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NMF-Based Super-Level Analysis and ECMSMA

Atsushi Kawaguchi

2026-08-26

1 Overview

Version 3.2 adds NMF and sparse NMF (sNMF) as super-level decomposition methods. They support clustering-oriented analyses of multiblock data while the default sprmethod = "PCA" preserves the version 3.1 computational path.

dat <- simdata(n = 45, rho = 0.8, Xps = c(5, 5, 5), Yps = 3, seed = 3)
X <- dat$X
names(X) <- paste0("block", seq_along(X))

2 Super-level methods

The principal options are:

sprmethod = "PCA": conventional super-level PCA;
sprmethod = "NMF": non-negative matrix factorization;
sprmethod = "sNMF": sparse non-negative matrix factorization.

fit_pca <- msma(X, comp = c(2, 2), sprmethod = "PCA", intseed = 1)
fit_nmf <- msma(X, comp = c(2, 2), sprmethod = "NMF", intseed = 1)
fit_snmf <- msma(
  X, comp = c(2, 2),
  sprmethod = "sNMF",
  lambdaXsup = 0.05,
  intseed = 1
)

3 Non-negative transformations

NMF requires non-negative input. The nneg argument controls transformation of the block scores:

"posneg": separates positive and negative parts;
"absolute": uses absolute values;
"min": shifts values to a non-negative range.

For a score vector \(s\), positive-negative decomposition is

\[ s^+ = \max(s,0), \qquad s^- = \max(-s,0), \qquad s=s^+-s^-. \]

fit_posneg <- msma(X, comp = c(2, 2), sprmethod = "NMF", nneg = "posneg")
fit_absolute <- msma(X, comp = c(2, 2), sprmethod = "NMF", nneg = "absolute")
fit_min <- msma(X, comp = c(2, 2), sprmethod = "NMF", nneg = "min")

4 Non-negativity and reproducibility

all(unlist(fit_snmf$ssX) >= 0)
#> [1] TRUE
all(unlist(fit_snmf$wsX) >= 0)
#> [1] TRUE

fit_snmf_2 <- msma(
  X, comp = c(2, 2),
  sprmethod = "sNMF",
  lambdaXsup = 0.05,
  intseed = 1
)
all.equal(fit_snmf$ssX, fit_snmf_2$ssX)
#> [1] TRUE
all.equal(fit_snmf$wsX, fit_snmf_2$wsX)
#> [1] TRUE

5 Clustering solutions

A simple cluster assignment is obtained from the largest super-score value for each observation. Each root component supplies one clustering solution.

cluster_matrix <- vapply(
  fit_snmf$ssX,
  function(score) max.col(score, ties.method = "first"),
  integer(nrow(fit_snmf$ssX[[1]]))
)
colnames(cluster_matrix) <- names(fit_snmf$ssX)
head(cluster_matrix)
#>      comp1 comp2
#> [1,]     2     2
#> [2,]     2     2
#> [3,]     1     1
#> [4,]     1     2
#> [5,]     1     1
#> [6,]     1     2
apply(cluster_matrix, 2, table)
#>   comp1 comp2
#> 1    28    23
#> 2    17    22

The resulting matrix may be supplied to a separate consensus or ensemble clustering procedure. The consensus step is not performed automatically by msma().

6 Multiple supervision variables

Z may be a numeric matrix. con4spv specifies weights used to combine its columns. This is a composite-supervision model rather than a multi-task model with a separate loading for every outcome.

set.seed(3)
z1 <- rnorm(nrow(X[[1]]))
z2 <- 0.5 * z1 + rnorm(nrow(X[[1]]), sd = 0.5)
Z <- cbind(clinical = z1, biomarker = z2)

fit_multi_z <- msma(
  X = X, Z = Z,
  con4spv = c(0.7, 0.3),
  comp = 2,
  muX = 0.20,
  intseed = 1
)
fit_multi_z$predictiv
#> [[1]]
#>         clinical   biomarker
#> [1,] 0.004986875 0.004354675
#> 
#> [[2]]
#>      clinical  biomarker
#> [1,] 0.036251 0.04306789

Conceptually, the combined supervision score is

\[ z_c = Zc, \]

where the supplied weights are normalized internally as required by the implementation.

7 One-column matrix compatibility

A vector and a one-column matrix produce the same result.

fit_z_vector <- msma(X, Z = z1, comp = 1, muX = 0.20, intseed = 1)
fit_z_matrix <- msma(X, Z = cbind(z1), comp = 1, muX = 0.20, intseed = 1)
fit_z_vector$call <- NULL
fit_z_matrix$call <- NULL
isTRUE(all.equal(fit_z_vector, fit_z_matrix, tolerance = 1e-8))
#> [1] TRUE

8 Practical recommendations

Fix intseed for reproducible NMF and sNMF results.
Inspect both ssX and wsX when interpreting clusters.
Compare several values of lambdaXsup for sNMF.
Treat clustering from different root components as distinct candidate solutions before ensemble integration.

9 Session information

sessionInfo()
#> R Under development (unstable) (2026-08-25 r90447 ucrt)
#> Platform: x86_64-w64-mingw32/x64
#> Running under: Windows 11 x64 (build 26200)
#> 
#> Matrix products: default
#>   LAPACK version 3.12.1
#> 
#> locale:
#> [1] LC_COLLATE=C                    LC_CTYPE=Japanese_Japan.utf8   
#> [3] LC_MONETARY=Japanese_Japan.utf8 LC_NUMERIC=C                   
#> [5] LC_TIME=Japanese_Japan.utf8    
#> 
#> time zone: Asia/Tokyo
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] msma_3.2
#> 
#> loaded via a namespace (and not attached):
#>  [1] digest_0.6.39   R6_2.6.1        fastmap_1.2.0   xfun_0.60      
#>  [5] cachem_1.1.0    knitr_1.51      htmltools_0.5.9 rmarkdown_2.31 
#>  [9] lifecycle_1.0.5 cli_3.6.6       sass_0.4.10     jquerylib_0.1.4
#> [13] compiler_4.7.0  tools_4.7.0     evaluate_1.0.5  bslib_0.12.0   
#> [17] yaml_2.3.12     otel_0.2.0      rlang_1.3.0     jsonlite_2.0.0

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