Q and T-squared Contributions of a PCA Model

Computes how much each variable (wavelength) contributes to the Q residual or to Hotelling's \(T^2\) of each sample, keeping the sign of the deviation, to find the variables that make a sample an outlier. Relative contributions, computed against reference samples, show what differs between a sample and the reference ones.

Usage

contributions(
  model,
  k = NULL,
  statistic = c("q", "t2"),
  data = NULL,
  samples = NULL,
  reference = NULL
)

Arguments

model: A stats::prcomp() fit, or an object returned by robpca(), rospca() or macropca().
k: The number of components. Required for a stats::prcomp() fit; for a robust fit, at most its number of components (the default).
statistic: "q" (default) for the Q contributions, or "t2" for the \(T^2\) contributions.
data: The samples whose contributions are computed: a numeric matrix or data frame with the variables of the model (other columns are ignored when the columns are named), preprocessed like the data the model was fitted on (for example, with center() if the model was fitted on center(spectra)). Default is the calibration data of a stats::prcomp() fit (which must keep all its components), or the imputed data of a macropca() fit; it is required for robpca() and rospca() fits, which do not keep their data.
samples: Optional indices or names of the rows of data to return. Default is all.
reference: Optional indices or names of rows of data whose mean contribution is subtracted (relative contributions), or "regular" for all the regular samples.

Value

A tibble of class specproc_contributions with one row per sample: sample (the row name, or number) and one column per variable. The attribute total holds the statistic of each sample (Q or \(T^2\), before any reference is subtracted). Draw it with plot_contributions().

Details

For a sample \(x\) (centered and scaled as in the model), with scores \(t = xP\) on the first \(k\) components (loadings \(P\), variances of the scores \(\lambda\)):

the Q contributions are the residuals \(e = x - tP^T\), whose sum of squares is Q (the squared orthogonal distance of robust fits);
the \(T^2\) contributions are \(t \Lambda^{-1/2} P^T\), the scaled scores projected back onto the variables, whose sum of squares is \(T^2\) (the squared score distance of robust fits). For rospca() fits, whose loadings are not exactly orthogonal, the sum of squares is close to, but not exactly, \(T^2\).

Q contributions show which variables the model does not describe for a sample: large contributions grouped on a few emission lines point to a systematic deviation (a contamination, a matrix effect, a saturated or shifted line), small ones spread over all channels to random noise. \(T^2\) contributions show which variables place a sample far from the center in the score space.

Relative contributions. Normally, contributions are relative to the model (its center, for \(T^2\)). With reference, the mean contribution of the reference samples is subtracted from the contribution of each sample, which shows what differs between them: for example, whether two samples have large Q residuals for the same reason, or what moves a sample from the regular samples to where it lies in the score space. reference = "regular" uses all the regular samples: of the robust fit, or at the highest confidence level of q_residuals() for a stats::prcomp() fit.

The contributions are in the units of the preprocessed data of the model: centered (and scaled, if the model scaled the variables).

References

Westerhuis, J.A., Gurden, S.P., Smilde, A.K. (2000). Generalized contribution plots in multivariate statistical process monitoring. Chemometrics and Intelligent Laboratory Systems, 51(1):95-114.

Author

Christian L. Goueguel

Examples

spectra_id <- names(forageLIBS)[1:2]
minerals <- names(forageLIBS)[3:14]
spectra <- forageLIBS |> dplyr::select(-dplyr::all_of(c(spectra_id, minerals)))
set.seed(1)
fit <- robpca(spectra, k = 3)

# Q contributions of two outlying samples, relative to the regular ones
q <- contributions(fit, data = spectra, samples = c(49, 127), reference = "regular")
q[, 1:5]
#> # Q contributions of 2 sample(s) to 4 variables (3 components), relative to reference samples
#> # A tibble: 2 × 5
#>   sample `199.3771616` `199.4644141` `199.5516666` `199.6389192`
#>   <chr>          <dbl>         <dbl>         <dbl>         <dbl>
#> 1 49              22.2          3.58          18.8        -15.3 
#> 2 127             11.1          5.69         -31.7         -3.89