Detecting nonlinearity in the light curves of active galactic nuclei

Abstract

This paper discusses a statistical method of detecting nonlinearity in the light curves of active galactic nuclei (AGN). We model the light curves of AGN by two-variate stochastic differential equation (SDE) in which one variable is observable but not the other. Applying a nonparametric model of the SDE as well as its parametric models of linear and quadratic functions to the light curves provided by the Kepler satellite, we estimate the three models and thereby compare their prediction accuracy to detect nonlinearity. The results suggest that there exist quadratic or other nonlinearities in the light curves, while the others exhibit linearity.

Appendix

This appendix presents an outline of derivation for NPF. More detail discussion can be seen in Shoji [18, 19].

First suppose the truncated Taylor’s expansion of f around ${{\varvec{x}}}_0=(x_0,y_0)$ up to second order for simplicity although higher order expansion is possible. Then, we get

$$\begin{aligned} f({{\varvec{x}}})\approx & {} {\tilde{f}}({{\varvec{x}}}),\\= & {} f({{\varvec{x}}}_0)+{\partial f\over \partial x}({{\varvec{x}}}_0)(x-x_0)+{\partial f\over \partial y} ({{\varvec{x}}}_0)(y-y_0),\\&+{1\over 2}{\partial ^2 f\over \partial x^2}({{\varvec{x}}}_0)(x-x_0)^2+{\partial ^2 f\over \partial x\partial y}({{\varvec{x}}}_0)(x-x_0)(y-y_0), \\&+{1\over 2}{\partial ^2 f\over \partial y^2}f({{\varvec{x}}}_0)(y-y_0)^2. \end{aligned}$$

Let $X_t=(X_{1,t},X_{2,t}){^T}$ and define $Y_t={\tilde{f}}(X_t)$ by putting $X_s$ into ${{\varvec{x}}}_0$ where $s\le t$. Because ${\tilde{f}}$ is at most quadratic, repeated application of the Ito’s formula gives

$$\begin{aligned} Y_t^{(0,0)}- Y_s^{(0,0)}= & {} \int _s^t Y_u^{(1,0)}\mathrm{d}X_{1,u}+\int _s^t Y_u^{(0,1)}\mathrm{d}X_{2,u}, \\&+{\sigma ^2\over 2}\int _s^t Y_u^{(0,2)}\mathrm{d}u,\\ Y_t^{(1,0)}- Y_s^{(1,0)}= & {} \int _s^t Y_u^{(2,0)}\mathrm{d}X_{1,u}+\int _s^t Y_u^{(1,1)}\mathrm{d}X_{2,u},\\ Y_t^{(0,1)}- Y_s^{(0,1)}= & {} \int _s^t Y_u^{(1,1)}\mathrm{d}X_{1,u}+\int _s^t Y_u^{(0,2)}\mathrm{d}X_{2,u},\\ Y_t^{(2,0)}- Y_s^{(2,0)}= & {} 0,\\ Y_t^{(1,1)}- Y_s^{(1,1)}= & {} 0,\\ Y_t^{(0,2)}- Y_s^{(0,2)}= & {} 0. \end{aligned}$$

where

$$\begin{aligned} Y_t^{(i,j)}={\partial ^{i+j}{\tilde{f}}\over \partial ^i x\partial ^j y}(X_t). \end{aligned}$$

To discretize the process at discrete times $\{t_k\}_{1\le k\le n}$, we assume that each integrand is a constant over the time interval $[t_k,t_{k+1})$. However, because we assume no function form of f, we must estimate $Y_t^{(i,j)}$ from discrete observation $\{Z_{t_k}\}_{1\le k\le n}$. To this end, we replace $Y_{t_k}^{(i,j)}$ with its filter value, or $Y_{t_k|t_k}^{(i,j)}=E[Y_{t_k}^{(i,j)}|\{Z_{t_j}\}_{1\le j\le k}]$. This replacement is also used for the formulation of the extended Kalman filter algorithm; see Anderson and Moor [2] and Jazwinski [6]. Furthermore, the last 3 equations imply that $Y_{t_k}^{(i,j)}(i+j=2)$ are constant. Denoting them by $\theta _j$ $(0\le j\le 2)$, we estimate $\theta _j$ as nuisance parameters from the data. Under the above setting, the following equations in discrete time are obtained:

$$\begin{aligned} Y_{t_{k+1}}^{(0,0)}- Y_{t_k}^{(0,0)}= & {} Y_{t_k|t_k}^{(1,0)}(X_{1,t_{k+1}}- X_{1,t_k}) + Y_{t_k|t_k}^{(0,1)} (X_{2,t_{k+1}}- X_{2,t_k}),\\&+{\sigma ^2\over 2}\theta _2 (t_{k+1}-t_k),\\ Y_{t_{k+1}}^{(1,0)}- Y_{t_k}^{(1,0)}= & {} \theta _0(X_{1,t_{k+1}}- X_{1,t_k}) + \theta _1 (X_{2,t_{k+1}}- X_{2,t_k}), \\ Y_{t_{k+1}}^{(0,1)}- Y_{t_k}^{(0,1)}= & {} \theta _{1}(X_{1,t_{k+1}}- X_{1,t_k}) +\theta _2(X_{2,t_{k+1}}- X_{2,t_k}). \end{aligned}$$

By substituting $f(X_{t})$ for ${\tilde{f}}(X_{t})=Y_{t}^{(0,0)}$ together with,

$$\begin{aligned} Y_{t}^{(0,0)}- Y_{t_k}^{(0,0)}= Y_{t_k|t_k}^{(1,0)}(X_{1,t}-X_{1,t_k})+ Y_{t_k|t_k}^{(0,1)}(X_{2,t}-X_{2,t_k}) +{\sigma ^2\over 2} \theta _2(t-t_k), \end{aligned}$$

we get a linear approximate SDE of (1)–(2). Since a linear SDE has an exact form of the solution, the approximate discretization at $t=t_{k+1}$ is given by

$$\begin{aligned} X_{t_{k+1}}= & {} X_{t_k} +J_{t_k}^{-1}\left\{ \exp (J_{t_k}\Delta t)-I\right\} \Phi _{t_k}\\&+ (J_{t_k}^{-1})^2\left\{ \exp (J_{t_k}\Delta t)-I-J_{t_k}\Delta t\right\} M_{t_k}\\&+\int _{t_k}^{t_{k+1}}\exp \{J_{t_k}(t_{k+1}-u)\}S dB_u. \end{aligned}$$

where

$$\begin{aligned} \Phi _{t_k}= & {} (X_{2,t_k},Y^{(0,0)}_{t_k})^T,\\ J_t= & {} \left( \begin{array}{cc} 0&{} 1\\ Y^{(1,0)}_{t|t}&{} Y^{(0,1)}_{t|t} \end{array} \right) ,\\ M_t= & {} (0,{\sigma ^2\over 2}\theta _2)^T. \end{aligned}$$

See [14] and Shoji and Ozaki [17] for details. Using this and the equations of $Y^{(i,j)}_{t_{k+1}}-Y^{(i,j)}_{t_{k}}$ above, we obtain the formula for NPF.