a and astar being equal

[llrs]

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:

  library("RGCCA")
#> 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

^{Created on 2021-11-15 by the reprex package (v2.0.1)}

Session info

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[GFabien]

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!

[llrs]

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)}

[GFabien]

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   

[llrs]

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.

[GFabien]

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.