def townsend_lombscargle_value(times, mags, omega):
'''
This calculates the periodogram value for each omega (= 2*pi*f). Mags must
be normalized to zero with variance scaled to unity.
'''
cos_omegat = npcos(omega * times)
sin_omegat = npsin(omega * times)
xc = npsum(mags * cos_omegat)
xs = npsum(mags * sin_omegat)
cc = npsum(cos_omegat * cos_omegat)
ss = npsum(sin_omegat * sin_omegat)
cs = npsum(cos_omegat * sin_omegat)
tau = nparctan(2 * cs / (cc - ss)) / (2 * omega)
ctau = npcos(omega * tau)
stau = npsin(omega * tau)
leftsumtop = (ctau * xc + stau * xs) * (ctau * xc + stau * xs)
leftsumbot = ctau * ctau * cc + 2.0 * ctau * stau * cs + stau * stau * ss
leftsum = leftsumtop / leftsumbot
rightsumtop = (ctau * xs - stau * xc) * (ctau * xs - stau * xc)
rightsumbot = ctau * ctau * ss - 2.0 * ctau * stau * cs + stau * stau * cc
rightsum = rightsumtop / rightsumbot
pval = 0.5 * (leftsum + rightsum)
return pval
def specwindow_lsp_value(times, mags, errs, omega):
'''
This calculates the peak associated with the spectral window function
for times and at the specified omega.
'''
norm_times = times - times.min()
tau = ((1.0 / (2.0 * omega)) * nparctan(
npsum(npsin(2.0 * omega * norm_times)) /
npsum(npcos(2.0 * omega * norm_times))))
lspval_top_cos = (npsum(1.0 * npcos(omega * (norm_times - tau))) *
npsum(1.0 * npcos(omega * (norm_times - tau))))
lspval_bot_cos = npsum((npcos(omega * (norm_times - tau))) *
(npcos(omega * (norm_times - tau))))
lspval_top_sin = (npsum(1.0 * npsin(omega * (norm_times - tau))) *
npsum(1.0 * npsin(omega * (norm_times - tau))))
lspval_bot_sin = npsum((npsin(omega * (norm_times - tau))) *
(npsin(omega * (norm_times - tau))))
lspval = 0.5 * ((lspval_top_cos / lspval_bot_cos) +
(lspval_top_sin / lspval_bot_sin))
return lspval
def specwindow_lsp_value(times, mags, errs, omega):
'''This calculates the peak associated with the spectral window function
for times and at the specified omega.
NOTE: this is classical Lomb-Scargle, not the Generalized
Lomb-Scargle. `mags` and `errs` are silently ignored since we're calculating
the periodogram of the observing window function. These are kept to present
a consistent external API so the `pgen_lsp` function below can call this
transparently.
Parameters
----------
times,mags,errs : np.array
The time-series to calculate the periodogram value for.
omega : float
The frequency to calculate the periodogram value at.
Returns
-------
periodogramvalue : float
The normalized periodogram at the specified test frequency `omega`.
'''
norm_times = times - times.min()
tau = ((1.0 / (2.0 * omega)) * nparctan(
npsum(npsin(2.0 * omega * norm_times)) /
npsum(npcos(2.0 * omega * norm_times))))
lspval_top_cos = (npsum(1.0 * npcos(omega * (norm_times - tau))) *
npsum(1.0 * npcos(omega * (norm_times - tau))))
lspval_bot_cos = npsum((npcos(omega * (norm_times - tau))) *
(npcos(omega * (norm_times - tau))))
lspval_top_sin = (npsum(1.0 * npsin(omega * (norm_times - tau))) *
npsum(1.0 * npsin(omega * (norm_times - tau))))
lspval_bot_sin = npsum((npsin(omega * (norm_times - tau))) *
(npsin(omega * (norm_times - tau))))
lspval = 0.5 * ((lspval_top_cos / lspval_bot_cos) +
(lspval_top_sin / lspval_bot_sin))
return lspval
def generalized_lsp_value(times, mags, errs, omega):
'''Generalized LSP value for a single omega.
P(w) = (1/YY) * (YC*YC/CC + YS*YS/SS)
where: YC, YS, CC, and SS are all calculated at T
and where: tan 2omegaT = 2*CS/(CC - SS)
and where:
Y = sum( w_i*y_i )
C = sum( w_i*cos(wT_i) )
S = sum( w_i*sin(wT_i) )
YY = sum( w_i*y_i*y_i ) - Y*Y
YC = sum( w_i*y_i*cos(wT_i) ) - Y*C
YS = sum( w_i*y_i*sin(wT_i) ) - Y*S
CpC = sum( w_i*cos(w_T_i)*cos(w_T_i) )
CC = CpC - C*C
SS = (1 - CpC) - S*S
CS = sum( w_i*cos(w_T_i)*sin(w_T_i) ) - C*S
'''
one_over_errs2 = 1.0 / (errs * errs)
W = npsum(one_over_errs2)
wi = one_over_errs2 / W
sin_omegat = npsin(omega * times)
cos_omegat = npcos(omega * times)
sin2_omegat = sin_omegat * sin_omegat
cos2_omegat = cos_omegat * cos_omegat
sincos_omegat = sin_omegat * cos_omegat
# calculate some more sums and terms
Y = npsum(wi * mags)
C = npsum(wi * cos_omegat)
S = npsum(wi * sin_omegat)
YpY = npsum(wi * mags * mags)
YpC = npsum(wi * mags * cos_omegat)
YpS = npsum(wi * mags * sin_omegat)
CpC = npsum(wi * cos2_omegat)
# SpS = npsum( wi*sin2_omegat )
CpS = npsum(wi * sincos_omegat)
# the final terms
YY = YpY - Y * Y
YC = YpC - Y * C
YS = YpS - Y * S
CC = CpC - C * C
SS = 1 - CpC - S * S # use SpS = 1 - CpC
CS = CpS - C * S
# calculate tau
tan_omega_tau_top = 2.0 * CS
tan_omega_tau_bottom = CC - SS
tan_omega_tau = tan_omega_tau_top / tan_omega_tau_bottom
tau = nparctan(tan_omega_tau / (2.0 * omega))
periodogramvalue = (YC * YC / CC + YS * YS / SS) / YY
return periodogramvalue
def generalized_lsp_value(times, mags, errs, omega):
'''Generalized LSP value for a single omega.
P(w) = (1/YY) * (YC*YC/CC + YS*YS/SS)
where: YC, YS, CC, and SS are all calculated at T
and where: tan 2omegaT = 2*CS/(CC - SS)
and where:
Y = sum( w_i*y_i )
C = sum( w_i*cos(wT_i) )
S = sum( w_i*sin(wT_i) )
YY = sum( w_i*y_i*y_i ) - Y*Y
YC = sum( w_i*y_i*cos(wT_i) ) - Y*C
YS = sum( w_i*y_i*sin(wT_i) ) - Y*S
CpC = sum( w_i*cos(w_T_i)*cos(w_T_i) )
CC = CpC - C*C
SS = (1 - CpC) - S*S
CS = sum( w_i*cos(w_T_i)*sin(w_T_i) ) - C*S
'''
one_over_errs2 = 1.0/(errs*errs)
W = npsum(one_over_errs2)
wi = one_over_errs2/W
sin_omegat = npsin(omega*times)
cos_omegat = npcos(omega*times)
sin2_omegat = sin_omegat*sin_omegat
cos2_omegat = cos_omegat*cos_omegat
sincos_omegat = sin_omegat*cos_omegat
# calculate some more sums and terms
Y = npsum( wi*mags )
C = npsum( wi*cos_omegat )
S = npsum( wi*sin_omegat )
YpY = npsum( wi*mags*mags)
YpC = npsum( wi*mags*cos_omegat )
YpS = npsum( wi*mags*sin_omegat )
CpC = npsum( wi*cos2_omegat )
# SpS = npsum( wi*sin2_omegat )
CpS = npsum( wi*sincos_omegat )
# the final terms
YY = YpY - Y*Y
YC = YpC - Y*C
YS = YpS - Y*S
CC = CpC - C*C
SS = 1 - CpC - S*S # use SpS = 1 - CpC
CS = CpS - C*S
# calculate tau
tan_omega_tau_top = 2.0*CS
tan_omega_tau_bottom = CC - SS
tan_omega_tau = tan_omega_tau_top/tan_omega_tau_bottom
tau = nparctan(tan_omega_tau/(2.0*omega))
periodogramvalue = (YC*YC/CC + YS*YS/SS)/YY
return periodogramvalue
def generalized_lsp_value(times, mags, errs, omega):
'''Generalized LSP value for a single omega.
The relations used are::
P(w) = (1/YY) * (YC*YC/CC + YS*YS/SS)
where: YC, YS, CC, and SS are all calculated at T
and where: tan 2omegaT = 2*CS/(CC - SS)
and where:
Y = sum( w_i*y_i )
C = sum( w_i*cos(wT_i) )
S = sum( w_i*sin(wT_i) )
YY = sum( w_i*y_i*y_i ) - Y*Y
YC = sum( w_i*y_i*cos(wT_i) ) - Y*C
YS = sum( w_i*y_i*sin(wT_i) ) - Y*S
CpC = sum( w_i*cos(w_T_i)*cos(w_T_i) )
CC = CpC - C*C
SS = (1 - CpC) - S*S
CS = sum( w_i*cos(w_T_i)*sin(w_T_i) ) - C*S
Parameters
----------
times,mags,errs : np.array
The time-series to calculate the periodogram value for.
omega : float
The frequency to calculate the periodogram value at.
Returns
-------
periodogramvalue : float
The normalized periodogram at the specified test frequency `omega`.
'''
one_over_errs2 = 1.0 / (errs * errs)
W = npsum(one_over_errs2)
wi = one_over_errs2 / W
sin_omegat = npsin(omega * times)
cos_omegat = npcos(omega * times)
sin2_omegat = sin_omegat * sin_omegat
cos2_omegat = cos_omegat * cos_omegat
sincos_omegat = sin_omegat * cos_omegat
# calculate some more sums and terms
Y = npsum(wi * mags)
C = npsum(wi * cos_omegat)
S = npsum(wi * sin_omegat)
CpS = npsum(wi * sincos_omegat)
CpC = npsum(wi * cos2_omegat)
CS = CpS - C * S
CC = CpC - C * C
SS = 1 - CpC - S * S # use SpS = 1 - CpC
# calculate tau
tan_omega_tau_top = 2.0 * CS
tan_omega_tau_bottom = CC - SS
tan_omega_tau = tan_omega_tau_top / tan_omega_tau_bottom
tau = nparctan(tan_omega_tau) / (2.0 * omega)
YpY = npsum(wi * mags * mags)
YpC = npsum(wi * mags * cos_omegat)
YpS = npsum(wi * mags * sin_omegat)
# SpS = npsum( wi*sin2_omegat )
# the final terms
YY = YpY - Y * Y
YC = YpC - Y * C
YS = YpS - Y * S
periodogramvalue = (YC * YC / CC + YS * YS / SS) / YY
return periodogramvalue