| PCAasymp {ICtest} | R Documentation |
Testing for Subsphericity using the Covariance Matrix or Tyler's Shape Matrix
Description
The function tests, assuming an elliptical model, that the last p-k eigenvalues of
a scatter matrix are equal and the k interesting components are those with a larger variance.
The scatter matrices that can be used here are the regular covariance matrix and Tyler's shape matrix.
Usage
PCAasymp(X, k, scatter = "cov", ...)
Arguments
X |
a numeric data matrix with p>1 columns. |
k |
the number of eigenvalues larger than the equal ones. Can be between 0 and p-2. |
scatter |
the scatter matrix to be used. Can be |
... |
arguments passed on to |
Details
The functions assumes an elliptical model and tests if the last p-k eigenvalues of PCA are equal. PCA can here be either be based on the regular covariance matrix or on Tyler's shape matrix.
For a sample of size n, the test statistic is
T = n / (2 \bar{d}^2 \sigma_1) \sum_{k+1}^p (d_i - \bar{d})^2,
where \bar{d} is the mean of the last p-k PCA eigenvalues.
The constant \sigma_1 is for the regular covariance matrix estimated from the data whereas for Tyler's shape matrix it is simply a function of the dimension of the data.
The test statistic has a limiting chisquare distribution with (p-k-1)(p-k+2)/2 degrees of freedom.
Note that the regular covariance matrix is here divided by n and not by n-1.
Value
A list of class ictest inheriting from class htest containing:
statistic |
the value of the test statistic. |
p.value |
the p-value of the test. |
parameter |
the degrees of freedom of the test. |
method |
character string which test was performed. |
data.name |
character string giving the name of the data. |
alternative |
character string specifying the alternative hypothesis. |
k |
the number or larger eigenvalues used in the testing problem. |
W |
the transformation matrix to the principal components. |
S |
data matrix with the centered principal components. |
D |
the underlying eigenvalues. |
MU |
the location of the data which was substracted before calculating the principal components. |
SCATTER |
the computed scatter matrix. |
sigma1 |
the asymptotic constant needed for the asymptotic test. |
Author(s)
Klaus Nordhausen
References
Nordhausen, K., Oja, H. and Tyler, D.E. (2022), Asymptotic and Bootstrap Tests for Subspace Dimension, Journal of Multivariate Analysis, 188, 104830. <doi:10.1016/j.jmva.2021.104830>.
See Also
Examples
n <- 200
X <- cbind(rnorm(n, sd = 2), rnorm(n, sd = 1.5), rnorm(n), rnorm(n), rnorm(n))
TestCov <- PCAasymp(X, k = 2)
TestCov
TestTyler <- PCAasymp(X, k = 1, scatter = "tyler")
TestTyler