1 - y_pred))
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# Derivative of loss w.r.t. theta1
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theta1_d = -2 * sum(X * (Y - y_pred) * y_pred * (1 - y_pred))
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theta0 = theta0 - learning_rate * theta0_d
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theta1 = theta1 - learning_rate * theta1_d
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return theta0, theta1
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# Training the model
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theta0, theta1 = logistic_regression(X_train, y_train)
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# Making predictions
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X_test_norm = normalize(X_test)
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y_pred = predict(X_test_norm, theta0, theta1)
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y_pred = [1 if p >= 0.5 else 0 for p in y_pred]
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# Evaluating the model
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print(list(y_test))
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print(y_pred)
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