centroids: Group Centroids and (Pooled) Variances

In sda: Shrinkage Discriminant Analysis and CAT Score Variable Selection

Description

centroids computes group centroids, the pooled mean and pooled variance, and optionally the group specific variances.

Usage

centroids(x, L, lambda.var, lambda.freqs, var.groups=FALSE, 
  centered.data=FALSE, verbose=TRUE)

Arguments

x	A matrix containing the data set. Note that the rows are sample observations and the columns are variables.
L	A factor with the group labels.
lambda.var	Shrinkage intensity for the variances. If not specified it is estimated from the data, see details below. lambda.var=0 implies no shrinkage and lambda.var=1 complete shrinkage.
lambda.freqs	Shrinkage intensity for the frequencies. If not specified it is estimated from the data. lambda.freqs=0 implies no shrinkage (i.e. empirical frequencies) and lambda.freqs=1 complete shrinkage (i.e. uniform frequencies).
var.groups	Estimate group-specific variances.
centered.data	Return column-centered data matrix.
verbose	Provide some messages while computing.

Details

As estimator of the variance we employ var.shrink as described in Opgen-Rhein and Strimmer (2007). For the estimates of frequencies we rely on freqs.shrink as described in Hausser and Strimmer (2009). Note that the pooled mean is computed using the estimated frequencies.

Value

centroids returns a list with the following components:

samples	a vector containing the samples sizes in each group,
freqs	a vector containing the estimated frequency in each group,
means	the group means and the pooled mean,
variances	the group-specific and the pooled variances, and
centered.data	a matrix containing the centered data.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

See Also

var.shrink, powcor.shrink.

Examples

# load sda library
library("sda")

## prepare data set
data(iris) # good old iris data
X = as.matrix(iris[,1:4])
Y = iris[,5]

## estimate centroids and empirical pooled variances
centroids(X, Y, lambda.var=0)
          
## also compute group-specific variances
centroids(X, Y, var.groups=TRUE, lambda.var=0)
   
## use shrinkage estimator for the variances
centroids(X, Y, var.groups=TRUE)

## return centered data
xc = centroids(X, Y, centered.data=TRUE)$centered.data
apply(xc, 2, mean)

## useful, e.g., to compute the inverse pooled correlation matrix
powcor.shrink(xc, alpha=-1)

Example output

Loading required package: entropy
Loading required package: corpcor
Loading required package: fdrtool
Number of variables: 4 
Number of observations: 150 
Number of classes: 3 

Estimating optimal shrinkage intensity lambda.freq (frequencies): 1 
Estimating variances (pooled across classes)
Specified shrinkage intensity lambda.var (variance vector): 0 

$samples
    setosa versicolor  virginica 
        50         50         50 

$freqs
    setosa versicolor  virginica 
 0.3333333  0.3333333  0.3333333 
attr(,"lambda.freqs")
[1] 1
attr(,"lambda.freqs.estimated")
[1] TRUE

$means
             setosa versicolor virginica (pooled)
Sepal.Length  5.006      5.936     6.588 5.843333
Sepal.Width   3.428      2.770     2.974 3.057333
Petal.Length  1.462      4.260     5.552 3.758000
Petal.Width   0.246      1.326     2.026 1.199333

$variances
               (pooled)
Sepal.Length 0.26500816
Sepal.Width  0.11538776
Petal.Length 0.18518776
Petal.Width  0.04188163
attr(,"lambda.var")
[1] 0
attr(,"lambda.var.estimated")
[1] FALSE

$centered.data
NULL

Number of variables: 4 
Number of observations: 150 
Number of classes: 3 

Estimating optimal shrinkage intensity lambda.freq (frequencies): 1 
Estimating variances (class #1)
Specified shrinkage intensity lambda.var (variance vector): 0 

Estimating variances (class #2)
Specified shrinkage intensity lambda.var (variance vector): 0 

Estimating variances (class #3)
Specified shrinkage intensity lambda.var (variance vector): 0 

Estimating variances (pooled across classes)
Specified shrinkage intensity lambda.var (variance vector): 0 

$samples
    setosa versicolor  virginica 
        50         50         50 

$freqs
    setosa versicolor  virginica 
 0.3333333  0.3333333  0.3333333 
attr(,"lambda.freqs")
[1] 1
attr(,"lambda.freqs.estimated")
[1] TRUE

$means
             setosa versicolor virginica (pooled)
Sepal.Length  5.006      5.936     6.588 5.843333
Sepal.Width   3.428      2.770     2.974 3.057333
Petal.Length  1.462      4.260     5.552 3.758000
Petal.Width   0.246      1.326     2.026 1.199333

$variances
                 setosa versicolor  virginica   (pooled)
Sepal.Length 0.12424898 0.26643265 0.40434286 0.26500816
Sepal.Width  0.14368980 0.09846939 0.10400408 0.11538776
Petal.Length 0.03015918 0.22081633 0.30458776 0.18518776
Petal.Width  0.01110612 0.03910612 0.07543265 0.04188163
attr(,"lambda.var")
[1] 0 0 0 0
attr(,"lambda.var.estimated")
[1] FALSE

$centered.data
NULL

Number of variables: 4 
Number of observations: 150 
Number of classes: 3 

Estimating optimal shrinkage intensity lambda.freq (frequencies): 1 
Estimating variances (class #1)
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.1315 

Estimating variances (class #2)
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.1287 

Estimating variances (class #3)
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.1354 

Estimating variances (pooled across classes)
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.0726 

$samples
    setosa versicolor  virginica 
        50         50         50 

$freqs
    setosa versicolor  virginica 
 0.3333333  0.3333333  0.3333333 
attr(,"lambda.freqs")
[1] 1
attr(,"lambda.freqs.estimated")
[1] TRUE

$means
             setosa versicolor virginica (pooled)
Sepal.Length  5.006      5.936     6.588 5.843333
Sepal.Width   3.428      2.770     2.974 3.057333
Petal.Length  1.462      4.260     5.552 3.758000
Petal.Width   0.246      1.326     2.026 1.199333

$variances
                 setosa versicolor  virginica  (pooled)
Sepal.Length 0.11806141 0.25269054 0.37725192 0.2566824
Sepal.Width  0.13494528 0.10634142 0.11758589 0.1179206
Petal.Length 0.03634675 0.21294430 0.29100594 0.1826549
Petal.Width  0.01979964 0.05461724 0.09288369 0.0497491
attr(,"lambda.var")
[1] 0.13152480 0.12868373 0.13542292 0.07257402
attr(,"lambda.var.estimated")
[1] TRUE

$centered.data
NULL

Number of variables: 4 
Number of observations: 150 
Number of classes: 3 

Estimating optimal shrinkage intensity lambda.freq (frequencies): 1 
Estimating variances (pooled across classes)
Estimating optimal shrinkage intensity lambda.var (variance vector): 0.0726 

 Sepal.Length   Sepal.Width  Petal.Length   Petal.Width 
-7.105174e-17 -5.624438e-17  1.909493e-16  4.533618e-17 
Estimating optimal shrinkage intensity lambda (correlation matrix): 0.0335 

             Sepal.Length Sepal.Width Petal.Length Petal.Width
Sepal.Length    2.5698469  -0.8141157   -1.7037483   0.2626377
Sepal.Width    -0.8141157   1.5677251    0.2842077  -0.5592243
Petal.Length   -1.7037483   0.2842077    2.4602011  -0.6809681
Petal.Width     0.2626377  -0.5592243   -0.6809681   1.4806445
attr(,"lambda")
[1] 0.03349646
attr(,"lambda.estimated")
[1] TRUE
attr(,"class")
[1] "shrinkage"

sda documentation built on Nov. 22, 2021, 1:07 a.m.