Create plot routine for extractor ``feets.extractors.ext_car.CAR``

[leliel12]

Create plot routine for extractor CAR.

Path: feets.extractors.ext_car.py

Features

• CAR_mean

• CAR_sigma

• CAR_tau

Extractor Documentation

In order to model the irregular sampled times series we use CAR
(Brockwell and Davis, 2002), a continious time auto regressive model.

CAR process has three parameters, it provides a natural and consistent way of estimating a characteristic time scale and variance of light-curves. CAR process is described by the following stochastic differential equation:

$$dX(t) = - \frac{1}{\tau} X(t)dt + \sigma_C \sqrt{dt} \epsilon(t) + bdt, \\\ for \: \tau, \sigma_C, t \geq 0$$

where the mean value of the lightcurve X(t) is b**τ and the variance is $\frac{\tau\sigma_C^2}{2}$. τ is the relaxation time of the process X(t), it can be interpreted as describing the variability amplitude of the time series. σ_C can be interpreted as describing the variability of the time series on time scales shorter than τ. ϵ(t) is a white noise process with zero mean and variance equal to one.

The likelihood function of a CAR model for a light-curve with observations x − {x₁, …, x_n} observed at times {t₁, …, t_n} with measurements error variances {δ₁², …, δ_n²} is:

$$p (x|b,\sigma_C,\tau) = \prod_{i=1}^n \frac{1}{ [2 \pi (\Omega_i + \delta_i^2 )]^{1/2}} exp \{-\frac{1}{2} \frac{(\hat{x}_i - x^*_i )^2}{\Omega_i + \delta^2_i}\} \\$$ $$x_i^* = x_i - b\tau \\$$ $$\hat{x}_0 = 0 \\$$ $$\Omega_0 = \frac{\tau \sigma^2_C}{2} \\$$ $$\hat{x}_i = a_i\hat{x}_{i-1} + \frac{a_i \Omega_{i-1}}{\Omega_{i-1} + \delta^2_{i-1}} (x^*_{i-1} + \hat{x}_{i-1}) \\$$ $$\Omega_i = \Omega_0 (1- a_i^2 ) + a_i^2 \Omega_{i-1} (1 - \frac{\Omega_{i-1}}{\Omega_{i-1} + \delta^2_{i-1}} )

To find the optimal parameters we maximize the likelihood with respect to σ_C and τ and calculate b as the mean magnitude of the light-curve divided by τ.

>>> fs = feets.FeatureSpace(
...     only=['CAR_sigma', 'CAR_tau','CAR_mean'])
>>> features, values = fs.extract(**lc_periodic)
>>> dict(zip(features, values))
{'CAR_mean': -9.230698873903961,
 'CAR_sigma': -0.21928049298842511,
 'CAR_tau': 0.64112037377348619}

References