📚 Generalised Linear Models: Core Principles and Mathematical Applications | 广义线性模型核心原理与数学应用
Generalised linear models (GLMs) extend classical linear regression to handle response variables that are not normally distributed. They provide a unified framework for modelling binary, count, and continuous outcomes underpinned by the exponential family of distributions.
广义线性模型（GLM）将经典线性回归推广到能够处理非正态分布的响应变量。它们为基于指数族分布的二分类、计数和连续结果建模提供了统一框架。
1. From Linear Regression to Generalised Linear Models | 从线性回归到广义线性模型
In ordinary linear regression, we model E(Y|X) = Xβ, with Y assumed to follow a Normal distribution and the variance constant across all observations. This restricts our ability to model data that are binary, counts, or positive skew.
在普通线性回归中，我们假设 E(Y|X) = Xβ，且 Y 服从正态分布，方差对所有观测值恒定。这限制了对二分类、计数或正偏斜数据的建模能力。
A GLM generalises this structure in two ways: first, the response can follow any exponential-family distribution; second, a link function links the mean to the linear predictor, g(μ) = Xβ. The linear predictor can still include categorical and continuous covariates.
GLM 从两方面推广该结构：第一，响应变量可以服从任意指数族分布；第二，连接函数将均值与线性预测子连接起来，g(μ) = Xβ。线性预测子仍然可以包含分类和连续协变量。
2. The Exponential Family of Distributions | 指数族分布
Many common distributions belong to the exponential family, including the Normal, Binomial, Poisson, Gamma, and Inverse Gaussian. A distribution belongs to this family if its probability function can be written in the canonical form:
许多常见分布属于指数族，包括正态、二项、泊松、伽马和逆高斯分布。若某个分布的概率函数能写成如下标准形式，则它属于该族：
f(y; θ, φ) = exp( (yθ − b(θ)) / a(φ) + c(y, φ) )
Here θ is the natural parameter, φ is the dispersion parameter, b(θ) is the cumulant function, and a(φ) is typically φ/w. The mean and variance of Y are then E(Y) = b'(θ) and Var(Y) = a(φ) b''(θ).
其中 θ 是自然参数，φ 是离散参数，b(θ) 是累积量函数，a(φ) 通常为 φ/w。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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