Robust root mean squared error of cross-validation (R-RMSECV) of the
robust PLS (rsimpls()) or robust PCR (rpcr()) models with 1 to
kmax components, and the robust component selection (RCS) criterion of
Engelen and Hubert (2005), to choose the number of components when the
calibration data contain outliers.
Usage
robust_rmsecv(
x,
y,
method = c("rsimpls", "rpcr"),
kmax = 10,
folds = 10,
gamma = 0.5,
alpha = 0.75,
ndir = 250,
nsamp = 500,
...
)Arguments
- x
A numeric matrix or data frame of the predictors (spectra), one observation per row.
- y
A numeric vector, matrix or data frame of the responses, with one row per observation.
- method
The model:
"rsimpls"(default), robust PLS, or"rpcr", robust PCR.- kmax
The largest number of components. Default is 10.
- folds
The number of folds of the cross-validation, from 2 to the number of observations (leave-one-out). Default is 10.
- gamma
The weight of the cross-validated error in the RCS criterion, between 0 and 1. Default is 0.5.
- alpha
The robustness parameter of
robpca(): the fraction of observations assumed to be regular, between 0.5 and 1. Default is 0.75.- ndir
The number of random directions of the outlyingness in
robpca(). Default is 250.- nsamp
The number of random subsets of FAST-MCD (and, for
"rpcr", of the LTS or MCD regression). Default is 500.- ...
Not used.
Value
A tibble with one row per number of components: ncomp, the
robust R2 and RMSE of the fit to all the observations (as the
components of rsimpls() and rpcr()), the robust RMSECV and the
RCS criterion. The weights \(w_i\) are its "weights" attribute.
Details
The squared prediction errors of outliers would dominate the usual
RMSECV. The robust RMSECV leaves them out: the model is first fitted to
all the observations, and only those regular in every model with 1 to
kmax components (residual distance below its cut-off) enter the error,
$$R\text{-}RMSECV_k = \sqrt{\frac{1}{q \sum_i w_i} \sum_i w_i
\|y_i - \hat y_{-i,k}\|^2},$$
where \(w_i\) is 1 for these observations and 0 for the others, and
\(\hat y_{-i,k}\) is the prediction of observation \(i\) by the model
with \(k\) components fitted without it.
The robust root mean squared error of the fit to the calibration data,
RMSE (the square root of the robust residual sum of squares, R-RSS),
decreases with the number of components, while the cross-validated error
eventually increases. The RCS criterion combines them,
$$RCS_k = \sqrt{\gamma \, R\text{-}RMSECV_k^2 + (1 - \gamma) \,
R\text{-}RSS_k},$$
and its first local minimum, or the point after which it decreases
little, suggests the number of components. gamma = 1 gives the robust
RMSECV, and gamma = 0 the R-RSS.
Differences from the paper. The paper computes the leave-one-out
errors with a fast approximation that updates the robust fits instead of
refitting them. Here, each fold is refitted. Leave-one-out
(folds = nrow(x)) is then slow, and K-fold cross-validation (default
10 folds) is used instead.
The folds are fitted in parallel when a parallel plan is set with
future::plan() (and the future.apply package is installed). The folds
and the fits are random: use set.seed() for reproducible results.
References
Engelen, S., Hubert, M. (2005). Fast model selection for robust calibration methods. Analytica Chimica Acta, 544(1-2):219-228.
Examples
# \donttest{
data(forageLIBS)
wl <- suppressWarnings(as.numeric(names(forageLIBS)))
spectra <- forageLIBS[which(wl > 380 & wl < 430)]
set.seed(1)
robust_rmsecv(spectra[1:300, ], forageLIBS$Ca[1:300], kmax = 8, folds = 5)
#> # A tibble: 8 × 5
#> ncomp R2 RMSE RMSECV RCS
#> <int> <dbl> <dbl> <dbl> <dbl>
#> 1 1 0.169 0.140 0.142 0.141
#> 2 2 0.500 0.109 0.114 0.111
#> 3 3 0.661 0.0895 0.0933 0.0914
#> 4 4 0.740 0.0784 0.0811 0.0798
#> 5 5 0.756 0.0760 0.0830 0.0796
#> 6 6 0.755 0.0762 0.0838 0.0801
#> 7 7 0.755 0.0762 0.0835 0.0799
#> 8 8 0.753 0.0765 0.0815 0.0790
# }