def chi2sigma(chi0,chi1,nu0,nharm):
from scipy.stats import betai
from scipy.special import betaln
nu1 = nu0 - 2.*nharm
dfn = nu0-nu1
dfd = nu1
sigma = 0.
if (dfn>0 and dfd>0 and chi0>chi1):
fstat = (chi0/chi1-1.)*dfd/dfn
prob = betai( dfd/2., dfn/2., dfd/(dfd+dfn*fstat) )
if (dfd<=0 or dfn<=0): lprob=0.
elif (chi1==0): lprob=-999.
elif (prob==0): lprob = 0.5*dfd*log( dfd/(dfd+dfn*fstat) )-log(dfd/2.)-betaln(dfd/2.,dfn/2.)
else: lprob = log(prob)
sigma = lprob2sigma(lprob)
return sigma
def chi2sigma(chi0, chi1, nu0, nharm):
from scipy.stats import betai
from scipy.special import betaln
nu1 = nu0 - 2. * nharm
dfn = nu0 - nu1
dfd = nu1
sigma = 0.
if (dfn > 0 and dfd > 0 and chi0 > chi1):
fstat = (chi0 / chi1 - 1.) * dfd / dfn
prob = betai(dfd / 2., dfn / 2., dfd / (dfd + dfn * fstat))
if (dfd <= 0 or dfn <= 0): lprob = 0.
elif (chi1 == 0): lprob = -999.
elif (prob == 0):
lprob = 0.5 * dfd * log(dfd / (dfd + dfn * fstat)) - log(
dfd / 2.) - betaln(dfd / 2., dfn / 2.)
else:
lprob = log(prob)
sigma = lprob2sigma(lprob)
return sigma
def lomb(time, signal, error, f1, df, numf, nharm=8, psdmin=6., detrend_order=0,freq_zoom=10.,tone_control=1.,return_model=True,lambda0=1.,lambda0_range=[-8,6]):
"""
C version of lomb_scargle:
Simultaneous fit of a sum of sinusoids by weighted, linear least squares.
model(t) = Sum_k Ck*t^k + Sum_i Sum_j Aij sin(2*pi*j*fi*(t-t0)+phij), i=[1,nfreq], j=[1,nharm]
[t0 defined such that ph11=0]
Inputs:
time: time vector
signal: data vector
error: data uncertainty vector
df: frequency step
numf: number of frequencies to consider
detrend_order: order of polynomial detrending (Ck orthogonol polynomial terms above;
0 floating mean; <0 no detrending)
psdmin: refine periodogram values with larger psd using multi-harmonic fit
nharm: number of harmonics to use in refinement
lambda0: typical value for regularization parameter (expert parameter)
lambda0_range: allowable range for log10 of regularization parameter
Output:
psd: power spectrum on frequency grid: f1,f1+df,...,f1+numf*df
out_dict: dictionary describing various parameters of the multiharmonic fit at
the best-fit frequency
"""
numt = len(time)
freq_zoom = round(freq_zoom/2.)*2.
dord = detrend_order
if (detrend_order<0):
dord=0
if (tone_control<0):
tone_control=0.
# polynomial terms
coef = empty(dord+1,dtype='float64')
norm = empty(dord+1,dtype='float64')
wth0 = (1./error).astype('float64')
s0 = dot(wth0,wth0)
wth0 /= sqrt(s0)
cn = (signal*wth0).astype('float64')
coef[0] = dot(cn,wth0); cn0 = coef[0]; norm[0] = 1.
cn -= coef[0]*wth0
vcn = 1.
# sin's and cosin's for later
tt = 2*pi*time.astype('float64')
sinx,cosx = sin(tt*f1)*wth0,cos(tt*f1)*wth0
sinx_step,cosx_step = sin(tt*df),cos(tt*df)
sinx_back,cosx_back = -sin(tt*df/2.),cos(tt*df/2)
sinx_smallstep,cosx_smallstep = sin(tt*df/freq_zoom),cos(tt*df/freq_zoom)
npar=2*nharm
hat_matr = empty((npar,numt),dtype='float64')
hat0 = empty((npar,dord+1),dtype='float64')
hat_hat = empty((npar,npar),dtype='float64')
soln = empty(npar,dtype='float64')
psd = zeros(numf,dtype='float64')
# detrend the data and create the orthogonal detrending basis
if (dord>0):
wth = empty((dord+1,numt),dtype='float64')
wth[0,:] = wth0
else:
wth = wth0
for i in range(detrend_order):
f = wth[i,:]*tt/(2*pi)
for j in range(i+1):
f -= dot(f,wth[j,:])*wth[j,:]
norm[i+1] = sqrt(dot(f,f)); f /= norm[i+1]
coef[i+1] = dot(cn,f)
cn -= coef[i+1]*f
wth[i+1,:] = f
vcn += (f/wth0)**2
chi0 = dot(cn,cn)
varcn = chi0/(numt-1-dord)
psdmin *= 2*varcn
Tr = array(0.,dtype='float64')
ifreq = array(0,dtype='int32')
lambda0 = array(lambda0/s0,dtype='float64')
lambda0_range = 10**array(lambda0_range,dtype='float64')/s0
vars=['numt','numf','nharm','detrend_order','psd','cn','wth','sinx','cosx','sinx_step','cosx_step','sinx_back','cosx_back','sinx_smallstep','cosx_smallstep','hat_matr','hat_hat','hat0','soln','chi0','freq_zoom','psdmin','tone_control','lambda0','lambda0_range','Tr','ifreq']
weave.inline(lomb_code, vars, support_code = eigs_code + lomb_scargle_support,force=0)
hat_hat /= s0
ii = arange(nharm,dtype='int32')
soln[0:nharm] /= (1.+ii)**2; soln[nharm:] /= (1.+ii)**2
if (detrend_order>=0):
hat_matr0 = outer(hat0[:,0],wth0)
for i in range(detrend_order):
hat_matr0 += outer(hat0[:,i+1],wth[i+1,:])
modl = dot(hat_matr.T,soln); modl0 = dot(hat_matr0.T,soln)
coef0 = dot(soln,hat0)
coef -= coef0
if (detrend_order>=0):
hat_matr -= hat_matr0
out_dict={}
out_dict['chi0'] = chi0*s0
if (return_model):
if (dord>0):
out_dict['trend'] = dot(coef,wth)/wth0
else:
out_dict['trend'] = coef[0] + 0*wth0
out_dict['model'] = modl/wth0 + out_dict['trend']
j = psd.argmax()
freq = f1+df*j + (ifreq/freq_zoom - 1/2.)*df
tt = (time*freq) % 1. ; s =tt.argsort()
out_dict['freq'] = freq
out_dict['s0'] = s0
out_dict['chi2'] = (chi0 - psd[j])*s0
out_dict['psd'] = psd[j]*0.5/varcn
out_dict['lambda0'] = lambda0*s0
out_dict['gcv_weight'] = (1-3./numt)/Tr
out_dict['trace'] = Tr
out_dict['nu0'] = numt - npar
npars = (1-Tr)*numt/2
out_dict['nu'] = numt-npars
out_dict['npars'] = npars
A0, B0 = soln[0:nharm],soln[nharm:]
hat_hat /= outer( hstack(((1.+ii)**2,(1.+ii)**2)),hstack(((1.+ii)**2,(1.+ii)**2)) )
err2 = diag(hat_hat)
vA0, vB0 = err2[0:nharm], err2[nharm:]
covA0B0 = hat_hat[(ii,nharm+ii)]
if (return_model):
vmodl = vcn/s0 + dot( (hat_matr/wth0).T, dot(hat_hat, hat_matr/wth0) )
vmodl0 = vcn/s0 + dot( (hat_matr0/wth0).T, dot(hat_hat, hat_matr0/wth0) )
out_dict['model_error'] = sqrt(diag(vmodl))
out_dict['trend_error'] = sqrt(diag(vmodl0))
amp = sqrt(A0**2+B0**2)
damp = sqrt( A0**2*vA0 + B0**2*vB0 + 2.*A0*B0*covA0B0 )/amp
phase = arctan2( B0,A0 )
rel_phase = phase - phase[0]*(1.+ii)
rel_phase = arctan2( sin(rel_phase),cos(rel_phase) )
dphase = 0.*rel_phase
for i in range(nharm-1):
j=i+1
v = array([-A0[0]*(1.+j)/amp[0]**2,B0[0]*(1.+j)/amp[0]**2,A0[j]/amp[j]**2,-B0[j]/amp[j]**2])
jj=array([0,nharm,j,j+nharm])
m = hat_hat[ix_(jj,jj)]
dphase[j] = sqrt( dot(dot(v,m),v) )
out_dict['amplitude'] = amp
out_dict['amplitude_error'] = damp
out_dict['rel_phase'] = rel_phase
out_dict['rel_phase_error'] = dphase
out_dict['time0'] = -phase[0]/(2*pi*freq)
ncp = norm.cumprod()
out_dict['trend_coef'] = coef/ncp
out_dict['cn0'] = out_dict['trend_coef'][0] - cn0
out_dict['trend_coef_error'] = sqrt( ( 1./s0 + diag(dot(hat0.T,dot(hat_hat,hat0))) )/ncp**2 )
out_dict['cn0_error'] = out_dict['trend_coef_error'][0]
prob = fdist.sf( 0.5*(numt-1.-dord)*(1.-out_dict['chi2']/out_dict['chi0']), 2,numt-1-dord )
out_dict['signif'] = lprob2sigma(log(prob))
return 0.5*psd/varcn,out_dict
def lomb(time, signal, error, f1, df, numf, nharm=8, psdmin=6., detrend_order=0,freq_zoom=10.,tone_control=1.,return_model=True,lambda0=1.,lambda0_range=[-8,6]):
"""
C version of lomb_scargle:
Simultaneous fit of a sum of sinusoids by weighted, linear least squares.
model(t) = Sum_k Ck*t^k + Sum_i Sum_j Aij sin(2*pi*j*fi*(t-t0)+phij), i=[1,nfreq], j=[1,nharm]
[t0 defined such that ph11=0]
Inputs:
time: time vector
signal: data vector
error: data uncertainty vector
df: frequency step
numf: number of frequencies to consider
detrend_order: order of polynomial detrending (Ck orthogonol polynomial terms above;
0 floating mean; <0 no detrending)
psdmin: refine periodogram values with larger psd using multi-harmonic fit
nharm: number of harmonics to use in refinement
lambda0: typical value for regularization parameter (expert parameter)
lambda0_range: allowable range for log10 of regularization parameter
Output:
psd: power spectrum on frequency grid: f1,f1+df,...,f1+numf*df
out_dict: dictionary describing various parameters of the multiharmonic fit at
the best-fit frequency
"""
numt = len(time)
freq_zoom = round(freq_zoom/2.)*2.
dord = detrend_order
if (detrend_order<0):
dord=0
if (tone_control<0):
tone_control=0.
# polynomial terms
coef = empty(dord+1,dtype='float64')
norm = empty(dord+1,dtype='float64')
wth0 = (1./error).astype('float64')
s0 = dot(wth0,wth0)
wth0 /= sqrt(s0)
cn = (signal*wth0).astype('float64')
coef[0] = dot(cn,wth0); cn0 = coef[0]; norm[0] = 1.
cn -= coef[0]*wth0
vcn = 1.
# sin's and cosin's for later
tt = 2*pi*time.astype('float64')
sinx,cosx = sin(tt*f1)*wth0,cos(tt*f1)*wth0
sinx_step,cosx_step = sin(tt*df),cos(tt*df)
sinx_back,cosx_back = -sin(tt*df/2.),cos(tt*df/2)
sinx_smallstep,cosx_smallstep = sin(tt*df/freq_zoom),cos(tt*df/freq_zoom)
npar=2*nharm
hat_matr = empty((npar,numt),dtype='float64')
hat0 = empty((npar,dord+1),dtype='float64')
hat_hat = empty((npar,npar),dtype='float64')
soln = empty(npar,dtype='float64')
psd = zeros(numf,dtype='float64')
# detrend the data and create the orthogonal detrending basis
if (dord>0):
wth = empty((dord+1,numt),dtype='float64')
wth[0,:] = wth0
else:
wth = wth0
for i in xrange(detrend_order):
f = wth[i,:]*tt/(2*pi)
for j in xrange(i+1):
f -= dot(f,wth[j,:])*wth[j,:]
norm[i+1] = sqrt(dot(f,f)); f /= norm[i+1]
coef[i+1] = dot(cn,f)
cn -= coef[i+1]*f
wth[i+1,:] = f
vcn += (f/wth0)**2
chi0 = dot(cn,cn)
varcn = chi0/(numt-1-dord)
psdmin *= 2*varcn
Tr = array(0.,dtype='float64')
ifreq = array(0,dtype='int32')
lambda0 = array(lambda0/s0,dtype='float64')
lambda0_range = 10**array(lambda0_range,dtype='float64')/s0
vars=['numt','numf','nharm','detrend_order','psd','cn','wth','sinx','cosx','sinx_step','cosx_step','sinx_back','cosx_back','sinx_smallstep','cosx_smallstep','hat_matr','hat_hat','hat0','soln','chi0','freq_zoom','psdmin','tone_control','lambda0','lambda0_range','Tr','ifreq']
weave.inline(lomb_code, vars, support_code = eigs_code + lomb_scargle_support,force=0)
hat_hat /= s0
ii = arange(nharm,dtype='int32')
soln[0:nharm] /= (1.+ii)**2; soln[nharm:] /= (1.+ii)**2
if (detrend_order>=0):
hat_matr0 = outer(hat0[:,0],wth0)
for i in xrange(detrend_order):
hat_matr0 += outer(hat0[:,i+1],wth[i+1,:])
modl = dot(hat_matr.T,soln); modl0 = dot(hat_matr0.T,soln)
coef0 = dot(soln,hat0)
coef -= coef0
if (detrend_order>=0):
hat_matr -= hat_matr0
out_dict={}
out_dict['chi0'] = chi0*s0
if (return_model):
if (dord>0):
out_dict['trend'] = dot(coef,wth)/wth0
else:
out_dict['trend'] = coef[0] + 0*wth0
out_dict['model'] = modl/wth0 + out_dict['trend']
j = psd.argmax()
freq = f1+df*j + (ifreq/freq_zoom - 1/2.)*df
tt = (time*freq) % 1. ; s =tt.argsort()
out_dict['freq'] = freq
out_dict['s0'] = s0
out_dict['chi2'] = (chi0 - psd[j])*s0
out_dict['psd'] = psd[j]*0.5/varcn
out_dict['lambda0'] = lambda0*s0
out_dict['gcv_weight'] = (1-3./numt)/Tr
out_dict['trace'] = Tr
out_dict['nu0'] = numt - npar
npars = (1-Tr)*numt/2
out_dict['nu'] = numt-npars
out_dict['npars'] = npars
A0, B0 = soln[0:nharm],soln[nharm:]
hat_hat /= outer( hstack(((1.+ii)**2,(1.+ii)**2)),hstack(((1.+ii)**2,(1.+ii)**2)) )
err2 = diag(hat_hat)
vA0, vB0 = err2[0:nharm], err2[nharm:]
covA0B0 = hat_hat[(ii,nharm+ii)]
if (return_model):
vmodl = vcn/s0 + dot( (hat_matr/wth0).T, dot(hat_hat, hat_matr/wth0) )
vmodl0 = vcn/s0 + dot( (hat_matr0/wth0).T, dot(hat_hat, hat_matr0/wth0) )
out_dict['model_error'] = sqrt(diag(vmodl))
out_dict['trend_error'] = sqrt(diag(vmodl0))
amp = sqrt(A0**2+B0**2)
damp = sqrt( A0**2*vA0 + B0**2*vB0 + 2.*A0*B0*covA0B0 )/amp
phase = arctan2( B0,A0 )
rel_phase = phase - phase[0]*(1.+ii)
rel_phase = arctan2( sin(rel_phase),cos(rel_phase) )
dphase = 0.*rel_phase
for i in xrange(nharm-1):
j=i+1
v = array([-A0[0]*(1.+j)/amp[0]**2,B0[0]*(1.+j)/amp[0]**2,A0[j]/amp[j]**2,-B0[j]/amp[j]**2])
jj=array([0,nharm,j,j+nharm])
m = hat_hat[ix_(jj,jj)]
dphase[j] = sqrt( dot(dot(v,m),v) )
out_dict['amplitude'] = amp
out_dict['amplitude_error'] = damp
out_dict['rel_phase'] = rel_phase
out_dict['rel_phase_error'] = dphase
out_dict['time0'] = -phase[0]/(2*pi*freq)
ncp = norm.cumprod()
out_dict['trend_coef'] = coef/ncp
out_dict['cn0'] = out_dict['trend_coef'][0] - cn0
out_dict['trend_coef_error'] = sqrt( ( 1./s0 + diag(dot(hat0.T,dot(hat_hat,hat0))) )/ncp**2 )
out_dict['cn0_error'] = out_dict['trend_coef_error'][0]
prob = fdist.sf( 0.5*(numt-1.-dord)*(1.-out_dict['chi2']/out_dict['chi0']), 2,numt-1-dord )
out_dict['signif'] = lprob2sigma(log(prob))
return 0.5*psd/varcn,out_dict