Handbook of Mixture Analysis
Mixture models have been around for over 150 years, and they are found in many branches of statistical modelling, as a versatile and multifaceted tool. They can be applied to a wide range of data: univariate or multivariate, continuous or categorical, cross-sectional, time series, networks, and much...
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| Hlavní autori: | , , |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Boca Raton
CRC Press
2019
CRC Press LLC Chapman & Hall |
| Vydanie: | 1 |
| Edícia: | Chapman & Hall/CRC Handbooks of Modern Statistical Methods |
| Predmet: | |
| ISBN: | 9781498763813, 1498763812, 9780367732066, 0367732068 |
| On-line prístup: | Získať plný text |
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Obsah:
- 6.3.2 Posterior simulation for normalized generalized gamma process mixtures -- 6.4 Bayesian Nonparametric Mixtures with Random Partitions -- 6.4.1 Locally weighted mixtures -- 6.4.2 Conditional regression -- 6.5 Repulsive Mixtures (Determinantal Point Process) -- 6.6 Concluding Remarks -- Bibliography -- 7: Model Selection for Mixture Models - Perspectives and Strategies -- 7.1 Introduction -- 7.2 Selecting G as a Density Estimation Problem -- 7.2.1 Testing the order of a finite mixture through likelihood ratio tests -- 7.2.2 Information criteria for order selection -- 7.2.2.1 AIC and BIC -- 7.2.2.2 The Slope Heuristics -- 7.2.2.3 DIC -- 7.2.2.4 The minimum message length -- 7.2.3 Bayesian model choice based on marginal likelihoods -- 7.2.3.1 Chib's method, limitations and extensions -- 7.2.3.2 Sampling-based approximations -- 7.3 Selecting G in the Framework of Model-Based Clustering -- 7.3.1 Mixtures as partition models -- 7.3.2 Classification-based information criteria -- 7.3.2.1 The integrated complete-data likelihood criterion -- 7.3.2.2 The conditional classification likelihood -- 7.3.2.3 Exact derivation of the ICL -- 7.3.3 Bayesian clustering -- 7.3.4 Selecting G under model misspecification -- 7.4 One-Sweep Methods for Cross-model Inference on G -- 7.4.1 Overfitting mixtures -- 7.4.2 Reversible jump MCMC -- 7.4.3 Allocation sampling -- 7.4.4 Bayesian nonparametric methods -- 7.4.5 Sparse finite mixtures for model-based clustering -- 7.5 Concluding Remarks -- Bibliography -- II: Mixture Modelling and Extensions -- 8: Model-Based Clustering -- 8.1 Introduction -- 8.1.1 Heuristic clustering -- 8.1.2 From k-means to Gaussian mixture modelling -- 8.1.3 Specifying the clustering problem -- 8.2 Specifying the Model -- 8.2.1 Components corresponding to clusters -- 8.2.2 Combining components into clusters -- 8.2.3 Selecting the clustering base
- Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- Preface -- Editors -- Contributors -- List of Symbols -- I: Foundations and Methods -- 1: Introduction to Finite Mixtures -- 1.1 Introduction and Motivation -- 1.1.1 Basic formulation -- 1.1.2 Likelihood -- 1.1.3 Latent allocation variables -- 1.1.4 A little history -- 1.2 Generalizations -- 1.2.1 Infinite mixtures -- 1.2.2 Continuous mixtures -- 1.2.3 Finite mixtures with nonparametric components -- 1.2.4 Covariates and mixtures of experts -- 1.2.5 Hidden Markov models -- 1.2.6 Spatial mixtures -- 1.3 Some Technical Concerns -- 1.3.1 Identifiability -- 1.3.2 Label switching -- 1.4 Inference -- 1.4.1 Frequentist inference, and the role of EM -- 1.4.2 Bayesian inference, and the role of MCMC -- 1.4.3 Variable number of components -- 1.4.4 Modes versus components -- 1.4.5 Clustering and classification -- 1.5 Concluding Remarks -- Bibliography -- 2: EM Methods for Finite Mixtures -- 2.1 Introduction -- 2.2 The EM Algorithm -- 2.2.1 Description of EM for finite mixtures -- 2.2.2 EM as an alternating-maximization algorithm -- 2.3 Convergence and Behavior of EM -- 2.4 Cousin Algorithms of EM -- 2.4.1 Stochastic versions of the EM algorithm -- 2.4.2 The Classification EM algorithm -- 2.5 Accelerating the EM Algorithm -- 2.6 Initializing the EM Algorithm -- 2.6.1 Random initialization -- 2.6.2 Hierarchical initialization -- 2.6.3 Recursive initialization -- 2.7 Avoiding Spurious Local Maximizers -- 2.8 Concluding Remarks -- Bibliography -- 3: An Expansive View of EM Algorithms -- 3.1 Introduction -- 3.2 The Product-of-Sums Formulation -- 3.2.1 Iterative algorithms and the ascent property -- 3.2.2 Creating a minorizing surrogate function -- 3.3 Likelihood as a Product of Sums -- 3.4 Non-standard Examples of EM Algorithms -- 3.4.1 Modes of a density -- 3.4.2 Gradient maxima
- 3.4.3 Two-step EM -- 3.5 Stopping Rules for EM Algorithms -- 3.6 Concluding Remarks -- Bibliography -- 4: Bayesian Mixture Models: Theory and Methods -- 4.1 Introduction -- 4.2 Bayesian Mixtures: From Priors to Posteriors -- 4.2.1 Models and representations -- 4.2.2 Impact of the prior distribution -- 4.2.2.1 Conjugate priors -- 4.2.2.2 Improper and non-informative priors -- 4.2.2.3 Data-dependent priors -- 4.2.2.4 Priors for overfitted mixtures -- 4.3 Asymptotic Properties of the Posterior Distribution in the Finite Case -- 4.3.1 Posterior concentration around the marginal density -- 4.3.2 Recovering the parameters in the well-behaved case -- 4.3.3 Boundary parameters: overfitted mixtures -- 4.3.4 Asymptotic behaviour of posterior estimates of the number of components -- 4.4 Concluding Remarks -- Bibliography -- 5: Computational Solutions for Bayesian Inference in Mixture Models -- 5.1 Introduction -- 5.2 Algorithms for Posterior Sampling -- 5.2.1 A computational problem? Which computational problem? -- 5.2.2 Gibbs sampling -- 5.2.3 Metropolis-Hastings schemes -- 5.2.4 Reversible jump MCMC -- 5.2.5 Sequential Monte Carlo -- 5.2.6 Nested sampling -- 5.3 Bayesian Inference in the Model-Based Clustering Context -- 5.4 Simulation Studies -- 5.4.1 Known number of components -- 5.4.2 Unknown number of components -- 5.5 Gibbs Sampling for High-Dimensional Mixtures -- 5.5.1 Determinant coefficient of determination -- 5.5.2 Simulation study using the determinant criterion -- 5.6 Concluding Remarks -- Bibliography -- 6: Bayesian Nonparametric Mixture Models -- 6.1 Introduction -- 6.2 Dirichlet Process Mixtures -- 6.2.1 The Dirichlet process prior -- 6.2.2 Posterior simulation in Dirichlet process mixture models -- 6.2.3 Dependent mixtures - the dependent Dirichlet process model -- 6.3 Normalized Generalized Gamma Process Mixtures -- 6.3.1 NRMI construction
- 10.5 Examples -- 10.5.1 Simulation study -- 10.5.2 Experimental data -- 10.6 Concluding Remarks -- Bibliography -- 11: Mixture Modelling of High-Dimensional Data -- 11.1 Introduction -- 11.2 High-Dimensional Data -- 11.2.1 Continuous data: Italian wine -- 11.2.2 Categorical data: lower back pain -- 11.2.3 Mixed data: prostate cancer -- 11.3 Mixtures for High-Dimensional Data -- 11.3.1 Curse of dimensionality/modeling issues -- 11.3.2 Data reduction -- 11.4 Mixtures for Continuous Data -- 11.4.1 Diagonal covariance -- 11.4.2 Eigendecomposed covariance -- 11.4.3 Mixtures of factor analyzers and probabilistic principal components analyzers -- 11.4.4 High-dimensional models -- 11.4.5 Sparse models -- 11.5 Mixtures for Categorical Data -- 11.5.1 Local independence models and latent class analysis -- 11.5.2 Other models -- 11.6 Mixtures for Mixed Data -- 11.7 Variable Selection -- 11.7.1 Wrapper-based methods -- 11.7.2 Stepwise approaches for continuous data -- 11.7.3 Stepwise approaches for categorical data -- 11.8 Examples -- 11.8.1 Continuous data: Italian wine -- 11.8.2 Categorical data: lower back pain -- 11.8.3 Mixed data: prostate cancer -- 11.9 Concluding Remarks -- Bibliography -- 12: Mixture of Experts Models -- 12.1 Introduction -- 12.2 The Mixture of Experts Framework -- 12.2.1 A mixture of experts model -- 12.2.2 An illustration -- 12.2.3 The suite of mixture of experts models -- 12.3 Statistical Inference for Mixture of Experts Models -- 12.3.1 Maximum likelihood estimation -- 12.3.2 Bayesian estimation -- 12.3.3 Model selection -- 12.4 Illustrative Applications -- 12.4.1 Analysing marijuana use through mixture of experts Markov chain models -- 12.4.2 A mixture of experts model for ranked preference data -- 12.4.3 A mixture of experts latent position cluster model -- 12.4.4 Software -- 12.5 Identifiability of Mixture of Experts Models
- 12.5.1 Identifiability of binomial mixtures
- 8.2.4 Selecting the number of clusters -- 8.3 Post-processing the Fitted Model -- 8.3.1 Identifying the model -- 8.3.2 Determining a partition -- 8.3.3 Characterizing clusters -- 8.3.4 Validating clusters -- 8.3.5 Visualizing cluster solutions -- 8.4 Illustrative Applications -- 8.4.1 Bioinformatics: Analysing gene expression data -- 8.4.2 Marketing: Determining market segments -- 8.4.3 Psychology and sociology: Revealing latent structures -- 8.4.4 Economics and finance: Clustering time series -- 8.4.5 Medicine and biostatistics: Unobserved heterogeneity -- 8.5 Concluding Remarks -- Bibliography -- 9: Mixture Modelling of Discrete Data -- 9.1 Introduction -- 9.2 Mixtures of Univariate Count Data -- 9.2.1 Introduction -- 9.2.2 Finite mixtures of Poisson and related distributions -- 9.2.3 Zero-inflated models -- 9.3 Extensions -- 9.3.1 Mixtures of time series count data -- 9.3.2 Hidden Markov models -- 9.3.3 Mixture of regression models for discrete data -- 9.3.4 Other models -- 9.4 Mixtures of Multivariate Count Data -- 9.4.1 Some models for multivariate counts -- 9.4.1.1 Multivariate reduction approach -- 9.4.1.2 Copulas approach -- 9.4.1.3 Other approaches -- 9.4.2 Finite mixture for multivariate counts -- 9.4.2.1 Conditional independence -- 9.4.2.2 Conditional dependence -- 9.4.2.3 Finite mixtures of multivariate Poisson distributions -- 9.4.3 Zero-inflated multivariate models -- 9.4.4 Copula-based models -- 9.4.5 Finite mixture of bivariate Poisson regression models -- 9.5 Other Mixtures for Discrete Data -- 9.5.1 Latent class models -- 9.5.2 Mixtures for ranking data -- 9.5.3 Mixtures of multinomial distributions -- 9.5.4 Mixtures of Markov chains -- 9.6 Concluding Remarks -- Bibliography -- 10: Continuous Mixtures with Skewness and Heavy Tails -- 10.1 Introduction -- 10.2 Skew-t Mixtures -- 10.3 Prior Formulation -- 10.4 Model Fitting