Closed llrs closed 2 years ago
Hi @llrs, this is totally normal in your example that a
and astar
are equal. They are always the same for the first canonical component (here ncomp = 1
), they differ for the next components.
To understand the differences between a
and astar
, we need to have a look at how components are extracted. Let denote X
one of the blocks we are analyzing, we are looking for a_1
that maximizes the S/RGCCA criterion, this yields y_1 = X a_1
. a_1
corresponds to a[[i]][, 1]
in R where i
is the index of block X
.
We want the next component to be extracted y_2
to be orthogonal (thus having no correlation) with y_1
. In order to do so, we using a deflation strategy by rerunning RGCCA on X_1
instead of X
with X_1 = X - y_1 (y_1^\top y_1)^{-1} y_1^\top X
. Therefore, we find a_2
, corresponding to a[[i]][, 2]
, such that y_2 = X_1 a_2
. Here you see that a_2
no longer interacts with the original block, hence if you want to apply the projection defined by a_2
to another dataset, you need to express a_2
in the original space, that is what astar
is for.
In fact if you denote k
the number of the component, and X_0 = X
, you have X_{k - 1} a_k = X astar_k
.
Hope this explanation is clear!
It is very clear, many thanks for the fast response. But shouldn't that also change due to using scaled data (now defaults scale = TRUE, scale_block = TRUE
)?
Multiplying the original (given as input) data by the astar values astar_1 on the first dimension doesn't return the final y_1 :
library("RGCCA")
#> Loading required package: MASS
#> Loading required package: MASS
# Create the dataset
data(Russett)
blocks = list(agriculture = Russett[, seq(3)],
industry = Russett[, 4:5],
politic = Russett[, 6:11])
# Blocks are fully connected, factorial scheme and tau =1 for all blocks is
# used by default
fit.rgcca = rgcca(blocks=blocks, method = "rgcca", connection = 1-diag(3),
scheme = "factorial", tau = rep(1, 3))
summary(fit.rgcca$a$agriculture[, 1] == fit.rgcca$astar$agriculture[, 1])
#> Mode TRUE
#> logical 3
summary(fit.rgcca$Y$agriculture == as.matrix(blocks$agriculture) %*% fit.rgcca$a$agriculture)
#> comp1
#> Mode :logical
#> FALSE:47
summary(fit.rgcca$Y$agriculture == as.matrix(blocks$agriculture) %*% fit.rgcca$astar$agriculture)
#> comp1
#> Mode :logical
#> FALSE:47
Created on 2021-11-15 by the reprex package (v2.0.1)
Indeed, we don't take into account the preprocessing of X
when computing astar
, so the equality is true only when multiplying the preprocessed block X
with astar
.
library("RGCCA")
#> Loading required package: MASS
#> Loading required package: MASS
# Create the dataset
data(Russett)
blocks = list(agriculture = Russett[, seq(3)],
industry = Russett[, 4:5],
politic = Russett[, 6:11])
# Blocks are fully connected, factorial scheme and tau =1 for all blocks is
# used by default
fit.rgcca = rgcca(blocks=blocks, method = "rgcca", connection = 1-diag(3),
scheme = "factorial", tau = rep(1, 3), ncomp = 2)
summary(fit.rgcca$a$agriculture[, 1] == fit.rgcca$astar$agriculture[, 1])
#> Mode TRUE
#> logical 3
summary(fit.rgcca$a$agriculture[, 2] == fit.rgcca$astar$agriculture[, 2])
#> Mode FALSE
#> logical 3
summary(abs(fit.rgcca$Y$agriculture - as.matrix(blocks$agriculture) %*% fit.rgcca$astar$agriculture) < 1e-14)
#> comp1 comp2
#> Mode :logical Mode :logical
#> FALSE:47 FALSE:47
summary(abs(fit.rgcca$Y$agriculture - as.matrix(fit.rgcca$call$blocks$agriculture) %*% fit.rgcca$astar$agriculture) < 1e-14)
#> comp1 comp2
#> Mode:logical Mode:logical
#> TRUE:47 TRUE:47
I was thinking to use the astar values as a measure of the importance of the variables on the data taking into account the specific values of the scaled data, but it seems I can not do that.
I would suggest to amend the documentation line "Each element of astar is a matrix defined as Y[[j]][, h] = A[[j]]%*%astar[[j]][, h]." that A[[j]] should be the scaled data obtained as you described.
I would suggest to amend the documentation line "Each element of astar is a matrix defined as Y[[j]][, h] = A[[j]]%*%astar[[j]][, h]." that A[[j]] should be the scaled data obtained as you described.
Yes indeed it can be confusing.
Perhaps I'm not understanding correctly how RGCCA works but shouldn't the weights and a* (or astar on the output list) be different?
I found that currently they are the same for both rgcca and sgcca methods:
Created on 2021-11-15 by the reprex package (v2.0.1)
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