Understanding Logistic Regression

Logistic regression is a statistical model that is used to predict a binary dependent variable based on one or more independent variables. It is a type of generalized linear model that uses the logistic function as the activation function.

The logistic function, also known as the sigmoid function, is a non-linear function that maps any real-valued input to a value between 0 and 1. It is defined as:

f(x) = 1 / (1 + exp(-x))

where x is the input and exp is the exponential function.

The output of the logistic function can be interpreted as the probability of the dependent variable belonging to the positive class (class 1). For example, if the output is 0.7, it means that there is a 70% probability of the dependent variable belonging to class 1.

In logistic regression, the input x is a linear combination of the independent variables and the model parameters (coefficients and intercept). The linear combination is computed as:

x = b0 + b1 * x1 + b2 * x2 + ... + bn * xn

where b0 is the intercept, b1, b2, ..., bn are the coefficients for the independent variables x1, x2, ..., xn, respectively.

The logistic regression model predicts the probability of the dependent variable belonging to class 1 using the following equation:

p(y=1|x) = f(x) = 1 / (1 + exp(-x))

where y is the dependent variable, and x is the linear combination of the independent variables and the model parameters.

The logistic regression model can be trained using a dataset by minimizing the cross-entropy loss between the predicted probabilities and the true labels. The cross-entropy loss is defined as:

Loss = -[y * log(p) + (1 - y) * log(1 - p)]

where y is the true label (0 or 1), p is the predicted probability, and log is the natural logarithm.

I hope this helps to clarify the concept of logistic regression.