Fit a matrix variate mixture model

Clustering by fitting a mixture model using EM with K groups and unconstrained covariance matrices for a matrix variate normal or matrix variate t distribution (with specified degrees of freedom nu).

matrixmixture(
  x,
  init = NULL,
  prior = NULL,
  K = length(prior),
  iter = 1000,
  model = "normal",
  method = NULL,
  row.mean = FALSE,
  col.mean = FALSE,
  tolerance = 0.1,
  nu = NULL,
  ...,
  verbose = 0,
  miniter = 5,
  convergence = TRUE
)

Arguments

x: data, \(p \times q \times n\) array
init: a list containing an array of K of \(p \times q\) means labeled centers, and optionally \(p \times p\) and \(q \times q\) positive definite variance matrices labeled U and V. By default, those are presumed to be identity if not provided. If init is missing, it will be provided using the prior or K by init_matrixmix.
prior: prior for the K classes, a vector that adds to unity
K: number of classes - provide either this or the prior. If this is provided, the prior will be of uniform distribution among the classes.
iter: maximum number of iterations.
model: whether to use the normal or t distribution.
method: what method to use to fit the distribution. Currently no options.
row.mean: By default, FALSE. If TRUE, will fit a common mean within each row. If both this and col.mean are TRUE, there will be a common mean for the entire matrix.
col.mean: By default, FALSE. If TRUE, will fit a common mean within each row. If both this and row.mean are TRUE, there will be a common mean for the entire matrix.
tolerance: convergence criterion, using Aitken acceleration of the log-likelihood by default.
nu: degrees of freedom parameter. Can be a vector of length K.
...: pass additional arguments to MLmatrixnorm or MLmatrixt
verbose: whether to print diagnostic output, by default 0. Higher numbers output more results.
miniter: minimum number of iterations
convergence: By default, TRUE, using Aitken acceleration to determine convergence. If false, it instead checks if the change in log-likelihood is less than tolerance. Aitken acceleration may prematurely end in the first few steps, so you may wish to set miniter or select FALSE if this is an issue.

Value

A list of class MixMatrixModel containing the following components:

prior: the prior probabilities used.
init: the initialization used.
K: the number of groups
N: the number of observations
centers: the group means.
U: the between-row covariance matrices
V: the between-column covariance matrix
posterior: the posterior probabilities for each observation
pi: the final proportions
nu: The degrees of freedom parameter if the t distribution was used.
convergence: whether the model converged
logLik: a vector of the log-likelihoods of each iteration ending in the final log-likelihood of the model
model: the model used
method: the method used
call: The (matched) function call.

References

Andrews, Jeffrey L., Paul D. McNicholas, and Sanjeena Subedi. 2011.
  "Model-Based Classification via Mixtures of Multivariate
  T-Distributions." Computational Statistics & Data Analysis 55 (1):
  520–29. \doi{10.1016/j.csda.2010.05.019}.

Fraley, Chris, and Adrian E Raftery. 2002. "Model-Based Clustering,
   Discriminant Analysis, and Density Estimation." Journal of the
   American Statistical Association 97 (458). Taylor & Francis: 611–31.
   \doi{10.1198/016214502760047131}.

McLachlan, Geoffrey J, Sharon X Lee, and Suren I Rathnayake. 2019.
      "Finite Mixture Models." Annual Review of Statistics and Its
      Application 6. Annual Reviews: 355–78.
      \doi{10.1146/annurev-statistics-031017-100325}.

Viroli, Cinzia. 2011. "Finite Mixtures of Matrix Normal Distributions
      for Classifying Three-Way Data." Statistics and Computing 21 (4):
      511–22. \doi{10.1007/s11222-010-9188-x}.

Examples

set.seed(20180221)
A <- rmatrixt(20,mean=matrix(0,nrow=3,ncol=4), df = 5)
# 3x4 matrices with mean 0
B <- rmatrixt(20,mean=matrix(1,nrow=3,ncol=4), df = 5)
# 3x4 matrices with mean 1
C <- array(c(A,B), dim=c(3,4,40)) # combine into one array
prior <- c(.5,.5) # equal probability prior
# create an intialization object, starts at the true parameters
init = list(centers = array(c(rep(0,12),rep(1,12)), dim = c(3,4,2)),
              U = array(c(diag(3), diag(3)), dim = c(3,3,2))*20,
              V = array(c(diag(4), diag(4)), dim = c(4,4,2))
 )
# fit model
 res<-matrixmixture(C, init = init, prior = prior, nu = 5,
                    model = "t", tolerance = 1e-3, convergence = FALSE)
print(res$centers) # the final centers
#> , , 1
#> 
#>              [,1]        [,2]        [,3]        [,4]
#> [1,]  0.117537755  0.05142832  0.03229732 -0.02163010
#> [2,]  0.005705083 -0.03489656 -0.03159498 -0.05027392
#> [3,] -0.041192795 -0.10381233 -0.07923590 -0.06302149
#> 
#> , , 2
#> 
#>          [,1]      [,2]      [,3]      [,4]
#> [1,] 1.065892 1.0573621 0.8615166 1.0380062
#> [2,] 1.069079 0.9432203 1.0902007 0.9915732
#> [3,] 1.080176 1.0671593 0.9880986 0.7968937
#> 
print(res$pi) # the final mixing proportion
#> [1] 0.4987695 0.5012305
plot(res) # the log likelihood by iteration

logLik(res) # log likelihood of final result
#> 'log Lik.' -414.8875 (df=54)
BIC(res) # BIC of final result
#> [1] 1028.975
predict(res, newdata = C[,,c(1,21)]) # predicted class membership
#> $class
#> [1] 1 2
#> 
#> $posterior
#>              [,1]         [,2]
#> [1,] 9.999883e-01 1.170306e-05
#> [2,] 6.703426e-07 9.999993e-01
#>

Arguments

Value

References

See also

Examples