An algorithm for partial SVD (or PCA) of a Filebacked Big Matrix through the eigen decomposition of the covariance between variables (primal) or observations (dual). Use this algorithm only if there is one dimension that is much smaller than the other. Otherwise use big_randomSVD.

big_SVD(
  X,
  fun.scaling = big_scale(center = FALSE, scale = FALSE),
  ind.row = rows_along(X),
  ind.col = cols_along(X),
  k = 10,
  block.size = block_size(nrow(X))
)

Arguments

X

An object of class FBM.

fun.scaling

A function that returns a named list of mean and sd for every column, to scale each of their elements such as followed: $$\frac{X_{i,j} - mean_j}{sd_j}.$$ Default doesn't use any scaling.

ind.row

An optional vector of the row indices that are used. If not specified, all rows are used. Don't use negative indices.

ind.col

An optional vector of the column indices that are used. If not specified, all columns are used. Don't use negative indices.

k

Number of singular vectors/values to compute. Default is 10.

block.size

Maximum number of columns read at once. Default uses block_size.

Value

A named list (an S3 class "big_SVD") of

  • d, the singular values,

  • u, the left singular vectors,

  • v, the right singular vectors,

  • center, the centering vector,

  • scale, the scaling vector.

Note that to obtain the Principal Components, you must use predict on the result. See examples.

Details

To get \(X = U \cdot D \cdot V^T\),

  • if the number of observations is small, this function computes \(K_(2) = X \cdot X^T \approx U \cdot D^2 \cdot U^T\) and then \(V = X^T \cdot U \cdot D^{-1}\),

  • if the number of variable is small, this function computes \(K_(1) = X^T \cdot X \approx V \cdot D^2 \cdot V^T\) and then \(U = X \cdot V \cdot D^{-1}\),

  • if both dimensions are large, use big_randomSVD instead.

Matrix parallelization

Large matrix computations are made block-wise and won't be parallelized in order to not have to reduce the size of these blocks. Instead, you may use Microsoft R Open or OpenBLAS in order to accelerate these block matrix computations. You can also control the number of cores used with bigparallelr::set_blas_ncores().

See also

Examples

set.seed(1) X <- big_attachExtdata() n <- nrow(X) # Using only half of the data ind <- sort(sample(n, n/2)) test <- big_SVD(X, fun.scaling = big_scale(), ind.row = ind) str(test)
#> List of 5 #> $ d : num [1:10] 178.5 114.5 91 87.1 86.3 ... #> $ u : num [1:258, 1:10] -0.1092 -0.0928 -0.0806 -0.0796 -0.1028 ... #> $ v : num [1:4542, 1:10] 0.00607 0.00739 0.02921 -0.01283 0.01473 ... #> $ center: num [1:4542] 1.34 1.63 1.51 1.64 1.09 ... #> $ scale : num [1:4542] 0.665 0.551 0.631 0.55 0.708 ... #> - attr(*, "class")= chr "big_SVD"
plot(test$u)
pca <- prcomp(X[ind, ], center = TRUE, scale. = TRUE) # same scaling all.equal(test$center, pca$center)
#> [1] TRUE
all.equal(test$scale, pca$scale)
#> [1] TRUE
# scores and loadings are the same or opposite # except for last eigenvalue which is equal to 0 # due to centering of columns scores <- test$u %*% diag(test$d) class(test)
#> [1] "big_SVD"
scores2 <- predict(test) # use this function to predict scores all.equal(scores, scores2)
#> [1] TRUE
dim(scores)
#> [1] 258 10
dim(pca$x)
#> [1] 258 258
tail(pca$sdev)
#> [1] 3.023287e+00 3.008386e+00 2.990514e+00 2.984375e+00 2.965688e+00 #> [6] 1.130391e-14
plot(scores2, pca$x[, 1:ncol(scores2)])
plot(test$v[1:100, ], pca$rotation[1:100, 1:ncol(scores2)])
# projecting on new data X2 <- sweep(sweep(X[-ind, ], 2, test$center, '-'), 2, test$scale, '/') scores.test <- X2 %*% test$v ind2 <- setdiff(rows_along(X), ind) scores.test2 <- predict(test, X, ind.row = ind2) # use this all.equal(scores.test, scores.test2)
#> [1] TRUE
scores.test3 <- predict(pca, X[-ind, ]) plot(scores.test2, scores.test3[, 1:ncol(scores.test2)])