Evaluation & Validation

Clustering.jl package provides a number of methods to evaluate the results of a clustering algorithm and/or to validate its correctness.

Cross tabulation

Cross tabulation, or contingency matrix, is a basis for many clustering quality measures. It shows how similar are the two clusterings on a cluster level.

Clustering.jl extends StatsBase.counts() with methods that accept ClusteringResult arguments:

StatsBase.counts — Method.

counts(a::ClusteringResult, b::ClusteringResult) -> Matrix{Int}
counts(a::ClusteringResult, b::AbstractVector{<:Integer}) -> Matrix{Int}
counts(a::AbstractVector{<:Integer}, b::ClusteringResult) -> Matrix{Int}

Calculate the cross tabulation (aka contingency matrix) for the two clusterings of the same data points.

Returns the $n_a × n_b$ matrix C, where $n_a$ and $n_b$ are the numbers of clusters in a and b, respectively, and C[i, j] is the size of the intersection of i-th cluster from a and j-th cluster from b.

The clusterings could be specified either as ClusteringResult instances or as vectors of data point assignments.

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Rand index

Rand index is a measure of the similarity between the two data clusterings. From a mathematical standpoint, Rand index is related to the prediction accuracy, but is applicable even when the original class labels are not used.

Clustering.randindex — Function.

randindex(a, b) -> NTuple{4, Float64}

Compute the tuple of Rand-related indices between the clusterings c1 and c2.

a and b can be either ClusteringResult instances or assignments vectors (AbstractVector{<:Integer}).

Returns a tuple of indices:

Hubert & Arabie Adjusted Rand index
Rand index (agreement probability)
Mirkin's index (disagreement probability)
Hubert's index ($P(\mathrm{agree}) - P(\mathrm{disagree})$)

References

Lawrence Hubert and Phipps Arabie (1985). Comparing partitions. Journal of Classification 2 (1): 193–218

Meila, Marina (2003). Comparing Clusterings by the Variation of Information. Learning Theory and Kernel Machines: 173–187.

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Silhouettes

Silhouettes is a method for evaluating the quality of clustering. Particularly, it provides a quantitative way to measure how well each point lies within its cluster in comparison to the other clusters.

The Silhouette value for the $i$-th data point is:

\[s_i = \frac{b_i - a_i}{\max(a_i, b_i)}, \ \text{where}\]

$a_i$ is the average distance from the $i$-th point to the other points in the same cluster $z_i$,
$b_i ≝ \min_{k \ne z_i} b_{ik}$, where $b_{ik}$ is the average distance from the $i$-th point to the points in the $k$-th cluster.

Note that $s_i \le 1$, and that $s_i$ is close to $1$ when the $i$-th point lies well within its own cluster. This property allows using mean(silhouettes(assignments, counts, X)) as a measure of clustering quality. Higher values indicate better separation of clusters w.r.t. point distances.

Clustering.silhouettes — Function.

silhouettes(assignments::AbstractVector, [counts,] dists) -> Vector{Float64}
silhouettes(clustering::ClusteringResult, dists) -> Vector{Float64}

Compute silhouette values for individual points w.r.t. given clustering.

Returns the $n$-length vector of silhouette values for each individual point.

Arguments

assignments::AbstractVector{Int}: the vector of point assignments (cluster indices)
counts::AbstractVector{Int}: the optional vector of cluster sizes (how many points assigned to each cluster; should match assignments)
clustering::ClusteringResult: the output of some clustering method
dists::AbstractMatrix: $n×n$ matrix of pairwise distances between the points

References

Peter J. Rousseeuw (1987). Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis. Computational and Applied Mathematics. 20: 53–65.

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Variation of Information

Variation of information (also known as shared information distance) is a measure of the distance between the two clusterings. It is devised from the mutual information, but it is a true metric, i.e. it is symmetric and satisfies the triangle inequality.

Clustering.varinfo — Function.

varinfo(a, b) -> Float64

Compute the variation of information between the two clusterings of the same data points.

a and b can be either ClusteringResult instances or assignments vectors (AbstractVector{<:Integer}).

References

Meila, Marina (2003). Comparing Clusterings by the Variation of Information. Learning Theory and Kernel Machines: 173–187.

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V-measure

V-measure can be used to compare the clustering results with the existing class labels of data points or with the alternative clustering. It is defined as the harmonic mean of homogeneity ($h$) and completeness ($c$) of the clustering:

\[V_{\beta} = (1+\beta)\frac{h \cdot c}{\beta \cdot h + c}.\]

Both $h$ and $c$ can be expressed in terms of the mutual information and entropy measures from the information theory. Homogeneity ($h$) is maximized when each cluster contains elements of as few different classes as possible. Completeness ($c$) aims to put all elements of each class in single clusters. The $\beta$ parameter ($\beta > 0$) could used to control the weights of $h$ and $c$ in the final measure. If $\beta > 1$, completeness has more weight, and when $\beta < 1$ it's homogeneity.

Clustering.vmeasure — Function.

vmeasure(a, b; [β = 1.0]) -> Float64

V-measure between the two clusterings.

a and b can be either ClusteringResult instances or assignments vectors (AbstractVector{<:Integer}).

The β parameter defines trade-off between homogeneity and completeness:

if $β > 1$, completeness is weighted more strongly,
if $β < 1$, homogeneity is weighted more strongly.

References

Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure

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Mutual information

Mutual information quantifies the "amount of information" obtained about one random variable through observing the other random variable. It is used in determining the similarity of two different clusterings of a dataset.

Clustering.mutualinfo — Function.

mutualinfo(a, b; normed=true) -> Float64

Compute the mutual information between the two clusterings of the same data points.

a and b can be either ClusteringResult instances or assignments vectors (AbstractVector{<:Integer}).

If normed parameter is true the return value is the normalized mutual information (symmetric uncertainty), see "Data Mining Practical Machine Tools and Techniques", Witten & Frank 2005.

References

Vinh, Epps, and Bailey, (2009). “Information theoretic measures for clusterings comparison”.

Proceedings of the 26th Annual International Conference on Machine Learning - ICML ‘09.

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