def trans(q, r, **kwargs):
# make transformation matrix given for FT
# the smearing construction is done following Steen Hansen's
# Bayesian IFT program writteon in Fortran
# I assume that the beam integral is equal to 0.5, and hence
# the 2 constant at the last step of Matrix construction
if len(kwargs) > 1:
smear = True
try:
y = kwargs['y']
Wy = kwargs['Wy']
dy = y[1] - y[0]
except NameError:
print("Please specify beam parameteres y and Wy")
raise
else:
smear = False
Nq = len(q)
Nr = len(r)
dr = r[1] - r[0]
qr = outer(q, r)
# sin(x)/x, when x=0, should be 1
qr[qr < 1.e-15] = 1.e-15
A = 4 * pi * sin(qr) / qr * dr
if smear:
# now this uses my fortran module 'transc' from 'trans_smear.f90'
A = transc(q, r, y, Wy)
# At = zeros((Nq,Nr))
# for i in range(Nq):
# for j in range(Nr):
# qy = sqrt(q[i]**2 + y**2) * r[j]
# qy[qy<1e-15] = 1e-15
# wrk = Wy * sin(qy)/qy * dy
# At[i,j] = 2*dr* wrk.sum()
# A = 4*pi*At
return A
def trans(q, r, **kwargs):
# make transformation matrix given for FT
# the smearing construction is done following Steen Hansen's
# Bayesian IFT program writteon in Fortran
# I assume that the beam integral is equal to 0.5, and hence
# the 2 constant at the last step of Matrix construction
if len(kwargs) > 1:
smear = True
try:
y = kwargs['y']
Wy = kwargs['Wy']
dy = y[1]-y[0]
except NameError:
print "Please specify beam parameteres y and Wy"
raise
else:
smear = False
Nq = len(q)
Nr = len(r)
dr = r[1]-r[0]
qr = outer(q,r)
# sin(x)/x, when x=0, should be 1
qr[qr<1.e-15] = 1.e-15
A = 4*pi* sin(qr)/qr * dr
if smear:
A = transc(q,r,y,Wy)
# At = zeros((Nq,Nr))
# for i in range(Nq):
# for j in range(Nr):
# qy = sqrt(q[i]**2 + y**2) * r[j]
# qy[qy<1e-15] = 1e-15
# wrk = Wy * sin(qy)/qy * dy
# At[i,j] = 2*dr* wrk.sum()
# A = 4*pi*At
return A
def iftv2(alpha, Dmax, q, Iq, sd, Nr, y, Wy, weightdata, smeared):
# calculates IFT solution and returns the complete set of solutions
# including extrapolation to zero angle
Nq = len(Iq)
# prepare q_full for extrapolation to zero angle
dq = diff(q).mean() # get average spacing
q_extra = arange(0, q.min(), dq)
q_full = append(q_extra, q)
r = linspace(0, Dmax, Nr)
sd_sq = sd**2
# Normalize variance / data weights
sdnorm = 1 / sd_sq
sdnorm = Nq * sdnorm / sdnorm.sum(
) # scale so it doesnt spread the eigenvalues
sdmat = diag(sdnorm)
# compute transformation matrix, uses FORTRAN module
# if smeared to speed up calculations
if smeared:
K = transc(q, r, y, Wy)
Kunsmeared = trans(q, r)
Kfull = transc(q_full, r, y, Wy)
Kfull_unsmeared = trans(q_full, r)
else:
K = trans(q, r)
Kunsmeared = K
Kfull = trans(q_full, r)
Kfull_unsmeared = Kfull
# construct matrix L and Z
# matrix L is the finite-difference 2nd derivative approximation
# matrix Z is the penalty matrix for having non-zero P(0) & P(Dmax)
L = -0.5 * eye(Nr, k=-1) + eye(Nr, k=0) - 0.5 * eye(Nr, k=1)
# L = -1 * eye(Nr,k=0) + eye(Nr,k=1)
Z = zeros((Nr, Nr))
Z[0, 0] = 1.0
Z[-1, -1] = 1.0
sqrtalpha = sqrt(alpha)
if weightdata:
Iq_obs = Iq.dot(sdmat) # normalize data by sigma
sdcol = sdnorm[:, newaxis]
# normalize transformation matrix along M-dimension by sigma
C = vstack([K * sdcol, sqrtalpha * L, 10. * sqrtalpha * Z])
else:
C = vstack([K, sqrtalpha * L, 10. * sqrtalpha * Z])
Iq_obs = Iq
# Use NNLS to solve P(r)
X = hstack([Iq_obs, zeros(Nr), zeros(Nr)])
sol, resnorm = nnls(C, X)
pr = sol
jreg = K.dot(sol)
ireg = Kunsmeared.dot(sol)
jreg_extrap = Kfull.dot(sol)
ireg_extrap = Kfull_unsmeared.dot(sol)
if weightdata:
chisq = ((Iq - jreg)**2 / sd_sq).mean()
B = (K.T).dot(K / sdcol) # hessian of Chi^2/2
else:
chisq = ((Iq - jreg)**2).sum() / (Nq - 1)
B = (K.T).dot(K)
S0 = sum(-L.dot(sol)**2)
# L is the hessian of S0
U = L.T @ L + B / alpha
detsign, rlogdet = slogdet(U)
logdetA = Nr * log(0.5) + log(Nr + 1)
Q = alpha * S0 - 0.5 * chisq * Nq
# compute evidence or Posterior probability (Likelihood * Prior)
# this score is in log space, so need to transform back when looking at distribution
# prior for alpha is 1/alpha, or in log(1/alpha) -> -log(alpha)
evidence = 0.5 * logdetA + Q - 0.5 * rlogdet - log(alpha) - log(Dmax)
print("Chisq: {0:10.4f}\tEvidence:{1:10.5E}".format(chisq, evidence))
return jreg, ireg, jreg_extrap, ireg_extrap, q_full, r, pr, evidence
def iftv2(alpha,Dmax,q,Iq,sd,Nr,y,Wy,weightdata,smeared):
# calculates IFT solution and returns the complete set of solutions
# including extrapolation to zero angle
Nq = len(Iq)
# prepare q_full for extrapolation to zero angle
dq = q[1]-q[0]
q_extra = arange(0,q.min(),dq)
q_full = append(q_extra, q)
if weightdata:
sdnorm = 1./sd**2
sdnorm = (sdnorm/sdnorm.sum()) * float(len(sdnorm))
sdnorm = sdnorm.reshape(sdnorm.size,1)
else:
sdnorm = ones((Nq,1))
r = linspace(0,Dmax,Nr)
if smeared:
K = transc(q,r,y,Wy)
Kunsmeared = trans(q,r)
Kfull = transc(q_full,r,y,Wy)
Kfull_unsmeared = trans(q_full,r)
else:
K = trans(q,r)
Kunsmeared = K
Kfull = trans(q_full,r)
Kfull_unsmeared = Kfull
Kweighted = repeat(sdnorm,K.shape[1],axis=1) * K
# construct matrix L and Z
L = -0.5*eye(Nr,k=-1) + eye(Nr,k=0) - 0.5*eye(Nr,k=1)
Z = zeros((Nr,Nr))
Z[0,0] = 1.0
Z[-1,-1] = 1.0
C = vstack([Kweighted, alpha*L, 10.*alpha*Z])
X = hstack([Iq*sdnorm[:,0], zeros(Nr), zeros(Nr)])
sol,resnorm = nnls(C,X)
pr = sol
jreg = K.dot(sol)
ireg = Kunsmeared.dot(sol)
jreg_extrap = Kfull.dot(sol)
ireg_extrap = Kfull_unsmeared.dot(sol)
if weightdata:
chisq = ((Iq-jreg)**2 / sd**2).mean()
else:
chisq = 1.0
S0 = sum(-L.dot(sol)**2)
U = L + (K.T).dot(K)/alpha
detsign,rlogdet = slogdet(U)
rnorm = 0.5*(Nr*log(0.5)) + log(float(Nr+1))
evidence = rnorm + (alpha*S0 - 0.5*chisq*Nq) - 0.5*rlogdet
print "Chisq: {0:10.4f}\tEvidence:{1:10.5E}".format(chisq,evidence)
return jreg, ireg, jreg_extrap, ireg_extrap, q_full, r, pr, evidence