Type:	Package
Title:	Fitting Shared Atoms Nested Models via Markov Chains Monte Carlo
Version:	0.1.1
Date:	2024-06-02
Maintainer:	Francesco Denti <francescodenti.personal@gmail.com>
URL:	https://github.com/laura-dangelo/SANple
BugReports:	https://github.com/laura-dangelo/SANple/issues
Description:	Estimate Bayesian nested mixture models via Markov Chain Monte Carlo methods. Specifically, the package implements the common atoms model (Denti et al., 2023), its finite version (D'Angelo et al., 2023), and a hybrid finite-infinite model. All models use Gaussian mixtures with a normal-inverse-gamma prior distribution on the parameters. Additional functions are provided to help analyzing the results of the fitting procedure. References: Denti, Camerlenghi, Guindani, Mira (2023) <doi:10.1080/01621459.2021.1933499>, D’Angelo, Canale, Yu, Guindani (2023) <doi:10.1111/biom.13626>.
License:	MIT + file LICENSE
Imports:	Rcpp, salso
Depends:	scales, RColorBrewer
LinkingTo:	Rcpp, RcppArmadillo, RcppProgress
RoxygenNote:	7.3.1
Encoding:	UTF-8
NeedsCompilation:	yes
Packaged:	2024-06-02 12:09:35 UTC; francescodenti
Author:	Francesco Denti [aut, cre], Laura D'Angelo [aut, cph]
Repository:	CRAN
Date/Publication:	2024-06-02 13:00:02 UTC

SANple: Fitting Shared Atoms Nested Models via Markov Chains Monte Carlo

Description

Estimate Bayesian nested mixture models via Markov Chain Monte Carlo methods. Specifically, the package implements the common atoms model (Denti et al., 2023), its finite version (D'Angelo et al., 2023), and a hybrid finite-infinite model. All models use Gaussian mixtures with a normal-inverse-gamma prior distribution on the parameters. Additional functions are provided to help analyzing the results of the fitting procedure. References: Denti, Camerlenghi, Guindani, Mira (2023) doi:10.1080/01621459.2021.1933499, D’Angelo, Canale, Yu, Guindani (2023) doi:10.1111/biom.13626.

Author(s)

Maintainer: Francesco Denti francescodenti.personal@gmail.com (ORCID)

Authors:

• Laura D'Angelo laura.dangelo@live.com (ORCID) [copyright holder]

See Also

Useful links:

• https://github.com/laura-dangelo/SANple

• Report bugs at https://github.com/laura-dangelo/SANple/issues

Relabel clusters

Description

Relabel clusters

Usage

.relabell(ix)

Arguments

Estimate observational and distributional clusters

Description

Given the MCMC output, estimate the observational and distributional partitions using salso::salso().

Usage

estimate_clusters(object, burnin = 0, ncores = 0)

Arguments

Value

Object of class SANclusters. The object contains:

est_oc estimated partition at the observational level. It is an object of class salso.estimate.

est_dc estimated partition at the distributional level. It is an object of class salso.estimate.

clus_means cluster-specific sample means of the estimated partition.

clus_vars cluster-specific sample variances of the estimated partition.

See Also

salso::salso(), print.SANmcmc, plot.SANmcmc, print.SANclusters

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
out <- sample_fiSAN(nrep = 500, burn = 200,
                     y = y, group = g, 
                    nclus_start = 2,
                    maxK = 20, maxL = 20,
                    beta = 1)
estimate_clusters(out)

Plotting MCMC output

Description

Plot method for objects of class SANmcmc. The function displays two graphs, meant to analyze the estimated distributional and observational clusters.

Usage

## S3 method for class 'SANmcmc'
plot(
  x,
  type = c("boxplot", "ecdf", "scatter"),
  estimated_clusters = NULL,
  burnin = 0,
  palette_brewed = FALSE,
  ncores = 1,
  ...
)

Arguments

Value

The function plots a summary of the fitted model.

See Also

print.SANmcmc, estimate_clusters

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
out <- sample_fiSAN(nrep = 500, burn = 200, 
                    y = y, group = g, 
                    nclus_start = 2,
                    maxK = 20, maxL = 20,
                    beta = 1)
plot(out, type = "ecdf", palette_brewed = TRUE)

Print cluster summary

Description

Print the cluster-specific sample means and variances of the estimated observational and distributional partition.

Usage

## S3 method for class 'SANclusters'
print(x, ...)

Arguments

Value

The function prints a summary of the estimated clusters.

Print MCMC output

Description

Print method for objects of class SANmcmc.

Usage

## S3 method for class 'SANmcmc'
print(x, ...)

Arguments

Value

The function prints a summary of the fitted model.

See Also

estimate_clusters, plot.SANmcmc

Sample CAM

Description

sample_CAM is used to perform posterior inference under the common atoms model (CAM) of Denti et al. (2023) with Gaussian likelihood. The model uses Dirichlet process mixtures (DPM) at both the observational and distributional levels. The implemented algorithm is based on the nested slice sampler of Denti et al. (2023), based on the algorithm of Kalli, Griffin and Walker (2011).

Usage

sample_CAM(nrep, burn, y, group, 
           maxK = 50, maxL = 50, 
           m0 = 0, tau0 = 0.1, lambda0 = 3, gamma0 = 2,
           hyp_alpha1 = 1, hyp_alpha2 = 1,
           hyp_beta1 = 1, hyp_beta2 = 1,
           alpha = NULL, beta = NULL,
           warmstart = TRUE, nclus_start = NULL,
           mu_start = NULL, sigma2_start = NULL,
           M_start = NULL, S_start = NULL,
           alpha_start = NULL, beta_start = NULL,
           progress = TRUE, seed = NULL)

Arguments

Details

Data structure

The common atoms mixture model is used to perform inference in nested settings, where the data are organized into J groups. The data should be continuous observations (Y_1,\dots,Y_J), where each Y_j = (y_{1,j},\dots,y_{n_j,j}) contains the n_j observations from group j, for j=1,\dots,J. The function takes as input the data as a numeric vector y in this concatenated form. Hence y should be a vector of length n_1+\dots+n_J. The group parameter is a numeric vector of the same size as y indicating the group membership for each individual observation. Notice that with this specification the observations in the same group need not be contiguous as long as the correspondence between the variables y and group is maintained.

Model

The data are modeled using a univariate Gaussian likelihood, where both the mean and the variance are observational-cluster-specific, i.e.,

y_{i,j}\mid M_{i,j} = l \sim N(\mu_l,\sigma^2_l)

where M_{i,j} \in \{1,2,\dots\} is the observational cluster indicator of observation i in group j. The prior on the model parameters is a Normal-Inverse-Gamma distribution (\mu_l,\sigma^2_l)\sim NIG (m_0,\tau_0,\lambda_0,\gamma_0), i.e., \mu_l\mid\sigma^2_l \sim N(m_0, \sigma^2_l / \tau_0), 1/\sigma^2_l \sim Gamma(\lambda_0, \gamma_0) (shape, rate).

Clustering

The model performs a clustering of both observations and groups. The clustering of groups (distributional clustering) is provided by the allocation variables S_j \in \{1,2,\dots\}, with

Pr(S_j = k \mid \dots ) = \pi_k \qquad \text{for } \: k = 1,2,\dots

The distribution of the probabilities is \{\pi_k\}_{k=1}^{\infty} \sim GEM(\alpha), where GEM is the Griffiths-Engen-McCloskey distribution of parameter \alpha, which characterizes the stick-breaking construction of the DP (Sethuraman, 1994).

The clustering of observations (observational clustering) is provided by the allocation variables M_{i,j} \in \{1,2,\dots\}, with

Pr(M_{i,j} = l \mid S_j = k, \dots ) = \omega_{l,k} \qquad \text{for } \: k = 1,2,\dots \, ; \: l = 1,2,\dots

The distribution of the probabilities is \{\omega_{l,k}\}_{l=1}^{\infty} \sim GEM(\beta) for all k = 1,2,\dots

Value

sample_CAM returns four objects:

• model: name of the fitted model.

• params: list containing the data and the parameters used in the simulation. Details below.

• sim: list containing the simulated values (MCMC chains). Details below.

• time: total computation time.

Data and parameters: params is a list with the following components:

nrep

Number of MCMC iterations.

y, group

Data and group vectors.

maxK, maxL

Maximum number of distributional and observational clusters.

m0, tau0, lambda0, gamma0

Model hyperparameters.

(hyp_alpha1,hyp_alpha2) or alpha

Either the hyperparameters on \alpha (if \alpha random), or the value for \alpha (if fixed).

(hyp_beta1,hyp_beta2) or beta

Either the hyperparameters on \beta (if \beta random), or the value for \beta (if fixed).

Simulated values: sim is a list with the following components:

mu

Matrix of size (nrep, maxL). Each row is a posterior sample of the mean parameter for each observational cluster (\mu_1,\dots\mu_L).

sigma2

Matrix of size (nrep, maxL). Each row is a posterior sample of the variance parameter for each observational cluster (\sigma^2_1,\dots\sigma^2_L).

obs_cluster

Matrix of size (nrep, n), with n = length(y). Each row is a posterior sample of the observational cluster allocation variables (M_{1,1},\dots,M_{n_J,J}).

distr_cluster

Matrix of size (nrep, J), with J = length(unique(group)). Each row is a posterior sample of the distributional cluster allocation variables (S_1,\dots,S_J).

pi

Matrix of size (nrep, maxK). Each row is a posterior sample of the distributional cluster probabilities (\pi_1,\dots,\pi_{maxK}).

omega

3-d array of size (maxL, maxK, nrep). Each slice is a posterior sample of the observational cluster probabilities. In each slice, each column k is a vector (of length maxL) observational cluster probabilities (\omega_{1,k},\dots,\omega_{maxL,k}) for distributional cluster k.

alpha

Vector of length nrep of posterior samples of the parameter \alpha.

beta

Vector of length nrep of posterior samples of the parameter \beta.

maxK

Vector of length nrep of the number of distributional DP components used by the slice sampler.

maxL

Vector of length nrep of the number of observational DP components used by the slice sampler.

References

Denti, F., Camerlenghi, F., Guindani, M., and Mira, A. (2023). A Common Atoms Model for the Bayesian Nonparametric Analysis of Nested Data. Journal of the American Statistical Association, 118(541), 405-416. <doi:10.1080/01621459.2021.1933499>

Kalli, M., Griffin, J.E., and Walker, S.G. (2011). Slice Sampling Mixture Models, Statistics and Computing, 21, 93–105. <doi:10.1007/s11222-009-9150-y>

Sethuraman, A.J. (1994). A Constructive Definition of Dirichlet Priors, Statistica Sinica, 4, 639–650.

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
plot(density(y[g==1]), xlim = c(-5,10))
lines(density(y[g==2]), col = 2)
out <- sample_CAM(nrep = 500, burn = 200, y = y, group = g,
                  nclus_start = 2,
                  maxL = 20, maxK = 20)
out 

Sample fSAN

Description

sample_fSAN is used to perform posterior inference under the finite shared atoms nested (fSAN) model with Gaussian likelihood (originally proposed in D'Angelo et al., 2023). The model uses Dirichlet mixtures with an unknown number of components (following the specification of Frühwirth-Schnatter et al., 2021) at both the observational and distributional level.

Usage

sample_fSAN(nrep, burn, y, group, 
            maxK = 50, maxL = 50, 
            m0 = 0, tau0 = 0.1, lambda0 = 3, gamma0 = 2,
            hyp_alpha1 = 6, hyp_alpha2 = 3, 
            hyp_beta1 = 6, hyp_beta2 = 3, 
            eps_alpha = NULL, eps_beta = NULL,
            alpha = NULL, beta = NULL,
            warmstart = TRUE, nclus_start = NULL,
            mu_start = NULL, sigma2_start = NULL,
            M_start = NULL, S_start = NULL,
            alpha_start = NULL, beta_start = NULL,
            progress = TRUE, seed = NULL)

Arguments

Details

Data structure

The finite common atoms mixture model is used to perform inference in nested settings, where the data are organized into J groups. The data should be continuous observations (Y_1,\dots,Y_J), where each Y_j = (y_{1,j},\dots,y_{n_j,j}) contains the n_j observations from group j, for j=1,\dots,J. The function takes as input the data as a numeric vector y in this concatenated form. Hence y should be a vector of length n_1+\dots+n_J. The group parameter is a numeric vector of the same size as y indicating the group membership for each individual observation. Notice that with this specification the observations in the same group need not be contiguous as long as the correspondence between the variables y and group is maintained.

Model

The data are modeled using a univariate Gaussian likelihood, where both the mean and the variance are observational-cluster-specific, i.e.,

y_{i,j}\mid M_{i,j} = l \sim N(\mu_l,\sigma^2_l)

where M_{i,j} \in \{1,\dots,L \} is the observational cluster indicator of observation i in group j. The prior on the model parameters is a Normal-Inverse-Gamma distribution (\mu_l,\sigma^2_l)\sim NIG (m_0,\tau_0,\lambda_0,\gamma_0), i.e., \mu_l\mid\sigma^2_l \sim N(m_0, \sigma^2_l / \tau_0), 1/\sigma^2_l \sim Gamma(\lambda_0, \gamma_0) (shape, rate).

Clustering

The model performs a clustering of both observations and groups. The clustering of groups (distributional clustering) is provided by the allocation variables S_j \in \{1,\dots,K\}, with

Pr(S_j = k \mid \dots ) = \pi_k \qquad \text{for } \: k = 1,\dots,K.

The distribution of the probabilities is (\pi_1,\dots,\pi_{K})\sim Dirichlet_K(\alpha/K,\dots,\alpha/K). Moreover, the dimension K is random (see Frühwirth-Schnatter et al., 2021).

The clustering of observations (observational clustering) is provided by the allocation variables M_{i,j} \in \{1,\dots,L\}, with

Pr(M_{i,j} = l \mid S_j = k, \dots ) = \omega_{l,k} \qquad \text{for } \: k = 1,\dots,K \, ; \: l = 1,\dots,L.

The distribution of the probabilities is (\omega_{1,k},\dots,\omega_{L,k})\sim Dirichlet_L(\beta/L,\dots,\beta/L) for all k = 1,\dots,K. Moreover, the dimension L is random (see Frühwirth-Schnatter et al., 2021).

Value

sample_fSAN returns four objects:

• model: name of the fitted model.

• params: list containing the data and the parameters used in the simulation. Details below.

• sim: list containing the simulated values (MCMC chains). Details below.

• time: total computation time.

Data and parameters: params is a list with the following components:

nrep

Number of MCMC iterations.

y, group

Data and group vectors.

maxK, maxL

Maximum number of distributional and observational clusters.

m0, tau0, lambda0, gamma0

Model hyperparameters.

(hyp_alpha1, hyp_alpha2, eps_alpha) or alpha

Either the hyperparameters on \alpha and MH step size (if \alpha random), or the value for \alpha (if fixed).

(hyp_beta1, hyp_beta2, eps_beta) or beta

Either the hyperparameters on \beta and MH step size (if \beta random), or the value for \beta (if fixed).

Simulated values: sim is a list with the following components:

mu

Matrix of size (nrep, maxL). Each row is a posterior sample of the mean parameter for each observational cluster (\mu_1,\dots\mu_L).

sigma2

Matrix of size (nrep, maxL). Each row is a posterior sample of the variance parameter for each observational cluster (\sigma^2_1,\dots\sigma^2_L).

obs_cluster

Matrix of size (nrep, n), with n = length(y). Each row is a posterior sample of the observational cluster allocation variables (M_{1,1},\dots,M_{n_J,J}).

distr_cluster

Matrix of size (nrep, J), with J = length(unique(group)). Each row is a posterior sample of the distributional cluster allocation variables (S_1,\dots,S_J).

pi

Matrix of size (nrep, maxK). Each row is a posterior sample of the distributional cluster probabilities (\pi_1,\dots,\pi_{K}).

omega

3-d array of size (maxL, maxK, nrep). Each slice is a posterior sample of the observational cluster probabilities. In each slice, each column k is a vector (of length maxL) observational cluster probabilities (\omega_{1,k},\dots,\omega_{L,k}) for distributional cluster k.

alpha

Vector of length nrep of posterior samples of the parameter \alpha.

beta

Vector of length nrep of posterior samples of the parameter \beta.

K_plus

Vector of length nrep of posterior samples of the number of distributional clusters.

L_plus

Vector of length nrep of posterior samples of the number of observational clusters.

K

Vector of length nrep of posterior samples of the distributional Dirichlet dimension.

L

Vector of length nrep of posterior samples of the observational Dirichlet dimension.

References

D’Angelo, L., Canale, A., Yu, Z., and Guindani, M. (2023). Bayesian nonparametric analysis for the detection of spikes in noisy calcium imaging data. Biometrics, 79(2), 1370–1382. <doi:10.1111/biom.13626>

Frühwirth-Schnatter, S., Malsiner-Walli, G. and Grün, B. (2021). Generalized mixtures of finite mixtures and telescoping sampling. Bayesian Analysis, 16(4), 1279–1307. <doi:10.1214/21-BA1294>

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
plot(density(y[g==1]), xlim = c(-5,10))
lines(density(y[g==2]), col = 2)
out <- sample_fSAN(nrep = 500, burn = 200, 
                    y = y, group = g, 
                   nclus_start = 2,
                   maxK = 20, maxL = 20,
                   alpha = 1, beta = 1)
out 

Sample fSAN with sparse mixtures

Description

sample_fSAN_sparsemix is used to perform posterior inference under the finite shared atoms nested (fSAN) model with Gaussian likelihood (originally proposed in D'Angelo et al., 2023). The model uses overfitted (sparse) Dirichlet mixtures (Malsiner-Walli et al., 2016) at both the observational and distributional level.

Usage

sample_fSAN_sparsemix(nrep, burn, y, group, 
               maxK = 50, maxL = 50, 
               m0 = 0, tau0 = 0.1, lambda0 = 3, gamma0 = 2,
               hyp_alpha = 10, hyp_beta = 10, 
               eps_alpha = NULL, eps_beta = NULL,
               alpha = NULL, beta = NULL,
               warmstart = TRUE, nclus_start = NULL,
               mu_start = NULL, sigma2_start = NULL, 
               M_start = NULL, S_start = NULL,
               alpha_start = NULL, beta_start = NULL,
               progress = TRUE, seed = NULL)

Arguments

Details

Data structure

The overfitted mixture common atoms model is used to perform inference in nested settings, where the data are organized into J groups. The data should be continuous observations (Y_1,\dots,Y_J), where each Y_j = (y_{1,j},\dots,y_{n_j,j}) contains the n_j observations from group j, for j=1,\dots,J. The function takes as input the data as a numeric vector y in this concatenated form. Hence y should be a vector of length n_1+\dots+n_J. The group parameter is a numeric vector of the same size as y indicating the group membership for each individual observation. Notice that with this specification the observations in the same group need not be contiguous as long as the correspondence between the variables y and group is maintained.

Model

The data are modeled using a univariate Gaussian likelihood, where both the mean and the variance are observational-cluster-specific, i.e.,

y_{i,j}\mid M_{i,j} = l \sim N(\mu_l,\sigma^2_l)

where M_{i,j} \in \{1,\dots,L \} is the observational cluster indicator of observation i in group j. The prior on the model parameters is a Normal-Inverse-Gamma distribution (\mu_l,\sigma^2_l)\sim NIG (m_0,\tau_0,\lambda_0,\gamma_0), i.e., \mu_l\mid\sigma^2_l \sim N(m_0, \sigma^2_l / \tau_0), 1/\sigma^2_l \sim Gamma(\lambda_0, \gamma_0) (shape, rate).

Clustering

The model performs a clustering of both observations and groups. The clustering of groups (distributional clustering) is provided by the allocation variables S_j \in \{1,\dots,K\}, with

Pr(S_j = k \mid \dots ) = \pi_k \qquad \text{for } \: k = 1,\dots,K.

The distribution of the probabilities is (\pi_1,\dots,\pi_{K})\sim Dirichlet_K(\alpha,\dots,\alpha).

The clustering of observations (observational clustering) is provided by the allocation variables M_{i,j} \in \{1,\dots,L\}, with

Pr(M_{i,j} = l \mid S_j = k, \dots ) = \omega_{l,k} \qquad \text{for } \: k = 1,\dots,K \, ; \: l = 1,\dots,L.

The distribution of the probabilities is (\omega_{1,k},\dots,\omega_{L,k})\sim Dirichlet_L(\beta,\dots,\beta) for all k = 1,\dots,K.

Value

sample_fSAN_sparsemix returns four objects:

• model: name of the fitted model.

• params: list containing the data and the parameters used in the simulation. Details below.

• sim: list containing the simulated values (MCMC chains). Details below.

• time: total computation time.

Data and parameters: params is a list with the following components:

nrep

Number of MCMC iterations.

y, group

Data and group vectors.

maxK, maxL

Maximum number of distributional and observational clusters.

m0, tau0, lambda0, gamma0

Model hyperparameters.

(hyp_alpha,eps_alpha) or alpha

Either the hyperparameter on \alpha and MH step size (if \alpha random), or the value for \alpha (if fixed).

(hyp_beta,eps_beta) or beta

Either the hyperparameter on \beta and MH step size (if \beta random), or the value for \beta (if fixed).

Simulated values: sim is a list with the following components:

mu

Matrix of size (nrep, maxL). Each row is a posterior sample of the mean parameter for each observational cluster (\mu_1,\dots\mu_L).

sigma2

Matrix of size (nrep, maxL). Each row is a posterior sample of the variance parameter for each observational cluster (\sigma^2_1,\dots\sigma^2_L).

obs_cluster

Matrix of size (nrep, n), with n = length(y). Each row is a posterior sample of the observational cluster allocation variables (M_{1,1},\dots,M_{n_J,J}).

distr_cluster

Matrix of size (nrep, J), with J = length(unique(group)). Each row is a posterior sample of the distributional cluster allocation variables (S_1,\dots,S_J).

pi

Matrix of size (nrep, maxK). Each row is a posterior sample of the distributional cluster probabilities (\pi_1,\dots,\pi_{K}).

omega

3-d array of size (maxL, maxK, nrep). Each slice is a posterior sample of the observational cluster probabilities. In each slice, each column k is a vector (of length maxL) observational cluster probabilities (\omega_{1,k},\dots,\omega_{L,k}) for distributional cluster k.

alpha

Vector of length nrep of posterior samples of the parameter \alpha.

beta

Vector of length nrep of posterior samples of the parameter \beta.

References

D’Angelo, L., Canale, A., Yu, Z., and Guindani, M. (2023). Bayesian nonparametric analysis for the detection of spikes in noisy calcium imaging data. Biometrics, 79(2), 1370–1382. <doi:10.1111/biom.13626>

Malsiner-Walli, G., Frühwirth-Schnatter, S. and Grün, B. (2016). Model-based clustering based on sparse finite Gaussian mixtures. Statistics and Computing 26, 303–324. <doi:10.1007/s11222-014-9500-2>

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
plot(density(y[g==1]), xlim = c(-5,10))
lines(density(y[g==2]), col = 2)
out <- sample_fSAN_sparsemix(nrep = 500, burn = 200, y = y, group = g, 
                             nclus_start = 2,
                             maxK = 20, maxL = 20,
                             alpha = 0.01, beta = 0.01)
out 

Sample fiSAN

Description

sample_fiSAN is used to perform posterior inference under the finite-infinite shared atoms nested (fiSAN) model with Gaussian likelihood. The model uses a Dirichlet process mixture prior at the distributional level, and Dirichlet mixture with an unknown number of components (following the specification of Frühwirth-Schnatter et al., 2021) at the observational level. The algorithm for the nonparametric component is based on the slice sampler for DPM of Kalli, Griffin and Walker (2011).

Usage

sample_fiSAN(nrep, burn, y, group, 
             maxK = 50, maxL = 50, 
             m0 = 0, tau0 = 0.1, lambda0 = 3, gamma0 = 2,
             hyp_alpha1 = 1, hyp_alpha2 = 1,
             hyp_beta1 = 6, hyp_beta2 = 3, 
             eps_beta = NULL,
             alpha = NULL, beta = NULL,
             warmstart = TRUE, nclus_start = NULL,
             mu_start = NULL, sigma2_start = NULL,
             M_start = NULL, S_start = NULL,
             alpha_start = NULL, beta_start = NULL,
             progress = TRUE, seed = NULL)

Arguments

Details

Data structure

The finite-infinite common atoms mixture model is used to perform inference in nested settings, where the data are organized into J groups. The data should be continuous observations (Y_1,\dots,Y_J), where each Y_j = (y_{1,j},\dots,y_{n_j,j}) contains the n_j observations from group j, for j=1,\dots,J. The function takes as input the data as a numeric vector y in this concatenated form. Hence y should be a vector of length n_1+\dots+n_J. The group parameter is a numeric vector of the same size as y indicating the group membership for each individual observation. Notice that with this specification the observations in the same group need not be contiguous as long as the correspondence between the variables y and group is maintained.

Model

The data are modeled using a univariate Gaussian likelihood, where both the mean and the variance are observational-cluster-specific, i.e.,

y_{i,j}\mid M_{i,j} = l \sim N(\mu_l,\sigma^2_l)

where M_{i,j} \in \{1,\dots,L \} is the observational cluster indicator of observation i in group j. The prior on the model parameters is a Normal-Inverse-Gamma distribution (\mu_l,\sigma^2_l)\sim NIG (m_0,\tau_0,\lambda_0,\gamma_0), i.e., \mu_l\mid\sigma^2_l \sim N(m_0, \sigma^2_l / \tau_0), 1/\sigma^2_l \sim Gamma(\lambda_0, \gamma_0) (shape, rate).

Clustering

The model performs a clustering of both observations and groups. The clustering of groups (distributional clustering) is provided by the allocation variables S_j \in \{1,2,\dots\}, with

Pr(S_j = k \mid \dots ) = \pi_k \qquad \text{for } \: k = 1,2,\dots

The distribution of the probabilities is \{\pi_k\}_{k=1}^{\infty} \sim GEM(\alpha), where GEM is the Griffiths-Engen-McCloskey distribution of parameter \alpha, which characterizes the stick-breaking construction of the DP (Sethuraman, 1994).

The clustering of observations (observational clustering) is provided by the allocation variables M_{i,j} \in \{1,\dots,L\}, with

Pr(M_{i,j} = l \mid S_j = k, \dots ) = \omega_{l,k} \qquad \text{for } \: k = 1,2,\dots \, ; \: l = 1,\dots,L.

The distribution of the probabilities is (\omega_{1,k},\dots,\omega_{L,k})\sim Dirichlet_L(\beta/L,\dots,\beta/L) for all k = 1,2,\dots. Moreover, the dimension L is random (see Frühwirth-Schnatter et al., 2021).

Value

sample_fiSAN returns four objects:

• model: name of the fitted model.

• params: list containing the data and the parameters used in the simulation. Details below.

• sim: list containing the simulated values (MCMC chains). Details below.

• time: total computation time.

Data and parameters: params is a list with the following components:

nrep

Number of MCMC iterations.

y, group

Data and group vectors.

maxK, maxL

Maximum number of distributional and observational clusters.

m0, tau0, lambda0, gamma0

Model hyperparameters.

(hyp_alpha1,hyp_alpha2) or alpha

Either the hyperparameters on \alpha (if \alpha random), or the value for \alpha (if fixed).

(hyp_beta1, hyp_beta2, eps_beta) or beta

Either the hyperparameters on \beta and MH step size (if \beta random), or the value for \beta (if fixed).

Simulated values: sim is a list with the following components:

mu

Matrix of size (nrep, maxL). Each row is a posterior sample of the mean parameter for each observational cluster (\mu_1,\dots\mu_L).

sigma2

Matrix of size (nrep, maxL). Each row is a posterior sample of the variance parameter for each observational cluster (\sigma^2_1,\dots\sigma^2_L).

obs_cluster

Matrix of size (nrep, n), with n = length(y). Each row is a posterior sample of the observational cluster allocation variables (M_{1,1},\dots,M_{n_J,J}).

distr_cluster

Matrix of size (nrep, J), with J = length(unique(group)). Each row is a posterior sample of the distributional cluster allocation variables (S_1,\dots,S_J).

pi

Matrix of size (nrep, maxK). Each row is a posterior sample of the distributional cluster probabilities (\pi_1,\dots,\pi_{maxK}).

omega

3-d array of size (maxL, maxK, nrep). Each slice is a posterior sample of the observational cluster probabilities. In each slice, each column k is a vector (of length maxL) observational cluster probabilities (\omega_{1,k},\dots,\omega_{L,k}) for distributional cluster k.

alpha

Vector of length nrep of posterior samples of the parameter \alpha.

beta

Vector of length nrep of posterior samples of the parameter \beta.

maxK

Vector of length nrep of the number of distributional DP components used by the slice sampler.

L_plus

Vector of length nrep of posterior samples of the number of observational clusters.

L

Vector of length nrep of posterior samples of the observational Dirichlet dimension.

References

Frühwirth-Schnatter, S., Malsiner-Walli, G. and Grün, B. (2021). Generalized mixtures of finite mixtures and telescoping sampling. Bayesian Analysis, 16(4), 1279–1307. <doi:10.1214/21-BA1294>

Kalli, M., Griffin, J.E., and Walker, S.G. (2011). Slice Sampling Mixture Models, Statistics and Computing, 21, 93–105. <doi:10.1007/s11222-009-9150-y>

Sethuraman, A.J. (1994). A Constructive Definition of Dirichlet Priors, Statistica Sinica, 4, 639–650.

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
plot(density(y[g==1]), xlim = c(-5,10))
lines(density(y[g==2]), col = 2)
out <- sample_fiSAN(nrep = 500, burn = 200, y = y, group = g, 
                    nclus_start = 2,
                    maxK = 20, maxL = 20,
                    beta = 1)
out 

Sample fiSAN with sparse mixtures

Description

sample_fiSAN_sparsemix is used to perform posterior inference under the finite-infinite shared atoms nested (fiSAN) model with Gaussian likelihood. The model uses a Dirichlet process mixture prior at the distributional level, and a sparse (overfitted) Dirichlet mixture (Malsiner-Walli et al., 2016) at the observational level. The algorithm for the nonparametric component is based on the slice sampler for DPM of Kalli, Griffin and Walker (2011).

Usage

sample_fiSAN_sparsemix(nrep, burn, y, group, 
                 maxK = 50, maxL = 50, 
                 m0 = 0, tau0 = 0.1, lambda0 = 3, gamma0 = 2,
                 hyp_alpha1 = 1, hyp_alpha2 = 1,
                 hyp_beta = 10, 
                 eps_beta = NULL, 
                 alpha = NULL, beta = NULL,
                 warmstart = TRUE, nclus_start = NULL,
                 mu_start = NULL, sigma2_start = NULL,
                 M_start = NULL, S_start = NULL,
                 alpha_start = NULL, beta_start = NULL,
                 progress = TRUE, seed = NULL)

Arguments

Details

Data structure

The finite-infinite common atoms mixture model is used to perform inference in nested settings, where the data are organized into J groups. The data should be continuous observations (Y_1,\dots,Y_J), where each Y_j = (y_{1,j},\dots,y_{n_j,j}) contains the n_j observations from group j, for j=1,\dots,J. The function takes as input the data as a numeric vector y in this concatenated form. Hence y should be a vector of length n_1+\dots+n_J. The group parameter is a numeric vector of the same size as y indicating the group membership for each individual observation. Notice that with this specification the observations in the same group need not be contiguous as long as the correspondence between the variables y and group is maintained.

Model

The data are modeled using a univariate Gaussian likelihood, where both the mean and the variance are observational-cluster-specific, i.e.,

y_{i,j}\mid M_{i,j} = l \sim N(\mu_l,\sigma^2_l)

where M_{i,j} \in \{1,\dots,L \} is the observational cluster indicator of observation i in group j. The prior on the model parameters is a Normal-Inverse-Gamma distribution (\mu_l,\sigma^2_l)\sim NIG (m_0,\tau_0,\lambda_0,\gamma_0), i.e., \mu_l\mid\sigma^2_l \sim N(m_0, \sigma^2_l / \tau_0), 1/\sigma^2_l \sim Gamma(\lambda_0, \gamma_0) (shape, rate).

Clustering

The model performs a clustering of both observations and groups. The clustering of groups (distributional clustering) is provided by the allocation variables S_j \in \{1,2,\dots\}, with

Pr(S_j = k \mid \dots ) = \pi_k \qquad \text{for } \: k = 1,2,\dots

The distribution of the probabilities is \{\pi_k\}_{k=1}^{\infty} \sim GEM(\alpha), where GEM is the Griffiths-Engen-McCloskey distribution of parameter \alpha, which characterizes the stick-breaking construction of the DP (Sethuraman, 1994).

The clustering of observations (observational clustering) is provided by the allocation variables M_{i,j} \in \{1,\dots,L\}, with

Pr(M_{i,j} = l \mid S_j = k, \dots ) = \omega_{l,k} \qquad \text{for } \: k = 1,2,\dots \, ; \: l = 1,\dots,L.

The distribution of the probabilities is (\omega_{1,k},\dots,\omega_{L,k})\sim Dirichlet_L(\beta,\dots,\beta) for all k = 1,2,\dots.

Value

sample_fiSAN_sparsemix returns four objects:

• model: name of the fitted model.

• params: list containing the data and the parameters used in the simulation. Details below.

• sim: list containing the simulated values (MCMC chains). Details below.

• time: total computation time.

Data and parameters: params is a list with the following components:

nrep

Number of MCMC iterations.

y, group

Data and group vectors.

maxK, maxL

Maximum number of distributional and observational clusters.

m0, tau0, lambda0, gamma0

Model hyperparameters.

(hyp_alpha1,hyp_alpha2) or alpha

Either the hyperparameters on \alpha (if \alpha random), or the value for \alpha (if fixed).

(hyp_beta,eps_beta) or beta

Either the hyperparameter on \beta and MH step size (if \beta random), or the value for \beta (if fixed).

Simulated values: sim is a list with the following components:

mu

Matrix of size (nrep, maxL). Each row is a posterior sample of the mean parameter for each observational cluster (\mu_1,\dots\mu_L).

sigma2

Matrix of size (nrep, maxL). Each row is a posterior sample of the variance parameter for each observational cluster (\sigma^2_1,\dots\sigma^2_L).

obs_cluster

Matrix of size (nrep, n), with n = length(y). Each row is a posterior sample of the observational cluster allocation variables (M_{1,1},\dots,M_{n_J,J}).

distr_cluster

Matrix of size (nrep, J), with J = length(unique(group)). Each row is a posterior sample of the distributional cluster allocation variables (S_1,\dots,S_J).

pi

Matrix of size (nrep, maxK). Each row is a posterior sample of the distributional cluster probabilities (\pi_1,\dots,\pi_{maxK}).

omega

3-d array of size (maxL, maxK, nrep). Each slice is a posterior sample of the observational cluster probabilities. In each slice, each column k is a vector (of length maxL) observational cluster probabilities (\omega_{1,k},\dots,\omega_{L,k}) for distributional cluster k.

alpha

Vector of length nrep of posterior samples of the parameter \alpha.

beta

Vector of length nrep of posterior samples of the parameter \beta.

maxK

Vector of length nrep of the number of distributional DP components used by the slice sampler.

References

Kalli, M., Griffin, J.E., and Walker, S.G. (2011). Slice Sampling Mixture Models, Statistics and Computing, 21, 93–105. <doi:10.1007/s11222-009-9150-y>

Malsiner-Walli, G., Frühwirth-Schnatter, S. and Grün, B. (2016). Model-based clustering based on sparse finite Gaussian mixtures. Statistics and Computing 26, 303–324. <doi:10.1007/s11222-014-9500-2>

Sethuraman, A.J. (1994). A Constructive Definition of Dirichlet Priors, Statistica Sinica, 4, 639–650.

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
plot(density(y[g==1]), xlim = c(-5,10))
lines(density(y[g==2]), col = 2)
out <- sample_fiSAN_sparsemix(nrep = 500, burn = 200, y = y, group = g, 
                              nclus_start = 2,
                              maxK = 20, maxL = 20,
                              beta = 0.01)
out 

Traceplot: plot MCMC chains

Description

Check the convergence of the MCMC through visual inspection of the chains.

Usage

traceplot(object, params, 
          show_density = TRUE, 
          show_burnin = TRUE, 
          length_burnin = NULL, 
          show_convergence = TRUE, 
          trunc_plot = 10)

Arguments

Value

The function displays the traceplots of the MCMC algorithm.

Note

The function is not available for the observational weights \omega.

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
out <- sample_fiSAN(nrep = 500, burn = 200, 
                    y = y, group = g, 
                    nclus_start = 2,
                    maxK = 20, maxL = 20,
                    beta = 1)
traceplot(out, params = c("mu", "sigma2"), trunc_plot = 2)