FRBpcaMM {FRB} | R Documentation |
Performs principal components analysis based on the robust MM-estimate of the shape matrix. Additionally uses the Fast and Robust Bootstrap method to compute inference measures such as standard errors and confidence intervals.
FRBpcaMM(Y, R = 999, conf = 0.95, control=MMcontrol(...), ...)
Y |
matrix or data frame |
R |
number of bootstrap samples |
conf |
level of the bootstrap confidence intervals. Default is conf=0.95 |
control |
a list with control parameters for tuning the MM-estimate and its computing algorithm,
see MMcontrol (). |
... |
allows for specifying control parameters directly instead of via control |
Multivariate MM-estimates are defined by first computing an S-estimate of location and covariance, then fixing its scale component and
re-estimating the location and the shape by a more efficient M-estimate, see Tatsuoka and Tyler (2000). Tukey's biweight is used for
the loss functions. By default, the first loss function (in the S-estimate) is tuned in order to obtain 50% breakdown point.
The default tuning of the second loss function (M-estimate) ensures 95% efficiency for the shape matrix estimate at the normal model.
This tuning is recommended but can be changed through argument control
if desired. (However, control parameter
shapeEff
will always be considered as TRUE
by this function, whichever value is specified.)
The MM-estimate is computed by a call to MMest_loccov
(), the result of which is returned as est
.
This function first performs the fast-S algorithm (see Sest_loccov
) and does the M-part by reweighted least
squares iteration. See MMcontrol
for some adjustable tuning parameters regarding the algorithm.
PCA is performed by computing the eigenvalues (eigval
) and eigenvectors (eigvec
) of the MM-estimate of shape, which is a rescaled version of the
MM-estimate of covariance (rescaled to have determinant equal to 1). With pvar
the function also provides the estimates for the percentage of
variance explained by the first k principal components, which are simply the cumulative proportions of the eigenvalues sum.
Here, k ranges from 1 to p-1 (with p the number of variables in Y
).
The eigenvectors are always given in the order of descending eigenvalues.
The Fast and Robust Bootstrap (Salibian-Barrera and Zamar 2002) is used to calculate standard errors, and also so-called
basic bootstrap confidence intervals and bias corrected and accelerated (BCa) confidence intervals (Davison and Hinkley 1997, p.194 and p.204 respectively) corresponding
to the estimates eigval
, eigvec
and pvar
.
The bootstrap is also used to estimate the average angles between true and estimated eigenvectors, returned as avgangle
.
See Salibian-Barrera, Van Aelst and Willems (2006).
The fast and robust bootstrap computations for the MM-estimates are performed by MMboot_loccov
() and its raw result can be found in bootest
.
The actual bootstrap recalculations for the PCA-related quantities can be found in eigval.boot
, eigvec.boot
and pvar.boot
, where each column
represents a bootstrap sample. For eigvec.boot
, the eigenvectors are stacked on top of each other and the same goes for
eigvec.CI.bca
and eigvec.CI.basic
which hold the confidence limits.
The two columns in the confidence limits always respectively represent the lower and upper limits. For the percentage of variance the function also provides one-sided confidence intervals ([-infty upper]), which can be used to test the hypothesis that the true percentage at least equals a certain value.
Bootstrap samples are discarded if the fast and robust shape estimate is not positive definite, such that the actual number
of recalculations used can be lower than R
. This actual number equals R
- failedsamples
.
However, if more than 0.75R
of the bootstrap shape estimates is non-positive definite, all bootstrap samples will be
used anyway, and the negative eigenvalues are simply set to zero (which may impact the confidence limits and standard errors
for the smallest eigenvalues in eigval
and pvar
).
An object of class FRBpca
, which is a list containing the following components:
shape |
(p x p) MM-estimate of the shape matrix of Y |
eigval |
(p x 1) eigenvalues of MM shape |
eigvec |
(p x p) eigenvectors of MM-shape |
pvar |
(p-1 x 1) percentages of variance for MM eigenvalues |
eigval.boot |
(p x R) eigenvalues of MM shape |
eigvec.boot |
(p*p x R) eigenvectors of MM-shape (vectorized) |
pvar.boot |
(p-1 x R) percentages of variance for MM eigenvalues |
eigval.SE |
(p x 1) bootstrap standard error for MM eigenvalues |
eigvec.SE |
(p x p) bootstrap standard error for MM eigenvectors |
pvar.SE |
(p-1 x 1) bootstrap standard error for percentage of variance for MM-eigenvalues |
angles |
(p x R) angles between bootstrap eigenvectors and original MM eigenvectors (in radians; in [0 pi/2]) |
avgangle |
(p x 1) average angles between bootstrap eigenvectors and original MM eigenvectors (in radians; in [0 pi/2]) |
eigval.CI.bca |
(p x 2) BCa intervals for MM eigenvalues |
eigvec.CI.bca |
(p*p x 2) BCa intervals for MM eigenvectors (vectorized) |
pvar.CI.bca |
(p-1 x 2) BCa intervals for percentage of variance for MM-eigenvalues |
pvar.CIone.bca |
(p-1 x 1) one-sided BCa intervals for percentage of variance for MM-eigenvalues ([-infty upper]) |
eigval.CI.basic |
(p x 2) basic bootstrap intervals for MM eigenvalues |
eigvec.CI.basic |
(p*p x 2) basic bootstrap intervals for MM eigenvectors (vectorized) |
pvar.CI.basic |
(p-1 x 2) basic bootstrap intervals for percentage of variance for MM-eigenvalues |
pvar.CIone.basic |
(p-1 x 1) one-sided basic bootstrap intervals for percentage of variance for MM-eigenvalues ([-infty upper]) |
est |
(list) result of MMest_loccov () |
bootest |
(list) result of MMboot_loccov () |
failedsamples |
number of bootstrap samples with non-positive definiteness of shape |
conf |
a copy of the conf argument |
method |
a character string giving the robust PCA method that was used |
Gert Willems and Ella Roelant
plot.FRBpca
, summary.FRBpca
, print.FRBpca
, FRBpcaS
,
MMest_loccov
, MMboot_loccov
, MMcontrol
data(ForgedBankNotes) MMpcares <- FRBpcaMM(ForgedBankNotes, R=999, conf=0.95) # the simple print method shows the standard deviations with confidence limits: MMpcares # the summary functions shows a lot more (see help(summary.FRBpca)): summary(MMpcares) # ask for the eigenvalues: MMpcares$eigval # or, in more pretty format, with confidence limits: summary(MMpcares)$eigvals # note that the standard deviations of the print-output can also be asked for by: sqrt( summary(MMpcares)$eigvals ) # the eigenvectors and their standard errors: MMpcares$eigvec # or prettier: summary(MMpcares)$eigvecs MMpcares$eigvec.SE # take a look at the bootstrap distribution of the first eigenvalue hist(MMpcares$eigval.boot[1,]) # that bootstrap distribution is used to compute confidence limits as depicted # by the screeplot function: plotFRBvars(MMpcares, cumul=0) # all plots for the FRB-PCA result: plot(MMpcares)