pcaPP (version 2.0-5)

PCAproj: Robust Principal Components using the algorithm of Croux and Ruiz-Gazen (2005)

Description

Computes a desired number of (robust) principal components using the algorithm of Croux and Ruiz-Gazen (JMVA, 2005).

Usage

PCAproj(x, k = 2, method = c("mad", "sd", "qn"), CalcMethod = c("eachobs",
"lincomb", "sphere"), nmax = 1000, update = TRUE, scores = TRUE, maxit = 5, 
maxhalf = 5, scale = NULL, center = l1median_NLM, zero.tol = 1e-16, control)

Value

The function returns a list of class "princomp", i.e. a list similar to the output of the function princomp.

sdev

the (robust) standard deviations of the principal components.

loadings

the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors). This is of class "loadings": see loadings for its print method.

center

the means that were subtracted.

scale

the scalings applied to each variable.

n.obs

the number of observations.

scores

if scores = TRUE, the scores of the supplied data on the principal components.

call

the matched call.

Arguments

x

a numeric matrix or data frame which provides the data for the principal components analysis.

k

desired number of components to compute

method

scale estimator used to detect the direction with the largest variance. Possible values are "sd", "mad" and "qn", the latter can be called "Qn" too. "mad" is the default value.

CalcMethod

the variant of the algorithm to be used. Possible values are "eachobs", "lincomb" and "sphere", with "eachobs" being the default.

nmax

maximum number of directions to search in each step (only when using "sphere" or "lincomb" as the CalcMethod).

update

a logical value indicating whether an update algorithm should be used.

scores

a logical value indicating whether the scores of the principal component should be calculated.

maxit

maximim number of iterations.

maxhalf

maximum number of steps for angle halving.

scale

this argument indicates how the data is to be rescaled. It can be a function like sd or mad or a vector of length ncol(x) containing the scale value of each column.

center

this argument indicates how the data is to be centered. It can be a function like mean or median or a vector of length ncol(x) containing the center value of each column.

zero.tol

the zero tolerance used internally for checking convergence, etc.

control

a list which elements must be the same as (or a subset of) the parameters above. If the control object is supplied, the parameters from it will be used and any other given parameters are overridden.

Author

Heinrich Fritz, Peter Filzmoser <>

Details

Basically, this algrithm considers the directions of each observation through the origin of the centered data as possible projection directions. As this algorithm has some drawbacks, especially if ncol(x) > nrow(x) in the data matrix, there are several improvements that can be used with this algorithm.

• update - An updating step basing on the algorithm for finding the eigenvectors is added to the algorithm. This can be used with any CalcMethod

• sphere - Additional search directions are added using random directions. The random directions are determined using random data points generated from a p-dimensional multivariate standard normal distribution. These new data points are projected to the unit sphere, giving the new search directions.

• lincomb - Additional search directions are added using linear combinations of the observations. It is similar to the "sphere" - algorithm, but the new data points are generated using linear combinations of the original data b_1*x_1 + ... + b_n*x_n where the coefficients b_i come from a uniform distribution in the interval [0, 1].

Similar to the function princomp, there is a print method for the these objects that prints the results in a nice format and the plot method produces a scree plot (screeplot). There is also a biplot method.

References

C. Croux, P. Filzmoser, M. Oliveira, (2007). Algorithms for Projection-Pursuit Robust Principal Component Analysis, Chemometrics and Intelligent Laboratory Systems, Vol. 87, pp. 218-225.

See Also

PCAgrid, ScaleAdv, princomp

Examples

  # multivariate data with outliers
  library(mvtnorm)
  x <- rbind(rmvnorm(200, rep(0, 6), diag(c(5, rep(1,5)))),
             rmvnorm( 15, c(0, rep(20, 5)), diag(rep(1, 6))))
  # Here we calculate the principal components with PCAgrid
  pc <- PCAproj(x, 6)
  # we could draw a biplot too:
  biplot(pc)

  # we could use another calculation method and another objective function, and 
  # maybe only calculate the first three principal components:
  pc <- PCAproj(x, 3, "qn", "sphere")
  biplot(pc)

  # now we want to compare the results with the non-robust principal components
  pc <- princomp(x)
  # again, a biplot for comparision:
  biplot(pc)