xmeans.py

"""!
 
@brief Cluster analysis algorithm: X-Means
@details Implementation based on papers @cite article::xmeans::1, @cite article::xmeans::mndl
 
@authors Andrei Novikov (pyclustering@yandex.ru)
@date 2014-2020
@copyright BSD-3-Clause
 
"""
 
 
import copy
import numpy
 
from enum import IntEnum
from math import log
 
from pyclustering.cluster.encoder import type_encoding
from pyclustering.cluster.kmeans import kmeans
from pyclustering.cluster.center_initializer import kmeans_plusplus_initializer
 
from pyclustering.core.metric_wrapper import metric_wrapper
from pyclustering.core.wrapper import ccore_library
 
import pyclustering.core.xmeans_wrapper as wrapper
 
from pyclustering.utils import distance_metric, type_metric
 
 
class splitting_type(IntEnum):
    """!
    @brief Enumeration of splitting types that can be used as splitting creation of cluster in X-Means algorithm.
    
    """
    
    
    BAYESIAN_INFORMATION_CRITERION = 0
    
    
    MINIMUM_NOISELESS_DESCRIPTION_LENGTH = 1
 
 
class xmeans:
    """!
    @brief Class represents clustering algorithm X-Means.
    @details X-means clustering method starts with the assumption of having a minimum number of clusters, 
             and then dynamically increases them. X-means uses specified splitting criterion to control 
             the process of splitting clusters. Method K-Means++ can be used for calculation of initial centers.
             
             CCORE implementation of the algorithm uses thread pool to parallelize the clustering process.
    
    Here example how to perform cluster analysis using X-Means algorithm:
    @code
        from pyclustering.cluster import cluster_visualizer
        from pyclustering.cluster.xmeans import xmeans
        from pyclustering.cluster.center_initializer import kmeans_plusplus_initializer
        from pyclustering.utils import read_sample
        from pyclustering.samples.definitions import SIMPLE_SAMPLES
 
        # Read sample 'simple3' from file.
        sample = read_sample(SIMPLE_SAMPLES.SAMPLE_SIMPLE3)
 
        # Prepare initial centers - amount of initial centers defines amount of clusters from which X-Means will
        # start analysis.
        amount_initial_centers = 2
        initial_centers = kmeans_plusplus_initializer(sample, amount_initial_centers).initialize()
 
        # Create instance of X-Means algorithm. The algorithm will start analysis from 2 clusters, the maximum
        # number of clusters that can be allocated is 20.
        xmeans_instance = xmeans(sample, initial_centers, 20)
        xmeans_instance.process()
 
        # Extract clustering results: clusters and their centers
        clusters = xmeans_instance.get_clusters()
        centers = xmeans_instance.get_centers()
 
        # Print total sum of metric errors
        print("Total WCE:", xmeans_instance.get_total_wce())
 
        # Visualize clustering results
        visualizer = cluster_visualizer()
        visualizer.append_clusters(clusters, sample)
        visualizer.append_cluster(centers, None, marker='*', markersize=10)
        visualizer.show()
    @endcode
 
    Visualization of clustering results that were obtained using code above and where X-Means algorithm allocates four clusters.
    @image html xmeans_clustering_simple3.png "Fig. 1. X-Means clustering results (data 'Simple3')."
 
    By default X-Means clustering algorithm uses Bayesian Information Criterion (BIC) to approximate the correct number
    of clusters. There is an example where another criterion Minimum Noiseless Description Length (MNDL) is used in order
    to find optimal amount of clusters:
    @code
        from pyclustering.cluster import cluster_visualizer
        from pyclustering.cluster.xmeans import xmeans, splitting_type
        from pyclustering.cluster.center_initializer import kmeans_plusplus_initializer
        from pyclustering.utils import read_sample
        from pyclustering.samples.definitions import FCPS_SAMPLES
 
        # Read sample 'Target'.
        sample = read_sample(FCPS_SAMPLES.SAMPLE_TARGET)
 
        # Prepare initial centers - amount of initial centers defines amount of clusters from which X-Means will start analysis.
        random_seed = 1000
        amount_initial_centers = 3
        initial_centers = kmeans_plusplus_initializer(sample, amount_initial_centers, random_state=random_seed).initialize()
 
        # Create instance of X-Means algorithm with MNDL splitting criterion.
        xmeans_mndl = xmeans(sample, initial_centers, 20, splitting_type=splitting_type.MINIMUM_NOISELESS_DESCRIPTION_LENGTH, random_state=random_seed)
        xmeans_mndl.process()
 
        # Extract X-Means MNDL clustering results.
        mndl_clusters = xmeans_mndl.get_clusters()
 
        # Visualize clustering results.
        visualizer = cluster_visualizer(titles=['X-Means with MNDL criterion'])
        visualizer.append_clusters(mndl_clusters, sample)
        visualizer.show()
    @endcode
 
    @image html xmeans_clustering_mndl_target.png "Fig. 2. X-Means MNDL clustering results (data 'Target')."
 
    As in many others algorithms, it is possible to specify metric that should be used for cluster analysis, for
    example, Chebyshev distance metric:
    @code
        # Create instance of X-Means algorithm with Chebyshev distance metric.
        chebyshev_metric = distance_metric(type_metric.CHEBYSHEV)
        xmeans_instance = xmeans(sample, initial_centers, max_clusters_amount, metric=chebyshev_metric).process()
    @endcode
 
    @see center_initializer
 
    """
    
    def __init__(self, data, initial_centers=None, kmax=20, tolerance=0.001, criterion=splitting_type.BAYESIAN_INFORMATION_CRITERION, ccore=True, **kwargs):
        """!
        @brief Constructor of clustering algorithm X-Means.
        
        @param[in] data (array_like): Input data that is presented as list of points (objects), each point should be represented by list or tuple.
        @param[in] initial_centers (list): Initial coordinates of centers of clusters that are represented by list: `[center1, center2, ...]`,
                    if it is not specified then X-Means starts from the random center.
        @param[in] kmax (uint): Maximum number of clusters that can be allocated.
        @param[in] tolerance (double): Stop condition for each iteration: if maximum value of change of centers of clusters is less than tolerance than algorithm will stop processing.
        @param[in] criterion (splitting_type): Type of splitting creation (by default `splitting_type.BAYESIAN_INFORMATION_CRITERION`).
        @param[in] ccore (bool): Defines if C++ pyclustering library should be used instead of Python implementation.
        @param[in] **kwargs: Arbitrary keyword arguments (available arguments: `repeat`, `random_state`, `metric`, `alpha`, `beta`).
 
        <b>Keyword Args:</b><br>
            - repeat (unit): How many times K-Means should be run to improve parameters (by default is `1`).
               With larger `repeat` values suggesting higher probability of finding global optimum.
            - random_state (int): Seed for random state (by default is `None`, current system time is used).
            - metric (distance_metric): Metric that is used for distance calculation between two points (by default
               euclidean square distance).
            - alpha (double): Parameter distributed [0.0, 1.0] for alpha probabilistic bound \f$Q\left(\alpha\right)\f$.
               The parameter is used only in case of MNDL splitting criterion, in all other cases this value is ignored.
            - beta (double): Parameter distributed [0.0, 1.0] for beta probabilistic bound \f$Q\left(\beta\right)\f$.
               The parameter is used only in case of MNDL splitting criterion, in all other cases this value is ignored.
 
        """
        
        self.__pointer_data = numpy.array(data)
        self.__clusters = []
        self.__random_state = kwargs.get('random_state', None)
        self.__metric = copy.copy(kwargs.get('metric', distance_metric(type_metric.EUCLIDEAN_SQUARE)))
        
        if initial_centers is not None:
            self.__centers = numpy.array(initial_centers)
        else:
            self.__centers = kmeans_plusplus_initializer(data, 2, random_state=self.__random_state).initialize()
        
        self.__kmax = kmax
        self.__tolerance = tolerance
        self.__criterion = criterion
        self.__total_wce = 0.0
        self.__repeat = kwargs.get('repeat', 1)
        self.__alpha = kwargs.get('alpha', 0.9)
        self.__beta = kwargs.get('beta', 0.9)
 
        self.__ccore = ccore and self.__metric.get_type() != type_metric.USER_DEFINED
        if self.__ccore is True:
            self.__ccore = ccore_library.workable()
 
        self.__verify_arguments()
 
 
    def process(self):
        """!
        @brief Performs cluster analysis in line with rules of X-Means algorithm.
        
        @return (xmeans) Returns itself (X-Means instance).
        
        @see get_clusters()
        @see get_centers()
        
        """
        
        if self.__ccore is True:
            self.__process_by_ccore()
 
        else:
            self.__process_by_python()
 
        return self
 
 
    def __process_by_ccore(self):
        """!
        @brief Performs cluster analysis using CCORE (C/C++ part of pyclustering library).
 
        """
 
        ccore_metric = metric_wrapper.create_instance(self.__metric)
 
        result = wrapper.xmeans(self.__pointer_data, self.__centers, self.__kmax, self.__tolerance, self.__criterion,
                                self.__alpha, self.__beta, self.__repeat, self.__random_state,
                                ccore_metric.get_pointer())
 
        self.__clusters = result[0]
        self.__centers = result[1]
        self.__total_wce = result[2][0]
 
 
    def __process_by_python(self):
        """!
        @brief Performs cluster analysis using python code.
 
        """
 
        self.__clusters = []
        while len(self.__centers) <= self.__kmax:
            current_cluster_number = len(self.__centers)
 
            self.__clusters, self.__centers, _ = self.__improve_parameters(self.__centers)
            allocated_centers = self.__improve_structure(self.__clusters, self.__centers)
 
            if current_cluster_number == len(allocated_centers):
                break
            else:
                self.__centers = allocated_centers
 
        self.__clusters, self.__centers, self.__total_wce = self.__improve_parameters(self.__centers)
 
 
    def predict(self, points):
        """!
        @brief Calculates the closest cluster to each point.
 
        @param[in] points (array_like): Points for which closest clusters are calculated.
 
        @return (list) List of closest clusters for each point. Each cluster is denoted by index. Return empty
                 collection if 'process()' method was not called.
 
        An example how to calculate (or predict) the closest cluster to specified points.
        @code
            from pyclustering.cluster.xmeans import xmeans
            from pyclustering.samples.definitions import SIMPLE_SAMPLES
            from pyclustering.utils import read_sample
 
            # Load list of points for cluster analysis.
            sample = read_sample(SIMPLE_SAMPLES.SAMPLE_SIMPLE3)
 
            # Initial centers for sample 'Simple3'.
            initial_centers = [[0.2, 0.1], [4.0, 1.0], [2.0, 2.0], [2.3, 3.9]]
 
            # Create instance of X-Means algorithm with prepared centers.
            xmeans_instance = xmeans(sample, initial_centers)
 
            # Run cluster analysis.
            xmeans_instance.process()
 
            # Calculate the closest cluster to following two points.
            points = [[0.25, 0.2], [2.5, 4.0]]
            closest_clusters = xmeans_instance.predict(points)
            print(closest_clusters)
        @endcode
 
        """
        nppoints = numpy.array(points)
        if len(self.__clusters) == 0:
            return []
 
        self.__metric.enable_numpy_usage()
 
        npcenters = numpy.array(self.__centers)
        differences = numpy.zeros((len(nppoints), len(npcenters)))
        for index_point in range(len(nppoints)):
            differences[index_point] = self.__metric(nppoints[index_point], npcenters)
 
        self.__metric.disable_numpy_usage()
 
        return numpy.argmin(differences, axis=1)
 
 
    def get_clusters(self):
        """!
        @brief Returns list of allocated clusters, each cluster contains indexes of objects in list of data.
        
        @return (list) List of allocated clusters.
        
        @see process()
        @see get_centers()
        @see get_total_wce()
        
        """
 
        return self.__clusters
 
 
    def get_centers(self):
        """!
        @brief Returns list of centers for allocated clusters.
        
        @return (list) List of centers for allocated clusters.
        
        @see process()
        @see get_clusters()
        @see get_total_wce()
        
        """
         
        return self.__centers
 
 
    def get_cluster_encoding(self):
        """!
        @brief Returns clustering result representation type that indicate how clusters are encoded.
        
        @return (type_encoding) Clustering result representation.
        
        @see get_clusters()
        
        """
        
        return type_encoding.CLUSTER_INDEX_LIST_SEPARATION
 
 
    def get_total_wce(self):
        """!
        @brief Returns sum of Euclidean Squared metric errors (SSE - Sum of Squared Errors).
        @details Sum of metric errors is calculated using distance between point and its center:
                 \f[error=\sum_{i=0}^{N}euclidean_square_distance(x_{i}-center(x_{i}))\f]
 
        @see process()
        @see get_clusters()
 
        """
 
        return self.__total_wce
 
 
    def __search_optimial_parameters(self, local_data):
        """!
        @brief Split data of the region into two cluster and tries to find global optimum by running k-means clustering
                several times (defined by 'repeat' argument).
 
        @param[in] local_data (list): Points of a region that should be split into two clusters.
 
        @return (tuple) List of allocated clusters, list of centers and total WCE (clusters, centers, wce).
 
        """
        optimal_wce, optimal_centers, optimal_clusters = float('+inf'), None, None
 
        for _ in range(self.__repeat):
            candidates = 5
            if len(local_data) < candidates:
                candidates = len(local_data)
 
            local_centers = kmeans_plusplus_initializer(local_data, 2, candidates, random_state=self.__random_state).initialize()
 
            kmeans_instance = kmeans(local_data, local_centers, tolerance=self.__tolerance, ccore=False, metric=self.__metric)
            kmeans_instance.process()
 
            local_wce = kmeans_instance.get_total_wce()
            if local_wce < optimal_wce:
                optimal_centers = kmeans_instance.get_centers()
                optimal_clusters = kmeans_instance.get_clusters()
                optimal_wce = local_wce
 
        return optimal_clusters, optimal_centers, optimal_wce
 
 
    def __improve_parameters(self, centers, available_indexes=None):
        """!
        @brief Performs k-means clustering in the specified region.
        
        @param[in] centers (list): Cluster centers, if None then automatically generated two centers using center initialization method.
        @param[in] available_indexes (list): Indexes that defines which points can be used for k-means clustering, if None then all points are used.
        
        @return (tuple) List of allocated clusters, list of centers and total WCE (clusters, centers, wce).
        
        """
 
        if available_indexes and len(available_indexes) == 1:
            index_center = available_indexes[0]
            return [available_indexes], self.__pointer_data[index_center], 0.0
 
        local_data = self.__pointer_data
        if available_indexes:
            local_data = [self.__pointer_data[i] for i in available_indexes]
 
        local_centers = centers
        if centers is None:
            clusters, local_centers, local_wce = self.__search_optimial_parameters(local_data)
        else:
            kmeans_instance = kmeans(local_data, local_centers, tolerance=self.__tolerance, ccore=False, metric=self.__metric).process()
 
            local_wce = kmeans_instance.get_total_wce()
            local_centers = kmeans_instance.get_centers()
            clusters = kmeans_instance.get_clusters()
 
        if available_indexes:
            clusters = self.__local_to_global_clusters(clusters, available_indexes)
        
        return clusters, local_centers, local_wce
 
 
    def __local_to_global_clusters(self, local_clusters, available_indexes):
        """!
        @brief Converts clusters in local region define by 'available_indexes' to global clusters.
 
        @param[in] local_clusters (list): Local clusters in specific region.
        @param[in] available_indexes (list): Map between local and global point's indexes.
 
        @return Global clusters.
 
        """
 
        clusters = []
        for local_cluster in local_clusters:
            current_cluster = []
            for index_point in local_cluster:
                current_cluster.append(available_indexes[index_point])
 
            clusters.append(current_cluster)
 
        return clusters
 
    
    def __improve_structure(self, clusters, centers):
        """!
        @brief Check for best structure: divides each cluster into two and checks for best results using splitting criterion.
        
        @param[in] clusters (list): Clusters that have been allocated (each cluster contains indexes of points from data).
        @param[in] centers (list): Centers of clusters.
        
        @return (list) Allocated centers for clustering.
        
        """
 
        allocated_centers = []
        amount_free_centers = self.__kmax - len(centers)
 
        for index_cluster in range(len(clusters)):
            # solve k-means problem for children where data of parent are used.
            (parent_child_clusters, parent_child_centers, _) = self.__improve_parameters(None, clusters[index_cluster])
 
            # If it's possible to split current data
            if len(parent_child_clusters) > 1:
                # Calculate splitting criterion
                parent_scores = self.__splitting_criterion([clusters[index_cluster]], [centers[index_cluster]])
                child_scores = self.__splitting_criterion([parent_child_clusters[0], parent_child_clusters[1]], parent_child_centers)
              
                split_require = False
                
                # Reallocate number of centers (clusters) in line with scores
                if self.__criterion == splitting_type.BAYESIAN_INFORMATION_CRITERION:
                    if parent_scores < child_scores:
                        split_require = True
                    
                elif self.__criterion == splitting_type.MINIMUM_NOISELESS_DESCRIPTION_LENGTH:
                    # If its score for the split structure with two children is smaller than that for the parent structure, 
                    # then representing the data samples with two clusters is more accurate in comparison to a single parent cluster.
                    if parent_scores > child_scores:
                        split_require = True
                
                if (split_require is True) and (amount_free_centers > 0):
                    allocated_centers.append(parent_child_centers[0])
                    allocated_centers.append(parent_child_centers[1])
                    
                    amount_free_centers -= 1
                else:
                    allocated_centers.append(centers[index_cluster])
 
            else:
                allocated_centers.append(centers[index_cluster])
          
        return allocated_centers
     
     
    def __splitting_criterion(self, clusters, centers):
        """!
        @brief Calculates splitting criterion for input clusters.
        
        @param[in] clusters (list): Clusters for which splitting criterion should be calculated.
        @param[in] centers (list): Centers of the clusters.
        
        @return (double) Returns splitting criterion. High value of splitting criterion means that current structure is
                 much better.
 
        @see __bayesian_information_criterion(clusters, centers)
        @see __minimum_noiseless_description_length(clusters, centers)
        
        """
        
        if self.__criterion == splitting_type.BAYESIAN_INFORMATION_CRITERION:
            return self.__bayesian_information_criterion(clusters, centers)
        
        elif self.__criterion == splitting_type.MINIMUM_NOISELESS_DESCRIPTION_LENGTH:
            return self.__minimum_noiseless_description_length(clusters, centers)
 
        else:
            assert 0
 
 
    def __minimum_noiseless_description_length(self, clusters, centers):
        """!
        @brief Calculates splitting criterion for input clusters using minimum noiseless description length criterion.
        
        @param[in] clusters (list): Clusters for which splitting criterion should be calculated.
        @param[in] centers (list): Centers of the clusters.
        
        @return (double) Returns splitting criterion in line with bayesian information criterion. 
                Low value of splitting cretion means that current structure is much better.
        
        @see __bayesian_information_criterion(clusters, centers)
        
        """
        
        score = float('inf')
        
        W = 0.0
        K = len(clusters)
        N = 0.0
 
        sigma_square = 0.0
        
        alpha = self.__alpha
        alpha_square = alpha * alpha
        beta = self.__beta
 
        for index_cluster in range(0, len(clusters), 1):
            Ni = len(clusters[index_cluster])
            if Ni == 0:
                return float('inf')
            
            Wi = 0.0
            for index_object in clusters[index_cluster]:
                Wi += self.__metric(self.__pointer_data[index_object], centers[index_cluster])
            
            sigma_square += Wi
            W += Wi / Ni
            N += Ni
        
        if N - K > 0:
            sigma_square /= (N - K)
            sigma = sigma_square ** 0.5
            
            Kw = (1.0 - K / N) * sigma_square
            Ksa = (2.0 * alpha * sigma / (N ** 0.5)) * (alpha_square * sigma_square / N + W - Kw / 2.0) ** 0.5
            UQa = W - Kw + 2.0 * alpha_square * sigma_square / N + Ksa
 
            score = sigma_square * K / N + UQa + sigma_square * beta * ((2.0 * K) ** 0.5) / N
        
        return score
 
 
    def __bayesian_information_criterion(self, clusters, centers):
        """!
        @brief Calculates splitting criterion for input clusters using bayesian information criterion.
        
        @param[in] clusters (list): Clusters for which splitting criterion should be calculated.
        @param[in] centers (list): Centers of the clusters.
        
        @return (double) Splitting criterion in line with bayesian information criterion.
                High value of splitting criterion means that current structure is much better.
                
        @see __minimum_noiseless_description_length(clusters, centers)
        
        """
 
        scores = [float('inf')] * len(clusters)     # splitting criterion
        dimension = len(self.__pointer_data[0])
          
        # estimation of the noise variance in the data set
        sigma_sqrt = 0.0
        K = len(clusters)
        N = 0.0
          
        for index_cluster in range(0, len(clusters), 1):
            for index_object in clusters[index_cluster]:
                sigma_sqrt += self.__metric(self.__pointer_data[index_object], centers[index_cluster])
 
            N += len(clusters[index_cluster])
      
        if N - K > 0:
            sigma_sqrt /= (N - K)
            p = (K - 1) + dimension * K + 1
 
            # in case of the same points, sigma_sqrt can be zero (issue: #407)
            sigma_multiplier = 0.0
            if sigma_sqrt <= 0.0:
                sigma_multiplier = float('-inf')
            else:
                sigma_multiplier = dimension * 0.5 * log(sigma_sqrt)
            
            # splitting criterion    
            for index_cluster in range(0, len(clusters), 1):
                n = len(clusters[index_cluster])
 
                L = n * log(n) - n * log(N) - n * 0.5 * log(2.0 * numpy.pi) - n * sigma_multiplier - (n - K) * 0.5
                
                # BIC calculation
                scores[index_cluster] = L - p * 0.5 * log(N)
                
        return sum(scores)
 
 
    def __verify_arguments(self):
        """!
        @brief Verify input parameters for the algorithm and throw exception in case of incorrectness.
 
        """
        if len(self.__pointer_data) == 0:
            raise ValueError("Input data is empty (size: '%d')." % len(self.__pointer_data))
 
        if len(self.__centers) == 0:
            raise ValueError("Initial centers are empty (size: '%d')." % len(self.__pointer_data))
 
        if self.__tolerance < 0:
            raise ValueError("Tolerance (current value: '%d') should be greater or equal to 0." %
                             self.__tolerance)
 
        if self.__repeat <= 0:
            raise ValueError("Repeat (current value: '%d') should be greater than 0." %
                             self.__repeat)
 
        if self.__alpha < 0.0 or self.__alpha > 1.0:
            raise ValueError("Parameter for the probabilistic bound Q(alpha) should in the following range [0, 1] "
                             "(current value: '%f')." % self.__alpha)
 
        if self.__beta < 0.0 or self.__beta > 1.0:
            raise ValueError("Parameter for the probabilistic bound Q(beta) should in the following range [0, 1] "
                             "(current value: '%f')." % self.__beta)