Side Note: Internal (In)Validity Measures

Internal cluster validity measures (see, e.g., [AGM+13, HBV01, MB02, MC85, XZLL20]) might aid in selecting the number of clusters a dataset should be partitioned to. They do not require the presence of expert labels. Instead, their aim is to quantify different aspects of the obtained partitions, e.g., the average degree of cluster compactness or point separability.

Sometimes internal measures are also used to compare the outputs of different clustering algorithms on the same dataset and determine which one is more correct.

However, in [GBC21] (also see the Clustering Results Repository (v1.1.0) section), we have pointed out that many measures promote some rather random groupings whilst other ones seem more suitable for… detecting outliers.

We should therefore not deem a high value of, say, the Silhouette or Davies–Bouldin index better than a lower one, at least not uncritically.

Overall, we do not recommend relying on such measures to judge whether a partition is meaningful or not.

Notation

For the sake of completeness, let us recall the definitions of some popular indices. Their implementation is included in the genieclust package for Python and R [Gag21]. Note that this section contains excerpts from [GBC21].

Let \(\mathbf{X}\in\mathbb{R}^{n\times d}\) denote the input dataset comprised of \(n\) points in a \(d\)-dimensional Euclidean space, with \(\mathbf{x}_i = (x_{i,1},\dots,x_{i,d})\) giving the coordinates of the \(i\)-th point, \(i\in\{1,2,\dots,n\}\).

A \(k\)-partition \(\{X_1,\dots,X_k\}\) of a set \(\{\mathbf{x}_1, \dots, \mathbf{x}_n\}\) can be encoded by means of a label vector[^footsurj] \(\mathbf{y}\), where \(y_i\in\{1,\dots,k\}\) gives the cluster number of the \(i\)-th point.

The measures listed below are based on Euclidean distances between all pairs of points, \(\|\mathbf{x}_i-\mathbf{x}_j\|\), or the input points and some other pivots, such as their corresponding cluster centroids, \(\|\mathbf{x}_i-\boldsymbol\mu_j\|,\) where for \(j\in\{1,\dots,k\}\) and \(l\in\{1,\dots,d\}\):

\[ \mu_{j,l} = \frac{1}{|X_j|} \sum_{\mathbf{x}_i\in X_j} x_{i,l}. \]

Indices Based on Cluster Centroids

Ball–Hall

The Ball–Hall index [BH65] is the within-cluster sum of squares weighted by the cluster cardinality:

\[\mathrm{BallHall}(\mathbf{y}) = -\sum_{i=1}^n \frac{1}{|X_{y_i}|} \| \mathbf{x}_i - \boldsymbol\mu_{y_i} \|^2. \]

Note the minus sign that accounts for the fact that we would rather have the index maximised.

Implementation: genieclust.cluster_validity.negated_ball_hall_index.

Caliński–Harabasz

The Caliński–Harabasz index (Eq. (3) in [CH74]; “variance ratio criterion”) is given by:

\[ \mathrm{CalińskiHarabasz}(\mathbf{y}) = \frac{n-k}{k-1} \frac{ \sum_{i=1}^n \| \boldsymbol\mu - \boldsymbol\mu_{y_i} \|^2 }{ \sum_{i=1}^n \| \mathbf{x}_i - \boldsymbol\mu_{y_i} \|^2 }, \]

where \(\boldsymbol\mu\) denotes the centroid of the whole dataset \(\mathbf{X}\), i.e., a vector such that \(\mu_{l} = \frac{1}{n} \sum_{x=1}^n x_{i,l}\) for \(l\in\{1,\dots,d\}\).

It may be shown that the task of minimising the (unweighted) within-cluster sum of squares is equivalent to maximising the Caliński–Harabasz index. Hence, this index is precisely the objective function in the \(k\)-means method [Llo57] and the algorithms by Ward, Edwards, and Cavalli–Sforza; see [CH74, ECS65, War63].

Implementation: genieclust.cluster_validity.calinski_harabasz_index.

Davies–Bouldin

The Davies-Bouldin (Def. 5 in [DB79]) is given as the average similarity between each cluster and its most similar counterpart (note the minus sign again):

\[ \mathrm{DaviesBouldin}(\mathbf{y}) = -\frac{1}{k} \sum_{i=1}^k\left( \max_{j\neq i} \frac{s_i+s_j}{m_{i,j}} \right), \]

where \(s_i\) is the dispersion of the \(i\)-th cluster: if \(|X_i|>1\), it is given by \(s_i=\frac{1}{|X_i|}\sum_{\mathbf{x}_u\in X_i} \|\mathbf{x}_u-\boldsymbol\mu_i\|\) and otherwise we set \(s_i=\infty\). Furthermore, \(m_{i,j}\) is the intra-cluster distance, \(m_{i,j}=\|\boldsymbol\mu_i-\boldsymbol\mu_j\|\).

On a side note, in [DB79], other choices of \(s_i\) and \(m_{i,j}\) are also suggested. We have recalled only the most popular setting here (used, e.g., in [AGM+13]).

Implementation: genieclust.cluster_validity.negated_davies_bouldin_index.

Silhouettes

Silhouette

In Sec. 2 of [Rou87], Rousseeuw proposes the notion of a silhouette as a graphical aid in cluster analysis.

Denote the average dissimilarity between the \(i\)-th point and all other points in its own cluster with:

\[ a_i = \frac{1}{|X_{y_i}|-1} \sum_{\mathbf{x}_u\in X_{y_i}} \| \mathbf{x}_i-\mathbf{x}_u \|, \]

and the average dissimilarity between the \(i\)-th point and all other entities in the “closest” cluster with:

\[ b_i = \min_{j\neq y_i} \left( \frac{1}{|X_j|} \sum_{\mathbf{x}_v\in X_{j}} \| \mathbf{x}_i-\mathbf{x}_v \| \right). \]

Then the Silhouette index is defined as the average silhouette score:

\[ \mathrm{Silhouette}(\mathbf{y}) = \frac{1}{n}\sum_{i=1}^n \frac{b_i-a_i}{\max\{ a_i, b_i \}}, \]

with convention \(\pm\infty/\infty=0\).

Implementation: genieclust.cluster_validity.silhouette_index.

SilhouetteW

The paper [Rou87] also defines what we call here the SilhouetteW index, being the mean of the cluster average silhouette widths:

\[ \mathrm{SilhouetteW}(\mathbf{y}) = \frac{1}{k-s} \sum_{i=1}^n \frac{1}{|X_{y_i}|} \frac{b_i-a_i}{\max\{ a_i, b_i \}}, \]

where \(s\) is the number of singletons, i.e., clusters of size 1. Note that SilhouetteW, just like BallHall, employs weighting by cluster cardinalities.

Implementation: genieclust.cluster_validity.silhouette_w_index.

Generalised Dunn Indices

In [Dun74] (see Eq. (3) therein), Dunn proposed an index defined as the ratio between the smallest between-cluster distance and the largest cluster diameter.

This index has been generalised by Bezdek and Pal in [BP98] as:

\[ \mathrm{GDunn}(\mathbf{y})= \frac{ \min_{i\neq j} d\left( X_i, X_j \right) }{ \max_{i} D\left( X_i \right) }. \]

The numerator measures the between-cluster separation whilst the denominator quantifies the cluster compactness.

The function \(d\) can be assumed one of:

  • \(d_1(X_i, X_j)=\mathrm{Min}\left( \left\{ \|\mathbf{x}_{u}-\mathbf{x}_{v}\|: \mathbf{x}_{u}\in X_i, \mathbf{x}_{v}\in X_j\right\} \right)\),

  • \(d_2(X_i, X_j)=\mathrm{Max}\left( \left\{ \|\mathbf{x}_{u}-\mathbf{x}_{v}\|: \mathbf{x}_{u}\in X_i, \mathbf{x}_{v}\in X_j \right\} \right)\),

  • \(d_3(X_i, X_j)=\mathrm{Mean}\left( \left\{ \|\mathbf{x}_{u}-\mathbf{x}_{v}\|: \mathbf{x}_{u}\in X_i, \mathbf{x}_{v}\in X_j \right\} \right)\),

  • \(d_4(X_i, X_j)= \|\boldsymbol\mu_i-\boldsymbol\mu_j\|\),

  • \(d_5(X_i, X_j)= \frac{ |X_i|\,\mathrm{Mean}\left( \left\{ \|\mathbf{x}_{u}-\boldsymbol\mu_i \|: \mathbf{x}_{u}\in X_i\right\} \right) + |X_j|\,\mathrm{Mean}\left( \left\{ \|\mathbf{x}_{v}-\boldsymbol\mu_j \|: \mathbf{x}_{v}\in X_j\right\} \right) }{ |X_i|+|X_j| }\).

Bezdek and Pal in [Dun74] also considered a function based on the Hausdorff metric, but it is overall quite slow to compute.

On the other hand, \(D\) can be, for example:

  • \(D_1(X_i)=\mathrm{Max}\left( \left\{ \|\mathbf{x}_{u}-\mathbf{x}_{v}\|: \mathbf{x}_{u},\mathbf{x}_{v}\in X_i\right\} \right)\),

  • \(D_2(X_i)=\mathrm{Mean}\left( \left\{ \|\mathbf{x}_{u}-\mathbf{x}_{v}\|: \mathbf{x}_{u},\mathbf{x}_{v}\in X_i\right\} \right)\),

  • \(D_3(X_i)=\mathrm{Mean}\left( \left\{ \|\mathbf{x}_{u}-\boldsymbol\mu_i\|: \mathbf{x}_{u}\in X_i\right\} \right)\).

There are thus 15 different combinations of the possible numerators and denominators, hence 15 different indices, which we may denote as GDunn_dX_DY. In particular, GDunn_d1_D1 gives the original Dunn [Dun74] index.

Implementation: genieclust.cluster_validity.generalised_dunn_index.