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Generalized Linear Models (GLMs) extend linear regression to handle different types of response variables by combining three components:
Random Component – specifies the probability distribution of the response (e.g., Gaussian, Bernoulli, Poisson) from the exponential family.
Systematic Component – a linear predictor, typically written as η = Xᵀβ.
Link Function – a function g(μ) = η that relates the mean response μ = E(Y|X) to the linear predictor.
Exponential Family Form A GLM assumes the density/mass function can be written as:
f(y; θ, φ) = exp{ [yθ - b(θ)] / a(φ) + c(y, φ) }Copied!✕Copyθ: natural parameter
φ: dispersion parameter
b(θ): relates θ to the mean μ
a(φ): scales variance
Common Examples
Gaussian: μ = η, identity link g(μ) = μ
Bernoulli: μ = p, logit link g(μ) = log(p/(1-p))
Poisson: μ > 0, log link g(μ) = log μ
The canonical link sets η = θ, often simplifying estimation.
Estimation via Maximum Likelihood Given samples (xᵢ, yᵢ), the log-likelihood for canonical link is:
At this point, for generalized linear models, you can just think of it as short form for a linear combination of the predictors, = TX. From a broader perspective, we’re aiming to model a transformation of the …
Introduction This short course provides an overview of generalized linear models (GLMs). We shall see that these models extend the linear modelling framework to variables that are not Normally …
Lectures 21-24: Generalized Linear Models - MIT …
Lecture Slides Lectures 21-24: Generalized Linear Models Resource Type: Lecture Notes pdf
Note, in general, most common analyses can be approached from a “modelling” approach. Some such as the log-linear and logistic are topics for this class. Why do we want to “model” data? The structural …
In this lecture we extend the ideas of linear regression to the more general idea of a generalized linear model (GLM).
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The lecture notes are offered in two formats: HTML and PDF. I expect most of you will want to print the notes, in which case you can use the links below to access the PDF file for each chapter.
Slides adapted from lectures by Prof. Yue Jiang. This is not a mathematical statistics class. There are semester-long (and multiple semester-long) courses on these topics, and so what we cover in just …
mation g(·) and focus on ηi = g(μi), where g(·) will be called the link function and ηi the linear predictor. We assume that the transformed mean follows a linear model, ηi = x0 iβ. Since the link function is …
Generalized Linear Models (GLMs) Dr. Guangliang Chen This lecture is based on the following textbook sections:
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