"""Utilities to evaluate the clustering performance of models. Functions named as *_score return a scalar value to maximize: the higher the better. """ # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause import warnings from math import log from numbers import Real import numpy as np from scipy import sparse as sp from ...utils._array_api import _max_precision_float_dtype, get_namespace_and_device from ...utils._param_validation import Interval, StrOptions, validate_params from ...utils.multiclass import type_of_target from ...utils.validation import check_array, check_consistent_length from ._expected_mutual_info_fast import expected_mutual_information def check_clusterings(labels_true, labels_pred): """Check that the labels arrays are 1D and of same dimension. Parameters ---------- labels_true : array-like of shape (n_samples,) The true labels. labels_pred : array-like of shape (n_samples,) The predicted labels. """ labels_true = check_array( labels_true, ensure_2d=False, ensure_min_samples=0, dtype=None, ) labels_pred = check_array( labels_pred, ensure_2d=False, ensure_min_samples=0, dtype=None, ) type_label = type_of_target(labels_true) type_pred = type_of_target(labels_pred) if "continuous" in (type_pred, type_label): msg = ( "Clustering metrics expects discrete values but received" f" {type_label} values for label, and {type_pred} values " "for target" ) warnings.warn(msg, UserWarning) # 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,)) check_consistent_length(labels_true, labels_pred) return labels_true, labels_pred def _generalized_average(U, V, average_method): """Return a particular mean of two numbers.""" if average_method == "min": return min(U, V) elif average_method == "geometric": return np.sqrt(U * V) elif average_method == "arithmetic": return np.mean([U, V]) elif average_method == "max": return max(U, V) else: raise ValueError( "'average_method' must be 'min', 'geometric', 'arithmetic', or 'max'" ) @validate_params( { "labels_true": ["array-like", None], "labels_pred": ["array-like", None], "eps": [Interval(Real, 0, None, closed="left"), None], "sparse": ["boolean"], "dtype": "no_validation", # delegate the validation to SciPy }, prefer_skip_nested_validation=True, ) def contingency_matrix( labels_true, labels_pred, *, eps=None, sparse=False, dtype=np.int64 ): """Build a contingency matrix describing the relationship between labels. Parameters ---------- labels_true : array-like of shape (n_samples,) Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,) Cluster labels to evaluate. eps : float, default=None 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 : bool, default=False If `True`, return a sparse CSR continency matrix. If `eps` is not `None` and `sparse` is `True` will raise ValueError. .. versionadded:: 0.18 dtype : numeric type, default=np.int64 Output dtype. Ignored if `eps` is not `None`. .. versionadded:: 0.24 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 unless set otherwise with the ``dtype`` argument. If ``eps`` is given, the dtype will be float. Will be a ``sklearn.sparse.csr_matrix`` if ``sparse=True``. Examples -------- >>> from sklearn.metrics.cluster import contingency_matrix >>> labels_true = [0, 0, 1, 1, 2, 2] >>> labels_pred = [1, 0, 2, 1, 0, 2] >>> contingency_matrix(labels_true, labels_pred) array([[1, 1, 0], [0, 1, 1], [1, 0, 1]]) """ 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=dtype, ) 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 @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], }, prefer_skip_nested_validation=True, ) def pair_confusion_matrix(labels_true, labels_pred): """Pair confusion matrix arising from two clusterings. The pair confusion matrix :math:`C` computes a 2 by 2 similarity matrix between two clusterings by considering all pairs of samples and counting pairs that are assigned into the same or into different clusters under the true and predicted clusterings [1]_. Considering a pair of samples that is clustered together a positive pair, then as in binary classification the count of true negatives is :math:`C_{00}`, false negatives is :math:`C_{10}`, true positives is :math:`C_{11}` and false positives is :math:`C_{01}`. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : array-like of shape (n_samples,), dtype=integral Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,), dtype=integral Cluster labels to evaluate. Returns ------- C : ndarray of shape (2, 2), dtype=np.int64 The contingency matrix. See Also -------- sklearn.metrics.rand_score : Rand Score. sklearn.metrics.adjusted_rand_score : Adjusted Rand Score. sklearn.metrics.adjusted_mutual_info_score : Adjusted Mutual Information. References ---------- .. [1] :doi:`Hubert, L., Arabie, P. "Comparing partitions." Journal of Classification 2, 193–218 (1985). <10.1007/BF01908075>` Examples -------- Perfectly matching labelings have all non-zero entries on the diagonal regardless of actual label values: >>> from sklearn.metrics.cluster import pair_confusion_matrix >>> pair_confusion_matrix([0, 0, 1, 1], [1, 1, 0, 0]) array([[8, 0], [0, 4]]... Labelings that assign all classes members to the same clusters are complete but may be not always pure, hence penalized, and have some off-diagonal non-zero entries: >>> pair_confusion_matrix([0, 0, 1, 2], [0, 0, 1, 1]) array([[8, 2], [0, 2]]... Note that the matrix is not symmetric. """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) n_samples = np.int64(labels_true.shape[0]) # Computation using the contingency data contingency = contingency_matrix( labels_true, labels_pred, sparse=True, dtype=np.int64 ) n_c = np.ravel(contingency.sum(axis=1)) n_k = np.ravel(contingency.sum(axis=0)) sum_squares = (contingency.data**2).sum() C = np.empty((2, 2), dtype=np.int64) C[1, 1] = sum_squares - n_samples C[0, 1] = contingency.dot(n_k).sum() - sum_squares C[1, 0] = contingency.transpose().dot(n_c).sum() - sum_squares C[0, 0] = n_samples**2 - C[0, 1] - C[1, 0] - sum_squares return C @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], }, prefer_skip_nested_validation=True, ) def rand_score(labels_true, labels_pred): """Rand index. 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 [1]_ [2]_. The raw RI score [3]_ is: .. code-block:: text RI = (number of agreeing pairs) / (number of pairs) Read more in the :ref:`User Guide `. Parameters ---------- labels_true : array-like of shape (n_samples,), dtype=integral Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,), dtype=integral Cluster labels to evaluate. Returns ------- RI : float Similarity score between 0.0 and 1.0, inclusive, 1.0 stands for perfect match. See Also -------- adjusted_rand_score: Adjusted Rand Score. adjusted_mutual_info_score: Adjusted Mutual Information. References ---------- .. [1] :doi:`Hubert, L., Arabie, P. "Comparing partitions." Journal of Classification 2, 193–218 (1985). <10.1007/BF01908075>`. .. [2] `Wikipedia: Simple Matching Coefficient `_ .. [3] `Wikipedia: Rand Index `_ Examples -------- Perfectly matching labelings have a score of 1 even >>> from sklearn.metrics.cluster import rand_score >>> rand_score([0, 0, 1, 1], [1, 1, 0, 0]) 1.0 Labelings that assign all classes members to the same clusters are complete but may not always be pure, hence penalized: >>> rand_score([0, 0, 1, 2], [0, 0, 1, 1]) np.float64(0.83...) """ contingency = pair_confusion_matrix(labels_true, labels_pred) numerator = contingency.diagonal().sum() denominator = contingency.sum() if numerator == denominator or denominator == 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. return 1.0 return numerator / denominator @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], }, prefer_skip_nested_validation=True, ) 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). The adjusted Rand index is bounded below by -0.5 for especially discordant clusterings. 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 : array-like of shape (n_samples,), dtype=int Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,), dtype=int Cluster labels to evaluate. Returns ------- ARI : float Similarity score between -0.5 and 1.0. Random labelings have an ARI close to 0.0. 1.0 stands for perfect match. See Also -------- adjusted_mutual_info_score : Adjusted Mutual Information. References ---------- .. [Hubert1985] L. Hubert and P. Arabie, Comparing Partitions, Journal of Classification 1985 https://link.springer.com/article/10.1007%2FBF01908075 .. [Steinley2004] D. Steinley, Properties of the Hubert-Arabie adjusted Rand index, Psychological Methods 2004 .. [wk] https://en.wikipedia.org/wiki/Rand_index#Adjusted_Rand_index .. [Chacon] :doi:`Minimum adjusted Rand index for two clusterings of a given size, 2022, J. E. Chacón and A. I. Rastrojo <10.1007/s11634-022-00491-w>` 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 but may not always be pure, hence penalized:: >>> adjusted_rand_score([0, 0, 1, 2], [0, 0, 1, 1]) 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]) 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 ARI may take a negative value for especially discordant labelings that are a worse choice than the expected value of random labels:: >>> adjusted_rand_score([0, 0, 1, 1], [0, 1, 0, 1]) -0.5 See :ref:`sphx_glr_auto_examples_cluster_plot_adjusted_for_chance_measures.py` for a more detailed example. """ (tn, fp), (fn, tp) = pair_confusion_matrix(labels_true, labels_pred) # convert to Python integer types, to avoid overflow or underflow tn, fp, fn, tp = int(tn), int(fp), int(fn), int(tp) # Special cases: empty data or full agreement if fn == 0 and fp == 0: return 1.0 return 2.0 * (tp * tn - fn * fp) / ((tp + fn) * (fn + tn) + (tp + fp) * (fp + tn)) @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], "beta": [Interval(Real, 0, None, closed="left")], }, prefer_skip_nested_validation=True, ) def homogeneity_completeness_v_measure(labels_true, labels_pred, *, beta=1.0): """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. V-Measure is identical to :func:`normalized_mutual_info_score` with the arithmetic averaging method. Read more in the :ref:`User Guide `. Parameters ---------- labels_true : array-like of shape (n_samples,) Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,) Cluster labels to evaluate. beta : float, default=1.0 Ratio of weight attributed to ``homogeneity`` vs ``completeness``. If ``beta`` is greater than 1, ``completeness`` is weighted more strongly in the calculation. If ``beta`` is less than 1, ``homogeneity`` is weighted more strongly. 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 : Homogeneity metric of cluster labeling. completeness_score : Completeness metric of cluster labeling. v_measure_score : V-Measure (NMI with arithmetic mean option). Examples -------- >>> from sklearn.metrics import homogeneity_completeness_v_measure >>> y_true, y_pred = [0, 0, 1, 1, 2, 2], [0, 0, 1, 2, 2, 2] >>> homogeneity_completeness_v_measure(y_true, y_pred) (np.float64(0.71...), np.float64(0.77...), np.float64(0.73...)) """ 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 = ( (1 + beta) * homogeneity * completeness / (beta * homogeneity + completeness) ) return homogeneity, completeness, v_measure_score @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], }, prefer_skip_nested_validation=True, ) 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 : array-like of shape (n_samples,) Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,) Cluster labels to evaluate. Returns ------- homogeneity : float Score between 0.0 and 1.0. 1.0 stands for perfectly homogeneous labeling. See Also -------- completeness_score : Completeness metric of cluster labeling. v_measure_score : V-Measure (NMI with arithmetic mean option). References ---------- .. [1] `Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure `_ Examples -------- Perfect labelings are homogeneous:: >>> from sklearn.metrics.cluster import homogeneity_score >>> homogeneity_score([0, 0, 1, 1], [1, 1, 0, 0]) np.float64(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])) 1.000000 >>> print("%.6f" % homogeneity_score([0, 0, 1, 1], [0, 1, 2, 3])) 1.000000 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])) 0.0... >>> print("%.6f" % homogeneity_score([0, 0, 1, 1], [0, 0, 0, 0])) 0.0... """ return homogeneity_completeness_v_measure(labels_true, labels_pred)[0] @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], }, prefer_skip_nested_validation=True, ) def completeness_score(labels_true, labels_pred): """Compute 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 : array-like of shape (n_samples,) Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,) Cluster labels to evaluate. Returns ------- completeness : float Score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling. See Also -------- homogeneity_score : Homogeneity metric of cluster labeling. v_measure_score : V-Measure (NMI with arithmetic mean option). References ---------- .. [1] `Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure `_ Examples -------- Perfect labelings are complete:: >>> from sklearn.metrics.cluster import completeness_score >>> completeness_score([0, 0, 1, 1], [1, 1, 0, 0]) np.float64(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])) 0.999... 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] @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], "beta": [Interval(Real, 0, None, closed="left")], }, prefer_skip_nested_validation=True, ) def v_measure_score(labels_true, labels_pred, *, beta=1.0): """V-measure cluster labeling given a ground truth. This score is identical to :func:`normalized_mutual_info_score` with the ``'arithmetic'`` option for averaging. The V-measure is the harmonic mean between homogeneity and completeness:: v = (1 + beta) * homogeneity * completeness / (beta * 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 : array-like of shape (n_samples,) Ground truth class labels to be used as a reference. labels_pred : array-like of shape (n_samples,) Cluster labels to evaluate. beta : float, default=1.0 Ratio of weight attributed to ``homogeneity`` vs ``completeness``. If ``beta`` is greater than 1, ``completeness`` is weighted more strongly in the calculation. If ``beta`` is less than 1, ``homogeneity`` is weighted more strongly. Returns ------- v_measure : float Score between 0.0 and 1.0. 1.0 stands for perfectly complete labeling. See Also -------- homogeneity_score : Homogeneity metric of cluster labeling. completeness_score : Completeness metric of cluster labeling. normalized_mutual_info_score : Normalized Mutual Information. References ---------- .. [1] `Andrew Rosenberg and Julia Hirschberg, 2007. V-Measure: A conditional entropy-based external cluster evaluation measure `_ 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]) np.float64(1.0) >>> v_measure_score([0, 0, 1, 1], [1, 1, 0, 0]) np.float64(1.0) Labelings that assign all classes members to the same clusters are complete but not homogeneous, hence penalized:: >>> print("%.6f" % v_measure_score([0, 0, 1, 2], [0, 0, 1, 1])) 0.8... >>> print("%.6f" % v_measure_score([0, 1, 2, 3], [0, 0, 1, 1])) 0.66... Labelings that have pure clusters with members coming from the same classes are homogeneous but un-necessary splits harm completeness and thus penalize V-measure as well:: >>> print("%.6f" % v_measure_score([0, 0, 1, 1], [0, 0, 1, 2])) 0.8... >>> print("%.6f" % v_measure_score([0, 0, 1, 1], [0, 1, 2, 3])) 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])) 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])) 0.0... """ return homogeneity_completeness_v_measure(labels_true, labels_pred, beta=beta)[2] @validate_params( { "labels_true": ["array-like", None], "labels_pred": ["array-like", None], "contingency": ["array-like", "sparse matrix", None], }, prefer_skip_nested_validation=True, ) 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 :math:`U` (i.e ``label_true``) with :math:`V` (i.e. ``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 : array-like of shape (n_samples,), dtype=integral A clustering of the data into disjoint subsets, called :math:`U` in the above formula. labels_pred : array-like of shape (n_samples,), dtype=integral A clustering of the data into disjoint subsets, called :math:`V` in the above formula. contingency : {array-like, sparse matrix} of shape \ (n_classes_true, n_classes_pred), default=None A contingency matrix given by the :func:`~sklearn.metrics.cluster.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, measured in nats using the natural logarithm. See Also -------- adjusted_mutual_info_score : Adjusted against chance Mutual Information. normalized_mutual_info_score : Normalized Mutual Information. Notes ----- The logarithm used is the natural logarithm (base-e). Examples -------- >>> from sklearn.metrics import mutual_info_score >>> labels_true = [0, 1, 1, 0, 1, 0] >>> labels_pred = [0, 1, 0, 0, 1, 1] >>> mutual_info_score(labels_true, labels_pred) np.float64(0.056...) """ 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] else: # For a sparse matrix nzx, nzy, nz_val = sp.find(contingency) contingency_sum = contingency.sum() pi = np.ravel(contingency.sum(axis=1)) pj = np.ravel(contingency.sum(axis=0)) # Since MI <= min(H(X), H(Y)), any labelling with zero entropy, i.e. containing a # single cluster, implies MI = 0 if pi.size == 1 or pj.size == 1: return 0.0 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).astype(np.int64, copy=False) * pj.take(nzy).astype( np.int64, copy=False ) log_outer = -np.log(outer) + log(pi.sum()) + log(pj.sum()) mi = ( contingency_nm * (log_contingency_nm - log(contingency_sum)) + contingency_nm * log_outer ) mi = np.where(np.abs(mi) < np.finfo(mi.dtype).eps, 0.0, mi) return np.clip(mi.sum(), 0.0, None) @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], "average_method": [StrOptions({"arithmetic", "max", "min", "geometric"})], }, prefer_skip_nested_validation=True, ) def adjusted_mutual_info_score( labels_true, labels_pred, *, average_method="arithmetic" ): """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))] / [avg(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 :math:`U` (``label_true``) with :math:`V` (``labels_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-like of shape (n_samples,) A clustering of the data into disjoint subsets, called :math:`U` in the above formula. labels_pred : int array-like of shape (n_samples,) A clustering of the data into disjoint subsets, called :math:`V` in the above formula. average_method : {'min', 'geometric', 'arithmetic', 'max'}, default='arithmetic' How to compute the normalizer in the denominator. .. versionadded:: 0.20 .. versionchanged:: 0.22 The default value of ``average_method`` changed from 'max' to 'arithmetic'. 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. The value is in adjusted nats (based on the natural logarithm). See Also -------- adjusted_rand_score : Adjusted Rand Index. mutual_info_score : Mutual Information (not adjusted for chance). 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 `_ 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]) ... # doctest: +SKIP 1.0 >>> adjusted_mutual_info_score([0, 0, 1, 1], [1, 1, 0, 0]) ... # doctest: +SKIP 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]) ... # doctest: +SKIP 0.0 """ 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. # It corresponds to both labellings having zero entropy. # 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) # 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) normalizer = _generalized_average(h_true, h_pred, average_method) denominator = normalizer - emi # Avoid 0.0 / 0.0 when expectation equals maximum, i.e. a perfect match. # normalizer should always be >= emi, but because of floating-point # representation, sometimes emi is slightly larger. Correct this # by preserving the sign. if denominator < 0: denominator = min(denominator, -np.finfo("float64").eps) else: denominator = max(denominator, np.finfo("float64").eps) ami = (mi - emi) / denominator return ami @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], "average_method": [StrOptions({"arithmetic", "max", "min", "geometric"})], }, prefer_skip_nested_validation=True, ) def normalized_mutual_info_score( labels_true, labels_pred, *, average_method="arithmetic" ): """Normalized Mutual Information between two clusterings. Normalized Mutual Information (NMI) is a 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 some generalized mean of ``H(labels_true)`` and ``H(labels_pred))``, defined by the `average_method`. This measure is not adjusted for chance. Therefore :func:`adjusted_mutual_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-like of shape (n_samples,) A clustering of the data into disjoint subsets. labels_pred : int array-like of shape (n_samples,) A clustering of the data into disjoint subsets. average_method : {'min', 'geometric', 'arithmetic', 'max'}, default='arithmetic' How to compute the normalizer in the denominator. .. versionadded:: 0.20 .. versionchanged:: 0.22 The default value of ``average_method`` changed from 'geometric' to 'arithmetic'. Returns ------- nmi : float Score between 0.0 and 1.0 in normalized nats (based on the natural logarithm). 1.0 stands for perfectly complete labeling. See Also -------- v_measure_score : V-Measure (NMI with arithmetic mean option). 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]) ... # doctest: +SKIP 1.0 >>> normalized_mutual_info_score([0, 0, 1, 1], [1, 1, 0, 0]) ... # doctest: +SKIP 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]) ... # doctest: +SKIP 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. # It corresponds to both labellings having zero entropy. # 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, copy=False) # Calculate the MI for the two clusterings mi = mutual_info_score(labels_true, labels_pred, contingency=contingency) # At this point mi = 0 can't be a perfect match (the special case of a single # cluster has been dealt with before). Hence, if mi = 0, the nmi must be 0 whatever # the normalization. if mi == 0: return 0.0 # Calculate entropy for each labeling h_true, h_pred = entropy(labels_true), entropy(labels_pred) normalizer = _generalized_average(h_true, h_pred, average_method) return mi / normalizer @validate_params( { "labels_true": ["array-like"], "labels_pred": ["array-like"], "sparse": ["boolean"], }, prefer_skip_nested_validation=True, ) def fowlkes_mallows_score(labels_true, labels_pred, *, sparse=False): """Measure the similarity of two clusterings of a set of points. .. versionadded:: 0.18 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 pairs of points that belong to the same cluster in both ``labels_true`` and ``labels_pred``), ``FP`` is the number of **False Positive** (i.e. the number of pairs of points that belong to the same cluster in ``labels_pred`` but not in ``labels_true``) and ``FN`` is the number of **False Negative** (i.e. the number of pairs of points that belong to the same cluster in ``labels_true`` but not in ``labels_pred``). 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 : array-like of shape (n_samples,), dtype=int A clustering of the data into disjoint subsets. labels_pred : array-like of shape (n_samples,), dtype=int A clustering of the data into disjoint subsets. sparse : bool, default=False Compute contingency matrix internally with sparse matrix. Returns ------- score : float The resulting Fowlkes-Mallows score. 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 `_ 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]) np.float64(1.0) >>> fowlkes_mallows_score([0, 0, 1, 1], [1, 1, 0, 0]) np.float64(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 """ labels_true, labels_pred = check_clusterings(labels_true, labels_pred) (n_samples,) = labels_true.shape c = contingency_matrix(labels_true, labels_pred, sparse=True) c = c.astype(np.int64, copy=False) 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 np.sqrt(tk / pk) * np.sqrt(tk / qk) if tk != 0.0 else 0.0 @validate_params( { "labels": ["array-like"], }, prefer_skip_nested_validation=True, ) def entropy(labels): """Calculate the entropy for a labeling. Parameters ---------- labels : array-like of shape (n_samples,), dtype=int The labels. Returns ------- entropy : float The entropy for a labeling. Notes ----- The logarithm used is the natural logarithm (base-e). """ xp, is_array_api_compliant, device_ = get_namespace_and_device(labels) labels_len = labels.shape[0] if is_array_api_compliant else len(labels) if labels_len == 0: return 1.0 pi = xp.astype(xp.unique_counts(labels)[1], _max_precision_float_dtype(xp, device_)) # single cluster => zero entropy if pi.size == 1: return 0.0 pi_sum = xp.sum(pi) # log(a / b) should be calculated as log(a) - log(b) for # possible loss of precision # Always convert the result as a Python scalar (on CPU) instead of a device # specific scalar array. return float(-xp.sum((pi / pi_sum) * (xp.log(pi) - log(pi_sum))))