big_SVD: Partial SVD

Description

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.

Usage

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

A 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.

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.

Examples

Run this code

# NOT RUN {
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)
plot(test$u)

pca <- prcomp(X[ind, ], center = TRUE, scale. = TRUE)

# same scaling
all.equal(test$center, pca$center)
all.equal(test$scale,  pca$scale)

# 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)
scores2 <- predict(test) # use this function to predict scores
all.equal(scores, scores2)
dim(scores)
dim(pca$x)
tail(pca$sdev)
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)
scores.test3 <- predict(pca, X[-ind, ])
plot(scores.test2, scores.test3[, 1:ncol(scores.test2)])
# }