ROBPCA - ROBust method for Principal Components Analysis
The ROBPCA algorithm was proposed by Hubert et al (2005) and stays for 'ROBust method for Principal Components Analysis'. It is resistant to outliers in the data. The robust loadings are computed using projection-pursuit techniques and the MCD method. Therefore ROBPCA can be applied to both low and high-dimensional data sets. In low dimensions, the MCD method is applied.
PcaHubert(x, ...) ## Default S3 method: PcaHubert(x, k = 0, kmax = 10, alpha = 0.75, mcd = TRUE, maxdir=250, scale = FALSE, signflip = TRUE, crit.pca.distances = 0.975, trace=FALSE, ...) ## S3 method for class 'formula' PcaHubert(formula, data = NULL, subset, na.action, ...)
formula |
a formula with no response variable, referring only to numeric variables. |
data |
an optional data frame (or similar: see
|
subset |
an optional vector used to select rows (observations) of the
data matrix |
na.action |
a function which indicates what should happen
when the data contain |
... |
arguments passed to or from other methods. |
x |
a numeric matrix (or data frame) which provides the data for the principal components analysis. |
k |
number of principal components to compute. If |
kmax |
maximal number of principal components to compute.
Default is |
alpha |
this parameter measures the fraction of outliers the algorithm should resist. In MCD alpha controls the size of the subsets over which the determinant is minimized, i.e. alpha*n observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.75. |
mcd |
Logical - when the number of variables is sufficiently small,
the loadings are computed as the eigenvectors of the MCD covariance matrix,
hence the function |
maxdir |
maximal number of random directions to use for computing the
outlyingness of the data points. Default is |
scale |
a value indicating whether and how the variables should be scaled.
If |
signflip |
a logical value indicating wheather to try to solve the sign indeterminancy of the loadings -
ad hoc approach setting the maximum element in a singular vector to be positive. Default is |
crit.pca.distances |
criterion to use for computing the cutoff values for the orthogonal and score distances. Default is 0.975. |
trace |
whether to print intermediate results. Default is |
PcaHubert, serving as a constructor for objects of class PcaHubert-class
is a generic function with "formula" and "default" methods.
The calculation is done using the ROBPCA method of Hubert et al (2005) which can
be described briefly as follows. For details see the relevant references.
Let n denote the number of observations, and p the
number of original variables in the input data matrix X. The
ROBPCA algorithm finds a robust center M (p x 1) of the data
and a loading matrix P which is (p x k) dimensional.
Its columns are orthogonal and define a new coordinate system. The
scores T, an (n x k) matrix, are the coordinates of the
centered observations with respect to the loadings:
T=(X-M)P
The ROBPCA algorithm also yields a robust covariance matrix (often singular) which can be computed as
S=PLP^t
where L is the diagonal matrix with the eigenvalues l_1, …, \l_k.
This is done in the following three main steps:
Step 1: The data are preprocessed by reducing their data space to
the subspace spanned by the n observations. This is done by
singular value decomposition of the input data matrix. As a result
the data are represented using at most n-1=rank(X) without
loss of information.
Step 2: In this step for each data point a measure of outlyingness
is computed. For this purpose the high-dimensional data points are
projected on many univariate directions, each time the univariate
MCD estimator of location and scale is computed and the
standardized distance to the center is measured. The largest of
these distances (over all considered directions) is the outlyingness
measure of the data point. The h data points with smallest
outlyingness measure are used to compute the covariance matrix
Σ_h and to select the number k of principal
components to retain. This is done by finding such k that
l_k/l_1 >= 10.E-3 and Σ_{j=1}^k l_j/Σ_{j=1}^r l_j >= 0.8
Alternatively the number of principal components k can be
specified by the user after inspecting the scree plot.
Step 3: The data points are projected on the k-dimensional subspace
spanned by the k eigenvectors corresponding to the largest
k eigenvalues of the matrix Σ_h. The location and
scatter of the projected data are computed using the
reweighted MCD estimator and the eigenvectors of this scatter matrix
yield the robust principal components.
An S4 object of class PcaHubert-class which is a subclass of the
virtual class PcaRobust-class.
The ROBPCA algorithm is implemented on the bases of the Matlab implementation, available as part of LIBRA, a Matlab Library for Robust Analysis to be found at www.wis.kuleuven.ac.be/stat/robust.html
Valentin Todorov valentin.todorov@chello.at
M. Hubert, P. J. Rousseeuw, K. Vanden Branden (2005), ROBPCA: a new approach to robust principal components analysis, Technometrics, 47, 64–79.
Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. URL http://www.jstatsoft.org/v32/i03/.
## PCA of the Hawkins Bradu Kass's Artificial Data
## using all 4 variables
data(hbk)
pca <- PcaHubert(hbk)
pca
## Compare with the classical PCA
prcomp(hbk)
## or
PcaClassic(hbk)
## If you want to print the scores too, use
print(pca, print.x=TRUE)
## Using the formula interface
PcaHubert(~., data=hbk)
## To plot the results:
plot(pca) # distance plot
pca2 <- PcaHubert(hbk, k=2)
plot(pca2) # PCA diagnostic plot (or outlier map)
## Use the standard plots available for prcomp and princomp
screeplot(pca)
biplot(pca)
## Restore the covraiance matrix
py <- PcaHubert(hbk)
cov.1 <- py@loadings %*% diag(py@eigenvalues) %*% t(py@loadings)
cov.1Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.