logistic regression编辑历史

# Logistic Regression

## 1 Introduction

### 1.1 Definition and purpose

Logistic regression is a statistical modeling technique used to estimate the probability of a binary outcome (e.g., success/failure, yes/no) based on one or more predictor variables. Its purpose is to model the relationship between predictors and a dichotomous dependent variable, providing interpretable measures of association and allowing classification into one of two categories.

### 1.2 Historical development

Logistic regression originated in the mid‑20th century, with early work by statisticians such as David Cox (1958) and developments in biostatistics for analyzing dose‑response relationships. The method was later popularized in epidemiology and social sciences through the work of researchers like Joseph Berkson and John Nelder. In the 1990s and 2000s, logistic regression became a cornerstone of machine learning for binary classification tasks.

### 1.3 Relationship to linear regression

Unlike linear regression, which predicts a continuous response, logistic regression predicts the probability of an event using the logistic (sigmoid) function. This ensures predicted probabilities are bounded between 0 and 1. Linear regression assumes a linear relationship between predictors and the outcome; logistic regression assumes a linear relationship between predictors and the log‑odds (logit) of the outcome.

## 2 Mathematical formulation

### 2.1 Logistic function (sigmoid)

The logistic function maps any real‑valued number to a value in (0, 1):
\[
\pi(x) = \frac{e^{x}}{1+e^{x}} = \frac{1}{1+e^{-x}}.
\]
In logistic regression, the argument is the linear combination of predictors.

### 2.2 Logit transformation

The logit (log‑odds) is the inverse of the logistic function. For a probability \(p\), the logit is given by
\[
\operatorname{logit}(p) = \ln\left(\frac{p}{1-p}\right).
\]

#### 2.2.1 Odds and log‑odds

Odds are defined as \(p/(1-p)\), ranging from 0 to ∞. The log‑odds