PcaLocantore: Spherical Principal Components

Description

The Spherical Principal Components procedure was proposed by Locantore et al., (1999) as a functional data analysis method. The idea is to perform classical PCA on the data, projected onto a unit sphere. The estimates of the eigenvectors are consistent and the procedure is extremely fast. The simulations of Maronna (2005) show that this method has very good performance.

Usage

PcaLocantore(x, ...)
# S3 method for default
PcaLocantore(x, k = ncol(x), kmax = ncol(x), delta = 0.001, 
    na.action = na.fail, scale = FALSE, signflip = TRUE, 
    crit.pca.distances = 0.975, trace=FALSE, ...)
# S3 method for formula
PcaLocantore(formula, data = NULL, subset, na.action, ...)

Value

An S4 object of class PcaLocantore-class which is a subclass of the virtual class PcaRobust-class.

Arguments

formula: a formula with no response variable, referring only to numeric variables.
data: an optional data frame (or similar: see model.frame) containing the variables in the formula formula.
subset: an optional vector used to select rows (observations) of the data matrix x.
na.action: a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The default is na.omit.
...: 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 k is missing, or k = 0, the algorithm itself will determine the number of components by finding such k that $l_{k} / l_{1} >= 10. E - 3$ and $Σ_{j = 1}^{k} l_{j} / Σ_{j = 1}^{r} l_{j} >= 0.8$ . It is preferable to investigate the scree plot in order to choose the number of components and then run again. Default is k=ncol(x).
kmax: maximal number of principal components to compute. Default is kmax=10. If k is provided, kmax does not need to be specified, unless k is larger than 10.
delta: an accuracy parameter
scale: a value indicating whether and how the variables should be scaled to have unit variance (only possible if there are no constant variables). If scale=FALSE (default) or scale=NULL no scaling is performed (a vector of 1s is returned in the scale slot). If scale=TRUE the data are scaled by mad. Alternatively it can be a function like sd or Qn or a vector of length equal the number of columns of x. The value is passed to the underlying function and the result returned is stored in the scale slot. Default is scale=FALSE.
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 signflip = TRUE
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 trace = FALSE

Author

Valentin Todorov valentin.todorov@chello.at The SPC algorithm is implemented on the bases of the available from the web site of the book Maronna et al. (2006) code https://www.wiley.com/legacy/wileychi/robust_statistics/

Details

PcaLocantore, serving as a constructor for objects of class PcaLocantore-class is a generic function with "formula" and "default" methods. For details see the relevant references.

References

N. Locantore, J. Marron, D. Simpson, N. Tripoli, J. Zhang and K. Cohen K. (1999), Robust principal components for functional data. Test, 8, 1-28.

R. Maronna, D. Martin and V. Yohai (2006), Robust Statistics: Theory and Methods. Wiley, New York.

R. Maronna (2005). Principal components and orthogonal regression based on robust scales. Technometrics, 47, 264-273.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1--47. tools:::Rd_expr_doi("10.18637/jss.v032.i03").

Examples

Run this code

## PCA of the Hawkins Bradu Kass's Artificial Data
##  using all 4 variables
    data(hbk)
    pca <- PcaLocantore(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
    PcaLocantore(~., data=hbk)

## To plot the results:

    plot(pca)                    # distance plot
    pca2 <- PcaLocantore(hbk, k=2)  
    plot(pca2)                   # PCA diagnostic plot (or outlier map)
    
## Use the standard plots available for for prcomp and princomp
    screeplot(pca)    
    biplot(pca)

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