puchwein {prospectr} | R Documentation

## Puchwein algorithm for calibration sampling

### Description

Select calibration samples from multivariate data using the Puchwein algorithm

### Usage

    puchwein(X,pc=0.95,k,min.sel,details=FALSE,.center = TRUE,.scale = FALSE)

### Arguments

`X` | input `data.frame` or `matrix` from which to select calibration samples
`pc` | number of principal components retained in the computation of the distance in the standardized Principal Component space (Mahalanobis distance). If `pc < 1`, the number of principal components kept corresponds to the number of components explaining at least (`pc * 100`) percent of the total variance (default = 0.95 as in the Puchwein paper).
`k` | initial limiting distance parameter, if not specified (default), set to 0.2. According to Puchwein, a good starting value for the limiting distance is d_{ini} = k(p-2) where p is the number of principal components
`min.sel` | minimum number of samples to select for calibration (default = 5).
`details` | logical value, if `TRUE`, adds a component in the output list with the indices of the objects kept in each loop (default to `FALSE`)
`.center` | logical value indicating whether the input matrix should be centered before Principal Component Analysis. Default set to TRUE.
`.scale` | logical value indicating whether the input matrix should be scaled before Principal Component Analysis. Default set to FALSE.

### Details

The Puchwein algorithm select samples from a data matrix by iteratively eliminating similar samples using the Mahalanobis distance. It starts by performing a PCA on the input `matrix` and extracts the score matrix truncated to A, the number of principal components. The score matrix is then normalized to unit variance and the Euclidean distance of each sample to the centre of the data is computed, which is identical to the Mahalanobis distance H. Additionally, the Mahalanobis distances between samples are comptuted. The algorithm then proceeds as follows:

1. Choose a initial limiting distance d_{ini}

2. Select the sample with the highest H distance to the centre

3. Remove all samples within the minimum distance d_{ini} from the the sample selected in step 2

4. Go back to step 2 and proceed until there are no samples/observations left in the dataset

5. Go back to step 1 and increase the minimum distance by multiplying the limiting distance by the loop number

It is not possible to obtain a pre-defined number of samples selected by the method. To choose the adequate number of samples, a `data.frame` is returned by `puchwein` function (`leverage`) giving the observed and theoretical cumulative sum of leverages of the points selected in each iteration. The theoretical cumulative sum of leverage is computed such as each point has the same leverage (the sum of leverages divided by the number of observations). The loop having the largest difference between the observed and theoretical sums is considered as producing the optimal selection of points (the subset that best reproduces the variability of the predictor space).

### Value

a `list` with components:

- '`model`' indices of the observations (row indices of the input data) selected for calibration

- '`test`' indices of the remaining observations (row indices of the input data)

- '`pc`'a numeric `matrix` of the scaled pc scores

- '`loop.optimal`' index of the loop producing the maximum difference between the observed and theoretical sum of leverages of the selected samples

- '`leverage`' `data.frame` giving the observed and theoretical cumulative sums of leverage of the points selected in each loop

- '`details`' list with the indices of the observations kept in each loop

### Note

The Puchwein algorithm is an iterative method and can be slow for large data matrices.

### Author(s)

Antoine Stevens

### References

Puchwein, G., 1988. Selection of calibration samples for near-infrared spectrometry by factor analysis of spectra. Analytical Chemystry 60, 569-573.

Shetty, N., Rinnan, A., and Gislum, R., 2012. Selection of representative calibration sample sets for near-infrared reflectance spectroscopy to predict nitrogen concentration in grasses. Chemometrics and Intelligent Laboratory Systems 111, 59-65.

### See Also

`kenStone`, `duplex`, `shenkWest`, `honigs`, `naes`

### Examples

    data(NIRsoil)
    sel <- puchwein(NIRsoil$spc,k=0.2,pc=.99)
    plot(sel$pc[,1:2])
    # points selected for calibration
    points(NIRsoil$spc[sel$model,1:2],col=2,pch=2)
    # Leverage plot
    opar <- par(no.readonly=TRUE)
    par(mar=c(4,5,2,2))
    plot(sel$leverage$loop,sel$leverage$diff,type='l',
         xlab='# loops',ylab='Difference between theoretical and \n observed sum of leverages')
    par(opar)

[Package prospectr version 0.1.3 Index]