2.3. Clustering

Clustering of unlabeled data can be performed with the module sklearn.cluster.

Each clustering algorithm comes in two variants: a class, that implements the fit method to learn the clusters on train data, and a function, that, given train data, returns an array of integer labels corresponding to the different clusters. For the class, the labels over the training data can be found in the labels_ attribute.

2.3.1. Overview of clustering methods

A comparison of the clustering algorithms in scikit-learn

Non-flat geometry clustering is useful when the clusters have a specific shape, i.e. a non-flat manifold, and the standard euclidean distance is not the right metric. This case arises in the two top rows of the figure above.

Gaussian mixture models, useful for clustering, are described in another chapter of the documentation dedicated to mixture models. KMeans can be seen as a special case of Gaussian mixture model with equal covariance per component.

2.3.2. K-means

The KMeans algorithm clusters data by trying to separate samples in n groups of equal variance, minimizing a criterion known as the inertia or within-cluster sum-of-squares. This algorithm requires the number of clusters to be specified. It scales well to large number of samples and has been used across a large range of application areas in many different fields.

The k-means algorithm divides a set of \(N\) samples \(X\) into \(K\) disjoint clusters \(C\), each described by the mean \(\mu_j\) of the samples in the cluster. The means are commonly called the cluster “centroids”; note that they are not, in general, points from \(X\), although they live in the same space. The K-means algorithm aims to choose centroids that minimise the inertia, or within-cluster sum of squared criterion:

Inertia, or the within-cluster sum of squares criterion, can be recognized as a measure of how internally coherent clusters are. It suffers from various drawbacks:

• Inertia makes the assumption that clusters are convex and isotropic, which is not always the case. It responds poorly to elongated clusters, or manifolds with irregular shapes.
• Inertia is not a normalized metric: we just know that lower values are better and zero is optimal. But in very high-dimensional spaces, Euclidean distances tend to become inflated (this is an instance of the so-called “curse of dimensionality”). Running a dimensionality reduction algorithm such as PCA prior to k-means clustering can alleviate this problem and speed up the computations.

K-means is often referred to as Lloyd’s algorithm. In basic terms, the algorithm has three steps. The first step chooses the initial centroids, with the most basic method being to choose \(k\) samples from the dataset \(X\). After initialization, K-means consists of looping between the two other steps. The first step assigns each sample to its nearest centroid. The second step creates new centroids by taking the mean value of all of the samples assigned to each previous centroid. The difference between the old and the new centroids are computed and the algorithm repeats these last two steps until this value is less than a threshold. In other words, it repeats until the centroids do not move significantly.

K-means is equivalent to the expectation-maximization algorithm with a small, all-equal, diagonal covariance matrix.

The algorithm can also be understood through the concept of Voronoi diagrams. First the Voronoi diagram of the points is calculated using the current centroids. Each segment in the Voronoi diagram becomes a separate cluster. Secondly, the centroids are updated to the mean of each segment. The algorithm then repeats this until a stopping criterion is fulfilled. Usually, the algorithm stops when the relative decrease in the objective function between iterations is less than the given tolerance value. This is not the case in this implementation: iteration stops when centroids move less than the tolerance.

Given enough time, K-means will always converge, however this may be to a local minimum. This is highly dependent on the initialization of the centroids. As a result, the computation is often done several times, with different initializations of the centroids. One method to help address this issue is the k-means++ initialization scheme, which has been implemented in scikit-learn (use the init='k-means++' parameter). This initializes the centroids to be (generally) distant from each other, leading to provably better results than random initialization, as shown in the reference.

The algorithm supports sample weights, which can be given by a parameter sample_weight. This allows to assign more weight to some samples when computing cluster centers and values of inertia. For example, assigning a weight of 2 to a sample is equivalent to adding a duplicate of that sample to the dataset \(X\).

A parameter can be given to allow K-means to be run in parallel, called n_jobs. Giving this parameter a positive value uses that many processors (default: 1). A value of -1 uses all available processors, with -2 using one less, and so on. Parallelization generally speeds up computation at the cost of memory (in this case, multiple copies of centroids need to be stored, one for each job).

K-means can be used for vector quantization. This is achieved using the transform method of a trained model of KMeans.

2.3.2.1. Mini Batch K-Means

The MiniBatchKMeans is a variant of the KMeans algorithm which uses mini-batches to reduce the computation time, while still attempting to optimise the same objective function. Mini-batches are subsets of the input data, randomly sampled in each training iteration. These mini-batches drastically reduce the amount of computation required to converge to a local solution. In contrast to other algorithms that reduce the convergence time of k-means, mini-batch k-means produces results that are generally only slightly worse than the standard algorithm.

The algorithm iterates between two major steps, similar to vanilla k-means. In the first step, \(b\) samples are drawn randomly from the dataset, to form a mini-batch. These are then assigned to the nearest centroid. In the second step, the centroids are updated. In contrast to k-means, this is done on a per-sample basis. For each sample in the mini-batch, the assigned centroid is updated by taking the streaming average of the sample and all previous samples assigned to that centroid. This has the effect of decreasing the rate of change for a centroid over time. These steps are performed until convergence or a predetermined number of iterations is reached.

MiniBatchKMeans converges faster than KMeans, but the quality of the results is reduced. In practice this difference in quality can be quite small, as shown in the example and cited reference.

2.3.3. Affinity Propagation

AffinityPropagation creates clusters by sending messages between pairs of samples until convergence. A dataset is then described using a small number of exemplars, which are identified as those most representative of other samples. The messages sent between pairs represent the suitability for one sample to be the exemplar of the other, which is updated in response to the values from other pairs. This updating happens iteratively until convergence, at which point the final exemplars are chosen, and hence the final clustering is given.

Affinity Propagation can be interesting as it chooses the number of clusters based on the data provided. For this purpose, the two important parameters are the preference, which controls how many exemplars are used, and the damping factor which damps the responsibility and availability messages to avoid numerical oscillations when updating these messages.

The main drawback of Affinity Propagation is its complexity. The algorithm has a time complexity of the order \(O(N^2 T)\), where \(N\) is the number of samples and \(T\) is the number of iterations until convergence. Further, the memory complexity is of the order \(O(N^2)\) if a dense similarity matrix is used, but reducible if a sparse similarity matrix is used. This makes Affinity Propagation most appropriate for small to medium sized datasets.

Algorithm description: The messages sent between points belong to one of two categories. The first is the responsibility \(r(i, k)\), which is the accumulated evidence that sample \(k\) should be the exemplar for sample \(i\). The second is the availability \(a(i, k)\) which is the accumulated evidence that sample \(i\) should choose sample \(k\) to be its exemplar, and considers the values for all other samples that \(k\) should be an exemplar. In this way, exemplars are chosen by samples if they are (1) similar enough to many samples and (2) chosen by many samples to be representative of themselves.

More formally, the responsibility of a sample \(k\) to be the exemplar of sample \(i\) is given by:

Where \(s(i, k)\) is the similarity between samples \(i\) and \(k\). The availability of sample \(k\) to be the exemplar of sample \(i\) is given by:

To begin with, all values for \(r\) and \(a\) are set to zero, and the calculation of each iterates until convergence. As discussed above, in order to avoid numerical oscillations when updating the messages, the damping factor \(\lambda\) is introduced to iteration process:

where \(t\) indicates the iteration times.

2.3.4. Mean Shift

MeanShift clustering aims to discover blobs in a smooth density of samples. It is a centroid based algorithm, which works by updating candidates for centroids to be the mean of the points within a given region. These candidates are then filtered in a post-processing stage to eliminate near-duplicates to form the final set of centroids.

Given a candidate centroid \(x_i\) for iteration \(t\), the candidate is updated according to the following equation:

Where \(N(x_i)\) is the neighborhood of samples within a given distance around \(x_i\) and \(m\) is the mean shift vector that is computed for each centroid that points towards a region of the maximum increase in the density of points. This is computed using the following equation, effectively updating a centroid to be the mean of the samples within its neighborhood:

The algorithm automatically sets the number of clusters, instead of relying on a parameter bandwidth, which dictates the size of the region to search through. This parameter can be set manually, but can be estimated using the provided estimate_bandwidth function, which is called if the bandwidth is not set.

The algorithm is not highly scalable, as it requires multiple nearest neighbor searches during the execution of the algorithm. The algorithm is guaranteed to converge, however the algorithm will stop iterating when the change in centroids is small.

Labelling a new sample is performed by finding the nearest centroid for a given sample.

2.3.5. Spectral clustering

SpectralClustering does a low-dimension embedding of the affinity matrix between samples, followed by a KMeans in the low dimensional space. It is especially efficient if the affinity matrix is sparse and the pyamg module is installed. SpectralClustering requires the number of clusters to be specified. It works well for a small number of clusters but is not advised when using many clusters.

For two clusters, it solves a convex relaxation of the normalised cuts problem on the similarity graph: cutting the graph in two so that the weight of the edges cut is small compared to the weights of the edges inside each cluster. This criteria is especially interesting when working on images: graph vertices are pixels, and edges of the similarity graph are a function of the gradient of the image.

similarity = np.exp(-beta * distance / distance.std())

2.3.5.1. Different label assignment strategies

Different label assignment strategies can be used, corresponding to the assign_labels parameter of SpectralClustering. The "kmeans" strategy can match finer details of the data, but it can be more unstable. In particular, unless you control the random_state, it may not be reproducible from run-to-run, as it depends on a random initialization. On the other hand, the "discretize" strategy is 100% reproducible, but it tends to create parcels of fairly even and geometrical shape.

assign_labels="kmeans"	assign_labels="discretize"

2.3.5.2. Spectral Clustering Graphs

Spectral Clustering can also be used to cluster graphs by their spectral embeddings. In this case, the affinity matrix is the adjacency matrix of the graph, and SpectralClustering is initialized with affinity=’precomputed’:

>>> from sklearn.cluster import SpectralClustering
>>> sc = SpectralClustering(3, affinity='precomputed', n_init=100,
...                         assign_labels='discretize')
>>> sc.fit_predict(adjacency_matrix)  

2.3.6. Hierarchical clustering

Hierarchical clustering is a general family of clustering algorithms that build nested clusters by merging or splitting them successively. This hierarchy of clusters is represented as a tree (or dendrogram). The root of the tree is the unique cluster that gathers all the samples, the leaves being the clusters with only one sample. See the Wikipedia page for more details.

The AgglomerativeClustering object performs a hierarchical clustering using a bottom up approach: each observation starts in its own cluster, and clusters are successively merged together. The linkage criteria determines the metric used for the merge strategy:

• Ward minimizes the sum of squared differences within all clusters. It is a variance-minimizing approach and in this sense is similar to the k-means objective function but tackled with an agglomerative hierarchical approach.
• Maximum or complete linkage minimizes the maximum distance between observations of pairs of clusters.
• Average linkage minimizes the average of the distances between all observations of pairs of clusters.
• Single linkage minimizes the distance between the closest observations of pairs of clusters.

AgglomerativeClustering can also scale to large number of samples when it is used jointly with a connectivity matrix, but is computationally expensive when no connectivity constraints are added between samples: it considers at each step all the possible merges.

2.3.6.1. Different linkage type: Ward, complete, average, and single linkage

AgglomerativeClustering supports Ward, single, average, and complete linkage strategies.

Agglomerative cluster has a “rich get richer” behavior that leads to uneven cluster sizes. In this regard, single linkage is the worst strategy, and Ward gives the most regular sizes. However, the affinity (or distance used in clustering) cannot be varied with Ward, thus for non Euclidean metrics, average linkage is a good alternative. Single linkage, while not robust to noisy data, can be computed very efficiently and can therefore be useful to provide hierarchical clustering of larger datasets. Single linkage can also perform well on non-globular data.

2.3.6.2. Adding connectivity constraints

An interesting aspect of AgglomerativeClustering is that connectivity constraints can be added to this algorithm (only adjacent clusters can be merged together), through a connectivity matrix that defines for each sample the neighboring samples following a given structure of the data. For instance, in the swiss-roll example below, the connectivity constraints forbid the merging of points that are not adjacent on the swiss roll, and thus avoid forming clusters that extend across overlapping folds of the roll.

These constraint are useful to impose a certain local structure, but they also make the algorithm faster, especially when the number of the samples is high.

The connectivity constraints are imposed via an connectivity matrix: a scipy sparse matrix that has elements only at the intersection of a row and a column with indices of the dataset that should be connected. This matrix can be constructed from a-priori information: for instance, you may wish to cluster web pages by only merging pages with a link pointing from one to another. It can also be learned from the data, for instance using sklearn.neighbors.kneighbors_graph to restrict merging to nearest neighbors as in this example, or using sklearn.feature_extraction.image.grid_to_graph to enable only merging of neighboring pixels on an image, as in the coin example.

2.3.6.3. Varying the metric

Single, average and complete linkage can be used with a variety of distances (or affinities), in particular Euclidean distance (l2), Manhattan distance (or Cityblock, or l1), cosine distance, or any precomputed affinity matrix.

• l1 distance is often good for sparse features, or sparse noise: i.e. many of the features are zero, as in text mining using occurrences of rare words.
• cosine distance is interesting because it is invariant to global scalings of the signal.

The guidelines for choosing a metric is to use one that maximizes the distance between samples in different classes, and minimizes that within each class.

2.3.7. DBSCAN

The DBSCAN algorithm views clusters as areas of high density separated by areas of low density. Due to this rather generic view, clusters found by DBSCAN can be any shape, as opposed to k-means which assumes that clusters are convex shaped. The central component to the DBSCAN is the concept of core samples, which are samples that are in areas of high density. A cluster is therefore a set of core samples, each close to each other (measured by some distance measure) and a set of non-core samples that are close to a core sample (but are not themselves core samples). There are two parameters to the algorithm, min_samples and eps, which define formally what we mean when we say dense. Higher min_samples or lower eps indicate higher density necessary to form a cluster.

More formally, we define a core sample as being a sample in the dataset such that there exist min_samples other samples within a distance of eps, which are defined as neighbors of the core sample. This tells us that the core sample is in a dense area of the vector space. A cluster is a set of core samples that can be built by recursively taking a core sample, finding all of its neighbors that are core samples, finding all of their neighbors that are core samples, and so on. A cluster also has a set of non-core samples, which are samples that are neighbors of a core sample in the cluster but are not themselves core samples. Intuitively, these samples are on the fringes of a cluster.

Any core sample is part of a cluster, by definition. Any sample that is not a core sample, and is at least eps in distance from any core sample, is considered an outlier by the algorithm.

In the figure below, the color indicates cluster membership, with large circles indicating core samples found by the algorithm. Smaller circles are non-core samples that are still part of a cluster. Moreover, the outliers are indicated by black points below.

2.3.8. Birch

The Birch builds a tree called the Characteristic Feature Tree (CFT) for the given data. The data is essentially lossy compressed to a set of Characteristic Feature nodes (CF Nodes). The CF Nodes have a number of subclusters called Characteristic Feature subclusters (CF Subclusters) and these CF Subclusters located in the non-terminal CF Nodes can have CF Nodes as children.

The CF Subclusters hold the necessary information for clustering which prevents the need to hold the entire input data in memory. This information includes:

• Number of samples in a subcluster.
• Linear Sum - A n-dimensional vector holding the sum of all samples
• Squared Sum - Sum of the squared L2 norm of all samples.
• Centroids - To avoid recalculation linear sum / n_samples.
• Squared norm of the centroids.

The Birch algorithm has two parameters, the threshold and the branching factor. The branching factor limits the number of subclusters in a node and the threshold limits the distance between the entering sample and the existing subclusters.

This algorithm can be viewed as an instance or data reduction method, since it reduces the input data to a set of subclusters which are obtained directly from the leaves of the CFT. This reduced data can be further processed by feeding it into a global clusterer. This global clusterer can be set by n_clusters. If n_clusters is set to None, the subclusters from the leaves are directly read off, otherwise a global clustering step labels these subclusters into global clusters (labels) and the samples are mapped to the global label of the nearest subcluster.

Algorithm description:

• A new sample is inserted into the root of the CF Tree which is a CF Node. It is then merged with the subcluster of the root, that has the smallest radius after merging, constrained by the threshold and branching factor conditions. If the subcluster has any child node, then this is done repeatedly till it reaches a leaf. After finding the nearest subcluster in the leaf, the properties of this subcluster and the parent subclusters are recursively updated.
• If the radius of the subcluster obtained by merging the new sample and the nearest subcluster is greater than the square of the threshold and if the number of subclusters is greater than the branching factor, then a space is temporarily allocated to this new sample. The two farthest subclusters are taken and the subclusters are divided into two groups on the basis of the distance between these subclusters.
• If this split node has a parent subcluster and there is room for a new subcluster, then the parent is split into two. If there is no room, then this node is again split into two and the process is continued recursively, till it reaches the root.

Birch or MiniBatchKMeans?

• Birch does not scale very well to high dimensional data. As a rule of thumb if n_features is greater than twenty, it is generally better to use MiniBatchKMeans.
• If the number of instances of data needs to be reduced, or if one wants a large number of subclusters either as a preprocessing step or otherwise, Birch is more useful than MiniBatchKMeans.

How to use partial_fit?

To avoid the computation of global clustering, for every call of partial_fit the user is advised

1. To set n_clusters=None initially
2. Train all data by multiple calls to partial_fit.
3. Set n_clusters to a required value using brc.set_params(n_clusters=n_clusters).
4. Call partial_fit finally with no arguments, i.e. brc.partial_fit() which performs the global clustering.

2.3.9. Clustering performance evaluation

Evaluating the performance of a clustering algorithm is not as trivial as counting the number of errors or the precision and recall of a supervised classification algorithm. In particular any evaluation metric should not take the absolute values of the cluster labels into account but rather if this clustering define separations of the data similar to some ground truth set of classes or satisfying some assumption such that members belong to the same class are more similar that members of different classes according to some similarity metric.

2.3.9.1. Adjusted Rand index

Given the knowledge of the ground truth class assignments labels_true and our clustering algorithm assignments of the same samples labels_pred, the adjusted Rand index is a function that measures the similarity of the two assignments, ignoring permutations and with chance normalization:

>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]

>>> metrics.adjusted_rand_score(labels_true, labels_pred)  
0.24...

One can permute 0 and 1 in the predicted labels, rename 2 to 3, and get the same score:

>>> labels_pred = [1, 1, 0, 0, 3, 3]
>>> metrics.adjusted_rand_score(labels_true, labels_pred)  
0.24...

Furthermore, adjusted_rand_score is symmetric: swapping the argument does not change the score. It can thus be used as a consensus measure:

>>> metrics.adjusted_rand_score(labels_pred, labels_true)  
0.24...

Perfect labeling is scored 1.0:

>>> labels_pred = labels_true[:]
>>> metrics.adjusted_rand_score(labels_true, labels_pred)
1.0

Bad (e.g. independent labelings) have negative or close to 0.0 scores:

>>> labels_true = [0, 1, 2, 0, 3, 4, 5, 1]
>>> labels_pred = [1, 1, 0, 0, 2, 2, 2, 2]
>>> metrics.adjusted_rand_score(labels_true, labels_pred)  
-0.12...

2.3.9.1.1. Advantages

• Random (uniform) label assignments have a ARI score close to 0.0 for any value of n_clusters and n_samples (which is not the case for raw Rand index or the V-measure for instance).
• Bounded range [-1, 1]: negative values are bad (independent labelings), similar clusterings have a positive ARI, 1.0 is the perfect match score.
• No assumption is made on the cluster structure: can be used to compare clustering algorithms such as k-means which assumes isotropic blob shapes with results of spectral clustering algorithms which can find cluster with “folded” shapes.

2.3.9.1.2. Drawbacks

• Contrary to inertia, ARI requires knowledge of the ground truth classes while is almost never available in practice or requires manual assignment by human annotators (as in the supervised learning setting).

  However ARI can also be useful in a purely unsupervised setting as a building block for a Consensus Index that can be used for clustering model selection (TODO).

2.3.9.1.3. Mathematical formulation

If C is a ground truth class assignment and K the clustering, let us define \(a\) and \(b\) as:

• \(a\), the number of pairs of elements that are in the same set in C and in the same set in K
• \(b\), the number of pairs of elements that are in different sets in C and in different sets in K

The raw (unadjusted) Rand index is then given by:

Where \(C_2^{n_{samples}}\) is the total number of possible pairs in the dataset (without ordering).

However the RI score does not guarantee that random label assignments will get a value close to zero (esp. if the number of clusters is in the same order of magnitude as the number of samples).

To counter this effect we can discount the expected RI \(E[\text{RI}]\) of random labelings by defining the adjusted Rand index as follows:

2.3.9.2. Mutual Information based scores

Given the knowledge of the ground truth class assignments labels_true and our clustering algorithm assignments of the same samples labels_pred, the Mutual Information is a function that measures the agreement of the two assignments, ignoring permutations. Two different normalized versions of this measure are available, Normalized Mutual Information (NMI) and Adjusted Mutual Information (AMI). NMI is often used in the literature, while AMI was proposed more recently and is normalized against chance:

>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]

>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)  
0.22504...

One can permute 0 and 1 in the predicted labels, rename 2 to 3 and get the same score:

>>> labels_pred = [1, 1, 0, 0, 3, 3]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)  
0.22504...

All, mutual_info_score, adjusted_mutual_info_score and normalized_mutual_info_score are symmetric: swapping the argument does not change the score. Thus they can be used as a consensus measure:

>>> metrics.adjusted_mutual_info_score(labels_pred, labels_true)  
0.22504...

Perfect labeling is scored 1.0:

>>> labels_pred = labels_true[:]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)  
1.0

>>> metrics.normalized_mutual_info_score(labels_true, labels_pred)  
1.0

This is not true for mutual_info_score, which is therefore harder to judge:

>>> metrics.mutual_info_score(labels_true, labels_pred)  
0.69...

Bad (e.g. independent labelings) have non-positive scores:

>>> labels_true = [0, 1, 2, 0, 3, 4, 5, 1]
>>> labels_pred = [1, 1, 0, 0, 2, 2, 2, 2]
>>> metrics.adjusted_mutual_info_score(labels_true, labels_pred)  
-0.10526...

2.3.9.2.1. Advantages

• Random (uniform) label assignments have a AMI score close to 0.0 for any value of n_clusters and n_samples (which is not the case for raw Mutual Information or the V-measure for instance).
• Upper bound of 1: Values close to zero indicate two label assignments that are largely independent, while values close to one indicate significant agreement. Further, an AMI of exactly 1 indicates that the two label assignments are equal (with or without permutation).

2.3.9.2.2. Drawbacks

• Contrary to inertia, MI-based measures require the knowledge of the ground truth classes while almost never available in practice or requires manual assignment by human annotators (as in the supervised learning setting).

  However MI-based measures can also be useful in purely unsupervised setting as a building block for a Consensus Index that can be used for clustering model selection.

• NMI and MI are not adjusted against chance.

2.3.9.2.3. Mathematical formulation

Assume two label assignments (of the same N objects), \(U\) and \(V\). Their entropy is the amount of uncertainty for a partition set, defined by:

where \(P(i) = |U_i| / N\) is the probability that an object picked at random from \(U\) falls into class \(U_i\). Likewise for \(V\):

With \(P'(j) = |V_j| / N\). The mutual information (MI) between \(U\) and \(V\) is calculated by:

where \(P(i, j) = |U_i \cap V_j| / N\) is the probability that an object picked at random falls into both classes \(U_i\) and \(V_j\).

It also can be expressed in set cardinality formulation:

The normalized mutual information is defined as

This value of the mutual information and also the normalized variant is not adjusted for chance and will tend to increase as the number of different labels (clusters) increases, regardless of the actual amount of “mutual information” between the label assignments.

The expected value for the mutual information can be calculated using the following equation [VEB2009]. In this equation, \(a_i = |U_i|\) (the number of elements in \(U_i\)) and \(b_j = |V_j|\) (the number of elements in \(V_j\)).

Using the expected value, the adjusted mutual information can then be calculated using a similar form to that of the adjusted Rand index:

For normalized mutual information and adjusted mutual information, the normalizing value is typically some generalized mean of the entropies of each clustering. Various generalized means exist, and no firm rules exist for preferring one over the others. The decision is largely a field-by-field basis; for instance, in community detection, the arithmetic mean is most common. Each normalizing method provides “qualitatively similar behaviours” [YAT2016]. In our implementation, this is controlled by the average_method parameter.

Vinh et al. (2010) named variants of NMI and AMI by their averaging method [VEB2010]. Their ‘sqrt’ and ‘sum’ averages are the geometric and arithmetic means; we use these more broadly common names.

2.3.9.3. Homogeneity, completeness and V-measure

Given the knowledge of the ground truth class assignments of the samples, it is possible to define some intuitive metric using conditional entropy analysis.

In particular Rosenberg and Hirschberg (2007) define the following two desirable objectives for any cluster assignment:

• homogeneity: each cluster contains only members of a single class.
• completeness: all members of a given class are assigned to the same cluster.

We can turn those concept as scores homogeneity_score and completeness_score. Both are bounded below by 0.0 and above by 1.0 (higher is better):

>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]

>>> metrics.homogeneity_score(labels_true, labels_pred)  
0.66...

>>> metrics.completeness_score(labels_true, labels_pred) 
0.42...

Their harmonic mean called V-measure is computed by v_measure_score:

>>> metrics.v_measure_score(labels_true, labels_pred)    
0.51...

The V-measure is actually equivalent to the mutual information (NMI) discussed above, with the aggregation function being the arithmetic mean [B2011].

Homogeneity, completeness and V-measure can be computed at once using homogeneity_completeness_v_measure as follows:

>>> metrics.homogeneity_completeness_v_measure(labels_true, labels_pred)
...                                                      
(0.66..., 0.42..., 0.51...)

The following clustering assignment is slightly better, since it is homogeneous but not complete:

>>> labels_pred = [0, 0, 0, 1, 2, 2]
>>> metrics.homogeneity_completeness_v_measure(labels_true, labels_pred)
...                                                      
(1.0, 0.68..., 0.81...)

homogeneity_score(a, b) == completeness_score(b, a)

2.3.9.3.1. Advantages

• Bounded scores: 0.0 is as bad as it can be, 1.0 is a perfect score.
• Intuitive interpretation: clustering with bad V-measure can be qualitatively analyzed in terms of homogeneity and completeness to better feel what ‘kind’ of mistakes is done by the assignment.
• No assumption is made on the cluster structure: can be used to compare clustering algorithms such as k-means which assumes isotropic blob shapes with results of spectral clustering algorithms which can find cluster with “folded” shapes.

2.3.9.3.2. Drawbacks

• The previously introduced metrics are not normalized with regards to random labeling: this means that depending on the number of samples, clusters and ground truth classes, a completely random labeling will not always yield the same values for homogeneity, completeness and hence v-measure. In particular random labeling won’t yield zero scores especially when the number of clusters is large.

  This problem can safely be ignored when the number of samples is more than a thousand and the number of clusters is less than 10. For smaller sample sizes or larger number of clusters it is safer to use an adjusted index such as the Adjusted Rand Index (ARI).

• These metrics require the knowledge of the ground truth classes while almost never available in practice or requires manual assignment by human annotators (as in the supervised learning setting).

2.3.9.3.3. Mathematical formulation

Homogeneity and completeness scores are formally given by:

where \(H(C|K)\) is the conditional entropy of the classes given the cluster assignments and is given by:

and \(H(C)\) is the entropy of the classes and is given by:

with \(n\) the total number of samples, \(n_c\) and \(n_k\) the number of samples respectively belonging to class \(c\) and cluster \(k\), and finally \(n_{c,k}\) the number of samples from class \(c\) assigned to cluster \(k\).

The conditional entropy of clusters given class \(H(K|C)\) and the entropy of clusters \(H(K)\) are defined in a symmetric manner.

Rosenberg and Hirschberg further define V-measure as the harmonic mean of homogeneity and completeness:

2.3.9.4. Fowlkes-Mallows scores

The Fowlkes-Mallows index (sklearn.metrics.fowlkes_mallows_score) can be used when the ground truth class assignments of the samples is known. The Fowlkes-Mallows score FMI is defined as the geometric mean of the pairwise precision and recall:

Where TP is the number of True Positive (i.e. the number of pair of points that belong to the same clusters in both the true labels and the predicted labels), FP is the number of False Positive (i.e. the number of pair of points that belong to the same clusters in the true labels and not in the predicted labels) and FN is the number of False Negative (i.e the number of pair of points that belongs in the same clusters in the predicted labels and not in the true labels).

The score ranges from 0 to 1. A high value indicates a good similarity between two clusters.

>>> from sklearn import metrics
>>> labels_true = [0, 0, 0, 1, 1, 1]
>>> labels_pred = [0, 0, 1, 1, 2, 2]

>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)  
0.47140...

One can permute 0 and 1 in the predicted labels, rename 2 to 3 and get the same score:

>>> labels_pred = [1, 1, 0, 0, 3, 3]

>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)  
0.47140...

Perfect labeling is scored 1.0:

>>> labels_pred = labels_true[:]
>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)  
1.0

Bad (e.g. independent labelings) have zero scores:

>>> labels_true = [0, 1, 2, 0, 3, 4, 5, 1]
>>> labels_pred = [1, 1, 0, 0, 2, 2, 2, 2]
>>> metrics.fowlkes_mallows_score(labels_true, labels_pred)  
0.0

2.3.9.4.1. Advantages

• Random (uniform) label assignments have a FMI score close to 0.0 for any value of n_clusters and n_samples (which is not the case for raw Mutual Information or the V-measure for instance).
• Upper-bounded at 1: Values close to zero indicate two label assignments that are largely independent, while values close to one indicate significant agreement. Further, values of exactly 0 indicate purely independent label assignments and a AMI of exactly 1 indicates that the two label assignments are equal (with or without permutation).
• No assumption is made on the cluster structure: can be used to compare clustering algorithms such as k-means which assumes isotropic blob shapes with results of spectral clustering algorithms which can find cluster with “folded” shapes.

2.3.9.4.2. Drawbacks

• Contrary to inertia, FMI-based measures require the knowledge of the ground truth classes while almost never available in practice or requires manual assignment by human annotators (as in the supervised learning setting).

2.3.9.5. Silhouette Coefficient

If the ground truth labels are not known, evaluation must be performed using the model itself. The Silhouette Coefficient (sklearn.metrics.silhouette_score) is an example of such an evaluation, where a higher Silhouette Coefficient score relates to a model with better defined clusters. The Silhouette Coefficient is defined for each sample and is composed of two scores:

• a: The mean distance between a sample and all other points in the same class.
• b: The mean distance between a sample and all other points in the next nearest cluster.

The Silhouette Coefficient s for a single sample is then given as:

The Silhouette Coefficient for a set of samples is given as the mean of the Silhouette Coefficient for each sample.

>>> from sklearn import metrics
>>> from sklearn.metrics import pairwise_distances
>>> from sklearn import datasets
>>> dataset = datasets.load_iris()
>>> X = dataset.data
>>> y = dataset.target

In normal usage, the Silhouette Coefficient is applied to the results of a cluster analysis.

>>> import numpy as np
>>> from sklearn.cluster import KMeans
>>> kmeans_model = KMeans(n_clusters=3, random_state=1).fit(X)
>>> labels = kmeans_model.labels_
>>> metrics.silhouette_score(X, labels, metric='euclidean')
...                                                      
0.55...

2.3.9.5.1. Advantages

• The score is bounded between -1 for incorrect clustering and +1 for highly dense clustering. Scores around zero indicate overlapping clusters.
• The score is higher when clusters are dense and well separated, which relates to a standard concept of a cluster.

2.3.9.5.2. Drawbacks

• The Silhouette Coefficient is generally higher for convex clusters than other concepts of clusters, such as density based clusters like those obtained through DBSCAN.

2.3.9.6. Calinski-Harabaz Index

If the ground truth labels are not known, the Calinski-Harabaz index (sklearn.metrics.calinski_harabaz_score) - also known as the Variance Ratio Criterion - can be used to evaluate the model, where a higher Calinski-Harabaz score relates to a model with better defined clusters.

For \(k\) clusters, the Calinski-Harabaz score \(s\) is given as the ratio of the between-clusters dispersion mean and the within-cluster dispersion:

where \(B_K\) is the between group dispersion matrix and \(W_K\) is the within-cluster dispersion matrix defined by:

with \(N\) be the number of points in our data, \(C_q\) be the set of points in cluster \(q\), \(c_q\) be the center of cluster \(q\), \(c\) be the center of \(E\), \(n_q\) be the number of points in cluster \(q\).

>>> from sklearn import metrics
>>> from sklearn.metrics import pairwise_distances
>>> from sklearn import datasets
>>> dataset = datasets.load_iris()
>>> X = dataset.data
>>> y = dataset.target

In normal usage, the Calinski-Harabaz index is applied to the results of a cluster analysis.

>>> import numpy as np
>>> from sklearn.cluster import KMeans
>>> kmeans_model = KMeans(n_clusters=3, random_state=1).fit(X)
>>> labels = kmeans_model.labels_
>>> metrics.calinski_harabaz_score(X, labels)  
561.62...

2.3.9.6.1. Advantages

• The score is higher when clusters are dense and well separated, which relates to a standard concept of a cluster.
• The score is fast to compute

2.3.9.6.2. Drawbacks

• The Calinski-Harabaz index is generally higher for convex clusters than other concepts of clusters, such as density based clusters like those obtained through DBSCAN.

2.3.9.7. Davies-Bouldin Index

If the ground truth labels are not known, the Davies-Bouldin index (sklearn.metrics.davies_bouldin_score) can be used to evaluate the model, where a lower Davies-Bouldin index relates to a model with better separation between the clusters.

The index is defined as the average similarity between each cluster \(C_i\) for \(i=1, ..., k\) and its most similar one \(C_j\). In the context of this index, similarity is defined as a measure \(R_{ij}\) that trades off:

• \(s_i\), the average distance between each point of cluster \(i\) and the centroid of that cluster – also know as cluster diameter.
• \(d_{ij}\), the distance between cluster centroids \(i\) and \(j\).

A simple choice to construct \(R_ij\) so that it is nonnegative and symmetric is:

Then the Davies-Bouldin index is defined as:

Zero is the lowest possible score. Values closer to zero indicate a better partition.

In normal usage, the Davies-Bouldin index is applied to the results of a cluster analysis as follows:

>>> from sklearn import datasets
>>> iris = datasets.load_iris()
>>> X = iris.data
>>> from sklearn.cluster import KMeans
>>> from sklearn.metrics import davies_bouldin_score
>>> kmeans = KMeans(n_clusters=3, random_state=1).fit(X)
>>> labels = kmeans.labels_
>>> davies_bouldin_score(X, labels)  
0.6619...

2.3.9.7.1. Advantages

• The computation of Davies-Bouldin is simpler than that of Silhouette scores.
• The index is computed only quantities and features inherent to the dataset.

2.3.9.7.2. Drawbacks

• The Davies-Boulding index is generally higher for convex clusters than other concepts of clusters, such as density based clusters like those obtained from DBSCAN.
• The usage of centroid distance limits the distance metric to Euclidean space.
• A good value reported by this method does not imply the best information retrieval.

2.3.9.8. Contingency Matrix

Contingency matrix (sklearn.metrics.cluster.contingency_matrix) reports the intersection cardinality for every true/predicted cluster pair. The contingency matrix provides sufficient statistics for all clustering metrics where the samples are independent and identically distributed and one doesn’t need to account for some instances not being clustered.

Here is an example:

>>> from sklearn.metrics.cluster import contingency_matrix
>>> x = ["a", "a", "a", "b", "b", "b"]
>>> y = [0, 0, 1, 1, 2, 2]
>>> contingency_matrix(x, y)
array([[2, 1, 0],
       [0, 1, 2]])

The first row of output array indicates that there are three samples whose true cluster is “a”. Of them, two are in predicted cluster 0, one is in 1, and none is in 2. And the second row indicates that there are three samples whose true cluster is “b”. Of them, none is in predicted cluster 0, one is in 1 and two are in 2.

A confusion matrix for classification is a square contingency matrix where the order of rows and columns correspond to a list of classes.

2.3.9.8.1. Advantages

• Allows to examine the spread of each true cluster across predicted clusters and vice versa.
• The contingency table calculated is typically utilized in the calculation of a similarity statistic (like the others listed in this document) between the two clusterings.

2.3.9.8.2. Drawbacks

• Contingency matrix is easy to interpret for a small number of clusters, but becomes very hard to interpret for a large number of clusters.
• It doesn’t give a single metric to use as an objective for clustering optimisation.