"""Utilities to evaluate the clustering performance of models. Functions named as *_score return a scalar value to maximize: the higher the better. """ # Authors: Olivier Grisel # Wei LI # Diego Molla # Arnaud Fouchet # Thierry Guillemot # Gregory Stupp # Joel Nothman # License: BSD 3 clause from __future__ import division from math import log import numpy as np from scipy import sparse as sp from .expected_mutual_info_fast import expected_mutual_information from ...utils.validation import check_array from ...utils.fixes import comb def comb2(n): # the exact version is faster for k == 2: use it by default globally in # this module instead of the float approximate variant return comb(n, 2, exact=1) def check_clusterings(labels_true, labels_pred): """Check that the two clusterings matching 1D integer arrays.""" labels_true = np.asarray(labels_true) labels_pred = np.asarray(labels_pred) # input checks if labels_true.ndim != 1: raise ValueError( "labels_true must be 1D: shape is %r" % (labels_true.shape,)) if labels_pred.ndim != 1: raise ValueError( "labels_pred must be 1D: shape is %r" % (labels_pred.shape,)) if labels_true.shape != labels_pred.shape: raise ValueError( "labels_true and labels_pred must have same size, got %d and %d" % (labels_true.shape[0], labels_pred.shape[0])) return labels_true, labels_pred def contingency_matrix(labels_true, labels_pred, eps=None, sparse=False): """Build a contingency matrix describing the relationship between labels. Parameters ---------- labels_true : int array, shape = [n_samples] Ground truth class labels to be used as a reference labels_pred : array, shape = [n_samples] Cluster labels to evaluate eps : None or float, optional. If a float, that value is added to all values in the contingency matrix. This helps to stop NaN propagation. If ``None``, nothing is adjusted. sparse : boolean, optional. If True, return a sparse CSR continency matrix. If ``eps is not None``, and ``sparse is True``, will throw ValueError. .. versionadded:: 0.18 Returns ------- contingency : {array-like, sparse}, shape=[n_classes_true, n_classes_pred] Matrix :math:`C` such that :math:`C_{i, j}` is the number of samples in true class :math:`i` and in predicted class :math:`j`. If ``eps is None``, the dtype of this array will be integer. If ``eps`` is given, the dtype will be float. Will be a ``scipy.sparse.csr_matrix`` if ``sparse=True``. """ if eps is not None and sparse: raise ValueError("Cannot set 'eps' when sparse=True") classes, class_idx = np.unique(labels_true, return_inverse=True) clusters, cluster_idx = np.unique(labels_pred, return_inverse=True) n_classes = classes.shape[0] n_clusters = clusters.shape[0] # Using coo_matrix to accelerate simple histogram calculation, # i.e. bins are consecutive integers # Currently, coo_matrix is faster than histogram2d for simple cases contingency = sp.coo_matrix((np.ones(class_idx.shape[0]), (class_idx, cluster_idx)), shape=(n_classes, n_clusters), dtype=np.int) if sparse: contingency = contingency.tocsr() contingency.sum_duplicates() else: contingency = contingency.toarray() if eps is not None: # don't use += as contingency is integer contingency = contingency + eps return contingency # clustering measures def adjusted_rand_score(labels_true, labels_pred): """Rand index adjusted for chance. The Rand Index computes a similarity measure between two clusterings by considering all pairs of samples and counting pairs that are assigned in the same or different clusters in the predicted and true clusterings. The raw RI score is then "adjusted for chance" into the ARI score using the following scheme:: ARI = (RI - Expected_RI) / (max(RI) - Expected_RI) The adjusted Rand index is thus ensured to have a value close to 0.0 for random labeling independently of the number of clusters and samples and exactly 1.0 when the clusterings are identical (up to a permutation). ARI is a symmetric measure:: adjusted_rand_score(a, b) == adjusted_rand_score(b, a) Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] Ground truth class labels to be used as a reference labels_pred : array, shape = [n_samples] Cluster labels to evaluate Returns ------- ari : float Similarity score between -1.0 and 1.0. Random labelings have an ARI close to 0.0. 1.0 stands for perfect match. Examples -------- Perfectly matching labelings have a score of 1 even >>> from sklearn.metrics.cluster import adjusted_rand_score >>> adjusted_rand_score([0, 0, 1, 1], [0, 0, 1, 1]) 1.0 >>> adjusted_rand_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 Labelings that assign all classes members to the same clusters are complete be not always pure, hence penalized:: >>> adjusted_rand_score([0, 0, 1, 2], [0, 0, 1, 1]) # doctest: +ELLIPSIS 0.57... ARI is symmetric, so labelings that have pure clusters with members coming from the same classes but unnecessary splits are penalized:: >>> adjusted_rand_score([0, 0, 1, 1], [0, 0, 1, 2]) # doctest: +ELLIPSIS 0.57... If classes members are completely split across different clusters, the assignment is totally incomplete, hence the ARI is very low:: >>> adjusted_rand_score([0, 0, 0, 0], [0, 1, 2, 3]) 0.0 References ---------- .. [Hubert1985] `L. Hubert and P. Arabie, Comparing Partitions, Journal of Classification 1985` http://link.springer.com/article/10.1007%2FBF01908075 .. [wk] https://en.wikipedia.org/wiki/Rand_index#Adjusted_Rand_index See also -------- adjusted_mutual_info_score: Adjusted Mutual Information """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) n_samples = labels_true.shape[0] n_classes = np.unique(labels_true).shape[0] n_clusters = np.unique(labels_pred).shape[0] # Special limit cases: no clustering since the data is not split; # or trivial clustering where each document is assigned a unique cluster. # These are perfect matches hence return 1.0. if (n_classes == n_clusters == 1 or n_classes == n_clusters == 0 or n_classes == n_clusters == n_samples): return 1.0 # Compute the ARI using the contingency data contingency = contingency_matrix(labels_true, labels_pred, sparse=True) sum_comb_c = sum(comb2(n_c) for n_c in np.ravel(contingency.sum(axis=1))) sum_comb_k = sum(comb2(n_k) for n_k in np.ravel(contingency.sum(axis=0))) sum_comb = sum(comb2(n_ij) for n_ij in contingency.data) prod_comb = (sum_comb_c * sum_comb_k) / comb(n_samples, 2) mean_comb = (sum_comb_k + sum_comb_c) / 2. return (sum_comb - prod_comb) / (mean_comb - prod_comb) def homogeneity_completeness_v_measure(labels_true, labels_pred): """Compute the homogeneity and completeness and V-Measure scores at once. Those metrics are based on normalized conditional entropy measures of the clustering labeling to evaluate given the knowledge of a Ground Truth class labels of the same samples. A clustering result satisfies homogeneity if all of its clusters contain only data points which are members of a single class. A clustering result satisfies completeness if all the data points that are members of a given class are elements of the same cluster. Both scores have positive values between 0.0 and 1.0, larger values being desirable. Those 3 metrics are independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score values in any way. V-Measure is furthermore symmetric: swapping ``labels_true`` and ``label_pred`` will give the same score. This does not hold for homogeneity and completeness. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] ground truth class labels to be used as a reference labels_pred : array, shape = [n_samples] cluster labels to evaluate Returns ------- homogeneity : float score between 0.0 and 1.0. 1.0 stands for perfectly homogeneous labeling completeness : float score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling v_measure : float harmonic mean of the first two See also -------- homogeneity_score completeness_score v_measure_score """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) if len(labels_true) == 0: return 1.0, 1.0, 1.0 entropy_C = entropy(labels_true) entropy_K = entropy(labels_pred) contingency = contingency_matrix(labels_true, labels_pred, sparse=True) MI = mutual_info_score(None, None, contingency=contingency) homogeneity = MI / (entropy_C) if entropy_C else 1.0 completeness = MI / (entropy_K) if entropy_K else 1.0 if homogeneity + completeness == 0.0: v_measure_score = 0.0 else: v_measure_score = (2.0 * homogeneity * completeness / (homogeneity + completeness)) return homogeneity, completeness, v_measure_score def homogeneity_score(labels_true, labels_pred): """Homogeneity metric of a cluster labeling given a ground truth. A clustering result satisfies homogeneity if all of its clusters contain only data points which are members of a single class. This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score value in any way. This metric is not symmetric: switching ``label_true`` with ``label_pred`` will return the :func:`completeness_score` which will be different in general. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] ground truth class labels to be used as a reference labels_pred : array, shape = [n_samples] cluster labels to evaluate Returns ------- homogeneity : float score between 0.0 and 1.0. 1.0 stands for perfectly homogeneous labeling References ---------- .. [1] `Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure `_ See also -------- completeness_score v_measure_score Examples -------- Perfect labelings are homogeneous:: >>> from sklearn.metrics.cluster import homogeneity_score >>> homogeneity_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 Non-perfect labelings that further split classes into more clusters can be perfectly homogeneous:: >>> print("%.6f" % homogeneity_score([0, 0, 1, 1], [0, 0, 1, 2])) ... # doctest: +ELLIPSIS 1.0... >>> print("%.6f" % homogeneity_score([0, 0, 1, 1], [0, 1, 2, 3])) ... # doctest: +ELLIPSIS 1.0... Clusters that include samples from different classes do not make for an homogeneous labeling:: >>> print("%.6f" % homogeneity_score([0, 0, 1, 1], [0, 1, 0, 1])) ... # doctest: +ELLIPSIS 0.0... >>> print("%.6f" % homogeneity_score([0, 0, 1, 1], [0, 0, 0, 0])) ... # doctest: +ELLIPSIS 0.0... """ return homogeneity_completeness_v_measure(labels_true, labels_pred)[0] def completeness_score(labels_true, labels_pred): """Completeness metric of a cluster labeling given a ground truth. A clustering result satisfies completeness if all the data points that are members of a given class are elements of the same cluster. This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score value in any way. This metric is not symmetric: switching ``label_true`` with ``label_pred`` will return the :func:`homogeneity_score` which will be different in general. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] ground truth class labels to be used as a reference labels_pred : array, shape = [n_samples] cluster labels to evaluate Returns ------- completeness : float score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling References ---------- .. [1] `Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure `_ See also -------- homogeneity_score v_measure_score Examples -------- Perfect labelings are complete:: >>> from sklearn.metrics.cluster import completeness_score >>> completeness_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 Non-perfect labelings that assign all classes members to the same clusters are still complete:: >>> print(completeness_score([0, 0, 1, 1], [0, 0, 0, 0])) 1.0 >>> print(completeness_score([0, 1, 2, 3], [0, 0, 1, 1])) 1.0 If classes members are split across different clusters, the assignment cannot be complete:: >>> print(completeness_score([0, 0, 1, 1], [0, 1, 0, 1])) 0.0 >>> print(completeness_score([0, 0, 0, 0], [0, 1, 2, 3])) 0.0 """ return homogeneity_completeness_v_measure(labels_true, labels_pred)[1] def v_measure_score(labels_true, labels_pred): """V-measure cluster labeling given a ground truth. This score is identical to :func:`normalized_mutual_info_score`. The V-measure is the harmonic mean between homogeneity and completeness:: v = 2 * (homogeneity * completeness) / (homogeneity + completeness) This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score value in any way. This metric is furthermore symmetric: switching ``label_true`` with ``label_pred`` will return the same score value. This can be useful to measure the agreement of two independent label assignments strategies on the same dataset when the real ground truth is not known. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] ground truth class labels to be used as a reference labels_pred : array, shape = [n_samples] cluster labels to evaluate Returns ------- v_measure : float score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling References ---------- .. [1] `Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure `_ See also -------- homogeneity_score completeness_score Examples -------- Perfect labelings are both homogeneous and complete, hence have score 1.0:: >>> from sklearn.metrics.cluster import v_measure_score >>> v_measure_score([0, 0, 1, 1], [0, 0, 1, 1]) 1.0 >>> v_measure_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 Labelings that assign all classes members to the same clusters are complete be not homogeneous, hence penalized:: >>> print("%.6f" % v_measure_score([0, 0, 1, 2], [0, 0, 1, 1])) ... # doctest: +ELLIPSIS 0.8... >>> print("%.6f" % v_measure_score([0, 1, 2, 3], [0, 0, 1, 1])) ... # doctest: +ELLIPSIS 0.66... Labelings that have pure clusters with members coming from the same classes are homogeneous but un-necessary splits harms completeness and thus penalize V-measure as well:: >>> print("%.6f" % v_measure_score([0, 0, 1, 1], [0, 0, 1, 2])) ... # doctest: +ELLIPSIS 0.8... >>> print("%.6f" % v_measure_score([0, 0, 1, 1], [0, 1, 2, 3])) ... # doctest: +ELLIPSIS 0.66... If classes members are completely split across different clusters, the assignment is totally incomplete, hence the V-Measure is null:: >>> print("%.6f" % v_measure_score([0, 0, 0, 0], [0, 1, 2, 3])) ... # doctest: +ELLIPSIS 0.0... Clusters that include samples from totally different classes totally destroy the homogeneity of the labeling, hence:: >>> print("%.6f" % v_measure_score([0, 0, 1, 1], [0, 0, 0, 0])) ... # doctest: +ELLIPSIS 0.0... """ return homogeneity_completeness_v_measure(labels_true, labels_pred)[2] def mutual_info_score(labels_true, labels_pred, contingency=None): """Mutual Information between two clusterings. The Mutual Information is a measure of the similarity between two labels of the same data. Where :math:`|U_i|` is the number of the samples in cluster :math:`U_i` and :math:`|V_j|` is the number of the samples in cluster :math:`V_j`, the Mutual Information between clusterings :math:`U` and :math:`V` is given as: .. math:: MI(U,V)=\sum_{i=1}^|U| \sum_{j=1}^|V| \\frac{|U_i\cap V_j|}{N} \log\\frac{N|U_i \cap V_j|}{|U_i||V_j|} This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score value in any way. This metric is furthermore symmetric: switching ``label_true`` with ``label_pred`` will return the same score value. This can be useful to measure the agreement of two independent label assignments strategies on the same dataset when the real ground truth is not known. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] A clustering of the data into disjoint subsets. labels_pred : array, shape = [n_samples] A clustering of the data into disjoint subsets. contingency : {None, array, sparse matrix}, shape = [n_classes_true, n_classes_pred] A contingency matrix given by the :func:`contingency_matrix` function. If value is ``None``, it will be computed, otherwise the given value is used, with ``labels_true`` and ``labels_pred`` ignored. Returns ------- mi : float Mutual information, a non-negative value See also -------- adjusted_mutual_info_score: Adjusted against chance Mutual Information normalized_mutual_info_score: Normalized Mutual Information """ if contingency is None: labels_true, labels_pred = check_clusterings(labels_true, labels_pred) contingency = contingency_matrix(labels_true, labels_pred, sparse=True) else: contingency = check_array(contingency, accept_sparse=['csr', 'csc', 'coo'], dtype=[int, np.int32, np.int64]) if isinstance(contingency, np.ndarray): # For an array nzx, nzy = np.nonzero(contingency) nz_val = contingency[nzx, nzy] elif sp.issparse(contingency): # For a sparse matrix nzx, nzy, nz_val = sp.find(contingency) else: raise ValueError("Unsupported type for 'contingency': %s" % type(contingency)) contingency_sum = contingency.sum() pi = np.ravel(contingency.sum(axis=1)) pj = np.ravel(contingency.sum(axis=0)) log_contingency_nm = np.log(nz_val) contingency_nm = nz_val / contingency_sum # Don't need to calculate the full outer product, just for non-zeroes outer = pi.take(nzx) * pj.take(nzy) log_outer = -np.log(outer) + log(pi.sum()) + log(pj.sum()) mi = (contingency_nm * (log_contingency_nm - log(contingency_sum)) + contingency_nm * log_outer) return mi.sum() def adjusted_mutual_info_score(labels_true, labels_pred): """Adjusted Mutual Information between two clusterings. Adjusted Mutual Information (AMI) is an adjustment of the Mutual Information (MI) score to account for chance. It accounts for the fact that the MI is generally higher for two clusterings with a larger number of clusters, regardless of whether there is actually more information shared. For two clusterings :math:`U` and :math:`V`, the AMI is given as:: AMI(U, V) = [MI(U, V) - E(MI(U, V))] / [max(H(U), H(V)) - E(MI(U, V))] This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score value in any way. This metric is furthermore symmetric: switching ``label_true`` with ``label_pred`` will return the same score value. This can be useful to measure the agreement of two independent label assignments strategies on the same dataset when the real ground truth is not known. Be mindful that this function is an order of magnitude slower than other metrics, such as the Adjusted Rand Index. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] A clustering of the data into disjoint subsets. labels_pred : array, shape = [n_samples] A clustering of the data into disjoint subsets. Returns ------- ami: float(upperlimited by 1.0) The AMI returns a value of 1 when the two partitions are identical (ie perfectly matched). Random partitions (independent labellings) have an expected AMI around 0 on average hence can be negative. See also -------- adjusted_rand_score: Adjusted Rand Index mutual_information_score: Mutual Information (not adjusted for chance) Examples -------- Perfect labelings are both homogeneous and complete, hence have score 1.0:: >>> from sklearn.metrics.cluster import adjusted_mutual_info_score >>> adjusted_mutual_info_score([0, 0, 1, 1], [0, 0, 1, 1]) 1.0 >>> adjusted_mutual_info_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 If classes members are completely split across different clusters, the assignment is totally in-complete, hence the AMI is null:: >>> adjusted_mutual_info_score([0, 0, 0, 0], [0, 1, 2, 3]) 0.0 References ---------- .. [1] `Vinh, Epps, and Bailey, (2010). Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance, JMLR `_ .. [2] `Wikipedia entry for the Adjusted Mutual Information `_ """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) n_samples = labels_true.shape[0] classes = np.unique(labels_true) clusters = np.unique(labels_pred) # Special limit cases: no clustering since the data is not split. # This is a perfect match hence return 1.0. if (classes.shape[0] == clusters.shape[0] == 1 or classes.shape[0] == clusters.shape[0] == 0): return 1.0 contingency = contingency_matrix(labels_true, labels_pred, sparse=True) contingency = contingency.astype(np.float64) # Calculate the MI for the two clusterings mi = mutual_info_score(labels_true, labels_pred, contingency=contingency) # Calculate the expected value for the mutual information emi = expected_mutual_information(contingency, n_samples) # Calculate entropy for each labeling h_true, h_pred = entropy(labels_true), entropy(labels_pred) ami = (mi - emi) / (max(h_true, h_pred) - emi) return ami def normalized_mutual_info_score(labels_true, labels_pred): """Normalized Mutual Information between two clusterings. Normalized Mutual Information (NMI) is an normalization of the Mutual Information (MI) score to scale the results between 0 (no mutual information) and 1 (perfect correlation). In this function, mutual information is normalized by ``sqrt(H(labels_true) * H(labels_pred))`` This measure is not adjusted for chance. Therefore :func:`adjusted_mustual_info_score` might be preferred. This metric is independent of the absolute values of the labels: a permutation of the class or cluster label values won't change the score value in any way. This metric is furthermore symmetric: switching ``label_true`` with ``label_pred`` will return the same score value. This can be useful to measure the agreement of two independent label assignments strategies on the same dataset when the real ground truth is not known. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = [n_samples] A clustering of the data into disjoint subsets. labels_pred : array, shape = [n_samples] A clustering of the data into disjoint subsets. Returns ------- nmi : float score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling See also -------- adjusted_rand_score: Adjusted Rand Index adjusted_mutual_info_score: Adjusted Mutual Information (adjusted against chance) Examples -------- Perfect labelings are both homogeneous and complete, hence have score 1.0:: >>> from sklearn.metrics.cluster import normalized_mutual_info_score >>> normalized_mutual_info_score([0, 0, 1, 1], [0, 0, 1, 1]) 1.0 >>> normalized_mutual_info_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 If classes members are completely split across different clusters, the assignment is totally in-complete, hence the NMI is null:: >>> normalized_mutual_info_score([0, 0, 0, 0], [0, 1, 2, 3]) 0.0 """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) classes = np.unique(labels_true) clusters = np.unique(labels_pred) # Special limit cases: no clustering since the data is not split. # This is a perfect match hence return 1.0. if (classes.shape[0] == clusters.shape[0] == 1 or classes.shape[0] == clusters.shape[0] == 0): return 1.0 contingency = contingency_matrix(labels_true, labels_pred, sparse=True) contingency = contingency.astype(np.float64) # Calculate the MI for the two clusterings mi = mutual_info_score(labels_true, labels_pred, contingency=contingency) # Calculate the expected value for the mutual information # Calculate entropy for each labeling h_true, h_pred = entropy(labels_true), entropy(labels_pred) nmi = mi / max(np.sqrt(h_true * h_pred), 1e-10) return nmi def fowlkes_mallows_score(labels_true, labels_pred, sparse=False): """Measure the similarity of two clusterings of a set of points. The Fowlkes-Mallows index (FMI) is defined as the geometric mean between of the precision and recall:: FMI = TP / sqrt((TP + FP) * (TP + FN)) Where ``TP`` is the number of **True Positive** (i.e. the number of pair of points that belongs in the same clusters in both ``labels_true`` and ``labels_pred``), ``FP`` is the number of **False Positive** (i.e. the number of pair of points that belongs in the same clusters in ``labels_true`` and not in ``labels_pred``) and ``FN`` is the number of **False Negative** (i.e the number of pair of points that belongs in the same clusters in ``labels_pred`` and not in ``labels_True``). The score ranges from 0 to 1. A high value indicates a good similarity between two clusters. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : int array, shape = (``n_samples``,) A clustering of the data into disjoint subsets. labels_pred : array, shape = (``n_samples``, ) A clustering of the data into disjoint subsets. sparse : bool Compute contingency matrix internally with sparse matrix. Returns ------- score : float The resulting Fowlkes-Mallows score. Examples -------- Perfect labelings are both homogeneous and complete, hence have score 1.0:: >>> from sklearn.metrics.cluster import fowlkes_mallows_score >>> fowlkes_mallows_score([0, 0, 1, 1], [0, 0, 1, 1]) 1.0 >>> fowlkes_mallows_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 If classes members are completely split across different clusters, the assignment is totally random, hence the FMI is null:: >>> fowlkes_mallows_score([0, 0, 0, 0], [0, 1, 2, 3]) 0.0 References ---------- .. [1] `E. B. Fowkles and C. L. Mallows, 1983. "A method for comparing two hierarchical clusterings". Journal of the American Statistical Association `_ .. [2] `Wikipedia entry for the Fowlkes-Mallows Index `_ """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) n_samples, = labels_true.shape c = contingency_matrix(labels_true, labels_pred, sparse=True) tk = np.dot(c.data, c.data) - n_samples pk = np.sum(np.asarray(c.sum(axis=0)).ravel() ** 2) - n_samples qk = np.sum(np.asarray(c.sum(axis=1)).ravel() ** 2) - n_samples return tk / np.sqrt(pk * qk) if tk != 0. else 0. def entropy(labels): """Calculates the entropy for a labeling.""" if len(labels) == 0: return 1.0 label_idx = np.unique(labels, return_inverse=True)[1] pi = np.bincount(label_idx).astype(np.float64) pi = pi[pi > 0] pi_sum = np.sum(pi) # log(a / b) should be calculated as log(a) - log(b) for # possible loss of precision return -np.sum((pi / pi_sum) * (np.log(pi) - log(pi_sum)))