def _hermnorm(N):
# return the negatively normalized hermite polynomials up to order N-1
# (inclusive)
# using the recursive relationship
# p_n+1 = p_n(x)' - x*p_n(x)
# and p_0(x) = 1
plist = [None]*N
plist[0] = poly1d(1)
for n in range(1,N):
plist[n] = plist[n-1].deriv() - poly1d([1,0])*plist[n-1]
return plist
def pdf_moments(cnt):
"""Return the Gaussian expanded pdf function given the list of central
moments (first one is mean).
"""
N = len(cnt)
if N < 2:
raise ValueError, "At least two moments must be given to" + \
"approximate the pdf."
totp = poly1d(1)
sig = sqrt(cnt[1])
mu = cnt[0]
if N > 2:
Dvals = _hermnorm(N+1)
for k in range(3,N+1):
# Find Ck
Ck = 0.0
for n in range((k-3)/2):
m = k-2*n
if m % 2: # m is odd
momdiff = cnt[m-1]
else:
momdiff = cnt[m-1] - sig*sig*scipy.factorial2(m-1)
Ck += Dvals[k][m] / sig**m * momdiff
# Add to totp
totp = totp + Ck*Dvals[k]
def thisfunc(x):
xn = (x-mu)/sig
return totp(xn)*exp(-xn*xn/2.0)/sqrt(2*pi)/sig
return thisfunc
def pade(an, m):
"""Given Taylor series coefficients in an, return a Pade approximation to
the function as the ratio of two polynomials p / q where the order of q is m.
"""
an = asarray(an)
N = len(an) - 1
n = N-m
if (n < 0):
raise ValueError, \
"Order of q <m> must be smaller than len(an)-1."
Akj = eye(N+1,n+1)
Bkj = zeros((N+1,m),'d')
for row in range(1,m+1):
Bkj[row,:row] = -(an[:row])[::-1]
for row in range(m+1,N+1):
Bkj[row,:] = -(an[row-m:row])[::-1]
C = hstack((Akj,Bkj))
pq = dot(linalg.inv(C),an)
p = pq[:n+1]
q = r_[1.0,pq[n+1:]]
return poly1d(p[::-1]), poly1d(q[::-1])
