Standardize an Observation Array
Standardizes an array of array-valued observations simultaneously from each mode. The method can be seen as a higher-order analogy for the regular multivariate standardization of random vectors.
Array of an order higher than two with the last dimension corresponding to the sampling units.
The algorithm first centers the n observed tensors Xi to have an element-wise mean of zero. Then it estimates the mth mode covariance matrix Cov_m(X) = E(X(m) X(m)^T)/(p1 ... p(m-1) p(m+1) ... pr), where X(m) is the centered m-flattening of X, for each mode and transforms the observations with the inverse square roots of the covariance matrices from the corresponding modes.
A list containing the following components:
Array of the same size as
List containing inverse square roots of the covariance matrices of different modes.
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# Generate sample data. n <- 100 x <- t(cbind(rnorm(n, mean = 0), rnorm(n, mean = 1), rnorm(n, mean = 2), rnorm(n, mean = 3), rnorm(n, mean = 4), rnorm(n, mean = 5))) dim(x) <- c(3, 2, n) # Standardize z <- tensorStandardize(x)$x # The m-mode covariance matrices of the standardized tensors mModeCovariance(z, 1) mModeCovariance(z, 2)
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