I feel like there must be some function in some package that does this, but for the life of me I cannot find one that calculates the gradient of an array of data points (not a function) by simply doing central differences centered on each data point.
Here's an example where I generate a 5x5x5 grid of data points with known x, y, and z coordinates, and calculate the gradient using central differences for the point 3,3,3:
response_data <- array(NA, dim = c(5,5,5))
for(i in 1:dim(response_data)[1]) {
for(j in 1:dim(response_data)[2]) {
for(k in 1:dim(response_data)[3]) {
response_data[i,j,k] <- i*j*k
}
}
}
x_values <- 1:5
y_values <- 1:5
z_values <- 1:5
#Central difference at 3,3,3:
partial_x <- (response_data[4, 3, 3] - response_data[2, 3, 3])/
(x_values[4] - x_values[2])
partial_y <- (response_data[3, 4, 3] - response_data[3, 2, 3])/
(y_values[4] - y_values[2])
partial_z <- (response_data[3, 3, 4] - response_data[3, 3, 2])/
(z_values[4] - z_values[2])
That's fairly straightforward. What I'm looking for is a function that can do so for all the points in the input dataset (or all the non-edge points at least), for an arbitrarily n-dimensional dataset.
Edit: pracma::gradient can do this for a 1- or 2-dimensional array. Still looking for one that can handle n-dimensional arrays
Edit 2: I was able to write code to calculate the gradient for a 3-dimensional array. It feels like there should be a straightforward way to loop through the dimensions for an arbitrarily n-dimensional array, but I'm not sure how to choose the dimension dynamically
response_data <- array(NA, dim = c(5,5,5))
for(i in 1:dim(response_data)[1]) {
for(j in 1:dim(response_data)[2]) {
for(k in 1:dim(response_data)[3]) {
response_data[i,j,k] <- i*j*k
}
}
}
#Put all coordinates into one list
coord_vals <- list(x_values = 1:5, y_values = 1:5, z_values = 1:5)
#Create empty array to hold partial derivatives
# (each list is partial derivatives wrt that pred variable)
pds <- rep(list(array(NA, dim = sapply(coord_vals, length))),
length(pred_vars))
names(pds) <- names(coord_vals)
#Calculate gradient
#First dimension
dimension <- 1
for(j in 1:length(coord_vals[[2]])) {
for(k in 1:length(coord_vals[[3]])) {
pds[[dimension]][ , j, k] <-
pracma::gradient(F = response_data[, j, k],
h1 = coord_vals[[dimension]])
}
}
#Second dimension
dimension <- 2
for(i in 1:length(coord_vals[[1]])) {
for(k in 1:length(coord_vals[[3]])) {
pds[[dimension]][i, , k] <-
pracma::gradient(F = response_data[i, , k],
h1 = coord_vals[[dimension]])
}
}
#Third dimension
dimension <- 3
for(i in 1:length(coord_vals[[1]])) {
for(j in 1:length(coord_vals[[2]])) {
pds[[dimension]][i, j, ] <-
pracma::gradient(F = response_data[i, j, ],
h1 = coord_vals[[dimension]])
}
}
If you don't want to import utilities from Python environments but would like to stay with R only, you can customize a recursive function to compute the N-dim matrix gradient, where pracma::gradient can be applied as an "atom" helper function.
For example, you can design your function like below
library(pracma)
f <- function(arr, coords) {
if (is.null(dim(arr))) {
return(gradient(arr, coords[[1]]))
}
idx <- seq_along(dim(arr))
lapply(idx, \(k) apply(arr, k, \(x) f(x, coords[-k]), simplify = FALSE))
}
arr and its coordinates coordsset.seed(0)
dims <- c(3, 4)
arr <- array(rnorm(prod(dims)), dims)
coords <- lapply(setNames(dims, seq_along(dims)), \(x) seq.int(x) / x)
you will obtain
> f(arr, coords)
[[1]]
[[1]][[1]]
[1] 0.03790015 -4.38304264 2.26444813 13.33288169
[[1]][[2]]
[1] 2.96349918 0.06302583 0.69790405 4.23325563
[[1]][[3]]
[1] -11.478997 -2.671133 1.481882 -3.172968
[[2]]
[[2]][[1]]
[1] -4.7675629 0.1002675 4.9680979
[[2]][[2]]
[1] -2.573364 -4.218569 -5.863774
[[2]][[3]]
[1] 1.9015398 1.3841998 0.8668598
[[2]][[4]]
[1] -4.923180 -4.805494 -4.687808
With 3-D matrix arr and its coordinates coords
dims <- c(3, 4, 5)
arr <- array(rnorm(prod(dims)), dims)
coords <- lapply(setNames(dims, seq_along(dims)), \(x) seq.int(x) / x)
you will obtain
> f(arr, coords)
[[1]]
[[1]][[1]]
[[1]][[1]][[1]]
[[1]][[1]][[1]][[1]]
[1] -12.0530565 -3.3001526 5.3495434 2.2228848 -0.8005662
[[1]][[1]][[1]][[2]]
[1] -8.4197008 -4.9084579 0.3304114 0.7856278 -0.4867822
[[1]][[1]][[1]][[3]]
[1] 6.8212517 1.7321512 -2.2211682 0.4520727 1.9895325
[[1]][[1]][[1]][[4]]
[1] -10.136289 -7.635313 -3.607466 1.192120 4.464835
[[1]][[1]][[2]]
[[1]][[1]][[2]][[1]]
[1] 0.03790015 -4.38304264 2.26444813 13.33288169
[[1]][[1]][[2]][[2]]
[1] 2.9445847 3.1666806 1.5778130 -0.2331506
[[1]][[1]][[2]][[3]]
[1] -2.53538826 -0.35719956 0.08296439 -1.65506036
[[1]][[1]][[2]][[4]]
[1] -5.086027 -2.889889 -1.572489 -2.451226
[[1]][[1]][[2]][[5]]
[1] -4.8349994 -1.7738492 0.4081584 -0.4709841
[[1]][[2]]
[[1]][[2]][[1]]
[[1]][[2]][[1]][[1]]
[1] 0.1838589 2.0746033 -0.3501288 -1.8273417 1.0109221
[[1]][[2]][[1]][[2]]
[1] -0.8120899 -4.2481020 3.7641991 9.3149100 3.4173077
[[1]][[2]][[1]][[3]]
[1] -4.7140899 -0.6204195 1.0137378 1.9825739 5.4109231
[[1]][[2]][[1]][[4]]
[1] -3.15128550 -0.09210678 -4.24279603 -7.37662755 -3.30059111
[[1]][[2]][[2]]
[[1]][[2]][[2]][[1]]
[1] 2.96349918 0.06302583 0.69790405 4.23325563
[[1]][[2]][[2]][[2]]
[1] 2.1667401 -1.8961537 -0.2377742 5.4834991
[[1]][[2]][[2]][[3]]
[1] -7.152829 -2.092992 4.022700 5.078556
[[1]][[2]][[2]][[4]]
[1] 8.7496648 -0.8050604 -6.6433703 -2.9269550
[[1]][[2]][[2]][[5]]
[1] 10.674773 0.954940 -9.330530 -9.896166
[[1]][[3]]
[[1]][[3]][[1]]
[[1]][[3]][[1]][[1]]
[1] -8.1450719 -0.6100748 3.8437980 -2.0490800 -4.8608337
[[1]][[3]][[1]][[2]]
[1] 3.240145 3.966691 3.631668 -2.105163 -6.780426
[[1]][[3]][[1]][[3]]
[1] -1.092504 -1.068858 -2.355757 2.628884 8.924069
[[1]][[3]][[1]][[4]]
[1] 8.0159938 4.8773025 0.8808687 -6.0388153 -12.1007569
[[1]][[3]][[2]]
[[1]][[3]][[2]][[1]]
[1] -11.478997 -2.671133 1.481882 -3.172968
[[1]][[3]][[2]][[2]]
[1] -2.3708240 0.1498945 3.3922213 4.1138296
[[1]][[3]][[2]][[3]]
[1] -4.156173 -3.038159 2.210371 6.340888
[[1]][[3]][[2]][[4]]
[1] -2.710232 -4.809749 1.191582 9.292430
[[1]][[3]][[2]][[5]]
[1] -4.2459059 0.7042119 -0.9365505 -7.5274307
[[2]]
[[2]][[1]]
[[2]][[1]][[1]]
[[2]][[1]][[1]][[1]]
[1] -12.0530565 -3.3001526 5.3495434 2.2228848 -0.8005662
[[2]][[1]][[1]][[2]]
[1] 0.1838589 2.0746033 -0.3501288 -1.8273417 1.0109221
[[2]][[1]][[1]][[3]]
[1] -8.1450719 -0.6100748 3.8437980 -2.0490800 -4.8608337
[[2]][[1]][[2]]
[[2]][[1]][[2]][[1]]
[1] -4.7675629 0.1002675 4.9680979
[[2]][[1]][[2]][[2]]
[1] 2.57458631 1.27266284 -0.02926063
[[2]][[1]][[2]][[3]]
[1] 1.682144 1.714314 1.746484
[[2]][[1]][[2]][[4]]
[1] -4.2650204 0.3692156 5.0034516
[[2]][[1]][[2]][[5]]
[1] -3.1781275 -0.8488647 1.4803982
[[2]][[2]]
[[2]][[2]][[1]]
[[2]][[2]][[1]][[1]]
[1] -8.4197008 -4.9084579 0.3304114 0.7856278 -0.4867822
[[2]][[2]][[1]][[2]]
[1] -0.8120899 -4.2481020 3.7641991 9.3149100 3.4173077
[[2]][[2]][[1]][[3]]
[1] 3.240145 3.966691 3.631668 -2.105163 -6.780426
[[2]][[2]][[2]]
[[2]][[2]][[2]][[1]]
[1] -2.573364 -4.218569 -5.863774
[[2]][[2]][[2]][[2]]
[1] 1.9912028 -0.7206154 -3.4324337
[[2]][[2]][[2]][[3]]
[1] -1.780937 1.106520 3.993977
[[2]][[2]][[2]][[4]]
[1] 6.111748 1.260139 -3.591471
[[2]][[2]][[2]][[5]]
[1] 8.4542020 -0.6279546 -9.7101112
[[2]][[3]]
[[2]][[3]][[1]]
[[2]][[3]][[1]][[1]]
[1] 6.8212517 1.7321512 -2.2211682 0.4520727 1.9895325
[[2]][[3]][[1]][[2]]
[1] -4.7140899 -0.6204195 1.0137378 1.9825739 5.4109231
[[2]][[3]][[1]][[3]]
[1] -1.092504 -1.068858 -2.355757 2.628884 8.924069
[[2]][[3]][[2]]
[[2]][[3]][[2]][[1]]
[1] 1.9015398 1.3841998 0.8668598
[[2]][[3]][[2]][[2]]
[1] -5.0196652 -0.9899268 3.0398116
[[2]][[3]][[2]][[3]]
[1] -0.9215451 -0.2964056 0.3287338
[[2]][[3]][[2]][[4]]
[1] -1.137778 -1.070680 -1.003582
[[2]][[3]][[2]][[5]]
[1] 0.9150564 1.0096812 1.1043059
[[2]][[4]]
[[2]][[4]][[1]]
[[2]][[4]][[1]][[1]]
[1] -10.136289 -7.635313 -3.607466 1.192120 4.464835
[[2]][[4]][[1]][[2]]
[1] -3.15128550 -0.09210678 -4.24279603 -7.37662755 -3.30059111
[[2]][[4]][[1]][[3]]
[1] 8.0159938 4.8773025 0.8808687 -6.0388153 -12.1007569
[[2]][[4]][[2]]
[[2]][[4]][[2]][[1]]
[1] -4.923180 -4.805494 -4.687808
[[2]][[4]][[2]][[2]]
[1] -0.7321779 0.6401908 2.0125594
[[2]][[4]][[2]][[3]]
[1] 4.128667 2.702075 1.275483
[[2]][[4]][[2]][[4]]
[1] -1.494574 3.333191 8.160957
[[2]][[4]][[2]][[5]]
[1] -6.153830 -1.636486 2.880858
[[3]]
[[3]][[1]]
[[3]][[1]][[1]]
[[3]][[1]][[1]][[1]]
[1] 0.03790015 -4.38304264 2.26444813 13.33288169
[[3]][[1]][[1]][[2]]
[1] 2.96349918 0.06302583 0.69790405 4.23325563
[[3]][[1]][[1]][[3]]
[1] -11.478997 -2.671133 1.481882 -3.172968
[[3]][[1]][[2]]
[[3]][[1]][[2]][[1]]
[1] -4.7675629 0.1002675 4.9680979
[[3]][[1]][[2]][[2]]
[1] -2.573364 -4.218569 -5.863774
[[3]][[1]][[2]][[3]]
[1] 1.9015398 1.3841998 0.8668598
[[3]][[1]][[2]][[4]]
[1] -4.923180 -4.805494 -4.687808
[[3]][[2]]
[[3]][[2]][[1]]
[[3]][[2]][[1]][[1]]
[1] 2.9445847 3.1666806 1.5778130 -0.2331506
[[3]][[2]][[1]][[2]]
[1] 2.1667401 -1.8961537 -0.2377742 5.4834991
[[3]][[2]][[1]][[3]]
[1] -2.3708240 0.1498945 3.3922213 4.1138296
[[3]][[2]][[2]]
[[3]][[2]][[2]][[1]]
[1] 2.57458631 1.27266284 -0.02926063
[[3]][[2]][[2]][[2]]
[1] 1.9912028 -0.7206154 -3.4324337
[[3]][[2]][[2]][[3]]
[1] -5.0196652 -0.9899268 3.0398116
[[3]][[2]][[2]][[4]]
[1] -0.7321779 0.6401908 2.0125594
[[3]][[3]]
[[3]][[3]][[1]]
[[3]][[3]][[1]][[1]]
[1] -2.53538826 -0.35719956 0.08296439 -1.65506036
[[3]][[3]][[1]][[2]]
[1] -7.152829 -2.092992 4.022700 5.078556
[[3]][[3]][[1]][[3]]
[1] -4.156173 -3.038159 2.210371 6.340888
[[3]][[3]][[2]]
[[3]][[3]][[2]][[1]]
[1] 1.682144 1.714314 1.746484
[[3]][[3]][[2]][[2]]
[1] -1.780937 1.106520 3.993977
[[3]][[3]][[2]][[3]]
[1] -0.9215451 -0.2964056 0.3287338
[[3]][[3]][[2]][[4]]
[1] 4.128667 2.702075 1.275483
[[3]][[4]]
[[3]][[4]][[1]]
[[3]][[4]][[1]][[1]]
[1] -5.086027 -2.889889 -1.572489 -2.451226
[[3]][[4]][[1]][[2]]
[1] 8.7496648 -0.8050604 -6.6433703 -2.9269550
[[3]][[4]][[1]][[3]]
[1] -2.710232 -4.809749 1.191582 9.292430
[[3]][[4]][[2]]
[[3]][[4]][[2]][[1]]
[1] -4.2650204 0.3692156 5.0034516
[[3]][[4]][[2]][[2]]
[1] 6.111748 1.260139 -3.591471
[[3]][[4]][[2]][[3]]
[1] -1.137778 -1.070680 -1.003582
[[3]][[4]][[2]][[4]]
[1] -1.494574 3.333191 8.160957
[[3]][[5]]
[[3]][[5]][[1]]
[[3]][[5]][[1]][[1]]
[1] -4.8349994 -1.7738492 0.4081584 -0.4709841
[[3]][[5]][[1]][[2]]
[1] 10.674773 0.954940 -9.330530 -9.896166
[[3]][[5]][[1]][[3]]
[1] -4.2459059 0.7042119 -0.9365505 -7.5274307
[[3]][[5]][[2]]
[[3]][[5]][[2]][[1]]
[1] -3.1781275 -0.8488647 1.4803982
[[3]][[5]][[2]][[2]]
[1] 8.4542020 -0.6279546 -9.7101112
[[3]][[5]][[2]][[3]]
[1] 0.9150564 1.0096812 1.1043059
[[3]][[5]][[2]][[4]]
[1] -6.153830 -1.636486 2.880858
The previous solution applied the 1-D gradient vectors as the basic expressions cells, but the following variant is based on 2-D matrix, which looks more concise and compact.
library(pracma)
f <- function(arr, coords) {
if (length(dim(arr))==2) {
return(gradient(arr, coords[[2]], coords[[1]]))
}
idx <- seq_along(dim(arr))
lapply(idx, \(k) apply(arr, k, \(x) f(x, coords[-k])))
}
arr and its coordinates coordsset.seed(0)
dims <- c(3, 4)
arr <- array(rnorm(prod(dims)), dims)
coords <- lapply(setNames(dims, seq_along(dims)), \(x) seq.int(x) / x)
you will obtain
> f(arr, coords)
$X
[,1] [,2] [,3] [,4]
[1,] 0.03790015 -4.38304264 2.2644481 13.332882
[2,] 2.96349918 0.06302583 0.6979041 4.233256
[3,] -11.47899722 -2.67113287 1.4818816 -3.172968
$Y
[,1] [,2] [,3] [,4]
[1,] -4.7675629 -2.573364 1.9015398 -4.923180
[2,] 0.1002675 -4.218569 1.3841998 -4.805494
[3,] 4.9680979 -5.863774 0.8668598 -4.687808
arr and its coordinates coordsset.seed(0)
dims <- c(3, 4)
arr <- array(rnorm(prod(dims)), dims)
coords <- lapply(setNames(dims, seq_along(dims)), \(x) seq.int(x) / x)
you will obtain
> f(arr, coords)
[[1]]
[[1]][[1]]
[[1]][[1]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] -12.053056 -3.300153 5.3495434 2.2228848 -0.8005662
[2,] -8.419701 -4.908458 0.3304114 0.7856278 -0.4867822
[3,] 6.821252 1.732151 -2.2211682 0.4520727 1.9895325
[4,] -10.136289 -7.635313 -3.6074656 1.1921204 4.4648354
[[1]][[1]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] 0.03790015 2.9445847 -2.53538826 -5.086027 -4.8349994
[2,] -4.38304264 3.1666806 -0.35719956 -2.889889 -1.7738492
[3,] 2.26444813 1.5778130 0.08296439 -1.572489 0.4081584
[4,] 13.33288169 -0.2331506 -1.65506036 -2.451226 -0.4709841
[[1]][[2]]
[[1]][[2]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] 0.1838589 2.07460333 -0.3501288 -1.827342 1.010922
[2,] -0.8120899 -4.24810197 3.7641991 9.314910 3.417308
[3,] -4.7140899 -0.62041952 1.0137378 1.982574 5.410923
[4,] -3.1512855 -0.09210678 -4.2427960 -7.376628 -3.300591
[[1]][[2]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] 2.96349918 2.1667401 -7.152829 8.7496648 10.674773
[2,] 0.06302583 -1.8961537 -2.092992 -0.8050604 0.954940
[3,] 0.69790405 -0.2377742 4.022700 -6.6433703 -9.330530
[4,] 4.23325563 5.4834991 5.078556 -2.9269550 -9.896166
[[1]][[3]]
[[1]][[3]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] -8.145072 -0.6100748 3.8437980 -2.049080 -4.860834
[2,] 3.240145 3.9666905 3.6316680 -2.105163 -6.780426
[3,] -1.092504 -1.0688579 -2.3557567 2.628884 8.924069
[4,] 8.015994 4.8773025 0.8808687 -6.038815 -12.100757
[[1]][[3]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] -11.478997 -2.3708240 -4.156173 -2.710232 -4.2459059
[2,] -2.671133 0.1498945 -3.038159 -4.809749 0.7042119
[3,] 1.481882 3.3922213 2.210371 1.191582 -0.9365505
[4,] -3.172968 4.1138296 6.340888 9.292430 -7.5274307
[[2]]
[[2]][[1]]
[[2]][[1]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] -12.0530565 -3.3001526 5.3495434 2.222885 -0.8005662
[2,] 0.1838589 2.0746033 -0.3501288 -1.827342 1.0109221
[3,] -8.1450719 -0.6100748 3.8437980 -2.049080 -4.8608337
[[2]][[1]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] -4.7675629 2.57458631 1.682144 -4.2650204 -3.1781275
[2,] 0.1002675 1.27266284 1.714314 0.3692156 -0.8488647
[3,] 4.9680979 -0.02926063 1.746484 5.0034516 1.4803982
[[2]][[2]]
[[2]][[2]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] -8.4197008 -4.908458 0.3304114 0.7856278 -0.4867822
[2,] -0.8120899 -4.248102 3.7641991 9.3149100 3.4173077
[3,] 3.2401446 3.966691 3.6316680 -2.1051632 -6.7804260
[[2]][[2]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] -2.573364 1.9912028 -1.780937 6.111748 8.4542020
[2,] -4.218569 -0.7206154 1.106520 1.260139 -0.6279546
[3,] -5.863774 -3.4324337 3.993977 -3.591471 -9.7101112
[[2]][[3]]
[[2]][[3]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] 6.821252 1.7321512 -2.221168 0.4520727 1.989532
[2,] -4.714090 -0.6204195 1.013738 1.9825739 5.410923
[3,] -1.092504 -1.0688579 -2.355757 2.6288840 8.924069
[[2]][[3]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] 1.9015398 -5.0196652 -0.9215451 -1.137778 0.9150564
[2,] 1.3841998 -0.9899268 -0.2964056 -1.070680 1.0096812
[3,] 0.8668598 3.0398116 0.3287338 -1.003582 1.1043059
[[2]][[4]]
[[2]][[4]]$X
[,1] [,2] [,3] [,4] [,5]
[1,] -10.136289 -7.63531259 -3.6074656 1.192120 4.464835
[2,] -3.151286 -0.09210678 -4.2427960 -7.376628 -3.300591
[3,] 8.015994 4.87730251 0.8808687 -6.038815 -12.100757
[[2]][[4]]$Y
[,1] [,2] [,3] [,4] [,5]
[1,] -4.923180 -0.7321779 4.128667 -1.494574 -6.153830
[2,] -4.805494 0.6401908 2.702075 3.333191 -1.636486
[3,] -4.687808 2.0125594 1.275483 8.160957 2.880858
[[3]]
[[3]][[1]]
[[3]][[1]]$X
[,1] [,2] [,3] [,4]
[1,] 0.03790015 -4.38304264 2.2644481 13.332882
[2,] 2.96349918 0.06302583 0.6979041 4.233256
[3,] -11.47899722 -2.67113287 1.4818816 -3.172968
[[3]][[1]]$Y
[,1] [,2] [,3] [,4]
[1,] -4.7675629 -2.573364 1.9015398 -4.923180
[2,] 0.1002675 -4.218569 1.3841998 -4.805494
[3,] 4.9680979 -5.863774 0.8668598 -4.687808
[[3]][[2]]
[[3]][[2]]$X
[,1] [,2] [,3] [,4]
[1,] 2.944585 3.1666806 1.5778130 -0.2331506
[2,] 2.166740 -1.8961537 -0.2377742 5.4834991
[3,] -2.370824 0.1498945 3.3922213 4.1138296
[[3]][[2]]$Y
[,1] [,2] [,3] [,4]
[1,] 2.57458631 1.9912028 -5.0196652 -0.7321779
[2,] 1.27266284 -0.7206154 -0.9899268 0.6401908
[3,] -0.02926063 -3.4324337 3.0398116 2.0125594
[[3]][[3]]
[[3]][[3]]$X
[,1] [,2] [,3] [,4]
[1,] -2.535388 -0.3571996 0.08296439 -1.655060
[2,] -7.152829 -2.0929925 4.02270020 5.078556
[3,] -4.156173 -3.0381594 2.21037116 6.340888
[[3]][[3]]$Y
[,1] [,2] [,3] [,4]
[1,] 1.682144 -1.780937 -0.9215451 4.128667
[2,] 1.714314 1.106520 -0.2964056 2.702075
[3,] 1.746484 3.993977 0.3287338 1.275483
[[3]][[4]]
[[3]][[4]]$X
[,1] [,2] [,3] [,4]
[1,] -5.086027 -2.8898887 -1.572489 -2.451226
[2,] 8.749665 -0.8050604 -6.643370 -2.926955
[3,] -2.710232 -4.8097493 1.191582 9.292430
[[3]][[4]]$Y
[,1] [,2] [,3] [,4]
[1,] -4.2650204 6.111748 -1.137778 -1.494574
[2,] 0.3692156 1.260139 -1.070680 3.333191
[3,] 5.0034516 -3.591471 -1.003582 8.160957
[[3]][[5]]
[[3]][[5]]$X
[,1] [,2] [,3] [,4]
[1,] -4.834999 -1.7738492 0.4081584 -0.4709841
[2,] 10.674773 0.9549400 -9.3305298 -9.8961664
[3,] -4.245906 0.7042119 -0.9365505 -7.5274307
[[3]][[5]]$Y
[,1] [,2] [,3] [,4]
[1,] -3.1781275 8.4542020 0.9150564 -6.153830
[2,] -0.8488647 -0.6279546 1.0096812 -1.636486
[3,] 1.4803982 -9.7101112 1.1043059 2.880858