dpmj {BNSP} R Documentation

Dirichlet process mixtures of joint models

Description

Fits Dirichlet process mixtures of joint response-covariate models, where the covariates are of mixed type while the discrete responses are represented utilizing continuous latent variables. See ‘Details’ section for a full model description and Papageorgiou (2018) for all technical details.

Usage

dpmj(formula, Fcdf, data, offset, sampler = "truncated", Xpred, offsetPred,
StorageDir, ncomp, sweeps, burn, thin = 1, seed, H, Hdf, d, D,
Alpha.xi, Beta.xi, Alpha.alpha, Beta.alpha, Trunc.alpha, ...)


Arguments

 formula a formula defining the response and the covariates e.g. y ~ x. Fcdf a description of the kernel of the response variable. Currently five options are supported: 1. "poisson", 2. "negative binomial", 3. "generalized poisson", 4. "binomial" and 5. "beta binomial". The first three kernels are used for count data analysis, where the third kernel allows for both over- and under-dispersion relative to the Poisson distribution. The last two kernels are used for binomial data analysis. See ‘Details’ section for some of the kernel details. data an optional data frame, list or environment (or object coercible by ‘as.data.frame’ to a data frame) containing the variables in the model. If not found in ‘data’, the variables are taken from ‘environment(formula)’. offset this can be used to specify an a priori known component to be included in the model. This should be ‘NULL’ or a numeric vector of length equal to the sample size. One ‘offset’ term can be included in the formula, and if more are required, their sum should be used. sampler the MCMC algorithm to be utilized. The two options are sampler = "slice" which implements a slice sampler (Walker, 2007; Papaspiliopoulos, 2008) and sampler = "truncated" which proceeds by truncating the countable mixture at ncomp components (see argument ncomp). Xpred an optional design matrix the rows of which include the values of the covariates x for which the conditional distribution of Y|x,D (where D denotes the data) is calculated. These are treated as ‘new’ covariates i.e. they do not contribute to the likelihood. The matrix shouldn't include a column of 1's. NA's can be included to obtain averaged effects. offsetPred the offset term associated with the new covariates Xpred. It is of dimension one i.e. the same offset term is used for all rows of Xpred. If Fcdf is one of "poisson" or "negative binomial" or "generalized poisson", then offsetPred is the Poisson offset term. If Fcdf is one of "binomial" or "beta binomial", then offsetPred is the number of Binomial trials. If offsetPred is missing, it is taken to be the mean of offset, rounded to the nearest integer. StorageDir a directory to store files with the posterior samples of models parameters and other quantities of interest. If a directory is not provided, files are created in the current directory and removed when the sampler completes. ncomp number of mixture components. It defines where the countable mixture of densities [in (1) below] is truncated. Even if sampler="slice" is chosen, ncomp needs to be specified as it is used in the initialization process. sweeps total number of posterior samples, including those discarded in burn-in period (see argument burn) and those discarded by the thinning process (see argument thin). burn length of burn-in period. thin thinning parameter. seed optional seed for the random generator. H optional scale matrix of the Wishart-like prior assigned to the restricted covariance matrices Σ_h^*. See ‘Details’ section. Hdf optional degrees of freedom of the prior Wishart-like prior assigned to the restricted covariance matrices Σ_h^*. See ‘Details’ section. d optional prior mean of the mean vector μ_h. See ‘Details’ section. D optional prior covariance matrix of the mean vector μ_h. See ‘Details’ section. Alpha.xi an optional parameter that depends on the specified Fcdf argument. If Fcdf = "poisson", this argument is parameter α_{ξ} of the prior of the Poisson rate: ξ \sim Gamma(α_{ξ},β_{ξ}). If Fcdf = "negative binomial", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Negative Binomial pmf. If Fcdf = "generalized poisson", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim N(α_{2ξ},β_{2ξ})I[ξ_2 \in R_{ξ_2}], where ξ_1 and ξ_2 are the two parameters of the Generalized Poisson pmf. Parameter ξ_2 is restricted in the range R_{ξ_2} = (0.05,∞) as it is a dispersion parameter. If Fcdf = "binomial", this argument is parameter α_{ξ} of the prior of the Binomial probability: ξ \sim Beta(α_{ξ},β_{ξ}). If Fcdf = "beta binomial", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Beta Binomial pmf. See ‘Details’ section. Beta.xi an optional parameter that depends on the specified family. If Fcdf = "poisson", this argument is parameter β_{ξ} of the prior of the Poisson rate: ξ \sim Gamma(α_{ξ},β_{ξ}). If Fcdf = "negative binomial", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Negative Binomial pmf. If Fcdf = "generalized poisson", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Normal(α_{2ξ},β_{2ξ})I[ξ_2 \in R_{ξ_2}], where ξ_1 and ξ_2 are the two parameters of the Generalized Poisson pmf. Parameter ξ_2 is restricted in the range R_{ξ_2} = (0.05,∞) as it is a dispersion parameter. Note that β_{2ξ} is a standard deviation. If Fcdf = "binomial", this argument is parameter β_{ξ} of the prior of the Binomial probability: ξ \sim Beta(α_{ξ},β_{ξ}). If Fcdf = "beta binomial", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Beta Binomial pmf. See ‘Details’ section. Alpha.alpha optional shape parameter α_{α} of the Gamma prior assigned to the concentration parameter α. See ‘Details’ section. Beta.alpha optional rate parameter β_{α} of the Gamma prior assigned to concentration parameter α. See ‘Details’ section. Trunc.alpha optional truncation point c_{α} of the Gamma prior assigned to concentration parameter α. See ‘Details’ section. ... Other options that will be ignored.

Details

Function dpmj returns samples from the posterior distributions of the parameters of the model:

f(y_i,x_i) = ∑_{h=1}^{∞} π_h f(y_i,x_i|θ_h), \hspace{200pt} (1)

where y_i is a univariate discrete response, x_i is a p-dimensional vector of mixed type covariates, and π_h, h ≥q 1, are obtained according to Sethuraman's (1994) stick-breaking construction: π_1 = v_1, and for l ≥q 2, π_l = v_l ∏_{j=1}^{l-1} (1-v_j), where v_k are iid samples v_k \simBeta (1,α), k ≥q 1.

Let Z denote a discrete variable (response or covariate). It is represented as discretized version of a continuous latent variable Z^*. Observed discrete Z and continuous latent variable Z^* are connected by:

z = q \iff c_{q-1} < z^* < c_{q}, q=0,1,2,…,

where the cut-points are obtained as: c_{-1} = -∞, while for q ≥q 0, c_{q} = c_{q}(λ) = Φ^{-1}\{F(q;λ)\}. Here Φ(.) is the cumulative distribution function (cdf) of a standard normal variable and F() denotes an appropriate cdf. Further, latent variables are assumed to independently follow a N(0,1) distribution, where the mean and variance are restricted to be zero and one as they are non-identifiable by the data. Choices for F() are described next.

For counts, three options are supported. First, F(.;λ_i) can be specified as the cdf of a Poisson(H_i ξ_h) variable. Here λ_i=(ξ_h,H_i)^T, ξ_h denotes the Poisson rate associated with cluster h, and H_i the offset term associated with sampling unit i. Second, F(.;λ_i) can be specified as the negative binomial cdf, where λ_i= (ξ_{1h},ξ_{2h},H_i)^T. This option allows for overdispersion within each cluster relative to the Poisson distribution. Third, F(.;λ_i) can be specified as the Generalized Poisson cdf, where, again, λ_i=(ξ_{1h},ξ_{2h},H_i)^T. This option allows for both over- and under-dispersion within each cluster.

For Binomial data, two options are supported. First, F(.;λ_i) may be taken to be the cdf of a Binomial(H_i,ξ_h) variable, where ξ_h denotes the success probability of cluster h and H_i the number of trials associated with sampling unit i. Second, F(.;λ_i) may be specified to be the beta-binomial cdf, where λ=(ξ_{1h},ξ_{2h},H_i)^T.

The special case of Binomial data is treated as

Z = 0 \iff z^* < 0, z^* \sim N(μ_z^{*},1).

Details on all kernels are provided in the two tables below. The first table provides the probability mass functions and the mean in the presence of an offset term (which may be taken to be one). The column ‘Sample’ indicates for which parameters the routine provides posterior samples. The second table provides information on the assumed priors along with the default values of the parameters of the prior distributions and it also indicates the function arguments that allow the user to alter these.

 Kernel PMF Offset Mean Sample Poisson \exp(-Hξ) (Hξ)^y /y! H H ξ ξ Negative Binomial \frac{Γ(y+ξ_1)}{Γ(ξ_1)Γ(y+1)}(\frac{ξ_2}{H+ξ_2})^{ξ_1}(\frac{H}{H+ξ_2})^{y} H H ξ_1/ξ_2 ξ_1, ξ_2 Generalized Poisson ξ_1 \{ξ_1+(ξ_2-1)y\}^{y-1} ξ_2^{-y} \times H Hξ_1 ξ_1,ξ_2 ~~ \exp\{-[ξ_1+(ξ_2-1)y]/ξ_2\}/y! Binomial {N \choose y} ξ^y (1-ξ)^{N-y} N N ξ ξ Beta Binomial {N \choose y} \frac{{Beta}{(y+ξ_1,N-y+ξ_2)}}{{Beta}{(ξ_1,ξ_2)}} N N ξ_1/(ξ_1+ξ_2) ξ_1,ξ_2
 Kernel Priors Default Values Poisson ξ \sim Gamma(α_{ξ},β_{ξ}) Alpha.xi = 1.0, Beta.xi = 0.1 Negative Binomial ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2 Alpha.xi = c(1.0,1.0), Beta.xi = c(0.1,0.1) Generalized Poisson ξ_1 \sim Gamma(α_{ξ_1},β_{ξ_1}) ξ_2 \sim N(α_{ξ_2},β_{ξ_2})I[ξ_2 > 0.05] Alpha.xi = c(1.0,1.0), Beta.xi = c(0.1,1.0) where β_{ξ_2} denotes st.dev. Binomial ξ \sim Beta(α_{ξ},β_{ξ}) Alpha.xi = 1.0, Beta.xi = 1.0 Beta Binomial ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2 Alpha.xi = c(1.0,1.0), Beta.xi = c(0.1,0.1)

Let z_i = (y_i,x_{i}^T)^T denote the joint vector of observed continuous and discrete variables and z_i^* the corresponding vector of continuous observed and latent variables. With θ_h denoting model parameters associated with the hth cluster, the joint density f(z_{i}|θ_h) takes the form

f(z_i|θ_h) = \int_{R(y)} \int_{R(x_{d})} N_{q}(z_i^*;μ^*_h,Σ^*_h) dx_{d}^{*} dy^{*},

where

\begin{array}{ll} μ^*_h = ≤ft( \begin{array}{l} 0 \\ μ_h \\ \end{array} \right), & Σ^*_h=≤ft[ \begin{array}{ll} C_h & ν_h^T \\ ν_h & Σ_h \\ \end{array} \right] \end{array},

where C_h is the covariance matrix of the latent continuous variables and it has diagonal elements equal to one i.e. it is a correlation matrix.

In addition to the priors defined in the table above, we specify the following:

1. The restricted covariance matrix Σ^*_h is assigned a prior distribution that is based on the Wishart distribution with degrees of freedom set by default to dimension of matrix plus two and diagonal scale matrix, with the sub-matrix that corresponds to discrete variables taken to be the identity matrix and with sub-matrix that corresponds to continuous variables having entries equal to 1/8 of the square of the observed data range. Default values can be changed using arguments H and Hdf.

2. The prior on μ_h, the non-zero part of μ_h^*, is taken to be multivariate normal μ_h \sim N(d,D). The mean d is taken to be equal to the center of the dataset. The covariance matrix D is taken to be diagonal. Its elements that correspond to continuous variables are set equal to 1/8 of the square of the observed data range while the elements that correspond to binary variables are set equal to 5. Arguments Mu.mu and Sigma.mu allow the user to change the default values.

3. The concentration parameter α is assigned a Gamma(α_{α},β_{α}) prior over the range (c_{α},∞), that is, f(α) \propto α^{α_{α}-1} \exp\{-α β_{α}\} I[α > c_{α}], where I[.] is the indicator function. The default values are α_{α}=2.0, β_{α}=5.0, and c_{α}=0.25. Users can alter the default using using arguments Alpha.alpha, Beta.alpha and Turnc.alpha.

Value

Function dpmj returns the following:

 call the matched call. seed the seed that was used (in case replication of the results is needed). meanReg if Xpred is specified, the function returns the posterior mean of the conditional expectation of the response y given each new covariate x. medianReg if Xpred is specified, the function returns the posterior mean of the conditional 50% quantile of the response y given each new covariate x. q1Reg if Xpred is specified, the function returns the posterior mean of the conditional 25% quantile of the response y given each new covariate x. q3Reg if Xpred is specified, the function returns the posterior mean of the conditional 75% quantile of the response y given each new covariate x. modeReg if Xpred is specified, the function returns the posterior mean of the conditional mode of the response y given each new covariate x. denReg if Xpred is specified, the function returns the posterior mean conditional density of the response y given each new covariate x. Results are presented in a matrix the rows of which correspond to the different xs. denVar if Xpred is specified, the function returns the posterior variance of the conditional density of the response y given each new covariate x. Results are presented in a matrix the rows of which correspond to the different xs.

Further, function dpmj creates files where the posterior samples are written. These files are (with all file names preceded by ‘BNSP.’):

 alpha.txt this file contains samples from the posterior of the concentration parameters α. The file is arranged in (sweeps-burn)/thin lines and one column, each line including one posterior sample. compAlloc.txt this file contains the allocations to clusters obtained during posterior sampling. It consists of (sweeps-burn)/thin lines, that represent the posterior samples, and n columns, that represent the sampling units. Clusters are represented by integers ranging from 0 to ncomp-1. MeanReg.txt this file contains the conditional means of the response y given covariates x obtained during posterior sampling. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the means are obtained. MedianReg.txt this file contains the 50% conditional quantile of the response y given covariates x obtained during posterior sampling. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the medians are obtained. muh.txt this file contains samples from the posteriors of the p-dimensional mean vectors μ_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and p columns. In more detail, sweeps create ncomp lines representing samples μ_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of μ_h^{(sw)} are written in the columns of the file. nmembers.txt this file contains (sweeps-burn)/thin lines and ncomp columns, where the lines represent posterior samples while the columns represent the components or clusters. The entries represent the number of sampling units allocated to each component. Q05Reg.txt this file contains the 5% conditional quantile of the response y given covariates x obtained during posterior sampling. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained. Q10Reg.txt as above, for the 10% conditional quantile. Q15Reg.txt as above, for the 15% conditional quantile. Q20Reg.txt as above, for the 20% conditional quantile. Q25Reg.txt as above, for the 25% conditional quantile. Q75Reg.txt as above, for the 75% conditional quantile. Q80Reg.txt as above, for the 80% conditional quantile. Q85Reg.txt as above, for the 85% conditional quantile. Q90Reg.txt as above, for the 90% conditional quantile. Q95Reg.txt as above, for the 95% conditional quantile. Sigmah.txt this file contains samples from the posteriors of the q \times q restricted covariance matrices Σ_h^*, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and q^2 columns. In more detail, sweeps create ncomp lines representing samples Σ_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of Σ_h^{(sw)} are written in the columns of the file. xih.txt this file contains samples from the posteriors of parameters ξ_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and one or two columns, depending on the number of parameters in the selected Fcdf. Sweeps write in the file ncomp lines representing samples ξ_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. Updated.txt this file contains (sweeps-burn)/thin lines with the number of components updated at each iteration of the sampler (relevant for slice sampling).

Author(s)

Georgios Papageorgiou gpapageo@gmail.com

References

Consul, P. C. & Famoye, G. C. (1992). Generalized Poisson regression model. Communications in Statistics - Theory and Methods, 1992, 89-109.

Papageorgiou, G. (2018). Bayesian density regression for discrete outcomes. arXiv:1603.09706v3 [stat.ME].

Papaspiliopoulos, O. (2008). A note on posterior sampling from Dirichlet mixture models. Technical report, University of Warwick.

Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4, 639-650.

Walker, S. G. (2007). Sampling the Dirichlet mixture model with slices. Communications in Statistics Simulation and Computation, 36(1), 45-54.

Examples

#Bayesian nonparametric joint model with binomial response Y and one predictor X
data(simD)
pred<-seq(with(simD,min(X))+0.1,with(simD,max(X))-0.1,length.out=30)
npred<-length(pred)
# fit1 and fit2 define the same model but with different numbers of
# components and posterior samples
fit1 <- dpmj(cbind(Y,(E-Y))~X, Fcdf="binomial", data=simD, ncomp=10, sweeps=20,
burn=10, sampler="truncated", Xpred=pred, offsetPred=30)
fit2 <- dpmj(cbind(Y,(E-Y))~X, Fcdf="binomial", data=simD, ncomp=50, sweeps=5000,
burn=1000, sampler="truncated", Xpred=pred, offsetPred=30)
plot(with(simD,X),with(simD,Y)/with(simD,E))
lines(pred,fit2\$medianReg/30,col=3,lwd=2)
# with discrete covariate
simD<-data.frame(simD,Xd=sample(c(0,1),300,replace=TRUE))
pred<-c(0,1)
fit3 <- dpmj(cbind(Y,(E-Y))~Xd, Fcdf="binomial", data=simD, ncomp=10, sweeps=20,
burn=10, sampler="truncated", Xpred=pred, offsetPred=30)


[Package BNSP version 2.1.6 Index]