Purity and Entropy of a Clustering
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.
purity(x, y, ...)
entropy(x, y, ...)
## S4 method for signature 'NMFfitXn,ANY'
purity(x, y, method = "best",
...)
## S4 method for signature 'NMFfitXn,ANY'
entropy(x, y, method = "best",
...)x |
an object that can be interpreted as a factor or
can generate such an object, e.g. via a suitable method
|
y |
a factor or an object coerced into a factor that
gives the true class labels for each sample. It may be
missing if |
... |
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 |
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:
Purity = \frac{1}{n} ∑_{q=1}^k \max_{1 ≤q j ≤q 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 ≤q j ≤q 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:
- 1/(n log2(l) ) sum_q sum_j n(q,j) log2( n(q,j) / n_q )
,
where:
n is the total number of samples;
n_q is the total number of samples in cluster q (1 ≤q q ≤q k);
n(q,j) is the number of samples in cluster q that belongs to original class j (1 ≤q j ≤q l).
The smaller the entropy, the better the clustering performance.
a single numeric value
the entropy (i.e. a single numeric value)
signature(x = "table", y =
"missing"): Computes the purity directly from the
contingency table x.
This is the workhorse method that is eventually called by all other methods.
signature(x = "factor", y = "ANY"):
Computes the purity on the contingency table of x
and y, that is coerced into a factor if necessary.
signature(x = "ANY", y = "ANY"):
Default method that should work for results of clustering
algorithms, that have a suitable predict method
that returns the cluster membership vector: the purity is
computed between x and predict{y}
signature(x = "NMFfitXn", y =
"ANY"): Computes the best or mean entropy across all NMF
fits stored in x.
signature(x = "table", y =
"missing"): Computes the purity directly from the
contingency table x
signature(x = "factor", y = "ANY"):
Computes the purity on the contingency table of x
and y, that is coerced into a factor if necessary.
signature(x = "ANY", y = "ANY"):
Default method that should work for results of clustering
algorithms, that have a suitable predict method
that returns the cluster membership vector: the purity is
computed between x and predict{y}
signature(x = "NMFfitXn", y =
"ANY"): Computes the best or mean purity across all NMF
fits stored in x.
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, <URL: http://dx.doi.org/10.1093/bioinformatics/btm134>, <URL: http://www.ncbi.nlm.nih.gov/pubmed/17483501>.
Other assess: sparseness
# 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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