Introduction:
Logistic Regression is a supervised learning algorithm that is used when the target variable is categorical. Hypothetical function h(x) of linear regression predicts unbounded values. But in the case of Logistic Regression, where the target variable is categorical we have to strict the range of predicted values. Consider a classification problem, where we need to classify whether an email is a spam or not. So, the hypothetical function of linear regression could not be used here to predict as it predicts unbound values, but we have to predict either 0 or 1.
To do, so we apply the sigmoid activation function on the hypothetical function of linear regression. So the resultant hypothetical function for logistic regression is given below :
h( x ) = sigmoid( wx + b ) Here, w is the weight vector. x is the feature vector. b is the bias. sigmoid( z ) = 1 / ( 1 + e( - z ) )
Mathematical Intuition:
The cost function of linear regression ( or mean square error ) can’t be used in logistic regression because it is a non-convex function of weights. Optimizing algorithms like i.e gradient descent only converge convex function into a global minimum.
So, the simplified cost function we use :
J = - ylog( h(x) ) - ( 1 - y )log( 1 - h(x) ) here, y is the real target value h( x ) = sigmoid( wx + b ) For y = 0, J = - log( 1 - h(x) ) and y = 1, J = - log( h(x) )
This cost function is because when we train, we need to maximize the probability by minimizing the loss function.
Gradient Descent Calculation:
repeat until convergence {
tmpi = wi - alpha * dwi
wi = tmpi
}
where alpha is the learning rate.
The chain rule is used to calculate the gradients like i.e dw.
Chain rule for dw
here, a = sigmoid( z ) and z = wx + b.
Implementation:
Diabetes Dataset used in this implementation can be downloaded from link.
It has 8 features columns like i.e “Age“, “Glucose” e.t.c, and the target variable “Outcome” for 108 patients. So in this, we will train a Logistic Regression Classifier model to predict the presence of diabetes or not for patients with such information.
# Importing librariesimport numpy as npimport pandas as pdfrom sklearn.model_selection import train_test_splitimport warningswarnings.filterwarnings( "ignore" ) # to compare our model's accuracy with sklearn modelfrom sklearn.linear_model import LogisticRegression# Logistic Regressionclass LogitRegression() : def __init__( self, learning_rate, iterations ) : self.learning_rate = learning_rate self.iterations = iterations # Function for model training def fit( self, X, Y ) : # no_of_training_examples, no_of_features self.m, self.n = X.shape # weight initialization self.W = np.zeros( self.n ) self.b = 0 self.X = X self.Y = Y # gradient descent learning for i in range( self.iterations ) : self.update_weights() return self # Helper function to update weights in gradient descent def update_weights( self ) : A = 1 / ( 1 + np.exp( - ( self.X.dot( self.W ) + self.b ) ) ) # calculate 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 # update weights self.W = self.W - self.learning_rate * dW self.b = self.b - self.learning_rate * db return self # Hypothetical 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 # Driver code def main() : # Importing dataset df = pd.read_csv( "diabetes.csv" ) X = df.iloc[:,:-1].values Y = df.iloc[:,-1:].values # Splitting dataset into train and test set X_train, X_test, Y_train, Y_test = train_test_split( X, Y, test_size = 1/3, random_state = 0 ) # Model training model = LogitRegression( learning_rate = 0.01, iterations = 1000 ) model.fit( X_train, Y_train ) model1 = LogisticRegression() model1.fit( X_train, Y_train) # Prediction on test set Y_pred = model.predict( X_test ) Y_pred1 = model1.predict( X_test ) # measure performance correctly_classified = 0 correctly_classified1 = 0 # counter count = 0 for count in range( np.size( Y_pred ) ) : if Y_test[count] == Y_pred[count] : correctly_classified = correctly_classified + 1 if Y_test[count] == Y_pred1[count] : correctly_classified1 = correctly_classified1 + 1 count = count + 1 print( "Accuracy on test set by our model : ", ( correctly_classified / count ) * 100 ) print( "Accuracy on test set by sklearn model : ", ( correctly_classified1 / count ) * 100 ) if __name__ == "__main__" : main() |
Output :
Accuracy on test set by our model : 58.333333333333336 Accuracy on test set by sklearn model : 61.111111111111114
Note: The above-trained model is to implement the mathematical intuition not just for improving accuracies.
