# Source code for pyod.utils.stat_models

```# -*- coding: utf-8 -*-
""" A collection of statistical models
"""
# Author: Yue Zhao <zhaoy@cmu.edu>

from __future__ import division
from __future__ import print_function

import numpy as np
from numba import njit
from scipy.stats import pearsonr
from sklearn.utils.validation import check_array
# noinspection PyProtectedMember
from sklearn.utils.validation import check_consistent_length

# TODO: disable p value calculation due to python 2.7 break
# from scipy.special import betainc

"""Utility function to calculate row-wise euclidean distance of two matrix.
Different from pair-wise calculation, this function would not broadcast.

For instance, X and Y are both (4,3) matrices, the function would return
a distance vector with shape (4,), instead of (4,4).

Parameters
----------
X : array of shape (n_samples, n_features)
First input samples

Y : array of shape (n_samples, n_features)
Second input samples

Returns
-------
distance : array of shape (n_samples,)
Row-wise euclidean distance of X and Y
"""
X = check_array(X)
Y = check_array(Y)

if X.shape != Y.shape or X.shape != Y.shape:
"matrix with different shapes {0} and {1}".format(
X.shape, Y.shape))

@njit
def _pairwise_distances_no_broadcast_helper(X, Y):  # pragma: no cover
"""Internal function for calculating the distance with numba. Do not use.

Parameters
----------
X : array of shape (n_samples, n_features)
First input samples

Y : array of shape (n_samples, n_features)
Second input samples

Returns
-------
distance : array of shape (n_samples,)
Intermediate results. Do not use.

"""
euclidean_sq = np.square(Y - X)
return np.sqrt(np.sum(euclidean_sq, axis=1)).ravel()

[docs]def wpearsonr(x, y, w=None):
"""Utility function to calculate the weighted Pearson correlation of two
samples.

See https://stats.stackexchange.com/questions/221246/such-thing-as-a-weighted-correlation

Parameters
----------
x : array, shape (n,)
Input x.

y : array, shape (n,)
Input y.

w : array, shape (n,)
Weights w.

Returns
-------
scores : float in range of [-1,1]
Weighted Pearson Correlation between x and y.

"""

# unweighted version
# note the return is different
# TODO: fix output differences
if w is None:
return pearsonr(x, y)

x = np.asarray(x)
y = np.asarray(y)
w = np.asarray(w)

check_consistent_length([x, y, w])
# n = len(x)

w_sum = w.sum()
mx = np.sum(x * w) / w_sum
my = np.sum(y * w) / w_sum

xm, ym = (x - mx), (y - my)

r_num = np.sum(xm * ym * w) / w_sum

xm2 = np.sum(xm * xm * w) / w_sum
ym2 = np.sum(ym * ym * w) / w_sum

r_den = np.sqrt(xm2 * ym2)
r = r_num / r_den

r = max(min(r, 1.0), -1.0)

# TODO: disable p value calculation due to python 2.7 break
#    df = n_train_ - 2
#
#    if abs(r) == 1.0:
#        prob = 0.0
#    else:
#        t_squared = r ** 2 * (df / ((1.0 - r) * (1.0 + r)))
#        prob = _betai(0.5 * df, 0.5, df / (df + t_squared))
return r  # , prob

#####################################
#      PROBABILITY CALCULATIONS     #
#####################################

# TODO: disable p value calculation due to python 2.7 break
# def _betai(a, b, x):
#     x = np.asarray(x)
#     x = np.where(x < 1.0, x, 1.0)  # if x > 1 then return 1.0
#     return betainc(a, b, x)

[docs]def pearsonr_mat(mat, w=None):
"""Utility function to calculate pearson matrix (row-wise).

Parameters
----------
mat : numpy array of shape (n_samples, n_features)
Input matrix.

w : numpy array of shape (n_features,)
Weights.

Returns
-------
pear_mat : numpy array of shape (n_samples, n_samples)
Row-wise pearson score matrix.

"""
mat = check_array(mat)
n_row = mat.shape
n_col = mat.shape
pear_mat = np.full([n_row, n_row], 1).astype(float)

if w is not None:
for cx in range(n_row):
for cy in range(cx + 1, n_row):
curr_pear = wpearsonr(mat[cx, :], mat[cy, :], w)
pear_mat[cx, cy] = curr_pear
pear_mat[cy, cx] = curr_pear
else:
for cx in range(n_col):
for cy in range(cx + 1, n_row):
curr_pear = pearsonr(mat[cx, :], mat[cy, :])
pear_mat[cx, cy] = curr_pear
pear_mat[cy, cx] = curr_pear

return pear_mat

[docs]def column_ecdf(matrix: np.ndarray) -> np.ndarray:
"""
Utility function to compute the column wise empirical cumulative distribution of a 2D feature matrix,
where the rows are samples and the columns are features per sample. The accumulation is done in the positive
direction of the sample axis.

E.G.
p(1) = 0.2, p(0) = 0.3, p(2) = 0.1, p(6) = 0.4
ECDF E(5) = p(x <= 5)
ECDF E would be E(-1) = 0, E(0) = 0.3, E(1) = 0.5, E(2) = 0.6, E(3) = 0.6, E(4) = 0.6, E(5) = 0.6, E(6) = 1

Similar to and tested against:
https://www.statsmodels.org/stable/generated/statsmodels.distributions.empirical_distribution.ECDF.html

Returns
-------

"""
# check the matrix dimensions
assert len(matrix.shape) == 2, 'Matrix needs to be two dimensional for the ECDF computation.'

# create a probability array the same shape as the feature matrix which we will reorder to build
# the ecdf
probabilities = np.linspace(np.ones(matrix.shape) / matrix.shape, np.ones(matrix.shape), matrix.shape)

# get the sorting indices for a numpy array
sort_idx = np.argsort(matrix, axis=0)

# sort the numpy array, as we need to look for duplicates in the feature values (that would have different
# probabilities if we would just resort the probabilities array)
matrix = np.take_along_axis(matrix, sort_idx, axis=0)

# deal with equal values
ecdf_terminate_equals_inplace(matrix, probabilities)

# return the resorted accumulated probabilities (by reverting the sorting of the input matrix)
# looks a little complicated but is faster this way
reordered_probabilities = np.ones_like(probabilities)
np.put_along_axis(reordered_probabilities, sort_idx, probabilities, axis=0)
return reordered_probabilities

[docs]@njit
def ecdf_terminate_equals_inplace(matrix: np.ndarray, probabilities: np.ndarray):
"""
This is a helper function for computing the ecdf of an array. It has been outsourced from the original
function in order to be able to use the njit compiler of numpy for increased speeds, as it unfortunately
needs a loop over all rows and columns of a matrix. It acts in place on the probabilities' matrix.

Parameters
----------
matrix : a feature matrix where the rows are samples and each column is a feature !(expected to be sorted)!

probabilities : a probability matrix that will be used building the ecdf. It has values between 0 and 1 and
is also sorted.

Returns
-------

"""
for cx in range(probabilities.shape):
for rx in range(probabilities.shape - 2, -1, -1):
if matrix[rx, cx] == matrix[rx + 1, cx]:
probabilities[rx, cx] = probabilities[rx + 1, cx]
```