111 lines
2.4 KiB
Python
111 lines
2.4 KiB
Python
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"""
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Special functions for copulas not available in scipy
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Created on Jan. 27, 2023
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"""
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import numpy as np
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from scipy.special import factorial
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class Sterling1():
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"""Stirling numbers of the first kind
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"""
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# based on
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# https://rosettacode.org/wiki/Stirling_numbers_of_the_first_kind#Python
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def __init__(self):
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self._cache = {}
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def __call__(self, n, k):
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key = str(n) + "," + str(k)
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if key in self._cache.keys():
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return self._cache[key]
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if n == k == 0:
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return 1
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if n > 0 and k == 0:
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return 0
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if k > n:
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return 0
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result = sterling1(n - 1, k - 1) + (n - 1) * sterling1(n - 1, k)
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self._cache[key] = result
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return result
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def clear_cache(self):
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"""clear cache of Sterling numbers
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"""
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self._cache = {}
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sterling1 = Sterling1()
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class Sterling2():
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"""Stirling numbers of the second kind
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"""
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# based on
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# https://rosettacode.org/wiki/Stirling_numbers_of_the_second_kind#Python
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def __init__(self):
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self._cache = {}
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def __call__(self, n, k):
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key = str(n) + "," + str(k)
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if key in self._cache.keys():
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return self._cache[key]
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if n == k == 0:
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return 1
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if (n > 0 and k == 0) or (n == 0 and k > 0):
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return 0
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if n == k:
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return 1
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if k > n:
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return 0
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result = k * sterling2(n - 1, k) + sterling2(n - 1, k - 1)
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self._cache[key] = result
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return result
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def clear_cache(self):
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"""clear cache of Sterling numbers
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"""
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self._cache = {}
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sterling2 = Sterling2()
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def li3(z):
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"""Polylogarithm for negative integer order -3
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Li(-3, z)
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"""
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return z * (1 + 4 * z + z**2) / (1 - z)**4
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def li4(z):
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"""Polylogarithm for negative integer order -4
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Li(-4, z)
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"""
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return z * (1 + z) * (1 + 10 * z + z**2) / (1 - z)**5
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def lin(n, z):
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"""Polylogarithm for negative integer order -n
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Li(-n, z)
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https://en.wikipedia.org/wiki/Polylogarithm#Particular_values
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"""
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if np.size(z) > 1:
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z = np.array(z)[..., None]
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k = np.arange(n+1)
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st2 = np.array([sterling2(n + 1, ki + 1) for ki in k])
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res = (-1)**(n+1) * np.sum(factorial(k) * st2 * (-1 / (1 - z))**(k+1),
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axis=-1)
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return res
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