pca: Principal Component Analysis

In mdatools: Multivariate Data Analysis for Chemometrics

Principal Component Analysis

Description

pca is used to build and explore a principal component analysis (PCA) model.

Usage

pca(
  x,
  ncomp = min(nrow(x) - 1, ncol(x), 20),
  center = TRUE,
  scale = FALSE,
  exclrows = NULL,
  exclcols = NULL,
  x.test = NULL,
  method = "svd",
  rand = NULL,
  lim.type = "ddmoments",
  alpha = 0.05,
  gamma = 0.01,
  info = "",
  prep = NULL,
  do.round = FALSE
)

Arguments

x	calibration data (matrix or data frame).
ncomp	maximum number of components to calculate.
center	logical, do mean centering of data or not.
scale	logical, do standardization of data or not.
exclrows	rows to be excluded from calculations (numbers, names or vector with logical values)
exclcols	columns to be excluded from calculations (numbers, names or vector with logical values)
x.test	test data (matrix or data frame).
method	method to compute principal components ("svd", "nipals").
rand	vector with parameters for randomized PCA methods (if NULL, conventional PCA is used instead)
lim.type	which method to use for calculation of critical limits for residual distances (see details)
alpha	significance level for extreme limits for T2 and Q distances.
gamma	significance level for outlier limits for T2 and Q distances.
info	a short text with model description.
prep	optional list with preprocessing methods created using 'prep' function.
do.round	logical, round or not DoF for distances.

Details

If number of components is not specified, a minimum of number of objects - 1 and number of variables in calibration set is used. One can also specify an optimal number of components, once model is calibrated (ncomp.selected). The optimal number of components is used to build a residual distance plot, as well as for SIMCA classification.

If some of rows of calibration set should be excluded from calculations (e.g. because they are outliers) you can provide row numbers, names, or logical values as parameter exclrows. In this case they will be completely ignored when the model is calibrated. However, score and residual distances will be computed for these rows as well and then hidden. You can show them on corresponding plots by using parameter show.excluded = TRUE.

It is also possible to exclude selected columns from calculations by providing parameter exclcols in form of column numbers, names or logical values. In this case loading matrix will have zeros for these columns. This allows computing PCA models for selected variables without removing them physically from a dataset.

Take into account that if you use other packages to make plots (e.g. ggplot2) you will not be able to distinguish between hidden and normal objects.

By default loadings are computed for the original dataset using either SVD or NIPALS algorithm. However, for datasets with large number of rows (e.g. hyperspectral images), there is a possibility to run algorithms based on random permutations [1, 2]. In this case you have to define parameter rand as a vector with two values: p - oversampling parameter and q - number of power iterations. Usually rand = c(15, 0) or rand = c(5, 1) are good options, which give quite precise solutions but much faster.

There are several ways to calculate critical limits for orthogonal (Q, q) and score (T2, h) distances. In mdatools you can specify one of the following methods via parameter lim.type: "jm" Jackson-Mudholkar approach [3], "chisq" - method based on chi-square distribution [4], "ddmoments" and "ddrobust" - related to data driven method proposed in [5]. The "ddmoments" is based on method of moments for estimation of distribution parameters (also known as "classical" approach) while "ddrobust" is based on robust estimation.

If lim.type="chisq" or lim.type="jm" is used, only limits for Q-distances are computed based on corresponding approach, limits for T2-distances are computed using Hotelling's T-squared distribution. The methods utilizing the data driven approach calculate limits for combination of the distances based on chi-square distribution and parameters estimated from the calibration data.

The critical limits are calculated for a significance level defined by parameter 'alpha'. You can also specify another parameter, 'gamma', which is used to calculate acceptance limit for outliers (shown as dashed line on residual distance plot).

You can also recalculate the limits for existent model by using different values for alpha and gamma, without recomputing the model itself. In this case use the following code (it is assumed that you current PCA/SIMCA model is stored in variable m): m = setDistanceLimits(m, lim.type, alpha, gamma).

In case of PCA the critical limits are just shown on residual plot as lines and can be used for detection of extreme objects (solid line) and outliers (dashed line). When PCA model is used for classification in SIMCA (see simca) or DD-SIMCA (ddsimca) the limits are also employed for classification of objects.

If you provide a list with preprocessing methods, PCA will apply them to the training set before excluding the columns and rows (if specified). The list will be used to train a preprocessing model which becomes a part of the PCA model object. So when you use method 'predict()' the provided dataset will be automatically preprocessed by the preprocessing model.

Any PCA model (with or without preprocessing) developed in this package can be saved as JSON file using method writeJSON and then be loaded to interactive web-application for PCA available at https://mda.tools/pca. Likewise one can develop a model in the app, save it to JSON file and then load it to R by using method readJSON. In this case, however, the model object will not contain calibration/training results, so some of the plots and statistics will not be available.

Value

Returns an object of pca class with following fields:

ncomp	number of components included to the model.
ncomp.selected	selected (optimal) number of components.
loadings	matrix with loading values (nvar x ncomp).
eigenvals	vector with eigenvalues for all existent components.
expvar	vector with explained variance for each component (in percent).
cumexpvar	vector with cumulative explained variance for each component (in percent).
T2lim	statistical limit for T2 distance.
Qlim	statistical limit for Q residuals.
info	information about the model, provided by user when building the model.
prep	trained preprocessing model (if specified)
calres	an object of class pcares with PCA results for a calibration data.
testres	an object of class pcares with PCA results for a test data, if it was provided.

More details and examples can be found in the Bookdown tutorial.

Author(s)

Sergey Kucheryavskiy ([e-mail])

References

1. N. Halko, P.G. Martinsson, J.A. Tropp. Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53 (2010) pp. 217-288.

2. S. Kucheryavskiy, Blessing of randomness against the curse of dimensionality, Journal of Chemometrics, 32 (2018).

3. J.E. Jackson, A User's Guide to Principal Components, John Wiley & Sons, New York, NY (1991).

4. A.L. Pomerantsev, Acceptance areas for multivariate classification derived by projection methods, Journal of Chemometrics, 22 (2008) pp. 601-609.

5. A.L. Pomerantsev, O.Ye. Rodionova, Concept and role of extreme objects in PCA/SIMCA, Journal of Chemometrics, 28 (2014) pp. 429-438.

See Also

Methods for pca objects:

plot.pca	makes an overview of PCA model with four plots.
summary.pca	shows some statistics for the model.
categorize.pca	categorize data rows as "normal", "extreme" or "outliers".
selectCompNum.pca	set number of optimal components in the model
setDistanceLimits.pca	set critical limits for residuals
predict.pca	applies PCA model to a new data.
writeJSON.pca	saves PCA model to a JSON file so it can be used in web-applications.
pca.readJSON	loads PCA model from a JSON file to pca object.

Plotting methods for pca objects:

plotScores.pca	shows scores plot.
plotLoadings.pca	shows loadings plot.
plotVariance.pca	shows explained variance plot.
plotCumVariance.pca	shows cumulative explained variance plot.
plotEigenvalues.pca	shows eigenvalues plot.
plotDistances.pca	shows plot for residual distances (Q vs. T2).
plotBiplot.pca	shows bi-plot.
plotExtremes.pca	shows extreme plot.
plotT2DoF	plot with degrees of freedom for score distance.
plotQDoF	plot with degrees of freedom for orthogonal distance.
plotDistDoF	plot with degrees of freedom for both distances.

Most of the methods for plotting data are also available for PCA results (pcares) objects. Also check pca.mvreplace, which replaces missing values in a data matrix with approximated using iterative PCA decomposition.

Examples

library(mdatools)
### Examples for PCA class

## 1. Make PCA model for People data with autoscaling

data(people)
model = pca(people, scale = TRUE, info = "Simple PCA model")
model = selectCompNum(model, 4)
summary(model)
plot(model, show.labels = TRUE)

## 2. Show scores and loadings plots for the model

par(mfrow = c(2, 2))
plotScores(model, comp = c(1, 3), show.labels = TRUE)
plotScores(model, comp = 2, type = "h", show.labels = TRUE)
plotLoadings(model, comp = c(1, 3), show.labels = TRUE)
plotLoadings(model, comp = c(1, 2), type = "h", show.labels = TRUE)
par(mfrow = c(1, 1))

## 3. Show residual distance and variance plots for the model
par(mfrow = c(2, 2))
plotVariance(model, type = "h")
plotCumVariance(model, show.labels = TRUE, legend.position = "bottomright")
plotResiduals(model, show.labels = TRUE)
plotResiduals(model, ncomp = 2, show.labels = TRUE)
par(mfrow = c(1, 1))

## 4. Use list with preprocessing methods

# get the data (calibration and test set)
data(simdata)
Xc <- simdata$spectra.c
Xt <- simdata$spectra.t

# create a list with two preprocessing methods
p <- list(
   prep("savgol", width = 7, porder = 2, dorder = 2),
   prep("norm", type = "snv")
)

# build a PCA model with and without preprocessing
m1 <- pca(Xc, 5, prep = p)
m2 <- pca(Xc, 5)

# apply the models to test set
r1 <- predict(m1, Xt)
r2 <- predict(m2, Xt)

# check scores
par(mfrow = c(1, 2))
plotScores(m1, c(1, 2), res = list(cal = m1$calres, test = r1), main = "With preprocessing")
plotScores(m2, c(1, 2), res = list(cal = m2$calres, test = r2), main = "Without preprocessing")

mdatools documentation built on March 6, 2026, 5:08 p.m.