AI - Logistic Regression

Logistic Regression is a statistical method used for binary classification problems. It predicts the probability that a given input belongs to a particular class. Unlike linear regression, which predicts continuous values, logistic regression is used to predict categorical outcomes, typically binary outcomes such as 0 or 1, true or false, yes or no.

Key Components

Binary Classification:

• Logistic regression is primarily used for binary classification, where the target variable has two possible outcomes.

Logistic Function (Sigmoid Function):

• The logistic function, also known as the sigmoid function, is used to map predicted values to probabilities. It has the form:

$$\begin{array}{}\text{(1)}& \sigma (z)=\frac{1}{1+e^{-z}}\end{array}$$

• The output of the sigmoid function is always between 0 and 1, making it suitable for probability estimation.

Model Equation:

• The logistic regression model estimates the probability $P(Y=1)$ as:

$$\begin{array}{}\text{(2)}& P(Y=1\mid X)=\sigma ({\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_n{X}_n)\end{array}$$

Here, $({\beta }_0)$ is the intercept, and $({\beta }_1,{\beta }_2,\dots ,{\beta }_n)$ are the coefficients for the predictor variables $(X_1,X_2,\dots ,X_n)$.

Decision Boundary:

• The decision boundary is determined by the threshold value (commonly 0.5). If the predicted probability is greater than or equal to 0.5, the output is classified as 1; otherwise, it is classified as 0.

Gradient Descent:

• Gradient descent is often used to find the optimal parameters (coefficients) that minimize the cost function. The cost function in logistic regression is the log-loss or binary cross-entropy.

Applications

• Medical Diagnosis: Predicting the presence or absence of a disease.
• Spam Detection: Classifying emails as spam or not spam.
• Customer Churn: Predicting whether a customer will stay or leave a service.
• Credit Scoring: Assessing the probability of a customer defaulting on a loan.

Conclusion

Logistic regression is a simple yet powerful tool for binary classification problems. It provides probabilities and a clear decision boundary, making it useful for many real-world applications. Its foundation in statistics and straightforward implementation make it a popular choice for binary classification tasks in machine learning.