# Generalized linear models

March 24, 2016 — August 5, 2021

Using the machinery of linear regression to predict in somewhat more general regressions, using least-squares or quasi-likelihood approaches. This means you are still doing something *like* familiar linear regression, but outside the setting of e.g. linear response and possibly homoskedastic Gaussian noise.

## 1 TODO

Discover the magical powers of log-concavity and what they enable.

## 2 Classic linear models

See linear models.

## 3 Generalised linear models

The original extension. Kenneth Tay’s explanation is simple and efficient.

To learn:

- When we can do this? e.g. Must the response be from an exponential family for really real? What happens if not?
- Does anything funky happen with regularisation? what?
- model selection theory

### 3.1 Response distribution

🏗 What constraints do we have here?

### 3.2 Linear Predictor

🏗

### 3.3 Link function

An invertible (monotonic?) function relating the mean of the linear predictor and the mean of the response distribution.

### 3.4 Quaslilikelihood

A generalisation of likelihood of use in some tricky corners of GLMs. (Wedderburn 1976) used it to provide a unified GLM/ML rationale. I don’t yet understand it. Heyde says (Heyde 1997):

Historically there are two principal themes in statistical parameter estimation theory

It is now possible to unify these approaches under the general description of quasi-likelihood and to develop the theory of parameter estimation in a very general setting. […]

It turns out that the theory needs to be developed in terms of estimating functions (functions of both the data and the parameter) rather than the estimators themselves. Thus, our focus will be on functions that have the value of the parameter as a root rather than the parameter itself.

## 4 Hierarchical generalised linear models

GLM + hierarchical model = HGLM.

## 5 Generalised additive models

Generalised generalised linear models. Semiparametric simultaneous discovery of some non-linear predictors and their response curve under the assumption that the interaction is additive in the transformed predictors \[ g(\operatorname{E}(Y))=\beta_0 + f_1(x_1) + f_2(x_2)+ \cdots + f_m(x_m). \]

These have now also been generalised in the obvious way.

## 6 Generalised additive models for location, scale and shape

Folding GARCH and other regression models into GAMs.

GAMLSS is a modern distribution-based approach to (semiparametric) regression models, where all the parameters of the assumed distribution for the response can be modelled as additive functions of the explanatory variables

## 7 Vector generalised additive models

See Yee (2015).

## 8 Vector generalised hierarchical additive models for location, scale and shape

Exercise for the student.

## 9 Generalised estimating equations

🏗

But see Johnny Hong and Kellie Ottoboni. Is this just the quasi-likelihood thing again?

## 10 GGLLM

Generalized² Linear² models (Gordon 2002) unify GLMs with non-linear matrix factorisations.

## 11 References

*IEEE Signal Processing Magazine*.

*arXiv:1708.03395 [Cond-Mat, Physics:math-Ph]*.

*Trends in Ecology & Evolution*.

*arXiv:1609.06764 [Stat]*.

*Journal of the American Statistical Association*.

*Annals of Statistics*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Journal of Time Series Analysis*.

*arXiv:1606.08650 [Stat]*.

*Journal of Statistical Software*.

*Proceedings of the 15th International Conference on Neural Information Processing Systems*. NIPS’02.

*arXiv:1003.0848 [Math, Stat]*.

*Generalized Additive Models*.

*Quasi-likelihood and its application a general approach to optimal parameter estimation*.

*The American Statistician*.

*Generalized linear models with random effects*. Monographs on statistics and applied probability 106.

*Journal of the Royal Statistical Society: Series C (Applied Statistics)*.

*European Journal of Operational Research*.

*Encyclopedia of Statistical Sciences*.

*Journal of the Royal Statistical Society. Series A (General)*.

*International Journal of Financial Engineering and Risk Management*.

*Journal of Statistical Software*.

*Flexible Regression and Smoothing: Using GAMLSS in R*.

*Advances in Neural Information Processing Systems 28*.

*Fisheries Research*, Models in Fisheries Research: GLMs, GAMS and GLMMs,.

*Biometrika*.

*Biometrika*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Statistics*.

*Vector Generalized Linear and Additive Models*. Springer Series in Statistics.

*2007 5th International Symposium on Image and Signal Processing and Analysis*.