purity: Purity and Entropy of a Clustering

Description

The functions purity and entropy respectively compute the purity and the entropy of a clustering given a priori known classes.

The purity and entropy measure the ability of a clustering method, to recover known classes (e.g. one knows the true class labels of each sample), that are applicable even when the number of cluster is different from the number of known classes. Kim et al. (2007) used these measures to evaluate the performance of their alternate least-squares NMF algorithm.

Usage

purity(x, y, ...)
  entropy(x, y, ...)
  ## S3 method for class 'NMFfitXn,ANY':
purity(x, y, method = "best",
    ...)
  ## S3 method for class 'NMFfitXn,ANY':
entropy(x, y, method = "best",
    ...)

Arguments

an object that can be interpreted as a factor or can generate such an object, e.g. via a suitable method predict, which gives the cluster membership for each sample.

a factor or an object coerced into a factor that gives the true class labels for each sample. It may be missing if x is a contingency table.

...

extra arguments to allow extension, and usually passed to the next method.

method

a character string that specifies how the value is computed. It may be either 'best' or 'mean' to compute the best or mean purity respectively.

Value

a single numeric value
the entropy (i.e. a single numeric value)

Details

Suppose we are given $l$ categories, while the clustering method generates $k$ clusters.

The purity of the clustering with respect to the known categories is given by: $P u r i t y = \frac{1}{n} \sum_{q = 1}^{k} max_{1 \leq j \leq l} n_{q}^{j}$ ,

where:

$n$is the total number of samples;
$n_q^j$is the number of samples in cluster$q$that belongs to original class$j$($1 \leq j \leq l$).

The purity is therefore a real number in $[0,1]$. The larger the purity, the better the clustering performance.

The entropy of the clustering with respect to the known categories is given by: $E n t r o p y = - \frac{1}{n \log_{2} l} \sum_{q = 1}^{k} \sum_{j = 1}^{l} n_{q}^{j} \log_{2} \frac{n_{q}^{j}}{n_{q}}$ ,

where:

$n$is the total number of samples;
$n$is the total number of samples in cluster$q$($1 \leq q \leq k$);
$n_q^j$is the number of samples in cluster$q$that belongs to original class$j$($1 \leq j \leq l$).

The smaller the entropy, the better the clustering performance.

References

Kim H and Park H (2007). "Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis." _Bioinformatics (Oxford, England)_, *23*(12), pp. 1495-502. ISSN 1460-2059, , .

Examples

Run this code

# roxygen generated flag
options(R_CHECK_RUNNING_EXAMPLES_=TRUE)

# generate a synthetic dataset with known classes: 50 features, 18 samples (5+5+8)
n <- 50; counts <- c(5, 5, 8);
V <- syntheticNMF(n, counts)
cl <- unlist(mapply(rep, 1:3, counts))

# perform default NMF with rank=2
x2 <- nmf(V, 2)
purity(x2, cl)
entropy(x2, cl)
# perform default NMF with rank=2
x3 <- nmf(V, 3)
purity(x3, cl)
entropy(x3, cl)

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