PcaLocantore: Spherical Principal Components
In armstrtw/rrcov: Scalable Robust Estimators with High Breakdown Point

Description Usage Arguments Details Value Author(s) References Examples

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.

PcaLocantore(x, ...)
## Default S3 method:
PcaLocantore(x, k = 0, kmax = ncol(x), delta = 0.001, 
    na.action = na.fail, scale = FALSE, signflip = TRUE, trace=FALSE, ...)
## S3 method for class 'formula'
PcaLocantore(formula, data = NULL, subset, na.action, ...)

`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 `NA`s. 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=0`.
`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 logical value indicating whether the variables should be scaled to have unit variance (only possible if there are no constant variables). As a scale function `mad` is used but alternatively, a vector of length equal the number of columns of x can be supplied. The value is passed to scale and the result of the scaling 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 = FALSE`
`trace`	whether to print intermediate results. Default is `trace = FALSE`

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.

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

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 http://www.wiley.com/legacy/wileychi/robust_statistics/

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. URL http://www.jstatsoft.org/v32/i03/.

## 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)