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Aligned Multiblock Canonical Correlation Analysis

Source: vignettes/aligned_mcca.Rmd
aligned_mcca.Rmd

Why aligned MCCA?

aligned_mcca() solves the same row-mismatch problem as aligned_mfa(), but in the language of canonical correlation. You use it when each block has its own row set, yet all blocks should still line up on one shared score space over a common latent population.

It is the correlation-based counterpart to aligned_mfa(): same alignment idea, different objective.

What does a quick fit look like? 

set.seed(42)
N <- 75
k <- 2
Z <- scale(matrix(rnorm(N * k), nrow = N, ncol = k), center = TRUE, scale = FALSE)

idx1 <- sort(sample.int(N, 60, replace = FALSE))
idx2 <- sort(c(setdiff(seq_len(N), idx1), sample(idx1, 40, replace = FALSE)))
idx3 <- sort(sample.int(N, 55, replace = FALSE))

make_block <- function(idx, p, noise_sd = 0.1) {
  A <- matrix(rnorm(p * k), nrow = p, ncol = k)
  X <- Z[idx, , drop = FALSE] %*% t(A) +
    matrix(rnorm(length(idx) * p, sd = noise_sd), nrow = length(idx), ncol = p)
  scale(X, center = TRUE, scale = FALSE)
}

blocks <- list(
  transcriptomics = make_block(idx1, 20),
  imaging = make_block(idx2, 22),
  physiology = make_block(idx3, 18)
)
row_index <- list(transcriptomics = idx1, imaging = idx2, physiology = idx3)
coverage <- tabulate(c(idx1, idx2, idx3), nbins = N)

full_blocks <- list(
  transcriptomics = make_block(seq_len(N), 20),
  imaging = make_block(seq_len(N), 22)
)
c(
  n_transcriptomics_rows = nrow(blocks$transcriptomics),
  n_imaging_rows = nrow(blocks$imaging),
  n_physiology_rows = nrow(blocks$physiology)
)
#> n_transcriptomics_rows         n_imaging_rows      n_physiology_rows 
#>                     60                     55                     55
fit <- aligned_mcca(blocks, row_index, N = N, ncomp = 2, ridge = 1e-6)
S <- multivarious::scores(fit)

stopifnot(all(dim(S) == c(N, 2)))
stopifnot(all(is.finite(S)))
cc <- cancor(
  scale(Z, center = TRUE, scale = FALSE),
  scale(S, center = TRUE, scale = FALSE)
)$cor[1:2]

stopifnot(all(is.finite(cc)))
stopifnot(mean(cc) > 0.80)
stopifnot(min(cc) > 0.75)

round(cc, 3)
#> [1] 0.866 0.785
Aligned MCCA scores over the latent reference rows. Darker coverage means more blocks contribute to that row.

Aligned MCCA scores over the latent reference rows. Darker coverage means more blocks contribute to that row.

The shared components recover the latent signal even though no block covers the same row pattern as any other block.

How is this different from ordinary MCCA?

When all blocks share the same rows, aligned_mcca() should collapse back to mcca().

fit_aligned_full <- aligned_mcca(
  full_blocks,
  list(transcriptomics = seq_len(N), imaging = seq_len(N)),
  N = N,
  ncomp = 2,
  ridge = 1e-6
)
fit_mcca <- mcca(full_blocks, ncomp = 2, ridge = 1e-6)

P1 <- multivarious::scores(fit_aligned_full) %*%
  solve(crossprod(multivarious::scores(fit_aligned_full)), t(multivarious::scores(fit_aligned_full)))
P2 <- multivarious::scores(fit_mcca) %*%
  solve(crossprod(multivarious::scores(fit_mcca)), t(multivarious::scores(fit_mcca)))
rel <- norm(P1 - P2, type = "F") / (norm(P2, type = "F") + 1e-12)

stopifnot(is.finite(rel))
stopifnot(rel < 0.005)

rel
#> [1] 0.001293249

That reduction matters because it tells you the alignment layer is only solving the missing-row problem. When the problem is already fully aligned, the fitted subspace is numerically very close to the ordinary GCCA/MCCA solution.

What should you inspect after fitting? 

sapply(fit$partial_scores, dim)
#>      transcriptomics imaging physiology
#> [1,]              60      55         55
#> [2,]               2       2          2
sapply(fit$canonical_weights, dim)
#>      transcriptomics imaging physiology
#> [1,]              20      22         18
#> [2,]               2       2          2

The partial_scores live at the observed-row level for each block. The canonical_weights live in each block’s feature space. Together they tell you how the shared reference-space scores are supported by the observed data.

When should you use aligned MCCA?

Use aligned_mcca() when:

  • each block has its own row set,
  • you want correlation-based shared components,
  • repeated or subset row mappings are part of the data collection process, and
  • no block should be treated as the anchor.

If you do have a fully observed anchor block, anchored_mcca() is the more direct interface.

Where next?