202 lines
6.8 KiB
Python
202 lines
6.8 KiB
Python
|
from __future__ import division, print_function, absolute_import
|
||
|
|
||
|
import numpy as np
|
||
|
from numpy import poly1d
|
||
|
from scipy.special import beta
|
||
|
|
||
|
|
||
|
# The following code was used to generate the Pade coefficients for the
|
||
|
# Tukey Lambda variance function. Version 0.17 of mpmath was used.
|
||
|
#---------------------------------------------------------------------------
|
||
|
# import mpmath as mp
|
||
|
#
|
||
|
# mp.mp.dps = 60
|
||
|
#
|
||
|
# one = mp.mpf(1)
|
||
|
# two = mp.mpf(2)
|
||
|
#
|
||
|
# def mpvar(lam):
|
||
|
# if lam == 0:
|
||
|
# v = mp.pi**2 / three
|
||
|
# else:
|
||
|
# v = (two / lam**2) * (one / (one + two*lam) -
|
||
|
# mp.beta(lam + one, lam + one))
|
||
|
# return v
|
||
|
#
|
||
|
# t = mp.taylor(mpvar, 0, 8)
|
||
|
# p, q = mp.pade(t, 4, 4)
|
||
|
# print("p =", [mp.fp.mpf(c) for c in p])
|
||
|
# print("q =", [mp.fp.mpf(c) for c in q])
|
||
|
#---------------------------------------------------------------------------
|
||
|
|
||
|
# Pade coefficients for the Tukey Lambda variance function.
|
||
|
_tukeylambda_var_pc = [3.289868133696453, 0.7306125098871127,
|
||
|
-0.5370742306855439, 0.17292046290190008,
|
||
|
-0.02371146284628187]
|
||
|
_tukeylambda_var_qc = [1.0, 3.683605511659861, 4.184152498888124,
|
||
|
1.7660926747377275, 0.2643989311168465]
|
||
|
|
||
|
# numpy.poly1d instances for the numerator and denominator of the
|
||
|
# Pade approximation to the Tukey Lambda variance.
|
||
|
_tukeylambda_var_p = poly1d(_tukeylambda_var_pc[::-1])
|
||
|
_tukeylambda_var_q = poly1d(_tukeylambda_var_qc[::-1])
|
||
|
|
||
|
|
||
|
def tukeylambda_variance(lam):
|
||
|
"""Variance of the Tukey Lambda distribution.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
lam : array_like
|
||
|
The lambda values at which to compute the variance.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
v : ndarray
|
||
|
The variance. For lam < -0.5, the variance is not defined, so
|
||
|
np.nan is returned. For lam = 0.5, np.inf is returned.
|
||
|
|
||
|
Notes
|
||
|
-----
|
||
|
In an interval around lambda=0, this function uses the [4,4] Pade
|
||
|
approximation to compute the variance. Otherwise it uses the standard
|
||
|
formula (http://en.wikipedia.org/wiki/Tukey_lambda_distribution). The
|
||
|
Pade approximation is used because the standard formula has a removable
|
||
|
discontinuity at lambda = 0, and does not produce accurate numerical
|
||
|
results near lambda = 0.
|
||
|
"""
|
||
|
lam = np.asarray(lam)
|
||
|
shp = lam.shape
|
||
|
lam = np.atleast_1d(lam).astype(np.float64)
|
||
|
|
||
|
# For absolute values of lam less than threshold, use the Pade
|
||
|
# approximation.
|
||
|
threshold = 0.075
|
||
|
|
||
|
# Play games with masks to implement the conditional evaluation of
|
||
|
# the distribution.
|
||
|
# lambda < -0.5: var = nan
|
||
|
low_mask = lam < -0.5
|
||
|
# lambda == -0.5: var = inf
|
||
|
neghalf_mask = lam == -0.5
|
||
|
# abs(lambda) < threshold: use Pade approximation
|
||
|
small_mask = np.abs(lam) < threshold
|
||
|
# else the "regular" case: use the explicit formula.
|
||
|
reg_mask = ~(low_mask | neghalf_mask | small_mask)
|
||
|
|
||
|
# Get the 'lam' values for the cases where they are needed.
|
||
|
small = lam[small_mask]
|
||
|
reg = lam[reg_mask]
|
||
|
|
||
|
# Compute the function for each case.
|
||
|
v = np.empty_like(lam)
|
||
|
v[low_mask] = np.nan
|
||
|
v[neghalf_mask] = np.inf
|
||
|
if small.size > 0:
|
||
|
# Use the Pade approximation near lambda = 0.
|
||
|
v[small_mask] = _tukeylambda_var_p(small) / _tukeylambda_var_q(small)
|
||
|
if reg.size > 0:
|
||
|
v[reg_mask] = (2.0 / reg**2) * (1.0 / (1.0 + 2 * reg) -
|
||
|
beta(reg + 1, reg + 1))
|
||
|
v.shape = shp
|
||
|
return v
|
||
|
|
||
|
|
||
|
# The following code was used to generate the Pade coefficients for the
|
||
|
# Tukey Lambda kurtosis function. Version 0.17 of mpmath was used.
|
||
|
#---------------------------------------------------------------------------
|
||
|
# import mpmath as mp
|
||
|
#
|
||
|
# mp.mp.dps = 60
|
||
|
#
|
||
|
# one = mp.mpf(1)
|
||
|
# two = mp.mpf(2)
|
||
|
# three = mp.mpf(3)
|
||
|
# four = mp.mpf(4)
|
||
|
#
|
||
|
# def mpkurt(lam):
|
||
|
# if lam == 0:
|
||
|
# k = mp.mpf(6)/5
|
||
|
# else:
|
||
|
# numer = (one/(four*lam+one) - four*mp.beta(three*lam+one, lam+one) +
|
||
|
# three*mp.beta(two*lam+one, two*lam+one))
|
||
|
# denom = two*(one/(two*lam+one) - mp.beta(lam+one,lam+one))**2
|
||
|
# k = numer / denom - three
|
||
|
# return k
|
||
|
#
|
||
|
# # There is a bug in mpmath 0.17: when we use the 'method' keyword of the
|
||
|
# # taylor function and we request a degree 9 Taylor polynomial, we actually
|
||
|
# # get degree 8.
|
||
|
# t = mp.taylor(mpkurt, 0, 9, method='quad', radius=0.01)
|
||
|
# t = [mp.chop(c, tol=1e-15) for c in t]
|
||
|
# p, q = mp.pade(t, 4, 4)
|
||
|
# print("p =", [mp.fp.mpf(c) for c in p])
|
||
|
# print("q =", [mp.fp.mpf(c) for c in q])
|
||
|
#---------------------------------------------------------------------------
|
||
|
|
||
|
# Pade coefficients for the Tukey Lambda kurtosis function.
|
||
|
_tukeylambda_kurt_pc = [1.2, -5.853465139719495, -22.653447381131077,
|
||
|
0.20601184383406815, 4.59796302262789]
|
||
|
_tukeylambda_kurt_qc = [1.0, 7.171149192233599, 12.96663094361842,
|
||
|
0.43075235247853005, -2.789746758009912]
|
||
|
|
||
|
# numpy.poly1d instances for the numerator and denominator of the
|
||
|
# Pade approximation to the Tukey Lambda kurtosis.
|
||
|
_tukeylambda_kurt_p = poly1d(_tukeylambda_kurt_pc[::-1])
|
||
|
_tukeylambda_kurt_q = poly1d(_tukeylambda_kurt_qc[::-1])
|
||
|
|
||
|
|
||
|
def tukeylambda_kurtosis(lam):
|
||
|
"""Kurtosis of the Tukey Lambda distribution.
|
||
|
|
||
|
Parameters
|
||
|
----------
|
||
|
lam : array_like
|
||
|
The lambda values at which to compute the variance.
|
||
|
|
||
|
Returns
|
||
|
-------
|
||
|
v : ndarray
|
||
|
The variance. For lam < -0.25, the variance is not defined, so
|
||
|
np.nan is returned. For lam = 0.25, np.inf is returned.
|
||
|
|
||
|
"""
|
||
|
lam = np.asarray(lam)
|
||
|
shp = lam.shape
|
||
|
lam = np.atleast_1d(lam).astype(np.float64)
|
||
|
|
||
|
# For absolute values of lam less than threshold, use the Pade
|
||
|
# approximation.
|
||
|
threshold = 0.055
|
||
|
|
||
|
# Use masks to implement the conditional evaluation of the kurtosis.
|
||
|
# lambda < -0.25: kurtosis = nan
|
||
|
low_mask = lam < -0.25
|
||
|
# lambda == -0.25: kurtosis = inf
|
||
|
negqrtr_mask = lam == -0.25
|
||
|
# lambda near 0: use Pade approximation
|
||
|
small_mask = np.abs(lam) < threshold
|
||
|
# else the "regular" case: use the explicit formula.
|
||
|
reg_mask = ~(low_mask | negqrtr_mask | small_mask)
|
||
|
|
||
|
# Get the 'lam' values for the cases where they are needed.
|
||
|
small = lam[small_mask]
|
||
|
reg = lam[reg_mask]
|
||
|
|
||
|
# Compute the function for each case.
|
||
|
k = np.empty_like(lam)
|
||
|
k[low_mask] = np.nan
|
||
|
k[negqrtr_mask] = np.inf
|
||
|
if small.size > 0:
|
||
|
k[small_mask] = _tukeylambda_kurt_p(small) / _tukeylambda_kurt_q(small)
|
||
|
if reg.size > 0:
|
||
|
numer = (1.0 / (4 * reg + 1) - 4 * beta(3 * reg + 1, reg + 1) +
|
||
|
3 * beta(2 * reg + 1, 2 * reg + 1))
|
||
|
denom = 2 * (1.0/(2 * reg + 1) - beta(reg + 1, reg + 1))**2
|
||
|
k[reg_mask] = numer / denom - 3
|
||
|
|
||
|
# The return value will be a numpy array; resetting the shape ensures that
|
||
|
# if `lam` was a scalar, the return value is a 0-d array.
|
||
|
k.shape = shp
|
||
|
return k
|