import numpy as np
from sklearn.utils import check_random_state
[docs]def generate_power_law(N, dt, beta, generate_complex=False, random_state=None):
"""Generate a power-law light curve
This uses the method from Timmer & Koenig [1]_
Parameters
----------
N : integer
Number of equal-spaced time steps to generate
dt : float
Spacing between time-steps
beta : float
Power-law index. The spectrum will be (1 / f)^beta
generate_complex : boolean (optional)
if True, generate a complex time series rather than a real time series
random_state : None, int, or np.random.RandomState instance (optional)
random seed or random number generator
Returns
-------
x : ndarray
the length-N
References
----------
.. [1] Timmer, J. & Koenig, M. On Generating Power Law Noise. A&A 300:707
"""
random_state = check_random_state(random_state)
dt = float(dt)
N = int(N)
Npos = int(N / 2)
# Nneg = int((N - 1) / 2)
domega = (2 * np.pi / dt / N)
if generate_complex:
omega = domega * np.fft.ifftshift(np.arange(N) - int(N / 2))
else:
omega = domega * np.arange(Npos + 1)
x_fft = np.zeros(len(omega), dtype=complex)
x_fft.real[1:] = random_state.normal(0, 1, len(omega) - 1)
x_fft.imag[1:] = random_state.normal(0, 1, len(omega) - 1)
x_fft[1:] *= (1. / omega[1:]) ** (0.5 * beta)
x_fft[1:] *= (1. / np.sqrt(2))
# by symmetry, the Nyquist frequency is real if x is real
if (not generate_complex) and (N % 2 == 0):
x_fft.imag[-1] = 0
if generate_complex:
x = np.fft.ifft(x_fft)
else:
x = np.fft.irfft(x_fft, N)
return x
[docs]def generate_damped_RW(t_rest, tau=300., z=2.0,
xmean=0, SFinf=0.3, random_state=None):
"""Generate a damped random walk light curve
This uses a damped random walk model to generate a light curve similar
to that of a QSO [1]_.
Parameters
----------
t_rest : array_like
rest-frame time. Should be in increasing order
tau : float
relaxation time
z : float
redshift
xmean : float (optional)
mean value of random walk; default=0
SFinf : float (optional
Structure function at infinity; default=0.3
random_state : None, int, or np.random.RandomState instance (optional)
random seed or random number generator
Returns
-------
x : ndarray
the sampled values corresponding to times t_rest
Notes
-----
The differential equation is (with t = time/tau):
dX = -X(t) * dt + sigma * sqrt(tau) * e(t) * sqrt(dt) + b * tau * dt
where e(t) is white noise with zero mean and unit variance, and
Xmean = b * tau
SFinf = sigma * sqrt(tau / 2)
so
dX(t) = -X(t) * dt + sqrt(2) * SFint * e(t) * sqrt(dt) + Xmean * dt
References
----------
.. [1] Kelly, B., Bechtold, J. & Siemiginowska, A. (2009)
Are the Variations in Quasar Optical Flux Driven by Thermal
Fluctuations? ApJ 698:895 (2009)
"""
# Xmean = b * tau
# SFinf = sigma * sqrt(tau / 2)
t_rest = np.atleast_1d(t_rest)
if t_rest.ndim != 1:
raise ValueError('t_rest should be a 1D array')
random_state = check_random_state(random_state)
N = len(t_rest)
t_obs = t_rest * (1. + z) / tau
x = np.zeros(N)
x[0] = random_state.normal(xmean, SFinf)
E = random_state.normal(0, 1, N)
for i in range(1, N):
dt = t_obs[i] - t_obs[i - 1]
x[i] = (x[i - 1]
- dt * (x[i - 1] - xmean)
+ np.sqrt(2) * SFinf * E[i] * np.sqrt(dt))
return x