LDA for matrix variate distributions

Performs linear discriminant analysis on matrix variate data. This works slightly differently from the LDA function in MASS: it does not sphere the data or otherwise normalize it. It presumes equal variance matrices and probabilities are given as if the data are from a matrix variate normal distribution. The estimated variance matrices are weighted by the prior. However, if there are not enough members of a class to estimate a variance, this may be a problem. The function does not take the formula interface. If method = 't' is selected, this performs discrimination using the matrix variate t distribution, presuming equal covariances between classes.

matrixlda(
  x,
  grouping,
  prior,
  tol = 1e-04,
  method = "normal",
  nu = 10,
  ...,
  subset
)

Arguments

x

3-D array of matrix data indexed by the third dimension

grouping

vector

prior

a vector of prior probabilities of the same length as the number of classes

tol

by default, 1e-4. Tolerance parameter checks for 0 variance.

method

whether to use the normal distribution (normal) or the t distribution (t). By default, normal.

nu

If using the t-distribution, the degrees of freedom parameter. By default, 10.

...

Arguments passed to or from other methods, such as additional parameters to pass to MLmatrixnorm (e.g., row.mean)

subset

An index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)

Value

Returns a list of class matrixlda containing the following components:

prior

the prior probabilities used.

counts

the counts of group membership

means

the group means.

scaling

the scalar variance parameter

U

the between-row covariance matrix

V

the between-column covariance matrix

lev

levels of the grouping factor

N

The number of observations used.

method

The method used.

nu

The degrees of freedom parameter if the t distribution was used.

call

The (matched) function call.

References

G Z Thompson, R Maitra, W Q Meeker, A Bastawros (2019),
"Classification with the matrix-variate-t distribution", arXiv
e-prints arXiv:1907.09565 <https://arxiv.org/abs/1907.09565>

Ming Li, Baozong Yuan, "2D-LDA: A statistical linear discriminant
  analysis for image matrix", Pattern Recognition Letters, Volume 26,
  Issue 5, 2005, Pages 527-532, ISSN 0167-8655.

Aaron Molstad & Adam J. Rothman (2019), "A Penalized Likelihood Method for Classification With Matrix-Valued Predictors", Journal of Computational and Graphical Statistics, 28:1, 11-22, doi:10.1080/10618600.2018.1476249 MatrixLDA

Venables, W. N. & Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0 MASS

See also

predict.matrixlda(), MASS::lda(), MLmatrixnorm() and MLmatrixt() matrixqda(), and matrixmixture()

Examples

set.seed(20180221)
# construct two populations of 3x4 random matrices with different means
A <- rmatrixnorm(30, mean = matrix(0, nrow = 3, ncol = 4))
B <- rmatrixnorm(30, mean = matrix(1, nrow = 3, ncol = 4))
C <- array(c(A, B), dim = c(3, 4, 60)) # combine together
groups <- c(rep(1, 30), rep(2, 30)) # define groups
prior <- c(.5, .5) # set prior
D <- matrixlda(C, groups, prior) # fit model
logLik(D)
#> 'log Lik.' -962.6539 (df=39)
print(D)
#> $prior
#>   1   2 
#> 0.5 0.5 
#> 
#> $counts
#>  1  2 
#> 30 30 
#> 
#> $means
#> , , 1
#> 
#>             [,1]        [,2]        [,3]        [,4]
#> [1,]  0.07351694 -0.08704738  0.13826608  0.06657800
#> [2,] -0.05562527 -0.08631880 -0.12719766 -0.07388404
#> [3,] -0.12802865 -0.33410177 -0.05220913 -0.03813562
#> 
#> , , 2
#> 
#>           [,1]     [,2]      [,3]     [,4]
#> [1,] 1.1213360 1.230815 0.9142843 1.082488
#> [2,] 1.3283770 1.244675 0.5694279 1.000457
#> [3,] 0.8961279 0.800175 1.1439663 1.048916
#> 
#> 
#> $scaling
#> [1] 0.793638
#> 
#> $U
#>             [,1]        [,2]         [,3]
#> [1,] 1.000000000  0.06898319  0.007912669
#> [2,] 0.068983195  0.93223418 -0.033466058
#> [3,] 0.007912669 -0.03346606  0.967866385
#> 
#> $V
#>             [,1]         [,2]         [,3]        [,4]
#> [1,]  1.00000000 -0.071847232  0.065658541 -0.12269956
#> [2,] -0.07184723  0.949290705  0.006997484  0.09237596
#> [3,]  0.06565854  0.006997484  1.306664921 -0.02648614
#> [4,] -0.12269956  0.092375955 -0.026486138  1.25434970
#> 
#> $lev
#> [1] "1" "2"
#> 
#> $N
#> [1] 60
#> 
#> $method
#> [1] "normal"
#> 
#> $nu
#> NULL
#> 
#> $call
#> matrixlda(x = C, grouping = groups, prior = prior)
#> 
#> attr(,"class")
#> [1] "matrixlda"