Principal component analysis by NIPALS, non-linear iterative partial least squares

Used for finding principal components of a numeric matrix. Missing values in the matrix are allowed. Principal Components are extracted one a time. The algorithm computes x = TP', where T is the 'scores' matrix and P is the 'loadings' matrix.

Usage

nipals(
  x,
  ncomp = min(nrow(x), ncol(x)),
  center = TRUE,
  scale = TRUE,
  maxiter = 500,
  tol = 1e-06,
  startcol = 0,
  fitted = FALSE,
  force.na = FALSE,
  gramschmidt = TRUE,
  verbose = FALSE
)

Arguments

x: Numerical matrix for which to find principal compontents. Missing values are allowed.
ncomp: Maximum number of principal components to extract from x.
center: If TRUE, subtract the mean from each column of x before PCA.
scale: if TRUE, divide the standard deviation from each column of x before PCA.
maxiter: Maximum number of NIPALS iterations for each principal component.
tol: Default 1e-6 tolerance for testing convergence of the NIPALS iterations for each principal component.
startcol: Determine the starting column of x for the iterations of each principal component. If 0, use the column of x that has maximum absolute sum. If a number, use that column of x. If a function, apply the function to each column of x and choose the column with the maximum value of the function.
fitted: Default FALSE. If TRUE, return the fitted (reconstructed) value of x.
force.na: Default FALSE. If TRUE, force the function to use the method for missing values, even if there are no missing values in x.
gramschmidt: Default TRUE. If TRUE, perform Gram-Schmidt orthogonalization at each iteration.
verbose: Default FALSE. Use TRUE or 1 to show some diagnostics.

Value

A list with components eig, scores, loadings, fitted, ncomp, R2, iter, center, scale.

Details

The R2 values that are reported are marginal, not cumulative.

References

Wold, H. (1966) Estimation of principal components and related models by iterative least squares. In Multivariate Analysis (Ed., P.R. Krishnaiah), Academic Press, NY, 391-420.

Andrecut, Mircea (2009). Parallel GPU implementation of iterative PCA algorithms. Journal of Computational Biology, 16, 1593-1599.

Author

Kevin Wright

Examples

B <- matrix(c(50, 67, 90, 98, 120,
              55, 71, 93, 102, 129,
              65, 76, 95, 105, 134,
              50, 80, 102, 130, 138,
              60, 82, 97, 135, 151,
              65, 89, 106, 137, 153,
              75, 95, 117, 133, 155), ncol=5, byrow=TRUE)
rownames(B) <- c("G1","G2","G3","G4","G5","G6","G7")
colnames(B) <- c("E1","E2","E3","E4","E5")
dim(B) # 7 x 5
#> [1] 7 5
p1 <- nipals(B)
dim(p1$scores) # 7 x 5
#> [1] 7 5
dim(p1$loadings) # 5 x 5
#> [1] 5 5

B2 = B
B2[1,1] = B2[2,2] = NA
p2 = nipals(B2, fitted=TRUE)

# Two ways to make a biplot

# method 1
biplot(p2$scores, p2$loadings)

# method 2
class(p2) <- "princomp"
p2$sdev <- sqrt(p2$eig)
biplot(p2, scale=0)