What is MICA?

MICA stands for Multiple and Inhomogeneous Component Analysis. It is a non-parametric approach to analyze light curves in reverberation mapping and infer the transfer functions and time delays.

A transfer function or delay map relates a time series to its driving time series as

\[L(t) = \int \Psi(\tau) C(t-\tau) d\tau.\]

mica2 expresses the transfer function into a family of displaced components. If taking Gaussians as an example, the transfer function is written

\[\Psi(\tau) = \sum_{k=1}^{K} \frac{f_k}{\sqrt{2\pi}\omega_k} \exp\left[-\frac{(\tau-\tau_k)^2}{2\omega_k^2}\right].\]

Note

There is a factor \(1/\sqrt{2\pi}\omega_k\) before the exponential in the above transfer function, therefore, \(f_k\) represents the response amplitide.

Schematic of the transfer function for a system that consists of discrete clouds.

As alternative options, mica2 also supports several types of transfer functions and their mixtures.

• Top-hat transfer function.

  \[\Psi(\tau) = \sum_{k=1}^{K} \frac{f_k}{2\omega_k} H(\tau, \tau_k, \omega_k),\]

  where \(H(\tau, \tau_k, \omega_k)\) is the top-hat function

  \[ \begin{align}\begin{aligned}H(\tau, \tau_k, \omega_k) =~1~{if}~\tau_k-\omega_k \leqslant \tau \leqslant \tau_k + \omega_k\\ =~0~else~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{aligned}\end{align} \]

• Gamma transfer function.

  \[\Psi(\tau) = \sum_{k=1}^{K} \frac{f_k}{\omega_k^2} (\tau-\tau_{k}) \exp\left[-\frac{(\tau-\tau_{k})}{\omega_k}\right].\]

  For \(k\)-th component, the transfer function is zero if \(\tau < \tau_{k}\), the peak location is \(\tau_k+\omega_k\) and the centroid lag is \(\tau_k+2\omega_k\).

• Exponential transfer function.

  \[\Psi(\tau) = \sum_{k=1}^{K} \frac{f_k}{\omega_k} \exp\left[-\frac{(\tau-\tau_{k})}{\omega_k}\right].\]

  For \(k\)-th component, the transfer function is zero if \(\tau < \tau_{k}\), the peak location is \(\tau_k\), and the centriod lag is \(\tau_k+\omega_k\).

Different shapes of transfer funcitons in MICA.

Here is an example for reverberation mapping analysis of the light curves from Hu et al. (2020) using MICA,

Reverberation mapping analysis of the light curve data for PG 2130+099 (Hu et al. 2020, ApJ, 890, 71). The left panel shows transfer functions and the right panel shows the light curves and their reconstructions.