Delay estimation from noisy time series

We propose here a method to estimate a delay from a time series taking advantage of analysis of random walks with delay. This method is applicable to a time series coming out of a system which is or can be approximated as a linear feedback system with delay and noise. We successfully test the method with a time series generated by discrete Langevin equation with delay. Estimation of delay from a noisy time series has attracted much attention. Especially, when the time series is chaotic, estimation of delay has a practical motivation: time{delayed coordinates are typically used to estimate fractal dimensions and Lyapunov exponents. There are series of works considering the subject from this viewpoint [1, 2, 3, 4]. Another viewpoint is to consider that a noisy time series consists of underlying deterministic dynamics with past in uence and a noise term. Some statisticians have taken this stand and devised methods of analysis, for example, using the generalized Langevin equation [5, 6] and uctuation dissipation theorem[7]. In physiological elds, a more specialized case of the feedback delay associated with control system has attracted great deal of interest. A series of attempts has been made to estimate delay from physiological experimental data (see e.g., [8, 9, 10]). Against this background, we present here a method of estimating delay from a time series which is or is approximately generated by a noisy linear feedback system. We take advantage of analysis of random walks whose transition probability depends on the walker's position in a xed interval past. Such random walks are termed delayed random walks and were proposed recently as a platform on which to study systems with both noise and delay [9, 11]. We will describe each step of the method in a transparent manner for implementation into computer algorithms. The method is tested to show its e ectiveness on several test time series generated by discrete Langevin equation with delay [12]. Let us rst describe the delayed random walk, on whose analysis we base our method for delay estimation. We consider a random walk which takes a unit step in a unit time. The delayed random walk we start with is an extension of a position dependent random walk whose step toward the origin is more likely when no delay exists. Formally, it has the following de nition: P (X t+1 = n;X t+1 = s) = g(n 1; s 1)P (X t = n 1;X t+1 = s;X t = s 1) + g(n 1; s+ 1)P (X t = n 1;X t+1 = s;X t = s+ 1) + f(n+ 1; s 1)P (X t = n + 1;X t+1 = s;X t = s 1) + f(n+ 1; s+ 1)P (X t = n + 1;X t+1 = s;X t = s + 1); (1) f(x; y) = 1 2 (1 + x+ y) g(x; y) = 1 2 (1 x y) (2)