NMixMCMC {mixAK} | R Documentation

## MCMC estimation of (multivariate) normal mixtures with possibly censored data.

### Description

This function runs MCMC for a model in which unknown density is specified as a normal mixture with either known or unknown number of components. With a prespecified number of components, MCMC is implemented through Gibbs sampling (see Diebolt and Robert, 1994) and dimension of the data can be arbitrary. With unknown number of components, currently only univariate case is implemented using the reversible jump MCMC (Richardson and Green, 1997).

Further, the data are allowed to be censored in which case additional Gibbs step is used within the MCMC algorithm

### Usage

    NMixMCMC(y0, y1, censor, scale, prior,
             init, init2, RJMCMC,
             nMCMC=c(burn=10, keep=10, thin=1, info=10),
             PED, keep.chains=TRUE, onlyInit=FALSE, dens.zero=1e-300)

    ## S3 method for class 'NMixMCMC':
    print(x, dic, ...)

    ## S3 method for class 'NMixMCMClist':
    print(x, ped, dic, ...)

### Arguments

`y0` | numeric vector of length n or n x p matrix with observed data. It contains exactly observed, right-censored, left-censored data and lower limits for interval-censored data.
`y1` | numeric vector of length n or n x p matrix with upper limits for interval-censored data. Elements corresponding to exactly observed, right-censored or left-censored data are ignored and can be filled arbitrarily (by `NA`'s) as well. It does not have to be supplied if there are no interval-censored data.
`censor` | numeric vector of length n or n x p matrix with censoring indicators. The following values indicate: 0 right-censored observation, 1 exactly observed value, 2 left-censored observation, 3 interval-censored observation. If it is not supplied then it is assumed that all values are exactly observed.
`scale` | a list specifying how to scale the data before running MCMC. It should have two components: shift a vector of length 1 or p specifying shift vector m, scale a vector of length 1 or p specifying diagonal of the scaling matrix S. If there is no censoring, and argument `scale` is missing then the data are scaled to have zero mean and unit variances, i.e., `scale(y0)` is used for MCMC. In the case there is censoring and `scale` is missing, `scale$shift` is taken to be a sample mean of `init$y` and `scale$scale` are sample standard deviations of columns of `init$y`. If you do not wish to scale the data before running MCMC, specify `scale=list(shift=0, scale=1)`.
`prior` | a list with the parameters of the prior distribution. It should have the following components (for some of them, the program can assign default values and the user does not have to specify them if he/she wishes to use the defaults): priorK a character string which specifies the type of the prior for K (the number of mixture components). It should have one of the following values: “`fixed`” Number of mixture components is assumed to be fixed to K[max]. This is a default value. “`uniform`” A priori K ~ Unif{1,...,K[max]}. “`tpoisson`” A priori K ~ truncated-Poiss(lambda, K[max]). priormuQ a character string which specifies the type of the prior for mu[1], ..., mu[K[max]] (mixture means) and Q[1], ..., Q[K[max]] (inverted mixture covariance matrices). It should have one of the following values: “`independentC`” = independent conjugate prior (this is a default value). That is, a priori (mu[j], Q[j]) ~ N(xi[j], D[j]) * Wishart(zeta, Xi) independently for j=1,...,K, where normal means xi[1],...,xi[K], normal variances D[1],...,D[K], and Wishart degrees of freedom zeta are specified further as `xi`, `D`, `zeta` components of the list `prior`. “`naturalC`” = natural conjugate prior. That is, a priori (mu[j], Q[j]) ~ N(xi[j], (c[j]Q[j])^(-1)) * Wishart(zeta, Xi) independently for j=1,...,K, where normal means xi[1],...,xi[K], precisions c[1],...,c[K], and Wishart degrees of freedom zeta are specified further as `xi`, `ce`, `zeta` components of the list `prior`. For both, independent conjugate and natural conjugate prior, the Wishart scale matrix Xi is assumed to be diagonal with gamma[1],...,gamma[p] on a diagonal. For gamma[j]^(-1) (j=1,...,K) additional gamma hyperprior G(g[j], h[j]) is assumed. Values of g[1],...,g[p] and h[1],...,h[p] are further specified as `g` and `h` components of the `prior` list. Kmax maximal number of mixture components K[max]. It must always be specified by the user. lambda parameter lambda for the truncated Poisson prior on K. It must be positive and must always be specified if `priorK` is “`tpoisson`”. delta parameter delta for the Dirichlet prior on the mixture weights w[1],...,w[K]. It must be positive. Its default value is 1. xi a numeric value, vector or matrix which specifies xi[1], ..., xi[K[max]] (prior means for the mixture means mu[1], ..., mu[K[max]]). Default value is a matrix K[max] x p with midpoints of columns of `init$y` in rows which follows Richardson and Green (1997). If p=1 and `xi`=xi is a single value then xi[1]=...=xi[K[max]] = xi. If p=1 and `xi`=boldsymbol{xi} is a vector of length K[max] then the j-th element of `xi` gives xi[j] (j=1,...,K[max]). If p>1 and `xi`=xi is a vector of length p then xi[1]=...=xi[K[max]] = xi. If p>1 and `xi` is a K[max] x p matrix then the j-th row of `xi` gives xi[j] (j=1,...,K[max]). ce a numeric value or vector which specifies prior precision parameters c[1],...,c[K[max]] for the mixture means mu[1], ..., mu[K[max]] when `priormuQ` is “`naturalC`”. Its default value is a vector of ones which follows Cappe, Robert and Ryden (2003). If `ce`=c is a single value then c[1]=...=c[K[max]]=c. If `ce` c is a vector of length K[max] then the j-th element of `ce` gives c[j] (j=1,...,K[max]). D a numeric vector or matrix which specifies D[1], ..., D[K[max]] (prior variances or covariance matrices of the mixture means mu[1], ..., mu[K[max]] when `priormuQ` is “`independentC`”.) Its default value is a diagonal matrix with squared ranges of each column of `init$y` on a diagonal. If p=1 and `D`=d is a single value then d[1]=...=d[K[max]] = d. If p=1 and `D`=d is a vector of length K[max] then the j-th element of `D` gives d[j] (j=1,...,K[max]). If p>1 and `D`=D is a p x p matrix then D[1]=...=D[K[max]] = D. If p>1 and `D` is a (K[max]*p) x p matrix then the the first p rows of `D` give D[1], rows p+1,...,2p of `D` give D[2] etc. zeta degrees of freedom zeta for the Wishart prior on the inverted mixture variances Q[1], ...,Q[K[max]].. It must be higher then p-1. Its default value is p + 1. g a value or a vector of length p with the shape parameters g[1],...,g[p] for the Gamma hyperpriors on gamma[1],...,gamma[p]. It must be positive. Its default value is a vector (0.2,...,0.2)'. h a value or a vector of length p with the rate parameters h[1],...,h[p] for the Gamma hyperpriors on gamma[1],...,gamma[p]. It must be positive. Its default value is a vector containing 10/R[l]^2, where R[l] is a range of the l-th column of `init$y`.
`init` | a list with the initial values for the MCMC. All initials can be determined by the program if they are not specified. The list may have the following components: y a numeric vector or matrix with the initial values for the latent censored observations. K a numeric value with the initial value for the number of mixture components. w a numeric vector with the initial values for the mixture weights. mu a numeric vector or matrix with the initial values for the mixture means. Sigma a numeric vector or matrix with the initial values for the mixture variances. Li a numeric vector with the initial values for the Colesky decomposition of the mixture inverse variances. gammaInv a numeric vector with the initial values for the inverted components of the hyperparameter gamma. r a numeric vector with the initial values for the mixture allocations.
`init2` | a list with the initial values for the second chain needed to estimate the penalized expected deviance of Plummer (2008). The list `init2` has the same structure as the list `init`. All initials in `init2` can be determined by the program (differently than the values in `init`) if they are not specified. Ignored when `PED` is `FALSE`.
`RJMCMC` | a list with the parameters needed to run reversible jump MCMC for mixtures with varying number of components. It does not have to be specified if the number of components is fixed. Most of the parameters can be determined by the program if they are not specified. The list may have the following components: Paction probabilities (or proportionalit constants) which are used to choose an action of the sampler within each iteration of MCMC to update the mixture related parameters. Let `Paction` = (p[1], p[2], p[3])'. Then with probability p[1] only steps assuming fixed k (number of mixture components) are performed, with probability p[2] split-combine move is proposed and with probability p[3] birth-death move is proposed. If not specified (default) then in each iteration of MCMC, all sampler actions are performed. Psplit a numeric vector of length `prior$Kmax` giving conditional probabilities of the split move given k as opposite to the combine move. Default value is (1, 0.5, ..., 0.5, 0)'. Pbirth a numeric vector of length `prior$Kmax` giving conditional probabilities of the birth move given k as opposite to the death move. Default value is (1, 0.5, ..., 0.5, 0)'. par.u1 a two component vector with parameters of the beta distribution used to generate an auxiliary value u[1]. A default value is `par.u1` = (2, 2)', i.e., u[1] ~ Beta(2, 2). par.u2 a two component vector (for p=1) or a matrix (for p > 1) with two columns with parameters of the distributions of the auxiliary values u[2,1],...,u[2,p] in rows. A default value leads to u[2,d] ~ Unif(-1, 1) (d=1,...,p-1), u[2,p] ~ Beta(1, 2p). par.u3 a two component vector (for p=1) or a matrix (for p > 1) with two columns with parameters of the distributions of the auxiliary values u[3,1],...,u[3,p] in rows. A default value leads to u[3,d] ~ Unif(0, 1) (d=1,...,p-1), u[3,p] ~ Beta(1, p).
`nMCMC` | numeric vector of length 4 giving parameters of the MCMC simulation. Its components may be named (ordering is then unimportant) as: burn length of the burn-in (after discarding the thinned values), can be equal to zero as well. keep length of the kept chains (after discarding the thinned values), must be positive. thin thinning interval, must be positive. info interval in which the progress information is printed on the screen. In total (M[burn] + M[keep]) * M[thin] MCMC scans are performed.
`PED` | a logical value which indicates whether the penalized expected deviance (see Plummer, 2008 for more details) is to be computed (which requires two parallel chains). If not specified, `PED` is set to `TRUE` for models with fixed number of components and is set to `FALSE` for models with numbers of components estimated using RJ-MCMC.
`keep.chains` | logical. If `FALSE`, only summary statistics are returned in the resulting object. This might be useful in the model searching step to save some memory.
`onlyInit` | logical. If `TRUE` then the function only determines parameters of the prior distribution, initial values, values of `scale` and parameters for the reversible jump MCMC.
`dens.zero` | a small value used instead of zero when computing deviance related quantities.
`x` | an object of class `NMixMCMC` or `NMixMCMClist` to be printed.
`dic` | logical which indicates whether DIC should be printed. By default, DIC is printed only for models with a fixed number of mixture components.
`ped` | logical which indicates whether PED should be printed. By default, PED is printed only for models with a fixed number of mixture components.
`...` | additional arguments passed to the default `print` method.

### Details

See accompanying paper (Komárek, 2009). In the rest of the helpfile, the same notation is used as in the paper, namely, n denotes the number of observations, p is dimension of the data, K is the number of mixture components, w[1],...,w[K] are mixture weights, mu[1],...,mu[K] are mixture means, Sigma_1,...,Sigma_K are mixture variance-covariance matrices, Q_1,...,Q_K are their inverses.

For the data y[1],...,y[n] the following g(y) density is assumed

g(y) = |S|^(-1) * sum[j=1]^K w[j] phi(S^(-1)*(y - m) | mu[j], Sigma[j]),

where phi(. | mu, Sigma) denotes a density of the (multivariate) normal distribution with mean mu and a~variance-covariance matrix Σ. Finally, S is a pre-specified diagonal scale matrix and m is a pre-specified shift vector. Sometimes, by setting m to sample means of components of y and diagonal of S to sample standard deviations of y (considerable) improvement of the MCMC algorithm is achieved.

### Value

An object of class `NMixMCMC` or class `NMixMCMClist`. Object of class `NMixMCMC` is returned if `PED` is `FALSE`. Object of class `NMixMCMClist` is returned if `PED` is `TRUE`.
Objects of class `NMixMCMC` have the following components:

`iter` | index of the last iteration performed.
`nMCMC` | used value of the argument `nMCMC`.
`dim` | dimension p of the distribution of data
`prior` | a list containing the used value of the argument `prior`.
`init` | a list containing the used value of the argument `init`.
`RJMCMC` | a list containing the used value of the argument `RJMCMC`.
`scale` | a list containing the used value of the argument `scale`.
`state` | a list having the components labeled `y`, `K`, `w`, `mu`, `Li`, `Q`, `Sigma`, `gammaInv`, `r` containing the last sampled values of generic parameters.
`freqK` | frequency table of K based on the sampled chain.
`propK` | posterior distribution of K based on the sampled chain.
`DIC` | a `data.frame` having columns labeled `DIC`, `pD`, `D.bar`, `D.in.bar` containing values used to compute deviance information criterion (DIC). Currently only DIC[3] of Celeux et al. (2006) is implemented.
`moves` | a `data.frame` which summarizes the acceptance probabilities of different move types of the sampler.
`K` | numeric vector with a chain for K (number of mixture components).
`w` | numeric vector or matrix with a chain for w (mixture weights). It is a matrix with K columns when K is fixed. Otherwise, it is a vector with weights put sequentially after each other.
`mu` | numeric vector or matrix with a chain for mu (mixture means). It is a matrix with p*K columns when K is fixed. Otherwise, it is a vector with means put sequentially after each other.
`Q` | numeric vector or matrix with a chain for lower triangles of Q (mixture inverse variances). It is a matrix with (p*(p+1)2)*K columns when K is fixed. Otherwise, it is a vector with lower triangles of Q matrices put sequentially after each other.
`Sigma` | numeric vector or matrix with a chain for lower triangles of Sigma (mixture variances). It is a matrix with (p*(p+1)2)*K columns when K is fixed. Otherwise, it is a vector with lower triangles of Sigma matrices put sequentially after each other.
`Li` | numeric vector or matrix with a chain for lower triangles of Cholesky decompositions of Q matrices. It is a matrix with (p*(p+1)2)*K columns when K is fixed. Otherwise, it is a vector with lower triangles put sequentially after each other.
`gammaInv` | matrix with p columns with a chain for inverses of the hyperparameter gamma.
`order` | numeric vector or matrix with order indeces of mixture components. It is a matrix with K columns when K is fixed. Otherwise it is a vector with orders put sequentially after each other.
`rank` | numeric vector or matrix with rank indeces of mixture components. It is a matrix with K columns when K is fixed. Otherwise it is a vector with ranks put sequentially after each other.
`mixture` | `data.frame` with columns labeled `y.Mean.*`, `y.SD.*`, `y.Corr.*.*`, `z.Mean.*`, `z.SD.*`, `z.Corr.*.*` containing the chains for the means, standard deviations and correlations of the distribution of the original (`y`) and scaled (`z`) data based on a normal mixture at each iteration.
`deviance` | `data.frame` with columns labeles `LogL0`, `LogL1`, `dev.complete`, `dev.observed` containing the chains of quantities needed to compute DIC.
`pm.y` | a `data.frame` with p columns with posterior means for (latent) values of observed data (useful when there is censoring).
`pm.z` | a `data.frame` with p columns with posterior means for (latent) values of scaled observed data (useful when there is censoring).
`pm.indDev` | a `data.frame` with columns labeled `LogL0`, `LogL1`, `dev.complete`, `dev.observed`, `pred.dens` containing posterior means of individual contributions to the deviance.
`pred.dens` | a numeric vector with the predictive density of the data based on the MCMC sample evaluated at data points. Note that when there is censoring, this is not exactly the predictive density as it is computed as the average of densities at each iteration evaluated at sampled values of latent observations at iterations.
`poster.comp.prob1` | a matrix which is present in the output object if the number of mixture components in the distribution of random effects is fixed and equal to K. In that case, `poster.comp.prob1` is a matrix with K columns and n rows with estimated posterior component probabilities – posterior means of the components of the underlying 0/1 allocation vector. These can be used for possible clustering of the subjects.
`poster.comp.prob2` | a matrix which is present in the output object if the number of mixture components in the distribution of random effects is fixed and equal to K. In that case, `poster.comp.prob2` is a matrix with K columns and n rows with estimated posterior component probabilities – posterior mean over model parameters. These can also be used for possible clustering of the subjects
`summ.y.Mean` | Posterior summary statistics based on chains stored in `y.Mean.*` columns of the `data.frame` `mixture`.
`summ.y.SDCorr` | Posterior summary statistics based on chains stored in `y.SD.*` and `y.Corr.*.*` columns of the `data.frame` `mixture`.
`summ.z.Mean` | Posterior summary statistics based on chains stored in `z.Mean.*` columns of the `data.frame` `mixture`.
`summ.z.SDCorr` | Posterior summary statistics based on chains stored in `z.SD.*` and `z.Corr.*.*` columns of the `data.frame` `mixture`.
`poster.mean.w` | a numeric vector with posterior means of mixture weights after re-labeling. It is computed only if K is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
`poster.mean.mu` | a matrix with posterior means of mixture means after re-labeling. It is computed only if K is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
`poster.mean.Q` | a list with posterior means of mixture inverse variances after re-labeling. It is computed only if K is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
`poster.mean.Sigma` | a list with posterior means of mixture variances after re-labeling. It is computed only if K is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.
`poster.mean.Li` | a list with posterior means of Cholesky decompositions of mixture inverse variances after re-labeling. It is computed only if K is fixed and even then I am not convinced that these are useful posterior summary statistics. In any case, they should be used with care.

### Object of class NMixMCMClist

Objects of class `NMixMCMClist` is the list having two components of class `NMixMCMC` representing two parallel chains and additionally the following components:

PED
  values of penalized expected deviance and related quantities. It is a vector with five components: `D.expect` = estimated expected deviance, where the estimate is based on two parallel chains; `popt` = estimated penalty, where the estimate is based on simple MCMC average based on two parallel chains; `PED` = estimated penalized expected deviance = `D.expect` + `popt`; `wpopt` = estimated penalty, where the estimate is based on weighted MCMC average (through importance sampling) based on two parallel chains; `wPED` = estimated penalized expected deviance = `D.expect` + `wpopt`.
popt
  contributions to the unweighted penalty from each observation.
wpopt
  contributions to the weighted penalty from each observation.
inv.D
  for each observation, number of iterations (in both chains), where the deviance was in fact equal to infinity (when the corresponding density was lower than `dens.zero`) and was not taken into account when computing `D.expect`.
inv.popt
  for each observation, number of iterations, where the penalty was in fact equal to infinity and was not taken into account when computing `popt`.
inv.wpopt
  for each observation, number of iterations, where the importance sampling weight was in fact equal to infinity and was not taken into account when computing `wpopt`.
sumISw
  for each observation, sum of importance sampling weights.

### Author(s)

Arnošt Komárek arnost.komarek[AT]mff.cuni.cz

### References

Celeux, G., Forbes, F., Robert, C. P., and Titterington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1, 651-674.

Cappé, Robert and Rydén (2003). Reversible jump, birth-and-death and more general continuous time Markov chain Monte Carlo samplers. Journal of the Royal Statistical Society, Series B, 65, 679-700.

Diebolt, J. and Robert, C. P. (1994). Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56, 363–375.

Komárek, A. (2009). A new R package for Bayesian estimation of multivariate normal mixtures allowing for selection of the number of components and interval-censored data. Computational Statistics and Data Analysis, 53, 3932–3947.

Plummer, M. (2008). Penalized loss functions for Bayesian model comparison. Biostatistics, 9, 523-539.

Richardson, S. and Green, P. J. (1997). On Bayesian analysis of mixtures with unknown number of components (with Discussion). Journal of the Royal Statistical Society, Series B, 59, 731-792.

Spiegelhalter, D. J.,Best, N. G., Carlin, B. P., and van der Linde, A. (2002). Bayesian measures of model complexity and fit (with Discussion). Journal of the Royal Statistical Society, Series B, 64, 583-639.

### See Also

`NMixPredDensMarg`, `NMixPredDensJoint2`.

### Examples

    ## See also additional material available in
    ## YOUR_R_DIR/library/mixAK/doc/
    ## or YOUR_R_DIR/site-library/mixAK/doc/
    ## - files Galaxy.pdf, Faithful.pdf, Tandmob.pdf
    ## ==============================================

    ## Simple analysis of Anderson's iris data
    ## ==============================================
    library("colorspace")

    data(iris, package="datasets")
    summary(iris)
    VARS <- names(iris)[1:4]
    #COLS <- rainbow_hcl(3, start = 60, end = 240)
    COLS <- c("red", "darkblue", "darkgreen")
    names(COLS) <- levels(iris[, "Species"])

    ### Prior distribution and the length of MCMC
    Prior <- list(priorK = "fixed", Kmax = 3)
    nMCMC <- c(burn=5000, keep=10000, thin=5, info=1000)

    ### Run MCMC
    set.seed(20091230)
    fit <- NMixMCMC(y0 = iris[, VARS], prior = Prior, nMCMC = nMCMC)

    ### Basic posterior summary
    print(fit)

    ### Univariate marginal posterior predictive densities
    ### based on chain #1
    pdens1 <- NMixPredDensMarg(fit[[1]], lgrid=150)
    plot(pdens1)
    plot(pdens1, main=VARS, xlab=VARS)

    ### Bivariate (for each pair of margins) predictive densities
    ### based on chain #1
    pdens2a <- NMixPredDensJoint2(fit[[1]])
    plot(pdens2a)

    plot(pdens2a, xylab=VARS)
    plot(pdens2a, xylab=VARS, contour=TRUE)

    ### Determine the grid to compute bivariate densities
    grid <- list(Sepal.Length=seq(3.5, 8.5, length=75),
                 Sepal.Width=seq(1.8, 4.5, length=75),
                 Petal.Length=seq(0, 7, length=75),
                 Petal.Width=seq(-0.2, 3, length=75))
    pdens2b <- NMixPredDensJoint2(fit[[1]], grid=grid)
    plot(pdens2b, xylab=VARS)

    ### Plot with contours
    ICOL <- rev(heat_hcl(20, c=c(80, 30), l=c(30, 90), power=c(1/5, 2)))
    oldPar <- par(mfrow=c(2, 3), bty="n")
    for (i in 1:3){
      for (j in (i+1):4){
        NAME <- paste(i, "-", j, sep="")
        MAIN <- paste(VARS[i], "x", VARS[j])
        image(pdens2b$x[[i]], pdens2b$x[[j]], pdens2b$dens[[NAME]], col=ICOL,
              xlab=VARS[i], ylab=VARS[j], main=MAIN)
        contour(pdens2b$x[[i]], pdens2b$x[[j]], pdens2b$dens[[NAME]], add=TRUE, col="brown4")
      }
    }

    ### Plot with data
    for (i in 1:3){
      for (j in (i+1):4){
        NAME <- paste(i, "-", j, sep="")
        MAIN <- paste(VARS[i], "x", VARS[j])
        image(pdens2b$x[[i]], pdens2b$x[[j]], pdens2b$dens[[NAME]], col=ICOL,
              xlab=VARS[i], ylab=VARS[j], main=MAIN)
        for (spec in levels(iris[, "Species"])){
          Data <- subset(iris, Species==spec)
          points(Data[,i], Data[,j], pch=16, col=COLS[spec])
        }
      }
    }

    ### Set the graphical parameters back to their original values
    par(oldPar)

    ### Clustering based on posterior summary statistics of component allocations
    ### or on the posterior distribution of component allocations
    ### (these are two equivalent estimators of probabilities of belonging
    ###  to each mixture components for each observation)
    p1 <- fit[[1]]$poster.comp.prob1
    p2 <- fit[[1]]$poster.comp.prob2

    ### Clustering based on posterior summary statistics of mixture weight, means, variances
    p3 <- NMixPlugDA(fit[[1]], iris[, VARS])
    p3 <- p3[, paste("prob", 1:3, sep="")]

      ### Observations from "setosa" species (all would be allocated in component 1)
    apply(p1[1:50,], 2, quantile, prob=seq(0, 1, by=0.1))
    apply(p2[1:50,], 2, quantile, prob=seq(0, 1, by=0.1))
    apply(p3[1:50,], 2, quantile, prob=seq(0, 1, by=0.1))

      ### Observations from "versicolor" species (almost all would be allocated in component 2)
    apply(p1[51:100,], 2, quantile, prob=seq(0, 1, by=0.1))
    apply(p2[51:100,], 2, quantile, prob=seq(0, 1, by=0.1))
    apply(p3[51:100,], 2, quantile, prob=seq(0, 1, by=0.1))

      ### Observations from "virginica" species (all would be allocated in component 3)
    apply(p1[101:150,], 2, quantile, prob=seq(0, 1, by=0.1))
    apply(p2[101:150,], 2, quantile, prob=seq(0, 1, by=0.1))
    apply(p3[101:150,], 2, quantile, prob=seq(0, 1, by=0.1))

[Package mixAK version 0.6 Index]