def SpectralDensityFunction(f, H):
'''Spectral density function (-0.5<f<0.5) for fractional Gaussian noise with Hurst Exponent H
I don't know how to derive this formula. It is in Voichita's matlab code, and she refers to it as
"the almost exact formula from Percival and Walden".
I have a feeling that it might be a Taylor series approximation to the Fourier transform of the auto-covariance relation
for fractional Gaussian Noise.
>>> SpectralDensityFunction(0.26,0.2367)
1.1067908939278011
'''
if (H == 0):
return 2 * sin(pi * f)**2
else:
M = 100 #sum from k = -M to M of 1/|k+f|^(2H+1)
s = sum([1 / (abs(x + f)**(2 * H + 1)) for x in range(-M, M + 1)])
s += 1 / (2 * H * (M + 1 - f)**(2 * H))
s += 1 / (2 * H * (M + 1 + f)**(2 * H))
s -= (2 * H + 1) * (2 * H + 2) * (2 * H +
3) / (720 * (M + 1 - f)**(2 * H + 4))
s -= (2 * H + 1) * (2 * H + 2) * (2 * H +
3) / (720 * (M + 1 + f)**(2 * H + 4))
s += (2 * H + 1) / (12 * (M + 1 - f)**(2 * H + 2))
s += (2 * H + 1) / (12 * (M + 1 + f)**(2 * H + 2))
s += 1 / (2 * (M + 1 - f)**(2 * H + 1))
s += 1 / (2 * (M + 1 + f)**(2 * H + 1))
s *= 4 * gamma(2 * H + 1) * sin(pi * H) * (sin(pi * f)**2)
s /= (2 * pi)**(2 * H + 1)
return s
def IntegratedSpectralDensity(H, wsteps, quadraturesteps, i):
if(H==0.0):
if i==0:
return 1-2**wsteps/pi*sin(pi/2**wsteps)
else:
x=2**(wsteps+1-i)
return 1-(x/pi)*sin(2*pi/x)+(x/pi)*sin(pi/x)
elif(H==1.0):
if i==0:
return 2**wsteps
else:
return 1e-50
else:
if i==0:
return gamma(2*H+1)*sin(pi*H)/(2*pi)**(2*H-1)/(1-H)*2**((wsteps+1)*(2*H-1)-1)
else:
scale=2**(wsteps+2-i)
xold=1.0
yold=SpectralDensityFunction(xold/scale,H)
sum=0
for j in range(1,quadraturesteps+1):
xnew=(1+float(j)/quadraturesteps)
ynew=SpectralDensityFunction(xnew/scale,H)
sum+=(xnew-xold)*(ynew+yold)
xold=xnew
yold=ynew
return sum/2.0
def SpectralDensityFunction(f,H):
'''Spectral density function (-0.5<f<0.5) for fractional Gaussian noise with Hurst Exponent H
I don't know how to derive this formula. It is in Voichita's matlab code, and she refers to it as
"the almost exact formula from Percival and Walden".
I have a feeling that it might be a Taylor series approximation to the Fourier transform of the auto-covariance relation
for fractional Gaussian Noise.
>>> SpectralDensityFunction(0.26,0.2367)
1.1067908939278011
'''
if(H==0):
return 2*sin(pi*f)**2
else:
M=100 #sum from k = -M to M of 1/|k+f|^(2H+1)
s=sum( [1/(abs(x+f)**(2*H+1)) for x in range(-M,M+1)] )
s+=1/(2*H*(M+1-f)**(2*H))
s+=1/(2*H*(M+1+f)**(2*H))
s-=(2*H+1)*(2*H+2)*(2*H+3)/(720*(M+1-f)**(2*H+4))
s-=(2*H+1)*(2*H+2)*(2*H+3)/(720*(M+1+f)**(2*H+4))
s+=(2*H+1)/(12*(M+1-f)**(2*H+2))
s+=(2*H+1)/(12*(M+1+f)**(2*H+2))
s+=1/(2*(M+1-f)**(2*H+1))
s+=1/(2*(M+1+f)**(2*H+1))
s*=4*gamma(2*H+1)*sin(pi*H)*(sin(pi*f)**2)
s/=(2*pi)**(2*H+1)
return s
def IntegratedSpectralDensity(H, wsteps, quadraturesteps, i):
if (H == 0.0):
if i == 0:
return 1 - 2**wsteps / pi * sin(pi / 2**wsteps)
else:
x = 2**(wsteps + 1 - i)
return 1 - (x / pi) * sin(2 * pi / x) + (x / pi) * sin(pi / x)
elif (H == 1.0):
if i == 0:
return 2**wsteps
else:
return 1e-50
else:
if i == 0:
return gamma(2 * H + 1) * sin(pi * H) / (2 * pi)**(2 * H - 1) / (
1 - H) * 2**((wsteps + 1) * (2 * H - 1) - 1)
else:
scale = 2**(wsteps + 2 - i)
xold = 1.0
yold = SpectralDensityFunction(xold / scale, H)
sum = 0
for j in range(1, quadraturesteps + 1):
xnew = (1 + float(j) / quadraturesteps)
ynew = SpectralDensityFunction(xnew / scale, H)
sum += (xnew - xold) * (ynew + yold)
xold = xnew
yold = ynew
return sum / 2.0
def IntegratedSpectralDensityFunction(H, wsteps, quadraturesteps=20):
'''
Appears to be the expected values of the wavelet coefficients of a fractional Gaussian noise.
In the general 0<H<1 fine scale case these are calculated by doing a numerical integration
over a region of the Spectral Density. I have no idea where this formula comes from, and
have copied it from Voichita's matlab code (the function SDF_Int), and then clarified the calculation.
>>> IntegratedSpectralDensityFunction(0.0,5)
[0.0016056069643816118, 0.011220690747206885, 0.044596134335340376, 0.17386272609022058, 0.62707677142194329, 1.6366197723675813]
>>> IntegratedSpectralDensityFunction(1.0,5)
[32, 1e-50, 1e-50, 1e-50, 1e-50, 1e-50]
>>> IntegratedSpectralDensityFunction(0.76,5)
[6.4284808246144252, 2.5337868118662028, 1.7592398286153024, 1.2077211418590617, 0.8012983421742671, 0.51762057939023354]
>>> IntegratedSpectralDensityFunction(0.76,5,10)
[6.4284808246144252, 2.5344386424271996, 1.7596935438138646, 1.2080370431239666, 0.80152942631348534, 0.5178955503380388]
>>> IntegratedSpectralDensityFunction(0.35,8)
[0.16634193518773813, 0.24327887427188533, 0.29962683660583778, 0.36932052594955755, 0.45630849916626082, 0.56762788846911971, 0.71869843738552097, 0.94205671645702882, 1.2302926459188461]
'''
if (H == 0.0):
c = range(wsteps + 1, 0, -1)
c = [2.0**x for x in c]
c = [
1 - (x / pi) * sin(2 * pi / x) + (x / pi) * sin(pi / x) for x in c
]
c[0] = 1 - 2**wsteps / pi * sin(pi / 2**wsteps)
return c
elif (H == 1.0):
c = range(wsteps + 1, 0, -1)
c = [1e-50 for x in c]
c[0] = 2**wsteps
return c
else:
c = []
c.append(
gamma(2 * H + 1) * sin(pi * H) / (2 * pi)**(2 * H - 1) / (1 - H) *
2**((wsteps + 1) * (2 * H - 1) - 1))
grid = range(quadraturesteps + 1)
grid = [1 + x * 1.0 / quadraturesteps for x in grid]
for i in range(wsteps + 1, 1, -1):
fvals = [x / (2**i) for x in grid]
SDFvals = [SpectralDensityFunction(f, H) for f in fvals]
c.append(TrapeziumIntegration(grid, SDFvals))
return c
def IntegratedSpectralDensityFunction(H,wsteps,quadraturesteps=20):
'''
Appears to be the expected values of the wavelet coefficients of a fractional Gaussian noise.
In the general 0<H<1 fine scale case these are calculated by doing a numerical integration
over a region of the Spectral Density. I have no idea where this formula comes from, and
have copied it from Voichita's matlab code (the function SDF_Int), and then clarified the calculation.
>>> IntegratedSpectralDensityFunction(0.0,5)
[0.0016056069643816118, 0.011220690747206885, 0.044596134335340376, 0.17386272609022058, 0.62707677142194329, 1.6366197723675813]
>>> IntegratedSpectralDensityFunction(1.0,5)
[32, 1e-50, 1e-50, 1e-50, 1e-50, 1e-50]
>>> IntegratedSpectralDensityFunction(0.76,5)
[6.4284808246144252, 2.5337868118662028, 1.7592398286153024, 1.2077211418590617, 0.8012983421742671, 0.51762057939023354]
>>> IntegratedSpectralDensityFunction(0.76,5,10)
[6.4284808246144252, 2.5344386424271996, 1.7596935438138646, 1.2080370431239666, 0.80152942631348534, 0.5178955503380388]
>>> IntegratedSpectralDensityFunction(0.35,8)
[0.16634193518773813, 0.24327887427188533, 0.29962683660583778, 0.36932052594955755, 0.45630849916626082, 0.56762788846911971, 0.71869843738552097, 0.94205671645702882, 1.2302926459188461]
'''
if(H==0.0):
c=range(wsteps+1,0,-1)
c=[2.0**x for x in c]
c=[1-(x/pi)*sin(2*pi/x)+(x/pi)*sin(pi/x) for x in c]
c[0]=1-2**wsteps/pi*sin(pi/2**wsteps)
return c
elif(H==1.0):
c=range(wsteps+1,0,-1)
c=[1e-50 for x in c]
c[0]=2**wsteps
return c
else:
c=[]
c.append(gamma(2*H+1)*sin(pi*H)/(2*pi)**(2*H-1)/(1-H)*2**((wsteps+1)*(2*H-1)-1))
grid=range(quadraturesteps+1)
grid=[1+x*1.0/quadraturesteps for x in grid]
for i in range(wsteps+1,1,-1):
fvals=[x/(2**i) for x in grid]
SDFvals=[SpectralDensityFunction(f,H) for f in fvals]
c.append(TrapeziumIntegration(grid,SDFvals))
return c
