Dependent mixture models

For a short description of the package see depmixS4. See the vignette for an introduction to hidden Markov models and the package. The response to be modeled; either a formula or a list of formulae in the multivariate case ; this interfaces to the glm and other distributions. See 'Details'. An optional data. A family argument for the response. This must be a list of family 's if the response is multivariate. The number of rows of this data.

A vector specifying the lengths of individual, i. If not specified, the responses are assumed to form a single time series, i. If the data argument has an attribute ntimesthen this is used. The first example in fit uses this argument. The function depmix creates an S4 object of class depmixwhich needs to be fitted using fit to optimize the parameters.

The response model s are by default created by call s to GLMresponse using the formula and the family arguments, the latter specifying the error distribution. See GLMresponse for possible values of the family argument for glm -type responses ie a subset of the glm family options, and the multinomial.

Alternative response distributions are specified by using the makeDepmix function. Its help page has examples of specifying a model with a multivariate normal response, as well as an example of adding a user-defined response model, in this case for the ex-gauss distribution.

Mixture model

If response is a list of formulae, the response 's are assumed to be independent conditional on the latent state. The transitions are modeled as a multinomial logistic model for each state.

Hence, the transition matrix can be modeled using time-varying covariates.

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The prior density is also modeled as a multinomial logistic. Both of these models are created by calls to transInit.

dependent mixture models

Starting values for the initial, transition, and response models may be provided by their respective arguments.

NB: note that the starting values for the initial and transition models as well as of the multinomial logit response models are interpreted as probabilities, and internally converted to multinomial logit parameters.

The order in which parameters must be provided can be easily studied by using the setpars and getpars functions. The print function prints the formulae for the response, transition and prior models along with their parameter values.

A list of a list of response models; the first index runs over states; the second index runs over the independent responses in case a multivariate response is provided. A list of transInit models, ie multinomial logistic models with length the number of states. The total number of parameters of the model.

Note: this is not the degrees of freedom because there are redundancies in the parameters, in particular in the multinomial models for the transitions and prior probabilities. Models are not fitted; the return value of depmix is a model specification without optimized parameter values. Use the fit function to optimize parameters, and to specify additional constraints. Ingmar Visser and Maarten Speekenbrink Journal of Statistical Software, 36 7p.

Lawrence R. Rabiner Why Stata? Supported platforms. Stata Press books Books on Stata Books on statistics. Policy Contact. Bookstore Stata Journal Stata News. Contact us Hours of operation. Advanced search. Populations are often divided into groups or subpopulations—age groups, income brackets, levels of education. Regression models or distributions likely differ across these groups.

But sometimes we don't have a variable that identifies the groups. Perhaps the identifying variable is simply missing. Perhaps it is hard to collect—honest reporting of drug use, sex of goldfish, etc. Perhaps it is inherently unobservable—penchant for risky behavior, high propensity to save money, etc.

Mixed model

In such cases, we can use finite mixture models FMMs to model the probability of belonging to each unobserved group, to estimate distinct parameters of a regression model or distribution in each group, to classify individuals into the groups, and to draw inferences about how each group behaves.

For instance, we might want to model an individual's annual number of doctor visits based on age and medical conditions. However, the model is likely to differ for individuals who are inclined to schedule an appointment at the first sign of a problem compared with those who wait until conditions are more serious. An automobile insurance company might want to classify drivers into risk categories.

Those categories may be high and low risk, or they may be high, medium, and low risk. With FMMs, we can estimate the probability of belonging to a group and fit group-specific models. Let's continue with the insurance company example.

If we are interested in fitting a linear regression model, say. In the above example, y is a continuous outcome. If y were binary—it might stand for having an accident or not having one—we could type. We have fictional data on automobile insurance claims. Our data record the number of accidents drivers had in a year:. We want to model the number of accidents based on age, sex, and whether the individual lives in a metropolitan area.

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We are thinking about fitting the model. We hypothesize, however, that there are two groups of drivers: risky ones and cautious ones. If we are right, the Poisson model would differ across the two groups. We cannot include the driver risk group because risk group is inherently unobservable. There are three parts to the output: 1 results of a model for the unobserved group variable, 2 the Poisson model for accidents in the first group, and 3 the Poisson model for accidents in the second group.

The technical jargon for the two unobserved groups is latent class. That is why the first part of the output shows results for Class1. Classand 2.A mixed modelmixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects.

They are particularly useful in settings where repeated measurements are made on the same statistical units longitudinal studyor where measurements are made on clusters of related statistical units. Because of their advantage in dealing with missing values, mixed effects models are often preferred over more traditional approaches such as repeated measures ANOVA. Ronald Fisher introduced random effects models to study the correlations of trait values between relatives.

Mixed models are applied in many disciplines where multiple correlated measurements are made on each unit of interest. They are prominently used in research involving human and animal subjects in fields ranging from genetics to marketing, and have also been used in baseball [7] and industrial statistics.

dependent mixture models

In matrix notation a linear mixed model can be represented as. This is a consequence of the Gauss—Markov theorem when the conditional variance of the outcome is not scalable to the identity matrix.

When the conditional variance is known, then the inverse variance weighted least squares estimate is BLUE. However, the conditional variance is rarely, if ever, known.

So it is desirable to jointly estimate the variance and weighted parameter estimates when solving MMEs. One method used to fit such mixed models is that of the EM algorithm where the variance components are treated as unobserved nuisance parameters in the joint likelihood. The solution to the mixed model equations is a maximum likelihood estimate when the distribution of the errors is normal. From Wikipedia, the free encyclopedia.

Not to be confused with mixture model. Econometric Analysis of Panel Data Fourth ed.

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New York: Wiley. Transactions of the Royal Society of Edinburgh. Statistical Science. Henderson; Oscar Kempthorne; S. Searle; C. International Biometric Society. Dale Van Vleck. United States National Academy of Sciences. The American Statistician. American Statistical Association. Journal of Animal Science. American Society of Animal Science. Retrieved 17 August Applied Longitudinal Analysis. Categories : Regression models Analysis of variance. Namespaces Article Talk.

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View code.University of Turin and Collegio Carlo Alberto. Lancelot F. Hatjispyros, Spyridon J. Griffiths, R. You can help correct errors and omissions.

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The Dependent Random Measures with Independent Increments in Mixture Models

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dependent mixture models

Dependent mixture models: clustering and borrowing information. A comparison of two recently introduced classes of vectors of dependent nonparametric priors, based on the Dirichlet and the normalized s—stable processes respectively, is provided. These priors are used to define dependent hierarchical mixture models whose distributional properties are investigated. Furthermore, their inferential performance is examined through an extensive simulation study. The models exhibit different features, especially in terms of the clustering behavior and the borrowing of information across studies.

Compared to popular Dirichlet process based models, mixtures of dependent normalized s—stable processes turn out to be a valid choice being capable of more effectively detecting the clustering structure featured by the data.

Handle: RePEc:pav:demwpp:demwp as. Corrections All material on this site has been provided by the respective publishers and authors.Kristoffer S. Berlin, PhD, Natalie A. Williams, PhD, Gilbert R. Latent variable mixture modeling is an emerging person-centered statistical approach that models heterogeneity by classifying individuals into unobserved groupings latent classes with similar more homogenous patterns.

The purpose of this article is to offer a nontechnical introduction to cross-sectional mixture modeling. Latent variable mixture modeling LVMM is a flexible analytic tool that allows researchers to investigate questions about patterns of data and to determine the extent to which identified patterns relate to important variables. Do differential longitudinal trajectories of glycemic control exist among youth with type 1 diabetes Helgeson et al.

Each of these questions is relevant to pediatric psychology and has been explored using LVMM. We begin with a general overview of LVMM to highlight notable strengths of this analytic approach, and then provide step-by-step examples illustrating three prominent types of mixture modeling: Latent class, latent profile, and growth mixture modeling.

The primary goal of LVMM is to identify homogenous subgroups of individuals, with each subgroup possessing a unique set of characteristics that differentiates it from other subgroups.

In LVMM, subgroup membership is not observed and must be inferred from the data. Most broadly, LVMM refers to a collection of statistical approaches in which individuals are classified into unobserved subpopulations or latent classes.

The latent classes are represented by a categorical latent variable. Latent classes are based on these probabilities, and each individual is allowed fractional membership in all classes to reflect the varying degrees of certainty and precision of classification. Consequently, a diverse array of research questions involving latent classes can be investigated.

For example, hypotheses can focus on predicting class membership, identifying mean differences in outcomes across latent classes, or describing the extent to which latent class membership moderates the relationship between two or more variables. Names vary according to the type of data used for indicators continuous vs. Although there are many types of models that can be examined, we begin in Part 1 by focusing on cross-sectional examples using latent class analysis and latent profile analysis.

In Part 2, we focus on longitudinal LVMM and present examples of latent class growth modeling and growth mixture modeling.

dependent mixture models

For both articles, we organize our discussion and examples using the four steps recommended by Ram and Grimm : a problem definition, b model specification, c model estimation, and d model selection and interpretation.When observations are organized into groups where commonalties exist amongst them, the dependent random measures can be an ideal choice for modeling.

One of the propositions of the dependent random measures is that the atoms of the posterior distribution are shared amongst groups, and hence groups can borrow information from each other. When normalized dependent random measures prior with independent increments are applied, we can derive appropriate exchangeable probability partition function EPPFand subsequently also deduce its inference algorithm given any mixture model likelihood.

We provide all necessary derivation and solution to this framework. For demonstration, we used mixture of Gaussians likelihood in combination with a dependent structure constructed by linear combinations of CRMs.

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Our experiments show superior performance when using this framework, where the inferred values including the mixing weights and the number of clusters both respond appropriately to the number of completely random measure used.

Cheng Luo. Richard Yi Da Xu. Yang Xiang. Normalized compound random measures are flexible nonparametric priors fo We present a general construction for dependent random measures based on Fotiet al. We consider predictive inference using a class of temporally dependent D We characterize the unbiasedness of the score function, viewed as an inf This letter deals with a very simple issue: if we have grouped data with In this paper we propose a Bayesian nonparametric model for clustering p Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

Non-parametric statistical methods provide a very flexible framework for survival analysis and mixture models. Different from the classical Bayesian methods, the observations are assumed to be sampled from a probability random measure instead of a fixed probability distribution. One of the popular methods to define a probability random measure is using the normalization of a Completely Random Measure CRM.

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For observations organized in groups, a natural assumption is that some of the observations share the same atoms across groups whereas others do not. Obviously, defining a single random measure across all groups is insufficient in this setting.

Consequently, we should define a vector of dependent random measures. Hence we feel the need to provide a framework of mixture models when the observations are formed in groups, and each group is associated with a random measure.

However, in order for the inference algorithm to be practically implemented, a further assumption is required: For disjoint measurable sets A 1. This assumption has a direct consequence that the increments for the dependent random measures are independent, and hence the name CNRMI was used in [ 6 ].

Followed by the pioneering work of [ 5 ]Dirichlet Process is the first and one of most important stochastic processes introduced to Non-parametric Bayesian community. It is a special case of the class of normalized random measures.

Dirichlet Process Mixture Models and Gibbs Sampling

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