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Table of contents
1.
Introduction
2.
Vectorizing Logistic Regression
3.
Vectorizing Logistic Regression’s Gradient Computation
4.
Implementation
4.1.
Simple logistic regression
4.2.
Vectorized code for logistic regression
5.
Frequently Asked Questions
6.
Key Takeaways
Last Updated: Mar 27, 2024

Vectorizing Logistic Regression

Author soham Medewar
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Master Python: Predicting weather forecasts
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Ashwin Goyal
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Introduction

Before moving to the vectorized form of logistic regression, let us briefly discuss logistic regression. Logistic regression is the supervised machine learning algorithm that is used for both classification and regression purposes. The output of the logistic regression is the probability of the target value. Logistic regression can be used in many classifications like predicting malignant and benign tumor of a breast cancer patient, spam email prediction.

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Vectorizing Logistic Regression

Let us consider a training set having M training examples. To train a dataset having M training examples every time we need to go through the four propagation steps for each training example in the dataset.

Let x(i) be the first training example.

Here w and x will be of size (n by 1)

We will calculate the following steps for the dataset.

Apart from using an explicit loop for M training examples, we can do all the calculations using a single operation.

Let us declare X as M training inputs. The size of matrix X will be (Nby M). W is (Nby 1) matrix, where W represents weights of the logistic regression model. Nx is the number of features.

Now we will declare another matrix Z where we will calculate z(i) for all the training examples in the dataset.

Z consists of all the z(i). The size of Z will be of (1,m) numpy array.

In numpy, we can calculate the value of Z using a single operation, as shown below.

Now we will calculate activation for each z(i) using the sigmoid activation function. Storing the results of all the activation of Z in A matrix as shown below.

Here a(i) stores activation for the respective z(i).

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Vectorizing Logistic Regression’s Gradient Computation

Till now, we have seen the forward propagation of the logistic regression. Calculating activation of every training example and storing it in (1,m) numpy matrix simultaneously.

In this section, we will see the vectorized implementation of logistic regression, i.e., gradient computation for logistic regression.

So, dz(1) for the first training example will be a(i) - y(i), for the second training example, dz(2) will be a(2) - y(2), similarly for mth training example, dz(m) will be a(m) - y(m).

Let us consider a row matrix dZ of (1 by m) that will stores dz of all the training examples. dZ can be simplified as shown below.

After calculating dZ we need to calculate dw and db. The normal way to calculate is shown below.

In the above process of calculating “dw” and “db”, we need to iterate over m training examples.

As shown below, we can use the vectorization method to avoid this iteration.

Now we will update w and b as shown below.

Here 𝛂 is the learning rate in the gradient descent algorithm. If you want thousands of iterations, simply run a loop a thousand times for the above process to minimize the error and adjust the weights.

Above are the steps for vectorizing logistic regression.

Implementation

Simple logistic regression

Let us see the Standard and optimized technique of logistic regression.

source

The above code shows the standard way of implementing logistic regression.

Vectorized code for logistic regression

#logistic regression
class LogitRegression() :
    def __init__( self, learning_rate, iterations ) :        
        self.learning_rate = learning_rate        
        self.iterations = iterations
          
    # model training function   
    def fit( self, X, Y ) :        
        # m = number of training examples
        # n = number of features
        self.m, self.n = X.shape        
        # weights of the model       
        self.W = np.zeros( self.n )
        self.b = 0        
        self.X = X
        self.Y = Y
          
        # gradient descent
                  
        for i in range( self.iterations ) :            
            self.update_weights()            
        return self
      
    # updating weights
      
    def update_weights( self ) :           
        A = 1 / ( 1 + np.exp( - ( self.X.dot( self.W ) + self.b ) ) )
          
        # calculating gradients
        tmp = ( A - self.Y.T )
        tmp = np.reshape( tmp, self.m )  
        dW = np.dot( self.X.T, tmp ) / self.m
        db = np.sum( tmp ) / self.m 
          
        # updating weights
        self.W = self.W - self.learning_rate * dW    
        self.b = self.b - self.learning_rate * db
          
        return self
      
    # hypothesis function h(x)
    def predict( self, X ) :    
        Z = 1 / ( 1 + np.exp( - ( X.dot( self.W ) + self.b ) ) )        
        Y = np.where( Z > 0.5, 1, 0 )        
        return Y

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Frequently Asked Questions

  1. Why vectorize the logistic regression?
    Logistic regression in vectorized form takes less time. It does not have any loop for iterating over the dataset. It is easy to implement.
     
  2. Is logistic regression a generative or a descriptive classifier? Why?
    Logistic regression is a descriptive model. Logistic regression learns to classify by knowing what features differentiate two or more classes of objects.
     
  3. According to you, is the method to fit the data in logistic regression best?
    Maximum Likelihood Estimation to obtain the model coefficients related to the predictors and target.

Key Takeaways

  • In this article, we have learned vectorized form of logistic regression.
  • Implementation of logistic regression in vectorized form.


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