Author: Geoffrey Thompson
Date: 2024-09-30

# ML estimation of the Matrix Variate t Distribution

#### Geoffrey Thompson

#### 2024-09-30

    library(MixMatrix)

## Estimation of the Matrix Variate t Distribution

The parameters of the multivariate t distribution can be estimated using the EM algorithm. An EM algorithm for the multivariate \(t\)-distribution with no missing data was provided by Rubin (1983). This was variously extended and refined, as the EM algorithm can be quite slow to converge. One set of refinements split the M-step into a series of conditional maximization (CM) steps (Meng and Rubin 1993), and then a later refinement (called ECME - Expectation/Conditional Maximization Either) allowed for either conditional maximization steps or maximizing the constrained actual likelihood (Liu and Rubin 1994), which leads to dramatically faster convergence. The matrix variate \(t\) is an extension of the multivariate version (Dickey 1967); however, unlike the normal distribution, it cannot be treated as a rearrangement of the multivariate case, so the addition of the extra dimension poses a non-trivial problem for extending the results.

An \(n \times p\) random matrix \(\mathbf{X}\) is distributed as a matrix variate \(t\) random variable if it has probability density function as follows: \[f({\mathbf {X}};\nu,{\mathbf {M}}, {\boldsymbol \Omega }, {\boldsymbol \Sigma }) = {\frac {\Gamma _{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{{\frac {np}{2}}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol \Omega }|^{{-{\frac {n}{2}}}}|{\boldsymbol \Sigma }|^{{-{\frac {p}{2}}}}\times \left|{\mathbf {I}}_{n}+{\boldsymbol \Sigma }^{{-1}}({\mathbf {X}}-{\mathbf {M}}){\boldsymbol \Omega }^{{-1}}({\mathbf {X}}-{\mathbf {M}})^{{{\rm {T}}}}\right|^{{-{\frac {\nu +n+p-1}{2}}}}\]

With \(\Omega\) and \(\Sigma\) covariance matrices of appropriate dimension, \(\mathbf{M}\) a mean matrix, \(\nu\) the degrees of freedom parameter, and \(\Gamma_p()\) the multivariate gamma function, which is implemented in the `CholWishart` package available on CRAN.

For \(p = 1\) and \(\Sigma = \nu\) (or \(q = 1\) and \(\Omega = \nu\)), this reduces to the familiar multivariate \(t\) distribution. If the \(p \times q\) dimensional random variable \(X\) follows the matrix variate \(t\) distribution \(t(\nu, M, \Sigma, \Omega)\) with center \(M\), positive definite spread matrices \(\Sigma\) (\(p \times p\)) and \(\Omega\) (\(q \times q\)), and degrees of freedom \(\nu\), it can be shown that, given a certain weight matrix \(S\), \(X\) has a matrix variate normal distribution and that \(S\) is Wishart-distributed. Specifically, \[\begin{align*} X | M, \Sigma, \Omega, \nu, S & \sim N(M, S^{-1} \otimes \Omega) \\ S|M, \Omega, \Sigma, \nu & \sim W_p(\nu + p -1, \Sigma^{-1}) \end{align*}\] Additionally, it can be shown the Wishart is the conjugate prior distribution for the parameter \(\Sigma\), and thus the conditional posterior distribution of \(S\), i.e., its distribution given \((M, \Omega, \Sigma, \nu, X)\) is \[S | X, M, \Omega, \Sigma, \nu \sim \mathrm{W}_p(\nu+p+q-1, [(X-M)\Omega^{-1}(X-M)^T + \Sigma]^{-1}) \]

For an observed set of \(X_i\), \(i = 1, 2, \ldots, N\), then, we can construct an ECM-style algorithm by augmenting our data with a set of weights, \(S_i\), and use this to estimate the parameters of the matrix variate \(t\) distribution. Briefly, we define a set of complete data sufficient statistics based on the \(S_i\) and update them with their expectations in the E-step. Then, the other parameters are maximized in the CM steps.

The complete data sufficient statistics for the quantities to be estimated are: \[S_{SX} =\sum_{i = 1}^N S_i X_i; \quad S_S = \sum_{i = 1}^N S_i; \quad S_{XSX} = \sum_{i = 1}^N X_i^T S_i X_i \quad S_{|S|} = \sum_{i = 1}^N \log |Si|\]

### E-step

Define at step \(t\) the set \(\Theta^{(t)} = (X_{obs},\nu^{(t)}, M^{(t)}, \Sigma^{(t)}, \Omega^{(t)})\). Then, given these values, we have, based on the properties of the Wishart distribution, \[S_{i}^{(t+1)} = E(S_i | \Theta^{(t)}) = (\nu^{(t)}+p+q-1)[(X_i-M^{(t)})\Omega^{(t)^{-1 }}(X_i-M^{(t)})^T + \Sigma^{(t)}]^{-1} \] From this we can derive the other expected values: \[ \begin{align*} S_{S}^{(t+1)} &= E(S_S | \Theta^{(t)}) = \sum_{i=1}^N S_{i}^{(t+1)} \\ S_{SX}^{(t+1)} &= E(S_{SX} | \Theta^{(t)}) = \sum_{i=1}^NE(S_iX_i|\Theta^{(t)}) = \sum_{i=1}^N S_{i}^{(t+1)}X_i \\ S_{XSX}^{(t+1)} &= E(S_{XSX} | \Theta^{(t)}) = \sum_{i=1}^NE(X_i^TS_iX_i|\Theta^{(t)}) = \sum_{i=1}^N X_i^TS_{i}^{(t+1)}X_i \end{align*} \] This last value only needs to be computed if \(\nu\) is unknown: \[ \begin{align*} S_{|S|}^{(t+1)} &= E(S_{|S|} | \Theta^{(t)}) = E \left[\sum_{i = 1}^N \log| S_i| \big| \Theta^{(t)} \right]\\ &=N\psi_{p}\left({\frac {\nu^{(t)} + p + q -1}{2}}\right) + Np \log 2 +\sum_{i=1}^N \log \left|\frac{S_{i,obs}^{(t+1)}}{\nu^{(t)} + p + q -1}\right| \end{align*} \]

### CM Steps

Similarly to the case of the multivariate \(t\), it is more efficient to partition the maximization step into multiple steps. This is an ECME algorithm that first maximizes the expected log-likelihood for \((M, \Sigma, \Omega)\) and then maximizes the actual log-likelihood over \(\nu\) given \((M, \Sigma, \Omega)\), similar to .

Based on the updated weight matrices \(S_i^{(t+1)}\) and statistics based on \(\Theta^{(t)}\) and \(X_{obs}\),

\[ \begin{align*} \widehat{M} &= \left(\sum_{i = 1}^NS_i^{(t+1)}\right)^{-1}\sum_{i=1}^NS_iX_i = {S_{S}^{(t+1)}}^{-1} S_{SX}^{(t+1)} \\ \widehat{\Omega} &= \frac{1}{Np} \sum_{i = 1}^N (X_i - {M}^{(t)})^T S_i^{(t+1)} (X_i - {M}^{(t)}) =\frac{1}{Np} \left(S_{XSX}^{(t+1)} - {S_{SX}^{(t+1)}}^T {S_{S}^{(t+1)}}^{-1}{S_{SX}^{(t+1)}} \right) \\ \widehat{\Sigma}^{-1} &= \frac{1}{N(\nu^{(t)}+p-1)}\sum_{i = 1}^N S_i^{(t+1)} = \frac{S_S^{(t+1)}}{N(\nu^{(t)}+p-1)} \end{align*} \] And, again, with the set of \(S_i\) observed, the MLE of \(\nu\) can be obtained: \[ N \frac{d}{d\nu}\log \Gamma_p ((\nu+p-1)/2) - \frac{1}{2} (S_{|S|} - Np\log2 + N \log |\widehat{\Sigma}|) = 0\] Note that \(\nu > p -1\). Specifically, we have: \[ \begin{align*} 0 &= N \psi_p((\nu+p-1)/2) - \left(N\psi_{p}\left({\frac {\nu + p + q -1}{2}}\right) +\sum_{i=1}^N \log \left|\frac{S_{i,obs}^{(t+1)}}{\nu + p + q -1}\right| - N \log \left|\frac{S_S^{(t+1)}}{N(\nu+p-1)}\right|\right) \\* &= \psi_p((\nu+p-1)/2) - \left(\psi_{p}\left({\frac {\nu + p + q -1}{2}}\right) +\frac{1}{N}\sum_{i=1}^N \log \left|Z_{i,obs}^{(t+1)}\right| + p \log \frac{N(\nu + p -1)}{\nu + p +q -1} - \log \left|Z_S^{(t+1)}\right|\right) \end{align*}\] where \(Z_{*}\) is the appropriate \(S_{*}\) statistic with \((\nu + p + q -1)\) factored out and \(\psi_p\) is the \(p\)-dimensional digamma function. This can be solved for \(\nu\) using a 1-dimensional search.

## Applications and results

### With \(\nu\) known

If the degrees of freedom parameter, \(\nu\) known, the estimation is fairly straightforward. The procedure is similar to the multivariate \(t\) or the matrix variate normal. In this case, the interface is just like the interface for the `MLmatrixnorm()` function:

    set.seed(20190622)
    sigma = (1/7) * rWishart(1, 7, 1*diag(3:1))[,,1]
    A = rmatrixt(n=100,mean=matrix(c(100,0,-100,0,25,-1000),nrow=2),
       V = sigma, df = 7)
    results=MLmatrixt(A, df = 7)
    print(results)
    #> $mean
    #>            [,1]          [,2]        [,3]
    #> [1,] 99.8291911 -1.000708e+02    25.01042
    #> [2,] -0.0425028 -8.261356e-03 -1000.02357
    #>
    #> $U
    #>             [,1]        [,2]
    #> [1,]  1.00000000 -0.08039981
    #> [2,] -0.08039981  0.97069319
    #>
    #> $V
    #>           [,1]       [,2]       [,3]
    #> [1,] 1.0000000 0.21375613 0.11094767
    #> [2,] 0.2137561 0.14872070 0.04098332
    #> [3,] 0.1109477 0.04098332 0.12860660
    #>
    #> $var
    #> [1] 5.148569
    #>
    #> $nu
    #> [1] 7
    #>
    #> $iter
    #> [1] 44
    #>
    #> $tol
    #> [1] 1.155385e-08
    #>
    #> $logLik
    #> [1] -386.7623
    #>
    #> $convergence
    #> [1] TRUE
    #>
    #> $call
    #> MLmatrixt(data = A, df = 7)

There are two restrictions possible for the mean matrices: `row.mean = TRUE` will force a common mean within a row and `col.mean = TRUE` will force a common mean within a column. Setting both will ensure a constant mean for the entire system. Restrictions on \(\mathbf{U}\) and \(\mathbf{V}\), the row-wise variance and column-wise variance, are possible with `row.variance` and `col.variance` commands.

The options for variance restrictions are the same as for the `MLmatrixnorm()` function. Currently the options for variance restrictions are to impose an AR(1) structure by providing the `AR(1)` option, a compound symmetry structure by providing the `CS` option, to impose a correlation matrix structure by specifying `correlation` or `corr`, or to impose an identical and independent structure by specifying `Independent` or `I`. This works by using `uniroot` to find the appropriate \(\rho\) which sets the derivative of the log-likelihood to zero for the `AR` and `CS` options - it is not fast but if this is the true structure it will be better in some sense than an unstructured variance matrix. The \(\rho\) parameter should be \(>0\) and is forced to be non-negative. If the data behaves incompatibly with those restrictions, the function will provide a warning and exit with the current model fit.

### With \(\nu\) unknown

Estimation of \(\nu\), the degrees of freedom parameter, is slow and the principal mathematical difficulty of the matrix-variate \(t\) distribution. It is performed using ECME. Generally, a fair amount more data are needed in order to have good convergence properties for the estimator, but they have not been derived analytically. Here you can see the recovery of the parameter for a few sample sizes. Because of the relative slowness of running a longer simulation, this only includes one set of examples. I give code for a longer and larger simulation than what is plotted if you’re really interested below. What is plotted below is only the `df = 10` example with 75 trials and a maximum number of iterations `max.iter = 20`. The full simulation may take several minutes.

samplesize | min | max | median | mean | sd
--- | --- | --- | --- | --- | ---
25 | 7.059192 | 15.76929 | 10.316514 | 10.512701 | 2.0603677
50 | 7.220806 | 14.47148 | 10.424858 | 10.415426 | 1.6347640
100 | 8.281614 | 11.85454 | 9.826301 | 9.859941 | 0.8019631

As expected, increased sample size leads to better results in recovering the parameter. The results for smaller sample sizes would be more divergent if left to run until convergence.

## Use for classification

Using the \(t\) distribution works in both `matrixlda()` and `matrixqda()` as expected by specifying `method = "t"` and providing either a single parameter (for `lda` or `qda`) for the degrees of freedom or a vector as long as the number of classes (for `qda`). Additional parameters for fitting can be passed through the `...` to `MLmatrixt()` just as for the normal case, including estimating the degrees of freedom parameter. The `qda` will only estimate `nu` with it varying between groups, it will not estimate a common `nu`.

    A <- rmatrixt(30, mean = matrix(0, nrow=2, ncol=3), df = 10)
    B <- rmatrixt(30, mean = matrix(c(1,0), nrow=2, ncol=3), df = 10)
    C <- rmatrixt(30, mean = matrix(c(0,1), nrow=2, ncol=3), df = 10)
    ABC <- array(c(A,B,C), dim = c(2,3,90))
    groups <- factor(c(rep("A",30),rep("B",30),rep("C",30)))
    prior = c(30,30,30)/90
    matlda <- matrixlda(x = ABC,grouping = groups, prior = prior,
                        method = 't', nu = 10, fixed = TRUE)
    predict(matlda, newdata = ABC[,,c(1,31,61)])
    #> $class
    #> [1] A B C
    #> Levels: A B C
    #>
    #> $posterior
    #>              [,1]         [,2]         [,3]
    #> [1,] 9.997064e-01 2.849389e-04 8.693417e-06
    #> [2,] 4.174839e-05 9.999508e-01 7.439619e-06
    #> [3,] 8.908508e-06 4.857627e-06 9.999862e-01

## Session info

This vignette was built using `rmarkdown`.

    sessionInfo()
    #> R version 4.4.1 (2024-06-14)
    #> Platform: x86_64-redhat-linux-gnu
    #> Running under: Fedora Linux 40 (Xfce)
    #>
    #> Matrix products: default
    #> BLAS/LAPACK: FlexiBLAS OPENBLAS-OPENMP;  LAPACK version 3.11.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.utf8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.utf8        LC_COLLATE=C
    #>  [5] LC_MONETARY=en_US.utf8    LC_MESSAGES=en_US.utf8
    #>  [7] LC_PAPER=en_US.utf8       LC_NAME=C
    #>  [9] LC_ADDRESS=C              LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.utf8 LC_IDENTIFICATION=C
    #>
    #> time zone: America/Chicago
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] magrittr_2.0.3  dplyr_1.1.4     ggplot2_3.5.1   MixMatrix_0.2.8
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] vctrs_0.6.5         CholWishart_1.1.2.1 cli_3.6.3
    #>  [4] knitr_1.48          rlang_1.1.4         xfun_0.47
    #>  [7] highr_0.11          generics_0.1.3      jsonlite_1.8.9
    #> [10] labeling_0.4.3      glue_1.7.0          colorspace_2.1-1
    #> [13] htmltools_0.5.8.1   sass_0.4.9          fansi_1.0.6
    #> [16] scales_1.3.0        rmarkdown_2.28      grid_4.4.1
    #> [19] evaluate_1.0.0      munsell_0.5.1       jquerylib_0.1.4
    #> [22] tibble_3.2.1        fastmap_1.2.0       yaml_2.3.10
    #> [25] lifecycle_1.0.4     compiler_4.4.1      pkgconfig_2.0.3
    #> [28] Rcpp_1.0.13         farver_2.1.2        digest_0.6.37
    #> [31] R6_2.5.1            tidyselect_1.2.1    utf8_1.2.4
    #> [34] pillar_1.9.0        bslib_0.8.0         withr_3.0.1
    #> [37] tools_4.4.1         gtable_0.3.5        cachem_1.1.0

## All the code for easy copying

    knitr::opts_chunk$set(
      collapse = TRUE,
      comment = "#>"
    )
    set.seed(20190622)
    sigma = (1/7) * rWishart(1, 7, 1*diag(3:1))[,,1]
    A = rmatrixt(n=100,mean=matrix(c(100,0,-100,0,25,-1000),nrow=2),
       V = sigma, df = 7)
    results=MLmatrixt(A, df = 7)
    print(results)
    ### Here is the long simulation
    library(ggplot2)

    set.seed(20181102)

    df = c(5, 10, 20)
    df5 <- rep(0,200)
    df10 <- rep(0,200)
    df100 <- rep(0,200)
    df550 <-  rep(0,200)
    df1050 <-  rep(0,200)
    df2050 <- rep(0,200)
    df5100 <-  rep(0,200)
    df10100 <-  rep(0,200)
    df20100 <- rep(0,200)

    meanmat = matrix(0,5,3)
    U = diag(5)
    V = diag(3)

    for(i in 1:200){
        df5[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 5, n = 35, U =U, V =V), fixed = FALSE)$nu
        df10[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 10, n = 35, U =U, V =V), fixed = FALSE)$nu
        df100[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 20, n = 35, U =U, V =V), fixed = FALSE)$nu
        df550[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 5, n = 50, U =U, V =V), fixed = FALSE)$nu
        df1050[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 10, n = 50, U =U, V =V), fixed = FALSE)$nu
        df2050[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 20, n = 50, U =U, V =V), fixed = FALSE)$nu
        df5100[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 5, n = 100, U =U, V =V), fixed = FALSE)$nu
        df10100[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 10, n = 100, U =U, V =V), fixed = FALSE)$nu
        df20100[i] = MLmatrixt(rmatrixt(mean = meanmat,
            df = 20, n = 100, U =U, V =V), fixed = FALSE)$nu
    }

    truedataframe = data.frame(truedf = factor(c(5,10,20),
                                               label = c('5 df', '10 df', '20 df')),
                               estdf = c(5,10,20))

    dfdataframe = data.frame(truedf = factor(rep(rep(c(5,10,20), each = 200),3),
                                             label = c('5 df', '10 df', '20 df')),
                             estdf = c(df5, df10, df100, df550, df1050, df2050, df5100, df10100, df20100),
                             samplesize = factor(rep(c(35,50,100), each = 600)))
    library(tidyverse)

    denseplot <- ggplot(data = subset(dfdataframe, estdf < 200),
                        aes(x=estdf, fill=samplesize)) +
        geom_density(alpha = .5) +
        geom_vline(data = truedataframe,
                   mapping = aes(xintercept = estdf),
                   size = .5) +
        theme_bw() +
        theme(axis.ticks.y=element_blank(), axis.text.y=element_blank(),
              strip.text = element_text(size = 8),
              legend.justification=c(1,0), legend.position=c(.95,.4),
              legend.background = element_blank(),
              legend.text =element_text(size = 8), legend.title = element_text(size = 8)) +
        ggtitle("Density plot of estimated degrees of freedom compared to actual") +
        xlab(NULL) +
        ylab(NULL) +
        scale_fill_manual(values = c("#050505", "#E69F00", "#56B4E9"),
            name = "Sample Size") +
        facet_wrap(factor(truedf)~.,  scales="free") +
        NULL
    denseplot

    knitr::kable(dfdataframe %>% group_by(truedf, samplesize) %>%
        summarize(min = min(estdf), max = max(estdf),
        median = median(estdf),
        mean=mean(estdf),
        sd = sd(estdf)))
    ### Here ends the long simulation

    ##### Here is what is really run

    set.seed(20190621)
    df10 <- rep(0,50)
    df1050 <-  rep(0,50)
    df10100 <-  rep(0,50)

    for(i in 1:50){
       df10[i] = suppressWarnings(MLmatrixt(rmatrixt(mean = matrix(0,5,3),df = 10, n = 25), fixed = FALSE, df = 5, max.iter = 20)$nu)
       df1050[i] = suppressWarnings(MLmatrixt(rmatrixt(mean = matrix(0,5,3),df = 10, n = 50), fixed = FALSE, df = 5, max.iter = 20)$nu)
       df10100[i] = suppressWarnings(MLmatrixt(rmatrixt(mean = matrix(0,5,3),df = 10, n = 100), fixed = FALSE, df = 5, max.iter = 20)$nu)
    }

    dfdataframe = data.frame(label = c('10 df'),
                             estdf = c(df10, df1050, df10100),
                             samplesize = factor(rep(c(25,50,100), each = 50)))
    library(ggplot2)
    library(dplyr)
    library(magrittr)
    denseplot <- ggplot(data = subset(dfdataframe, estdf < 200),aes(x=estdf, fill=samplesize)) +
        geom_density(alpha = .5) +
        geom_vline(mapping = aes(xintercept = 10),
                   size = .5) +
        theme_bw() +
        theme(axis.ticks.y=element_blank(), axis.text.y=element_blank(), strip.text = element_text(size = 8),
              legend.justification=c(1,0), legend.position=c(.95,.4),
              legend.background = element_blank(),
              legend.text =element_text(size = 8), legend.title = element_text(size = 8)) +
        ggtitle("Density plot of estimated degrees of freedom compared to actual") +
        xlab(NULL) +
        ylab(NULL) +
        scale_fill_manual(values = c("#050505", "#E69F00", "#56B4E9"), name = "Sample Size") +
     #   facet_wrap(factor(truedf)~.,  scales="free") +
        NULL
    denseplot

    knitr::kable(dfdataframe %>% group_by(samplesize) %>%
        summarize(min = min(estdf), max = max(estdf),
        median = median(estdf),
        mean=mean(estdf),
        sd = sd(estdf)))

    #### Here ends what is really run

    A <- rmatrixt(30, mean = matrix(0, nrow=2, ncol=3), df = 10)
    B <- rmatrixt(30, mean = matrix(c(1,0), nrow=2, ncol=3), df = 10)
    C <- rmatrixt(30, mean = matrix(c(0,1), nrow=2, ncol=3), df = 10)
    ABC <- array(c(A,B,C), dim = c(2,3,90))
    groups <- factor(c(rep("A",30),rep("B",30),rep("C",30)))
    prior = c(30,30,30)/90
    matlda <- matrixlda(x = ABC,grouping = groups, prior = prior,
                        method = 't', nu = 10, fixed = TRUE)
    predict(matlda, newdata = ABC[,,c(1,31,61)])

    sessionInfo()

## References

Dickey, James M. 1967. “Matricvariate Generalizations of the Multivariate \(t\) Distribution and the Inverted Multivariate \(t\) Distribution.” Ann. Math. Statist. 38 (2): 511–18. https://doi.org/10.1214/aoms/1177698967.
Liu, Chuanhai, and Donald B. Rubin. 1994. “The ECME Algorithm: A Simple Extension of EM and ECM with Faster Monotone Convergence.” Biometrika 81 (4): 633–48. https://doi.org/10.2307/2337067.
Meng, Xiao-Li, and Donald B. Rubin. 1993. “Maximum Likelihood Estimation via the ECM Algorithm: A General Framework.” Biometrika 80 (2): 267–78. https://doi.org/10.1093/biomet/80.2.267.
Rubin, D. B. 1983. “Encyclopedia of Statistical Sciences.” In, 4th ed., 272–75. John Wiley.