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longtail.py
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longtail.py
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# -*- coding: utf-8 -*-
"""
Transforms RV from the given empirical distribution to the standard normal distribution
Author: Dmitry Mottl (https://github.com/Mottl)
License: MIT
"""
import sys
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
try:
import pandas as pd
except:
pass
def fit_distributions(X, distributions=None, verbose=False):
"""
fit distributions to data `X`
Parameters
----------
X : 1d-array
distributions : array of strings (['norm', 'laplace', etc..])
defaults distributions are: ['norm', 'laplace', 'cauchy']
verbose: bool, default False
"""
if distributions is None:
if min(X) >= 0:
distributions = [
'halfnorm', 'halfcauchy'
]
else:
distributions = [
'norm', 'laplace', 'cauchy'
]
params = {}
for name in distributions:
distr = getattr(stats, name)
params[name] = distr.fit(X)
if verbose:
print(name, params[name])
return params
def plot(X, X_name=None, params=None, **kwargs):
"""
plot probability distribution function for `y`
and overlay distributions calculated with `params`
Parameters
----------
x : array
params: list of best-fit parameters returned by fit_distributions() function
Return value
------------
params from fit_distributions() function
"""
if X is not np.ndarray:
X = np.array(X)
if params is None:
print("Estimating distributions parameters...")
params = fit_distributions(X, verbose=True)
label = X_name or "data"
# plot PDF
x_min = np.percentile(X, 0.9)
x_max = -np.percentile(-X, 0.9)
X_ = X[(X>=x_min) & (X<=x_max)]
num_bins = int(np.log(len(X_))*5)
x_space = np.linspace(x_min, x_max, 1000)
f, ax = plt.subplots(**kwargs)
ax.hist(X_, bins=num_bins, density=True, alpha=0.33, color="dodgerblue", label=label)
for name, param in params.items():
distr = getattr(stats, name)
ax.plot(x_space, distr.pdf(x_space, loc=param[0], scale=param[1]), label=name)
ax.legend()
ax.set_ylabel('pdf')
if X_name is not None:
ax.set_xlabel(X_name)
ax.grid(True)
plt.show()
# plot LOG PDF
x_min, x_max = X.min(), X.max()
num_bins = int(np.log(len(X))*5)
x_space = np.linspace(x_min, x_max, 1000)
bins_means = [] # mean of bin interval
bins_xs = [] # number of ys in interval
x_step = (x_max - x_min) / num_bins
for x_left in np.arange(x_min, x_max, x_step):
bins_means.append(x_left + x_step/2.)
bins_xs.append(np.sum((X>=x_left) & (X<x_left+x_step)))
bins_xs = np.array(bins_xs) / len(X) / x_step # normalize
f, ax = plt.subplots(**kwargs)
ax.scatter(bins_means, bins_xs, s=5., color="dodgerblue", label=label)
for name, param in params.items():
distr = getattr(stats, name)
ax.plot(x_space, distr.pdf(x_space, loc=param[0], scale=param[1]), label=name)
ax.legend()
ax.set_ylabel('pdf')
ax.set_yscale('log')
if X_name is not None:
ax.set_xlabel(X_name)
ax.grid(True)
plt.show()
return params
class GaussianScaler():
"""Transform data to make it Gaussian distributed."""
def __init__(self):
self.transform_table = None
self.features_names = None
self.__num_vars = None
def fit(self, X, y=None):
"""Compute empirical parameters for transforming the data to Gaussian distribution.
Parameters:
-----------
X : array-like (1 or 2 dim. np.ndarray, pandas.Series or pandas.DataFrame)
features to fit
"""
if len(X.shape)>2:
raise NotImplementedError("X must be an 1d-array or a 2d-matrix of observations x features")
# convert from pd.DataFrame to np.ndarrray:
if "pandas.core.frame" in sys.modules.keys() and \
type(X) in (pd.core.series.Series, pd.core.frame.DataFrame):
if type(X) == pd.core.frame.DataFrame:
self.features_names = X.columns.values
else:
self.features_names = np.array([X.name])
X = X.values
if X.dtype not in (float, np.float32, np.float64):
raise Exception("X.dtype is {}, but should be float".format(X.dtype))
if len(X.shape) == 2:
if X.shape[1] == 1:
X = X.ravel()
self.__num_vars = 1
else:
self.__num_vars = X.shape[1]
else:
self.__num_vars = 1
self.transform_table = []
for j in range(self.__num_vars):
if self.__num_vars > 1:
X_sorted = np.array(np.unique(X[:, j], return_counts=True)).T
else:
X_sorted = np.array(np.unique(X, return_counts=True)).T
X_sorted[:, 1] = np.cumsum(X_sorted[:, 1])
total = X_sorted[-1,1]
# X_sorted[:, 0] is x, X_sorted[:, 1] is the number of occurences <= x
STEP_MULT = 0.1 # step multiplier
MIN_STEP = 5
step = MIN_STEP
i = step
prev_x = -np.inf
transform_table = []
transform_table.append((-np.inf, -np.inf, 0.))
while True:
index = np.argmax(X_sorted[:,1] >= i)
row = X_sorted[index]
x = row[0]
if x != prev_x:
cdf_empiric = row[1] / total
x_norm = stats.norm.ppf(cdf_empiric)
if x_norm == np.inf: # too large - stop
break
if x_norm != -np.inf:
transform_table.append((x, x_norm, 0.))
if cdf_empiric < 0.5:
step = int(row[1] * STEP_MULT)
else:
step = int((total - row[1]) * STEP_MULT)
step = max(step, MIN_STEP)
prev_x = x
i = i + step
if i >= total:
break
transform_table.append((np.inf, np.inf, 0.))
transform_table = np.array(transform_table)
# compute x -> x_norm coefficients
dx = transform_table[2:-1, 0] - transform_table[1:-2, 0]
dx_norm = transform_table[2:-1, 1] - transform_table[1:-2, 1]
transform_table[2:-1, 2] = dx_norm / dx
"""
# generic non-optimized code would look like this:
for i in range(2, len(transform_table) - 1):
dx = transform_table[i, 0] - transform_table[i-1, 0]
dx_norm = transform_table[i, 1] - transform_table[i-1, 1]
transform_table[i, 2] = dx_norm / dx
"""
# fill boundary bins (plus/minus infinity) intervals:
transform_table[0, 2] = transform_table[2, 2]
transform_table[1, 2] = transform_table[2, 2]
transform_table[-1, 2] = transform_table[-2, 2]
# add current transform table for the feature to self.transform_table
self.transform_table.append(transform_table)
def transform(self, X, y=None):
"""Transform X to Gaussian distributed (standard normal).
Parameters
----------
X : array-like (1 or 2 dim. np.ndarray, pandas.Series or pandas.DataFrame)
features to transform
"""
if self.transform_table is None:
raise Exception(("This GaussianScaler instance is not fitted yet."
"Call 'fit' with appropriate arguments before using this method."))
if len(X.shape)>2:
raise NotImplementedError("X must be an 1d-array or a 2d-matrix of observations x features")
# convert from pd.DataFrame to np.ndarrray:
if "pandas.core.frame" in sys.modules.keys() and \
type(X) in (pd.core.series.Series, pd.core.frame.DataFrame):
if type(X) == pd.core.frame.DataFrame:
features_names = X.columns.values
else:
features_names = np.array([X.name])
if (features_names != self.features_names).any():
raise Exception("Feature names mismatch.\nFeatures for fit():{}\nFeatures for transform:{}".format(
self.features_names, features_names))
save_index = X.index.copy()
X = X.values.copy()
if X.dtype not in (float, np.float32, np.float64):
raise Exception("X.dtype is {}, but should be float".format(X.dtype))
if len(X.shape) == 2:
if X.shape[1] == 1:
X = X.ravel()
num_vars = 1
else:
num_vars = X.shape[1]
else:
num_vars = 1
if self.__num_vars != num_vars:
raise Exception("Number of features mismatch for fit() and transform(): {} vs {}".format(
self.__num_vars, num_vars))
def _transform(x, j):
# x(empirical) -> x(normaly distributed)
transform_table = self.transform_table[j]
lefts = transform_table[transform_table[:, 0] < x]
rights = transform_table[transform_table[:, 0] >= x]
left_boundary = lefts[-1]
right_boundary = rights[0]
k = right_boundary[2]
if right_boundary[0] == np.inf:
x_norm = left_boundary[1] + k * (x - left_boundary[0])
else:
x_norm = right_boundary[1] + k * (x - right_boundary[0])
return x_norm
vtransform = np.vectorize(_transform)
if num_vars > 1:
# transform all features:
for j in range(self.__num_vars):
X[:, j] = vtransform(X[:, j], j)
else:
X = vtransform(X, 0)
# reconstruct X as a Series or a DataFrame:
if self.features_names is not None:
if num_vars > 1:
X = pd.DataFrame(X, columns=self.features_names, index=save_index)
else:
X = pd.Series(X, name=self.features_names[0], index=save_index)
return X
def fit_transform(self, X, y=None):
""" Fit to data, then transform it.
Parameters
----------
X : array-like (1 or 2 dim. np.ndarray, pandas.Series or pandas.DataFrame)
features to transform
"""
self.fit(X)
return self.transform(X)
def inverse_transform(self, X):
"""Transform back the data from Gaussian to the original empirical distribution.
Parameters
----------
X : array-like (1 or 2 dim. np.ndarray, pandas.Series or pandas.DataFrame)
features to inverse transform
"""
if self.transform_table is None:
raise Exception(("This GaussianScaler instance is not fitted yet."
"Call 'fit' with appropriate arguments before using this method."))
if len(X.shape)>2:
raise NotImplementedError("X must be an 1d-array or a 2d-matrix of observations x features")
# convert from pd.DataFrame to np.ndarrray:
if "pandas.core.frame" in sys.modules.keys() and \
type(X) in (pd.core.series.Series, pd.core.frame.DataFrame):
if type(X) == pd.core.frame.DataFrame:
features_names = X.columns.values
else:
features_names = np.array([X.name])
if (features_names != self.features_names).any():
raise Exception("Feature names mismatch.\nFeatures for fit():{}\nFeatures for transform:{}".format(
self.features_names, features_names))
save_index = X.index.copy()
X = X.values.copy()
if X.dtype not in (float, np.float32, np.float64):
raise Exception("X.dtype is {}, but should be float".format(X.dtype))
if len(X.shape) == 2:
if X.shape[1] == 1:
X = X.ravel()
num_vars = 1
else:
num_vars = X.shape[1]
else:
num_vars = 1
if self.__num_vars != num_vars:
raise Exception("Number of features mismatch for fit() and transform(): {} vs {}".format(
self.__num_vars, num_vars))
def _inverse_transform(x, j):
# x(normaly distributed) -> x(empirical)
transform_table = self.transform_table[j]
lefts = transform_table[transform_table[:, 1] < x]
rights = transform_table[transform_table[:, 1] >= x]
left_boundary = lefts[-1]
right_boundary = rights[0]
k = right_boundary[2]
if right_boundary[1] == np.inf:
x_emp = left_boundary[0] + (x - left_boundary[1]) / k
else:
x_emp = right_boundary[0] + (x - right_boundary[1]) / k
return x_emp
vinverse_transform = np.vectorize(_inverse_transform)
if num_vars > 1:
# transform all features:
for j in range(self.__num_vars):
X[:, j] = vinverse_transform(X[:, j], j)
else:
X = vinverse_transform(X, 0)
# reconstruct X as a Series or a DataFrame:
if self.features_names is not None:
if num_vars > 1:
X = pd.DataFrame(X, columns=self.features_names, index=save_index)
else:
X = pd.Series(X, name=self.features_names[0], index=save_index)
return X