R: Sample fSAN with sparse mixtures

sample_fSAN_sparsemix {SANple}

R Documentation

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

`nrep`	Number of MCMC iterations.
`burn`	Number of discarded iterations.
`y`	Vector of observations.
`group`	Vector of the same length of y indicating the group membership (numeric).
`maxK`	Maximum number of distributional clusters `K` (default = 50).
`maxL`	Maximum number of observational clusters `L` (default = 50).
`m0`, `tau0`, `lambda0`, `gamma0`	Hyperparameters on `(\mu, \sigma^2) \sim NIG(m_0, \tau_0, \lambda_0,\gamma_0)`. Default is (0, 0.1, 3, 2).
`hyp_alpha`, `eps_alpha`	If a random `\alpha` is used, `hyp_alpha` specifies the hyperparameter (default = 10). The prior is `\alpha` ~ Gamma(`hyp_alpha`, `hyp_alpha*maxK`), following Malsiner-Walli et al. (2016). In this case, `eps_alpha` is the tuning parameter of the MH step.
`hyp_beta`, `eps_beta`	If a random `\beta` is used, `hyp_beta` specifies the hyperparameter (default = 10). The prior is `\beta` ~ Gamma(`hyp_beta`, `hyp_beta*maxL`), following Malsiner-Walli et al. (2016). In this case, `eps_beta` is the tuning parameter of the MH step.
`alpha`	Distributional Dirichlet parameter if fixed (optional). The distribution is Dirichlet( `rep(alpha, maxK)` ).
`beta`	Observational Dirichlet parameter if fixed (optional). The distribution is Dirichlet( `rep(beta, maxL)` ).
`warmstart`, `nclus_start`	Initialization of the observational clustering. `warmstart` is logical parameter (default = `TRUE`) of whether a kmeans clustering should be used to initialize the chains. An initial guess of the number of observational clusters can be passed via the `nclus_start` parameter (optional)
`mu_start`, `sigma2_start`, `M_start`, `S_start`, `alpha_start`, `beta_start`	Starting points of the MCMC chains (optional). Default is `nclus_start = min(c(maxL, 30))`. `mu_start, sigma2_start` are vectors of length `maxL`. `M_start` is a vector of observational cluster allocation of length N. `S_start` is a vector of observational cluster allocation of length J. `alpha_start, alpha_start` are numeric.
`progress`	show a progress bar? (logical, default TRUE).
`seed`	set a fixed seed.

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

[Package SANple version 0.1.1 Index]