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cluster_stats_new.py
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cluster_stats_new.py
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import numpy as np
from numpy.linalg import inv, slogdet
from scipy.special import gammaln
from scipy.stats import invwishart
from scipy.stats import norm
from scipy.stats import invgamma
from scipy.stats import multivariate_t
# class for statistics of the Gaussian clusters with general covariance matrix
class gaussianClusters(object):
"""
A class to represent and manage the statistics of Gaussian clusters with a general covariance matrix.
Attributes:
X (np.ndarray): Data points, a N x D array.
prior: Prior parameters for the Gaussian clusters.
alpha (float): Alpha parameter for the Dirichlet Process.
K_max (int): Maximum number of clusters.
assignments (np.ndarray): Initial assignments of clusters for each data point.
sum_X (np.ndarray): Sum of data points for each cluster, a K x D array.
outer_prod_X (np.ndarray): Outer product of data points for each cluster, a K x D x D array.
counts (np.ndarray): Number of data points in each cluster, a K-dimensional vector.
log_det_covariances (np.ndarray): Log determinant of covariance matrices for each cluster, a K-dimensional vector.
inv_covariances (np.ndarray): Inverse of covariance matrices for each cluster, a K x D x D array.
K (int): Current number of clusters.
"""
def __init__(self, X: np.ndarray, prior, alpha: float, K: int, assignments: np.ndarray = None):
"""
Initializes the GaussianClusters object with the given data and parameters.
Args:
X (np.ndarray): Data points, a N x D array.
prior: Prior parameters for the Gaussian clusters.
alpha (float): Alpha parameter for the Dirichlet Process.
K (int): Maximum number of clusters.
assignments (np.ndarray, optional): Initial assignments of clusters for each data point.
"""
self.X = X
self.N, self.D = X.shape
self.prior = prior
self.K_max = K
self.alpha = alpha
self.sum_X = np.zeros((self.K_max, self.D), float)
self.outer_prod_X = np.zeros((self.K_max, self.D, self.D), float)
self.counts = np.zeros(self.K_max, int)
self.log_det_covariances = np.zeros(self.K_max)
self.inv_covariances = np.zeros((self.K_max, self.D, self.D))
self._cache()
self.K = 0
if assignments is None:
self.assignments = -1 * np.ones(self.N, int)
else:
self.assignments = assignments
for k in range(self.assignments.max() + 1):
for i in np.where(self.assignments == k)[0]:
self.add_assignment(i, k)
def _cache(self):
"""
Caches various precomputed values to optimize computations.
"""
self._cache_outer_X = np.zeros((self.N, self.D, self.D))
for i in range(self.N):
self._cache_outer_X[i, :, :] = np.outer(self.X[i], self.X[i])
self._cache_prior_outer_m_0 = np.outer(self.prior.m_0, self.prior.m_0)
Ns = np.concatenate([[1], np.arange(1, self.prior.v_0 + 2 * self.N + 4)])
self._cache_gammaln_by_2 = gammaln(Ns / 2.0)
self._cache_log_pi = np.log(np.pi)
Ks = self.prior.k_0 + np.arange(0, 2 * self.N + 1)
self._cache_log_Ks = np.log(Ks)
self._cache_log_Vs = np.log(Ns)
self._cache_gammaln_alpha = gammaln(self.alpha)
self._cache_prod_k0m0 = self.prior.k_0 * self.prior.m_0
self._cache_partial_S_sum = self.prior.S_0 + self.prior.k_0 * np.outer(self.prior.m_0, self.prior.m_0)
covar_prior = (self.prior.k_0 + 1.0) / (self.prior.k_0 * (self.prior.v_0 - self.D + 1.0)) * self.prior.S_0
self._cache_inv_covariance_prior = inv(covar_prior)
self._cache_logdet_covariance_prior = slogdet(covar_prior)[1]
self._cache_post_pred_prior_coeff = (
self._cache_gammaln_by_2[self.prior.v_0 + 1] - self._cache_gammaln_by_2[self.prior.v_0 - self.D + 1]
- self.D / 2.0 * (self._cache_log_Vs[self.prior.v_0 - self.D + 1] + self._cache_log_pi)
- 0.5 * self._cache_logdet_covariance_prior
)
def cache_cluster_stats(self, k: int):
"""
Caches the statistics for the k-th cluster.
Args:
k (int): Index of the cluster.
Returns:
tuple: Cached statistics (log_det_covariance, inv_covariance, count, sum_X, outer_prod_X).
"""
return (
self.log_det_covariances[k].copy(),
self.inv_covariances[k].copy(),
self.counts[k].copy(),
self.sum_X[k].copy(),
self.outer_prod_X[k].copy()
)
def restore_cluster_stats(self, k: int, log_det_covariance: float, inv_covariance: np.ndarray, count: int, sum_X: np.ndarray, outer_prod_X: np.ndarray):
"""
Restores the cached statistics for the k-th cluster.
Args:
k (int): Index of the cluster.
log_det_covariance (float): Log determinant of the covariance matrix.
inv_covariance (np.ndarray): Inverse of the covariance matrix.
count (int): Number of data points in the cluster.
sum_X (np.ndarray): Sum of data points in the cluster.
outer_prod_X (np.ndarray): Outer product of data points in the cluster.
"""
self.log_det_covariances[k] = log_det_covariance
self.inv_covariances[k, :, :] = inv_covariance
self.counts[k] = count
self.sum_X[k] = sum_X
self.outer_prod_X[k] = outer_prod_X
def add_assignment(self, i: int, k: int):
"""
Assigns the i-th data point to the k-th cluster and updates the cluster statistics.
Args:
i (int): Index of the data point.
k (int): Index of the cluster.
"""
if k == self.K:
self.K += 1
self.sum_X[k, :] = np.zeros(self.D)
self.outer_prod_X[k, :, :] = np.zeros((self.D, self.D))
self.assignments[i] = k
self.sum_X[k, :] += self.X[i]
self.outer_prod_X[k, :, :] += self._cache_outer_X[i]
self.counts[k] += 1
self._update_log_det_covariance_and_inv_covariance(k)
def del_assignment(self, i: int):
"""
Deletes the assignment of the i-th data point and updates the cluster statistics.
Args:
i (int): Index of the data point.
"""
k = self.assignments[i]
if k != -1:
self.assignments[i] = -1
self.counts[k] -= 1
if self.counts[k] == 0:
self.empty_cluster(k)
else:
self.sum_X[k, :] -= self.X[i]
self.outer_prod_X[k, :, :] -= self._cache_outer_X[i]
self._update_log_det_covariance_and_inv_covariance(k)
def empty_cluster(self, k: int):
"""
Empties the k-th cluster by removing it and updating the remaining clusters.
Args:
k (int): Index of the cluster to be emptied.
"""
self.K -= 1
if k != self.K:
self.sum_X[k, :] = self.sum_X[self.K, :]
self.outer_prod_X[k, :, :] = self.outer_prod_X[self.K, :, :]
self.counts[k] = self.counts[self.K]
self.log_det_covariances[k] = self.log_det_covariances[self.K]
self.inv_covariances[k, :, :] = self.inv_covariances[self.K, :, :]
self.assignments[np.where(self.assignments == self.K)] = k
self._update_log_det_covariance_and_inv_covariance_priors(self.K)
self.counts[self.K] = 0
self.sum_X[self.K, :] = np.zeros(self.D)
self.outer_prod_X[self.K, :, :] = np.zeros((self.D, self.D))
def log_post_pred_prior(self, i: int):
"""
Computes the log posterior predictive probability of the i-th data point under the prior alone.
Args:
i (int): Index of the data point.
Returns:
float: Log posterior predictive probability.
"""
return self._multivariate_students_t_prior(i)
def log_post_pred(self, i: int):
"""
Computes the log posterior predictive probabilities for the i-th data point.
Equation 22 in the thesis.
Args:
i (int): Index of the data point.
Returns:
np.ndarray: Log posterior predictive probabilities for each cluster.
"""
k_Ns = self.prior.k_0 + self.counts[:self.K]
v_Ns = self.prior.v_0 + self.counts[:self.K]
m_Ns = (self.sum_X[:self.K] + self._cache_prod_k0m0) / k_Ns[:, np.newaxis]
Vs = v_Ns - self.D + 1
deltas = m_Ns - self.X[i]
mahalonabis_dist = np.zeros(self.K)
for k in range(self.K):
mahalonabis_dist[k] = np.matmul(np.matmul(deltas[k], self.inv_covariances[k]), deltas[k])
prob = np.zeros(self.K_max)
prob[:self.K] = (
self._cache_gammaln_by_2[Vs + self.D] - self._cache_gammaln_by_2[Vs]
- (self.D / 2) * (self._cache_log_Vs[Vs] + self._cache_log_pi)
- 0.5 * self.log_det_covariances[:self.K]
- 0.5 * (Vs + self.D) * np.log(1 + mahalonabis_dist / Vs)
)
prob[self.K:] = self.log_post_pred_prior(i)
return prob
def get_post_S_N(self, k: int):
"""
Returns the posterior hyperparameters for the k-th cluster.
Equation 19 in the thesis.
Args:
k (int): Index of the cluster.
Returns:
np.ndarray: Posterior hyperparameters S_N.
"""
k_N = self.prior.k_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
S_N = self._cache_partial_S_sum + self.outer_prod_X[k] - k_N * np.outer(m_N, m_N)
return S_N
def get_post_posterior_S_N(self, k: int):
"""
Returns the posterior posterior hyperparameters for the k-th cluster.
Args:
k (int): Index of the cluster.
Returns:
np.ndarray: Posterior posterior hyperparameters S_N.
"""
k_N = self.prior.k_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
post_k_N = k_N + self.counts[k]
post_m_N = (self.sum_X[k] + m_N * k_N) / post_k_N
post_S_N = (
self.prior.S_0 + self._cache_prior_outer_m_0 + 2 * self.outer_prod_X[k]
- post_k_N * np.outer(post_m_N, post_m_N)
)
return post_S_N
def get_posterior_probability_Z_k(self, k: int):
"""
Returns the posterior probability for the k-th cluster.
Equation 20 in the thesis.
Args:
k (int): Index of the cluster.
Returns:
float: Posterior probability.
"""
if k >= self.K:
return gammaln(self.alpha / self.K_max)
else:
v_N = self.prior.v_0 + self.counts[k]
post_v_N = self.prior.v_0 + 2 * self.counts[k]
S_N = self.get_post_S_N(k)
post_S_N = self.get_post_posterior_S_N(k)
log_post_Z = (
-1 * self.counts[k] * (self.D / 2) * self._cache_log_pi
+ self._cache_gammaln_by_2[post_v_N] - self._cache_gammaln_by_2[v_N]
+ (v_N / 2) * np.log(slogdet(S_N)[1]) - (post_v_N / 2) * np.log(slogdet(post_S_N)[1])
- (self.D / 2) * (self._cache_log_Ks[self.counts[k]] - self._cache_log_Ks[2 * self.counts[k]])
+ gammaln(self.alpha / self.K_max + self.counts[k])
)
return log_post_Z
def get_log_marginal(self, k: int):
"""
Returns the log marginal likelihood for the k-th cluster.
Equation 20 in the thesis.
Args:
k (int): Index of the cluster.
Returns:
float: Log marginal likelihood.
"""
k_N, v_N, m_N, S_N = self.get_post_hyperparameters(k)
gammas = [
gammaln((v_N + 1 - i) / 2) - gammaln((self.prior.v_0 + 1 - i) / 2)
for i in range(1, self.D + 1)
]
return (
(-1 * self.counts[k] * self.D / 2) * self._cache_log_pi
+ (self.D / 2) * (np.log(self.prior.k_0) - np.log(self.k_N))
+ self.prior.v_0 * np.log(slogdet(self.prior.S_0[k])[1])
- v_N * np.log(slogdet(S_N[k])[1])
+ gammas
)
def random_cluster_params(self, k: int):
"""
Returns random mean vector and covariance matrix from the posterior NIW distribution for cluster k.
Equation 19 in the thesis.
Args:
k (int): Index of the cluster.
Returns:
tuple: Random mean vector (np.ndarray) and covariance matrix (np.ndarray).
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
S_N = self.S_N_partials[k] - k_N * np.outer(m_N, m_N)
sigma = invwishart.rvs(df=v_N, scale=S_N)
if self.D == 1:
mu = np.random.normal(m_N, sigma / k_N)
else:
mu = np.random.multivariate_normal(m_N, sigma / k_N)
return mu, sigma
def map_cluster_params(self, k: int):
"""
Returns the MAP estimates of the mean vector and covariance matrix for cluster k.
Args:
k (int): Index of the cluster.
Returns:
tuple: Mean vector (np.ndarray) and covariance matrix (np.ndarray).
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
sigma = (self._cache_partial_S_sum + self.outer_prod_X[k] - k_N * np.outer(m_N, m_N)) / (v_N + self.D + 2)
return m_N, sigma
def _multivariate_students_t_prior(self, i: int):
"""
Computes the log of the multivariate Student's t-distribution prior probability for the i-th data point.
Args:
i (int): Index of the data point.
Returns:
float: Log probability under the multivariate Student's t-distribution prior.
"""
mu = self.prior.m_0
inv_covariance = self._cache_inv_covariance_prior
v = self.prior.v_0 - self.D + 1
delta = self.X[i, :] - mu
return (
self._cache_post_pred_prior_coeff
- (v + self.D) / 2.0 * np.log(1 + 1.0 / v * np.dot(np.dot(delta, inv_covariance), delta))
)
def _update_log_det_covariance_and_inv_covariance_priors(self, k: int):
"""
Updates the log determinant and inverse covariance matrix for the prior of the k-th cluster.
Args:
k (int): Index of the cluster.
"""
self.log_det_covariances[k] = self._cache_logdet_covariance_prior
self.inv_covariances[k, :, :] = self._cache_inv_covariance_prior
def _update_log_det_covariance_and_inv_covariance(self, k: int):
"""
Updates the log determinant and inverse covariance matrix for the k-th cluster.
Refer to Equation 20 in the thesis.
Args:
k (int): Index of the cluster.
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
covar = (
(k_N + 1.0) / (k_N * (v_N - self.D + 1.0))
* (self._cache_partial_S_sum + self.outer_prod_X[k] - k_N * np.outer(m_N, m_N))
)
self.log_det_covariances[k] = slogdet(covar)[1]
self.inv_covariances[k, :, :] = inv(covar)
# class for statistics of the Gaussian clusters with diagonal covariance matrix
class gaussianClustersDiag:
"""
Class for managing Gaussian clusters with diagonal covariance matrices.
Attributes:
X (np.ndarray): Data matrix of shape (N, D).
prior (object): Prior distribution parameters.
alpha (float): Dirichlet process concentration parameter.
K_max (int): Maximum number of clusters.
assignments (np.ndarray): Initial assignments of clusters.
sum_X (np.ndarray): Sum of data points for each cluster.
square_prod_X (np.ndarray): Sum of squares of data points for each cluster.
counts (np.ndarray): Number of data points in each cluster.
log_det_covariances (np.ndarray): Log determinants of covariance matrices.
inv_covariances (np.ndarray): Inverse covariance matrices for each cluster.
"""
def __init__(self, X, prior, alpha, K_max, assignments=None):
"""
Initialize a gaussianClustersDiag object.
Args:
X (np.ndarray): Data matrix of shape (N, D).
prior (object): Prior distribution parameters.
alpha (float): Dirichlet process concentration parameter.
K_max (int): Maximum number of clusters.
assignments (np.ndarray, optional): Initial assignments of clusters. Defaults to None.
"""
self.X = X
self.N, self.D = X.shape
self.prior = prior
self.alpha = alpha
self.K_max = K_max
# Initialize arrays for cluster statistics
self.sum_X = np.zeros((self.K_max, self.D), dtype=float)
self.square_prod_X = np.zeros((self.K_max, self.D), dtype=float)
self.counts = np.zeros(self.K_max, dtype=int)
# Initialize arrays for hyperparameters
self.log_det_covariances = np.zeros(self.K_max, dtype=float)
self.inv_covariances = np.zeros((self.K_max, self.D), dtype=float)
# Cache some values for efficiency
self._cache()
# Initialize number of clusters
self.K = 0
# Assign initial cluster assignments
self.assignments = assignments
if assignments is not None:
for k in range(self.assignments.max() + 1):
for i in np.where(self.assignments == k)[0]:
self.add_assignment(i, k)
def _cache(self):
"""
Pre-compute and cache values to avoid redundant computations.
"""
self._cache_square_X = np.square(self.X)
self._cache_prior_square_m_0 = np.square(self.prior.m_0)
Ns = np.concatenate([[1], np.arange(1, self.prior.v_0 + 2 * self.N + 4)])
self._cache_gammaln_by_2 = gammaln(Ns / 2.0)
self._cache_log_pi = np.log(np.pi)
self._cache_log_Vs = np.log(Ns)
self._cache_gammaln_alpha = gammaln(self.alpha)
self._cache_prod_k0m0 = self.prior.k_0 * self.prior.m_0
self._cache_partial_S_sum = self.prior.S_0 + self.prior.k_0 * np.square(self.prior.m_0)
var = self.prior.S_0 * (self.prior.k_0 + 1.0) / (self.prior.k_0 * self.prior.v_0)
self._cache_inv_var_prior = 1.0 / var
self._cache_log_var_prod_prior = 0.5 * np.log(var).sum()
self._cache_post_pred_coeff_prior = (
self.D
* (
self._cache_gammaln_by_2[self.prior.v_0 + 1]
- self._cache_gammaln_by_2[self.prior.v_0]
- 0.5 * self._cache_log_Vs[self.prior.v_0]
- 0.5 * self._cache_log_pi
)
- self._cache_log_var_prod_prior
)
def cache_cluster_stats(self, k):
"""
Cache statistics of cluster k.
Args:
k (int): Cluster index.
Returns:
tuple: Cached statistics of the cluster (log_det_covariance, inv_covariance, count, sum_X, square_prod_X).
"""
return (
self.log_det_covariances[k].copy(),
self.inv_covariances[k].copy(),
self.counts[k].copy(),
self.sum_X[k].copy(),
self.square_prod_X[k].copy()
)
def restore_cluster_stats(self, k, log_det_covariance, inv_covariance, count, sum_X, square_prod_X):
"""
Restore cached statistics for cluster k.
Args:
k (int): Cluster index.
log_det_covariance (float): Log determinant of covariance.
inv_covariance (np.ndarray): Inverse covariance matrix.
count (int): Number of data points in the cluster.
sum_X (np.ndarray): Sum of data points in the cluster.
square_prod_X (np.ndarray): Sum of squares of data points in the cluster.
"""
self.log_det_covariances[k] = log_det_covariance
self.inv_covariances[k] = inv_covariance
self.counts[k] = count
self.sum_X[k] = sum_X
self.square_prod_X[k] = square_prod_X
def add_assignment(self, i, k):
"""
Add assignment of data point i to cluster k.
Args:
i (int): Data point index.
k (int): Cluster index.
"""
if k == self.K:
self.K += 1
self.sum_X[k, :] = np.zeros(self.D)
self.square_prod_X[k, :] = np.zeros(self.D)
self.assignments[i] = k
self.sum_X[k, :] += self.X[i]
self.square_prod_X[k, :] += self._cache_square_X[i]
self.counts[k] += 1
self._update_log_det_covariance_and_inv_covariance(k)
def del_assignment(self, i):
"""
Delete assignment of data point i from its cluster.
Args:
i (int): Data point index.
"""
k = self.assignments[i]
if k != -1:
self.assignments[i] = -1
self.counts[k] -= 1
if self.counts[k] == 0:
self.empty_cluster(k)
else:
self.sum_X[k, :] -= self.X[i]
self.square_prod_X[k, :] -= self._cache_square_X[i]
self._update_log_det_covariance_and_inv_covariance(k)
def empty_cluster(self, k):
"""
Remove an empty cluster and adjust the cluster statistics.
Args:
k (int): Cluster index to be removed.
"""
self.K -= 1
if k != self.K:
self.sum_X[k, :] = self.sum_X[self.K, :]
self.square_prod_X[k, :] = self.square_prod_X[self.K, :]
self.counts[k] = self.counts[self.K]
self.log_det_covariances[k] = self.log_det_covariances[self.K]
self.inv_covariances[k, :] = self.inv_covariances[self.K, :]
self.assignments[np.where(self.assignments == self.K)] = k
self.counts[self.K] = 0
self._update_log_det_covariance_and_inv_covariance_priors(self.K)
self.sum_X[self.K, :] = np.zeros(self.D)
self.square_prod_X[self.K, :] = np.zeros(self.D)
def log_post_pred(self, i):
"""
Compute log posterior predictive probability for data point i.
Refer to equation 32 in the thesis.
Args:
i (int): Data point index.
Returns:
np.ndarray: Log posterior predictive probabilities for all clusters.
"""
k_Ns = self.prior.k_0 + self.counts[:self.K]
v_Ns = self.prior.v_0 + self.counts[:self.K]
m_Ns = (self.sum_X[:self.K] + self._cache_prod_k0m0) / k_Ns[:, np.newaxis]
deltas = m_Ns - self.X[i]
res = np.zeros(self.K_max)
res[:self.K] = self.D * (
self._cache_gammaln_by_2[v_Ns + 1] - self._cache_gammaln_by_2[v_Ns]
- 0.5 * self._cache_log_Vs[v_Ns] - 0.5 * self._cache_log_pi
) - 0.5 * self.log_det_covariances[:self.K] - (v_Ns + 1) / 2.0 * (
np.log(1 + np.square(deltas) * self.inv_covariances[:self.K] * (1.0 / v_Ns[:, np.newaxis]))
).sum(axis=1)
res[self.K:] = self._students_t_prior(i)
return res
def _update_log_det_covariance_and_inv_covariance_priors(self, k):
"""
Update log determinant of covariance and inverse covariance matrix for prior.
Args:
k (int): Cluster index.
"""
self.log_det_covariances[k] = self._cache_log_var_prod_prior
self.inv_covariances[k, :] = self._cache_inv_var_prior
def _update_log_det_covariance_and_inv_covariance(self, k):
"""
Update log determinant of covariance and inverse covariance matrix for cluster k.
Refer to equation 30 in the thesis.
Args:
k (int): Cluster index.
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
var = (k_N + 1) / (k_N * v_N) * (self._cache_partial_S_sum + self.square_prod_X[k] - k_N * np.square(m_N))
self.log_det_covariances[k] = np.log(var).sum()
self.inv_covariances[k, :] = 1.0 / var
def get_post_hyperparameters(self, k):
"""
Get posterior hyperparameters for cluster k.
Refer to equation 30 in the thesis.
Args:
k (int): Cluster index.
Returns:
tuple: Posterior hyperparameters (k_N, v_N, m_N, S_N).
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
S_N = self._cache_partial_S_sum + self.square_prod_X[k] - k_N * np.square(m_N)
return k_N, v_N, m_N, S_N
def get_post_posterior_hyperparameters(self, k):
"""
Get posterior predictive hyperparameters for cluster k.
Args:
k (int): Cluster index.
Returns:
tuple: Posterior predictive hyperparameters (post_k_N, post_v_N, post_m_N, post_S_N).
"""
k_N = self.prior.k_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
post_k_N = k_N + self.counts[k]
post_v_N = self.prior.v_0 + 2 * self.counts[k]
post_m_N = (self.sum_X[k] + m_N * k_N) / post_k_N
post_S_N = self.prior.S_0 + self._cache_prior_square_m_0 + 2 * self.square_prod_X[k] - post_k_N * np.square(post_m_N)
return post_k_N, post_v_N, post_m_N, post_S_N
def get_posterior_probability_Z_k(self, k):
"""
Compute logarithm of posterior probability of cluster k.
Refer to equation 31 in the thesis.
Args:
k (int): Cluster index.
Returns:
float: Logarithm of posterior probability of cluster k.
"""
if k >= self.K:
return gammaln(self.alpha / self.K_max)
else:
k_N, v_N, m_N, S_N = self.prior.k_0, self.prior.v_0, self.prior.m_0, self.prior.S_0
post_k_N, post_v_N, post_m_N, post_S_N = self.get_post_hyperparameters(k)
log_post_Z = (
self.D * (
(-1.0 * self.counts[k] / 2) * self._cache_log_pi
+ self._cache_gammaln_by_2[post_v_N]
- self._cache_gammaln_by_2[v_N]
+ (1.0 / 2) * (np.log(k_N) - np.log(post_k_N))
)
+ np.sum((v_N / 2) * np.log(S_N) - (post_v_N / 2) * np.log(post_S_N))
+ gammaln(self.alpha / self.K_max + self.counts[k])
)
return log_post_Z
def random_cluster_params(self, k):
"""
Generate random mean vector and covariance matrix from the posterior NIW distribution for cluster k.
Refer to equation 31 in the thesis.
Args:
k (int): Cluster index.
Returns:
tuple: Random mean vector and covariance matrix (mu, sigma).
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
S_N = self._cache_partial_S_sum[k] - k_N * np.outer(m_N, m_N)
sigma = invwishart.rvs(df=v_N, scale=S_N)
if self.D == 1:
mu = np.random.normal(m_N, sigma / k_N)
else:
mu = np.random.multivariate_normal(m_N, sigma / k_N)
return mu, sigma
def map_cluster_params(self, k):
"""
Compute MAP estimates of cluster's mean vector and covariance matrix.
Args:
k (int): Cluster index.
Returns:
tuple: MAP estimates of mean vector and covariance matrix (m_N, sigma).
"""
k_N = self.prior.k_0 + self.counts[k]
v_N = self.prior.v_0 + self.counts[k]
m_N = (self.sum_X[k] + self._cache_prod_k0m0) / k_N
sigma = (self._cache_partial_S_sum + self.outer_prod_X[k] - k_N * np.outer(m_N, m_N)) / (v_N + self.D + 2)
return m_N, sigma
def _students_t_prior(self, i):
"""
Compute log prior probability of data point i under Student's t-distribution prior.
Args:
i (int): Data point index.
Returns:
float: Log prior probability of data point i.
"""
inv_var = self._cache_inv_var_prior
v = self.prior.v_0
mu = self.prior.m_0
delta = self.X[i, :] - mu
return (
self._cache_post_pred_coeff_prior
- ((v + 1.0) / 2.0 * (np.log(1.0 + 1.0 / v * np.square(delta) * inv_var)).sum())
)
# class for statistics of the categorical clusters
class categoricalClusters(object):
"""
Class for handling categorical data clustering using a specified prior.
Attributes:
N (int): Number of data points.
D (int): Number of dimensions (attributes) in the categorical data.
alpha (float): Dirichlet process parameter.
C (numpy.ndarray): Categorical data matrix of shape (N, D).
gamma (float): Hyperparameter for the Dirichlet distribution.
K_max (int): Maximum number of clusters including the empty ones.
Ms (numpy.ndarray): Array containing unique counts of categories for each dimension.
counts (numpy.ndarray): Array to store counts of data points assigned to each cluster.
catCounts (numpy.ndarray): Array to store categorical counts for each cluster and dimension.
assignments (numpy.ndarray): Initial assignments of data points to clusters.
K (int): Current number of clusters.
"""
def __init__(self, C, alpha, gamma, K, assignments=None):
"""
Initialize categorical clustering instance.
Args:
C (numpy.ndarray): Categorical data matrix of shape (N, D).
alpha (float): Dirichlet process parameter.
gamma (float): Hyperparameter for the Dirichlet distribution.
K (int): Maximum number of clusters including the empty ones.
assignments (numpy.ndarray or None): Initial assignments of data points to clusters.
If None, no initial assignments are made.
"""
self.N, self.D = C.shape
self.alpha = alpha
self.C = C
self.gamma = gamma
self.K_max = K
self.Ms = np.zeros(self.D, int)
for d in range(self.D):
self.Ms[d] = len(set(C[:, d]))
self.counts = np.zeros(self.K_max, int)
self.catCounts = np.zeros((self.K_max, self.Ms.max(), self.D), int)
self._cache()
self.K = 0
if assignments is None:
self.assignments = -1 * np.ones(self.N, int)
else:
self.assignments = assignments
for k in range(self.assignments.max() + 1):
for i in np.where(self.assignments == k)[0]:
self.add_assignment(i, k)
def _cache(self):
"""
Cache some precomputed values for efficiency.
"""
self._cache_log_pi = np.log(np.pi)
self._cache_gammaln_alpha = gammaln(self.alpha)
def cache_cluster_stats(self, k):
"""
Cache cluster k's statistics.
Args:
k (int): Cluster index.
Returns:
tuple: Cached cluster statistics (counts, catCounts).
"""
return (
self.counts[k].copy(),
self.catCounts[k].copy()
)
def restore_cluster_stats(self, k, count_N, catCount_N):
"""
Restore cluster k's statistics.
Args:
k (int): Cluster index.
count_N (numpy.ndarray): New counts for cluster k.
catCount_N (numpy.ndarray): New categorical counts for cluster k.
"""
self.counts[k] = count_N
self.catCounts[k] = catCount_N
def add_assignment(self, i, k):
"""
Assign data point i to cluster k.
Args:
i (int): Data point index.
k (int): Cluster index.
"""
if k == self.K:
self.K += 1
self.assignments[i] = k
for d in range(self.D):
self.catCounts[k][self.C[i][d]][d] += 1
self.counts[k] += 1
def del_assignment(self, i):
"""
Delete assignment of data point i.
Args:
i (int): Data point index.
"""
k = self.assignments[i]
if k != -1:
self.assignments[i] = -1
self.counts[k] -= 1
for d in range(self.D):
self.catCounts[k][self.C[i][d]][d] -= 1
if self.counts[k] == 0:
self.empty_cluster(k)
def empty_cluster(self, k):
"""
Empty out cluster k.
Args:
k (int): Cluster index.
"""
self.K -= 1
if k != self.K:
self.counts[k] = self.counts[self.K]
self.assignments[np.where(self.assignments == self.K)] = k
self.catCounts[k, :] = self.catCounts[self.K, :]
self.counts[self.K] = 0
self.catCounts[self.K, :] = np.zeros((self.Ms.max(), self.D), int)
def log_post_pred(self, i):
"""
Compute log posterior predictive probability for data point i.
Refer to equation 38 in the thesis.
Args:
i (int): Data point index.
Returns:
numpy.ndarray: Array of log posterior predictive probabilities for each cluster.
"""
gamma_Ns = self.gamma + self.catCounts
res = np.zeros((self.K_max, self.D))
for d in range(self.D):
res[:, d] = np.log(gamma_Ns[:, self.C[i][d], d]) - np.log(self.counts + self.gamma * self.Ms[d])
return res.sum(axis=1)
def get_posterior_probability_Z_k(self, k):
"""
Compute log posterior probability of cluster k.
Refer to equation 37 in the thesis.
Args:
k (int): Cluster index.
Returns:
float: Log posterior probability of cluster k.
"""
gamma_N = self.gamma + self.catCounts[k]
post_gamma_N = gamma_N + self.catCounts[k]
log_post_Z = np.zeros((self.D))
for d in range(self.D):
log_post_Z[d] = gammaln(gamma_N[:, d].sum()) + gammaln(post_gamma_N[:, d]).sum() \
- gammaln(gamma_N[:, d]).sum() - gammaln(self.counts[k] + gamma_N[:, d].sum())
return log_post_Z.sum() + gammaln(self.alpha / self.K_max + self.counts[k])
# return 0
if k >= self.K:
return gammaln(self.alpha/self.K_max)
else:
gamma_N = self.gamma + self.catCounts[k]
log_post_Z = np.zeros((self.D))
for d in range(self.D):
# log_post_Z[d] = gammaln(self.gamma * self.Ms[d]) + gammaln(gamma_N[:, d]).sum() - self.Ms[d] * gammaln(self.gamma) - gammaln(self.gamma * self.Ms[d] + self.counts[k])
log_post_Z[d] = gammaln(gamma_N[:, d]).sum() - gammaln(self.gamma * self.Ms[d] + self.counts[k])
return log_post_Z.sum()
# def random_cluster_params(self, k):
# return 1
# def map_cluster_params(self, k):
# return 1
# class for statistics of the clusters
# class for the statistics of the mixed data with categorical and Gaussian features combined
class categoricalGaussianClusters(object):
"""
Class for managing clusters of mixed categorical and Gaussian data.
This class handles the statistics and operations for clusters containing
both categorical and Gaussian features. It manages cluster assignments,
updates cluster statistics, and computes various probabilities.
Attributes:
X (numpy.ndarray): Input data matrix for Gaussian features.
C (numpy.ndarray): Input data matrix for categorical features.
N (int): Number of data points.
gD (int): Number of Gaussian features.
cD (int): Number of categorical features.
gamma (float): Concentration prior hyperparameter for categorical distribution.
prior (object): Prior hyperparameter for Gaussian features.
K_max (int): Maximum number of clusters.
alpha (float): Concentration prior hyperparameter for the Dirichlet process.
Ms (numpy.ndarray): Number of categories for each categorical feature.
sum_X (numpy.ndarray): Sum of Gaussian data for each cluster.
square_prod_X (numpy.ndarray): Sum of squared Gaussian data for each cluster.