Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Model selection is a topic of special relevance in molecular phylogenetics that affects many, if not all, stages of phylogenetic inference. Here we discuss some fundamental concepts and techniques of model selection in the context of phylogenetics. We start by reviewing different aspects of the selection of substitution models in phylogenetics from a theoretical, philosophical and practical point of view, and summarize this comparison in table format. We argue that the most commonly implemented model selection approach, the hierarchical likelihood ratio test, is not the optimal strategy for model selection in phylogenetics, and that approaches like the Akaike Information Criterion (AIC) and Bayesian methods offer important advantages. In particular, the latter two methods are able to simultaneously compare multiple nested or nonnested models, assess model selection uncertainty, and allow for the estimation of phylogenies and model parameters using all available models (model-averaged inference or multimodel inference). We also describe how the relative importance of the different parameters included in substitution models can be depicted. To illustrate some of these points, we have applied AIC-based model averaging to 37 mitochondrial DNA sequences from the subgenus Ohomopterus (genus Carabus) ground beetles described by Sota and Vogler (2001). (AIC; Bayes factors; BIC; likelihood ratio tests; model averaging; model uncertainty; model selection; multimodel inference.) It is clear that models of nucleotide substitution (henceforth models of evolution) play a significant role in molecular phylogenetics, particularly in the context of distance, maximum likelihood (ML), and Bayesian es- timation. We know that the use of one or other model affects many, if not all, stages of phylogenetic inference. For example, estimates of phylogeny, substitution rates, bootstrap values, posterior probabilities, or tests of the molecular clock are clearly influenced by the model of evolution used in the analysis (Buckley, 2002; Buckley

Model Selection and Model Averaging in Phylogenetics: Advantages of Akaike Information Criterion and Bayesian Approaches Over Likelihood Ratio Tests

Recently, researchers in several areas of ecology and evolution have begun to change the way in which they analyze data and make biological inferences. Rather than the traditional null hypothesis testing approach, they have adopted an approach called model selection, in which several competing hypotheses are simultaneously confronted with data. Model selection can be used to identify a single best model, thus lending support to one particular hypothesis, or it can be used to make inferences based on weighted support from a complete set of competing models. Model selection is widely accepted and well developed in certain fields, most notably in molecular systematics and mark-recapture analysis. However, it is now gaining support in several other areas, from molecular evolution to landscape ecology. Here, we outline the steps of model selection and highlight several ways that it is now being implemented. By adopting this approach, researchers in ecology and evolution will find a valuable alternative to traditional null hypothesis testing, especially when more than one hypothesis is plausible.

https://faculty.washington.edu/skalski/classes/QERM597/papers/Johnson%20and%20Omland.pdf

Model selection in ecology and evolution

8. Subset Selection in Regression (Monographs on Statistics and Applied Probability, no. 40). By A. J. Miller. ISBN 0 412 35380 6. Chapman and Hall, London, 1990. 240 pp. £25.00.

Subset Selection in Regression

Features of Financial Returns Modelling Price Volatility Forecasting Standard Deviations The Accuracy of Autocorrelation Estimates Testing the Random Walk Hypothesis Forecasting Trends in Prices Evidence Against the Efficiency of Futures Markets Valuing Options Appendix: A Computer Program for Modelling Financial Time Series.

Modelling Financial Time Series

MODEL STRUCTURE, PROPERTIES, AND METHODS Mixture Distributions and Markov Chains Introduction Independent mixture models Markov chains Hidden Markov Models: Definition and Properties A simple hidden Markov model The basics The likelihood Estimation by Direct Maximization of the Likelihood Introduction Scaling the likelihood computation Maximization subject to constraints Other problems Example: earthquakes Standard errors and confidence intervals Example: parametric bootstrap Estimation by the EM Algorithm Forward and backward probabilities The EM algorithm Examples of EM applied to Poisson HMMs Discussion Forecasting, Decoding, and State Prediction Conditional distributions Forecast distributions Decoding State prediction Model Selection and Checking Model selection by AIC and BIC Model checking with pseudo-residuals Examples Discussion Bayesian Inference for Poisson HMMs Applying the Gibbs sampler to Poisson HMMs Bayesian estimation of the number of states Example: earthquakes Discussion Extensions of the Basic Hidden Markov Model Introduction HMMs with general univariate state-dependent distribution HMMs based on a second-order Markov chain HMMs for multivariate series Series which depend on covariates Models with additional dependencies APPLICATIONS Epileptic Seizures Introduction Models fitted Model checking by pseudo-residuals Eruptions of the Old Faithful Geyser Introduction Binary time series of short and long eruptions Normal HMMs for durations and waiting times Bivariate model for durations and waiting times Drosophila Speed and Change of Direction Introduction Von Mises distributions Von Mises HMMs for the two subjects Circular autocorrelation functions Bivariate model Wind Direction at Koeberg Introduction Wind direction as classified into 16 categories Wind direction as a circular variable Models for Financial Series Thinly traded shares Multivariate HMM for returns on four shares Stochastic volatility models Births at Edendale Hospital Introduction Models for the proportion Caesarean Models for the total number of deliveries Conclusion Cape Town Homicides and Suicides Introduction Firearm homicides as a proportion of all homicides, suicides, and legal intervention homicides The number of firearm homicides Firearm homicide and suicide proportions Proportion in each of the five categories Animal-Behavior Model with Feedback Introduction The model Likelihood evaluation Parameter estimation by maximum likelihood Model checking Inferring the underlying state Models for a heterogeneous group of subjects Other modifications or extensions Application to caterpillar feeding behavior Discussion Appendix A: Examples of R code Stationary Poisson HMM, numerical maximization More on Poisson HMMs, including EM Bivariate normal state-dependent distributions Categorical HMM, constrained optimization Appendix B: Some Proofs Factorization needed for forward probabilities Two results for backward probabilities Conditional independence of Xt1 and XTt+1 References Author Index Subject Index Exercises appear at the end of most chapters.

https://www.jstatsoft.org/article/view/v043b04/v43b04.pdf

Hidden Markov Models for Time Series: An Introduction Using R

http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/Context/ModelSelection/Zucchini.pdf

An introduction to model selection

Estimating Animal Abundance

1 Introduction.- 2 Using likelihood for estimation.- 3 Building blocks.- 4 Plot sampling.- 5 Removal, catch-effort and change-in-ratio.- 6 Simple mark-recapture.- 7 Distance sampling.- 8 Nearest neighbour and point-to-nearest-object.- 9 Further building blocks.- 10 Spatial/temporal models with certain detection.- 11 Dealing with heterogeneity.- 12 Integrated models.- 13 Dynamic and open population models.- 14 Which method?.- A Notation and Glossary.- A.1 Notation.- A.2 Glossary.- B Statistical formulation for observation models.- B.1 Detection function.- B.2 Multiple surveys.- C The asymptotic variance of MLEs.- C.1 Estimating the variance of an MLE.- C.2 Estimating the variance of a function of an MLE.- C.3 A one-parameter example.- C.3.1 Fisher information version 1.- C.3.2 Fisher information version 2.- C.3.3 Observed information.- D State models for mark-recapture and removal methods.- D.1 Static population.- D.2 Independent dynamics.- D.3 Markov dynamics.- References.

Walter Zucchini

Papers

Hidden Markov Models for Time Series: An Introduction Using R

An introduction to model selection

Estimating Animal Abundance

Estimating Animal Abundance: Closed Populations

Hidden Markov Models for Time Series: An Introduction Using R, Second Edition