Logistic Regression

1. Logistic Model
Consider a model with features x1,x2,x3...xnx_1, x_2, x_3 ... x_nx1​,x2​,x3​...xn​
 . Let the binary output be denoted by yyy
 , that can take the values 0 or 1. Let ppp
 be the probability of y=1y = 1y=1
 , we can denote it as p=P(y=1)p = P(y=1)p=P(y=1)
 . The mathematical relationship between these variables can be denoted as:
ln(p1−p)=θ0+θ1x1+θ2x2+θ3x3ln(\frac{p}{1-p}) = \theta_0+\theta_1x_1+\theta_2x_2+\theta_3x_3\\ln(1−pp​)=θ0​+θ1​x1​+θ2​x2​+θ3​x3​
Here the term p1−p\frac{p}{1−p}1−pp​
 is known as the odds and denotes the likelihood of the event taking place. Thus ln(p1−p)ln(\frac{p}{1−p})ln(1−pp​)
 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 θ1,θ2,θ3,...\theta_1,\theta_2,\theta_3,...θ1​,θ2​,θ3​,...
are parameters (or weights) that we will estimate during training.
It is actually Sigmoid!
​
ln(p1−p)=θ0+θ1x1+θ2x2+θ3x3p1−p=eθ0+θ1x1+θ2x2+θ3x3eθ0+θ1x1+θ2x2+θ3x3−p(eθ0+θ1x1+θ2x2+θ3x3)=pp+p(eθ0+θ1x1+θ2x2+θ3x3)=eθ0+θ1x1+θ2x2+θ3x3p(1+eθ0+θ1x1+θ2x2+θ3x3)=eθ0+θ1x1+θ2x2+θ3x3p=eθ0+θ1x1+θ2x2+θ3x31+eθ0+θ1x1+θ2x2+θ3x3p=eθ0+θ1x1+θ2x2+θ3x3eθ0+θ1x1+θ2x2+θ3x31+eb0+b1x1+b2x2+b3x3eθ0+θ1x1+θ2x2+θ3x3p=11+1eθ0+θ1x1+θ2x2+θ3x3p=11+e−(θ0+θ1x1+θ2x2+θ3x3)S(x)=11+e−xln(\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}}ln(1−pp​)=θ0​+θ1​x1​+θ2​x2​+θ3​x3​1−pp​=eθ0​+θ1​x1​+θ2​x2​+θ3​x3​eθ0​+θ1​x1​+θ2​x2​+θ3​x3​−p(eθ0​+θ1​x1​+θ2​x2​+θ3​x3​)=pp+p(eθ0​+θ1​x1​+θ2​x2​+θ3​x3​)=eθ0​+θ1​x1​+θ2​x2​+θ3​x3​p(1+eθ0​+θ1​x1​+θ2​x2​+θ3​x3​)=eθ0​+θ1​x1​+θ2​x2​+θ3​x3​p=1+eθ0​+θ1​x1​+θ2​x2​+θ3​x3​eθ0​+θ1​x1​+θ2​x2​+θ3​x3​​p=eθ0​+θ1​x1​+θ2​x2​+θ3​x3​1+eb0​+b1​x1​+b2​x2​+b3​x3​​eθ0​+θ1​x1​+θ2​x2​+θ3​x3​eθ0​+θ1​x1​+θ2​x2​+θ3​x3​​​p=1+eθ0​+θ1​x1​+θ2​x2​+θ3​x3​1​1​p=1+e−(θ0​+θ1​x1​+θ2​x2​+θ3​x3​)1​S(x)=1+e−x1​
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 θ1,θ2,θ3,...\theta_1,\theta_2,\theta_3,...θ1​,θ2​,θ3​,...
 that result in least error.
2. Define the Loss Function
A L2 Loss function such as Least Squared Error will do the job.
L=∑i=1n(ytrue−ypredicted)2L = \sum_{i=1}^n (y_{true}-y_{predicted})^2L=i=1∑n​(ytrue​−ypredicted​)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, θ0=0\theta_0=0θ0​=0
 and θ1=0\theta_1=0θ1​=0
 .
2.
Calculate the partial derivative with respect to θ0\theta_0θ0​
 and θ1\theta_1θ1​
 dθ0=−2∑i=1n(yi−yiˉ)×yiˉ×(1−yiˉ)dθ1=−2∑i=1n(yi−yiˉ)×yiˉ×(1−yiˉ)×xid_{\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_idθ0​​=−2∑i=1n​(yi​−yi​ˉ​)×yi​ˉ​×(1−yi​ˉ​)dθ1​​=−2∑i=1n​(yi​−yi​ˉ​)×yi​ˉ​×(1−yi​ˉ​)×xi​
 
3.
Update the weights - values of b0b_0b0​
 and b1b_1b1​
 
θ0=θ0−l×dθ0θ1=θ1−l×dθ1\theta_0 = \theta_0 - l \times d_{\theta_0} \\ \theta_1 = \theta_1 - l \times d_{\theta_1}θ0​=θ0​−l×dθ0​​θ1​=θ1​−l×dθ1​​
 
Python Implementation
1
# Importing libraries
2
import numpy as np
3
import pandas as pd
4
from sklearn.model_selection import train_test_split
5
from math import exp
6
​
7
# Preparing the dataset
8
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]})
9
# Divide the data to training set and test set
10
X_train, X_test, y_train, y_test = train_test_split(data['feature'], data['label'], test_size=0.30)
11
​
12
## Logistic Regression Model
13
# Helper function to normalize data
14
def normalize(X):
15
    return X - X.mean()
16
​
17
# Method to make predictions
18
def predict(X, theta0, theta1):
19
    # Here the predict function is: 1/(1+e^(-x))
20
    return np.array([1 / (1 + exp(-(theta0 + theta1*x))) for x in X])
21
​
22
# Method to train the model
23
def logistic_regression(X, Y):
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    # Normalizing the data
25
    X = normalize(X)
26
​
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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
32
​
33
    # Training iteration
34
    for epoch in range(epochs):
35
        y_pred = predict(X, theta0, theta1)
36
​
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        ## Here the loss function is: sum(y-y_pred)^2 a.k.a least squared error (LSE)
38
        # Derivative of loss w.r.t. theta0
39
        theta0_d = -2 * sum((Y - y_pred) * y_pred * (1 - y_pred))
40
        # Derivative of loss w.r.t. theta1
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        theta1_d = -2 * sum(X * (Y - y_pred) * y_pred * (1 - y_pred))
42
​
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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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​
48
# Training the model
49
theta0, theta1 = logistic_regression(X_train, y_train)   
50
​
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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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​
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# Evaluating the model
57
print(list(y_test))
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print(y_pred)
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Contents

1. Logistic Model

2. Define the Loss Function

3. Utilize the Gradient Descent Algorithm

Python Implementation