Author: Geoffrey Thompson
Date: 2024-09-30

# Discriminant Analysis for Matrix Variate Distributions

#### Geoffrey Thompson

#### 2024-09-30

    library(MixMatrix)

In the `MixMatrix` package, there are two functions for training a linear or quadratic classifier. The usage is fairly similar to the function `MASS::lda()` or `MASS::qda()`, however it requires the input as an array and the group variable provided as a vector (that is, it cannot handle data frames or the formula interface directly, which is reasonable, as there is no immediately clear way to make that work for a collection of matrices).

    set.seed(20180222)
    library('MixMatrix')
    A <- rmatrixnorm(30, mean = matrix(0, nrow=2, ncol=3)) # creating the three groups
    B <- rmatrixnorm(30, mean = matrix(c(1, 0), nrow = 2, ncol = 3))
    C <- rmatrixnorm(30, mean = matrix(c(0, 1), nrow = 2, ncol = 3))
    ABC <- array(c(A,B,C), dim = c(2,3,90)) # combining into on array
    groups <- factor(c(rep("A", 30), rep("B", 30), rep("C", 30))) # labels
    prior = c(30, 30, 30) / 90 # equal prior
    matlda <- matrixlda(x = ABC, grouping = groups, prior = prior) # perform LDA
    matqda <- matrixqda(x = ABC, grouping = groups, prior = prior) # perform QDA

This does not sphere the data or extract an SVD or Fisher discriminant scores - it is a simple linear/quadratic discriminant function based on the likelihood function.

The `matrixlda` function presumes equal covariance matrices among the groups while `matrixqda` fits separate covariance for each groups.

It is possible to set variance or mean restrictions from the `MLmatrixnorm` and `MLmatrixt` functions using the `...` argument. These restrictions are applied to all groups.

The `predict` method for these objects works in much the same way as for `lda` or `qda` objects: provide the function and the new data, then it will return the MAP class assignments and the posterior. If no new data is provided, it will attempt to run it on the original data if it is available in the environment.

    ABC[, , c(1, 31, 61)] # true class memberships: A, B, C
    #> , , 1
    #>
    #>             [,1]       [,2]      [,3]
    #> [1,] -0.07020924  0.8559433 0.9827602
    #> [2,]  1.30097854 -0.6893054 0.2830575
    #>
    #> , , 2
    #>
    #>           [,1]     [,2]        [,3]
    #> [1,] 2.2374662 2.408268 -0.37053822
    #> [2,] 0.9653824 0.630897 -0.08850761
    #>
    #> , , 3
    #>
    #>            [,1]       [,2]      [,3]
    #> [1,] -1.6961217 -1.5642705 0.2045711
    #> [2,]  0.7782369  0.1842012 0.3869149
    #predict the membership of the first observation of each group
    predict(matlda, ABC[, , c(1, 31, 61)])
    #> $class
    #> [1] B B A
    #> Levels: A B C
    #>
    #> $posterior
    #>            [,1]        [,2]       [,3]
    #> [1,] 0.27340107 0.690217317 0.03638161
    #> [2,] 0.03833953 0.949049289 0.01261118
    #> [3,] 0.54647035 0.001273576 0.45225607
    #predict the membership of the first observation of each group
    predict(matqda,  ABC[, , c(1, 31, 61)])
    #> $class
    #> [1] B B A
    #> Levels: A B C
    #>
    #> $posterior
    #>            [,1]        [,2]       [,3]
    #> [1,] 0.24302341 0.735815885 0.02116070
    #> [2,] 0.03295848 0.963641160 0.00340036
    #> [3,] 0.54611977 0.007871269 0.44600896

In this example, points from classes A, B, and C were selected and they ended up being classified as B, B, and A. The QDA and LDA rules had only minor disagreements, which is to be expected when they do truly have the same covariances.

## Details of the modeling choices

Suppose there are two populations, \(\pi_1\) and \(\pi_2\) with prior probabilities of belonging to these classes, \(p_1\) and \(p_2\). Define a function, \(c(1|2)\) as the cost of misclassifying a member of population \(\pi_2\) as a member of class \(1\) (and vice versa). Further, define \(P(1|2)\) as the probability of classifying a member of population \(\pi_2\) as a member of class \(1\) (and vice versa). Then we define the expected cost of misclassification as:

\[ECM = c(2|1)P(2|1)p_1 + c(1|2)P(1|2)p_2 \]

## Classification Rule

A reasonable classification rule based on ECM is the following:

Classify as class \(1\) if:

\[ \frac{f_1(x)}{f_2(x)} \geq \frac{c(1|2) p_2}{c(2|1)p_1} \]

Where \(f_i(x)\) is the probability density function for group \(\pi_i\) evaluated at \(x\).

## Matrix Variate Normal Populations

Recall the probability density function for a matrix variate normal distribution:

\[f(\mathbf{X};\mathbf{M}, \mathbf{U}, \mathbf{V}) = \frac{\exp\left( -\frac{1}{2} \, \mathrm{tr}\left[ \mathbf{V}^{-1} (\mathbf{X} - \mathbf{M})^{T} \mathbf{U}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |\mathbf{V}|^{n/2} |\mathbf{U}|^{p/2}} \]

\(\mathbf{X}\) and \(\mathbf{M}\) are \(n \times p\), \(\mathbf{U}\) is \(n \times n\) and describes the covariance relationship between the rows, and \(\mathbf{V}\) is \(p \times p\) and describes the covariance relationship between the columns.

## Estimated Minimum ECM Rule for Two Matrix Variate Normal Populations

A decision rule for this case:

\[\begin{eqnarray} R(\mathbf{X}) & = & \mathrm{trace}\big[ -\frac{1}{2}(\mathbf{V}_1^{-1} \mathbf{X}^{T} \mathbf{U}_1^{-1} \mathbf{X} - \mathbf{V}_2^{-1} \mathbf{X}^{T} \mathbf{U}_2^{-1} \mathbf{X}) \\ & & +(\mathbf{V}_1^{-1} \mathbf{M}_1^{T} \mathbf{U}_1^{-1} - \mathbf{V}_2^{-1} \mathbf{M}_2^{T} \mathbf{U}_2^{-1}) \mathbf{X} \\ & & -\frac{1}{2}(\mathbf{V}_1^{-1} \mathbf{M}_1^{T} \mathbf{U}_1^{-1} \mathbf{M}_1 - \mathbf{V}_2^{-1} \mathbf{M}_2^{T} \mathbf{U}_2^{-1} \mathbf{M}_2) \big] \\ & & -\frac{1}{2}(p (\log|\mathbf{U}\_1|-\log|\mathbf{U}\_2|)+ n(\log|\mathbf{V}\_1|-\log|\mathbf{V}\_2|) ) \end{eqnarray}\]

## How to classify based on this:

If \(R(\mathbf{X}) \geq \log(c(1|2)p_2) - \log(c(2|1)p_1)\), assign \(\mathbf{X}\) to group \(1\), otherwise assign to group \(2\).

In the multivariate case, this is equivalent to the LDA/QDA rules - term (1) is the quadratic form which vanishes in case of equal covariances between groups, term (2) is the LDA term, and terms (3) and (4) are constants which depend on the parameters and not \(\mathrm{X}\).

Typically, the models we have used have implicitly used an equal probability prior and an equal cost of misclassification, but other inputs can be specified. In case of equal priors and equal cost of misclassification, this term is 0.

## If there are equal covariances:

If the two groups have the same covariances, then this simplifies. The classification rule is then: \[\begin{eqnarray} R(\mathbf{X}) & = & \mathrm{trace}\big[ (\mathbf{V}^{-1} (\mathbf{M}_1 -\mathbf{M}_2)^{T}\mathbf{U}^{-1}) \mathbf{X} \\ & & -\frac{1}{2}(\mathbf{V}^{-1} \mathbf{M}_1^{T} \mathbf{U}^{-1} \mathbf{M}_1 - \mathbf{V}^{-1} \mathbf{M}_2^{T} \mathbf{U}^{-1} \mathbf{M}_2) \big] \\ & \geq & \log(c(1|2)p_2) - \log(c(2|1)p_1) \end{eqnarray}\]

Classify to group \(1\) if the last term is true. Note this is linear in \(\mathbf{X}\). This is directly analogous to LDA in the multivariate case.

## Generalizing to multiple classes

The factor \(R\) for each group \(g\) in a QDA setting is:

\[\begin{eqnarray} R_g(\mathbf{X}) & = & \mathrm{trace}\big[ -\frac{1}{2}(\mathbf{V}_g^{-1} \mathbf{X}^{T} \mathbf{U}_g^{-1} \mathbf{X} +(\mathbf{V}_g^{-1} \mathbf{M}_g^{T} \mathbf{U}_g^{-1}) \mathbf{X} -\frac{1}{2}(\mathbf{V}_g^{-1} \mathbf{M}_g^{T} \mathbf{U}_g^{-1} \mathbf{M}_g) \big] \\ & & -\frac{1}{2}(p (\log|\mathbf{U}\_g|)+ n(\log|\mathbf{V}\_g|) ) + \log p_g \end{eqnarray}\] When \(U_i = U_j\) for all groups \(i,j\), several of these terms cancel. The posterior probability is:

\[ P(\mathbf{X} \in g) = \frac{ \exp (R_g (\mathbf{X}))}{\sum_i \exp (R_i(\mathbf{X}))} \] with the bottom sum being over all groups \(i\).

## Structure of the objects

The objects returned by `matrixlda` and `matrixqda` are S3 classes. See below example output:

    matlda
    #> $prior
    #>         A         B         C
    #> 0.3333333 0.3333333 0.3333333
    #>
    #> $counts
    #>  A  B  C
    #> 30 30 30
    #>
    #> $means
    #> , , 1
    #>
    #>            [,1]        [,2]       [,3]
    #> [1,] 0.08629672 -0.06362916 -0.3045685
    #> [2,] 0.03851982 -0.01375580  0.3978759
    #>
    #> , , 2
    #>
    #>           [,1]       [,2]       [,3]
    #> [1,] 1.2872487  1.1199220 1.01282774
    #> [2,] 0.3044151 -0.3797203 0.06539076
    #>
    #> , , 3
    #>
    #>            [,1]       [,2]       [,3]
    #> [1,] 0.07377351 -0.3278233 -0.3031961
    #> [2,] 0.92568372  1.3738475  0.9353985
    #>
    #>
    #> $scaling
    #> [1] 1.002299
    #>
    #> $U
    #>            [,1]       [,2]
    #> [1,] 1.00000000 0.03243759
    #> [2,] 0.03243759 0.97819549
    #>
    #> $V
    #>             [,1]        [,2]        [,3]
    #> [1,]  1.00000000  0.01988966 -0.04879263
    #> [2,]  0.01988966  0.93839973 -0.04563610
    #> [3,] -0.04879263 -0.04563610  0.93321444
    #>
    #> $lev
    #> [1] "A" "B" "C"
    #>
    #> $N
    #> [1] 90
    #>
    #> $method
    #> [1] "normal"
    #>
    #> $nu
    #> NULL
    #>
    #> $call
    #> matrixlda(x = ABC, grouping = groups, prior = prior)
    #>
    #> attr(,"class")
    #> [1] "matrixlda"

    matqda
    #> $prior
    #>         A         B         C
    #> 0.3333333 0.3333333 0.3333333
    #>
    #> $counts
    #>  A  B  C
    #> 30 30 30
    #>
    #> $means
    #> , , 1
    #>
    #>            [,1]        [,2]       [,3]
    #> [1,] 0.08629672 -0.06362916 -0.3045685
    #> [2,] 0.03851982 -0.01375580  0.3978759
    #>
    #> , , 2
    #>
    #>           [,1]       [,2]       [,3]
    #> [1,] 1.2872487  1.1199220 1.01282774
    #> [2,] 0.3044151 -0.3797203 0.06539076
    #>
    #> , , 3
    #>
    #>            [,1]       [,2]       [,3]
    #> [1,] 0.07377351 -0.3278233 -0.3031961
    #> [2,] 0.92568372  1.3738475  0.9353985
    #>
    #>
    #> $U
    #> , , 1
    #>
    #>            [,1]       [,2]
    #> [1,] 1.00000000 0.02620514
    #> [2,] 0.02620514 0.95989647
    #>
    #> , , 2
    #>
    #>            [,1]       [,2]
    #> [1,]  1.0000000 -0.0173514
    #> [2,] -0.0173514  0.9456145
    #>
    #> , , 3
    #>
    #>          [,1]     [,2]
    #> [1,] 1.000000 0.129851
    #> [2,] 0.129851 1.061265
    #>
    #>
    #> $V
    #> , , 1
    #>
    #>             [,1]        [,2]        [,3]
    #> [1,]  1.05498245 -0.07418973  0.09476411
    #> [2,] -0.07418973  0.98800103 -0.04390729
    #> [3,]  0.09476411 -0.04390729  0.79903040
    #>
    #> , , 2
    #>
    #>             [,1]       [,2]        [,3]
    #> [1,]  1.09941477 0.12678075 -0.03952663
    #> [2,]  0.12678075 0.94606849  0.03910645
    #> [3,] -0.03952663 0.03910645  0.85913748
    #>
    #> , , 3
    #>
    #>              [,1]         [,2]       [,3]
    #> [1,]  0.833193637  0.008466708 -0.2084547
    #> [2,]  0.008466708  0.875328749 -0.1142510
    #> [3,] -0.208454704 -0.114250988  1.1520533
    #>
    #>
    #> $lev
    #> [1] "A" "B" "C"
    #>
    #> $N
    #> [1] 90
    #>
    #> $method
    #> [1] "normal"
    #>
    #> $nu
    #> NULL
    #>
    #> $call
    #> matrixqda(x = ABC, grouping = groups, prior = prior)
    #>
    #> attr(,"class")
    #> [1] "matrixqda"

## 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] MixMatrix_0.2.8
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] digest_0.6.37     R6_2.5.1          fastmap_1.2.0     xfun_0.47
    #>  [5] glue_1.7.0        cachem_1.1.0      knitr_1.48        htmltools_0.5.8.1
    #>  [9] rmarkdown_2.28    lifecycle_1.0.4   cli_3.6.3         sass_0.4.9
    #> [13] jquerylib_0.1.4   compiler_4.4.1    tools_4.4.1       evaluate_1.0.0
    #> [17] bslib_0.8.0       Rcpp_1.0.13       yaml_2.3.10       rlang_1.1.4
    #> [21] jsonlite_1.8.9

## All the code for easy copying

    knitr::opts_chunk$set(
      collapse = TRUE,
      comment = "#>"
    )
    set.seed(20180222)
    library('MixMatrix')
    A <- rmatrixnorm(30, mean = matrix(0, nrow=2, ncol=3)) # creating the three groups
    B <- rmatrixnorm(30, mean = matrix(c(1, 0), nrow = 2, ncol = 3))
    C <- rmatrixnorm(30, mean = matrix(c(0, 1), nrow = 2, ncol = 3))
    ABC <- array(c(A,B,C), dim = c(2,3,90)) # combining into on array
    groups <- factor(c(rep("A", 30), rep("B", 30), rep("C", 30))) # labels
    prior = c(30, 30, 30) / 90 # equal prior
    matlda <- matrixlda(x = ABC, grouping = groups, prior = prior) # perform LDA
    matqda <- matrixqda(x = ABC, grouping = groups, prior = prior) # perform QDA
    ABC[, , c(1, 31, 61)] # true class memberships: A, B, C
    #predict the membership of the first observation of each group
    predict(matlda, ABC[, , c(1, 31, 61)])
    #predict the membership of the first observation of each group
    predict(matqda,  ABC[, , c(1, 31, 61)])

    matlda

    matqda

    sessionInfo()