Filling gaps in chaotic time series

Abstract

We propose a method for filling arbitrarily wide gaps in deterministic time series. Crucial to the method is the ability to apply Takens' theorem in order to reconstruct the dynamics underlying the time series. We introduce a functional to evaluate the degree of compatibility of a filling sequence of data with the reconstructed dynamics. An algorithm for finding highly compatible filling sequences with a reasonable computational effort is then discussed.

Introduction

One problem faced by many practitioners in the applied sciences is the presence of gaps (i.e. sequences of missing data) in observed time series, which makes hard or impossible any analysis. The problem is routinely solved by interpolation if the gap width is very short, but it becomes a formidable one if the gap width is larger than some time scale characterizing the predictability of the time series.

If the physical system under study is described by a small set of coupled ordinary differential equations, then a theorem by Takens [1], [2] suggests that from a single time series it is possible to build-up a mathematical model whose dynamics is diffeomorph to that of the system under examination. In this Letter we leverage the dynamic reconstruction theorem of Takens for filling an arbitrarily wide gap in a time series.

It is important to stress that the goal of the method is not that of recovering a good approximation to the lost data. Sensitive dependence on initial conditions, and imperfections of the reconstructed dynamics, make this goal a practical impossibility, except for some special cases, such as small gap width, or periodic dynamics. We rather aim at giving one or more surrogate data which can be considered compatible with the observed dynamics, in a sense which will be made rigorous in the following.

We shall assume that an observable quantity s is a function of the state of a continuous-time, low-dimensional dynamical system, whose time evolution is confined on a strange attractor (that is, we explicitly discard transient behavior). Both the explicit form of the equations governing the dynamical system and the function which links its state to the signal $s(t)$ may be unknown. We also assume that an instrument samples $s(t)$ at regular intervals of length Δt, yielding an ordered set of $\overline{N}$ data

$$s_i=s((i-1)\mathrm{\Delta }t),i=1,\dots ,\overline{N}.$$

If, for any cause, the instrument is unable to record the value of s for a number of times, there will be some invalid entries in the time series $\{s_i\}$, for some values of the index i.

From the time series $\{s_i\}$ we reconstruct the underlying dynamics with the technique of delay coordinates. That is, we shall invoke Takens' theorem [1], [2] and claim that the m-dimensional vectors

$${\mathbf{x}}_i=(s_i,s_{i+\tau },\dots ,s_{i+(m-1)\tau })$$

lie on a curve in ${\mathbb{R}}^m$ which is diffeomorph to the curve followed in its (unknown) phase space by the state of the dynamical system which originated the signal $s(t)$. Here τ is a positive integer, and i now runs only up to $N=\overline{N}-(m-1)\tau$. Severals pitfalls have to be taken into account in order to choose the most appropriate values for m and τ. Strong constraints also come from the length of the time series, compared to the characteristic time scales of the dynamical system, and from the amount of instrumental noise which affects the data. We shall not review these issues here, but address the reader to Refs. [3], [4], [5].

We note that gaps (that is, invalid entries) in the time series $\{s_i\}$ do not prevent a successful reconstruction of a set $\mathcal{R}=\{{\mathbf{x}}_i\}$ of state vectors, unless the total width of the gaps is comparable with $\overline{N}$. We simply mark as “missing” any reconstructed vector ${\mathbf{x}}_i$ whose components are not all valid entries. If the gap in the signal s spans more than $(m-1)\ast \tau$ data points, then it will be mapped into a contiguous gap in the sequence of reconstructed vectors.

If the valid vectors of $\mathcal{R}$ sample well enough the underlying strange attractor embedded in ${\mathbb{R}}^m$, one may hope to find, by means of a suitable interpolation technique, a vector field $\mathbf{F}:U\to {\mathbb{R}}^m$, such that within an open set U of ${\mathbb{R}}^m$ containing all the vectors ${\mathbf{x}}_i$, the observed dynamics can be approximated by

$$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x}).$$

This very idea is at the base of several forecasting schemes, where one takes the last observed vector ${\mathbf{x}}_N$ as the initial condition for Eq. (2), and integrates it forward in time (see, e.g., [7], [8]).

The gap-filling problem was framed in terms of forecasts by Serre et al. [9]. Their method, which amounts to a special form of the shooting algorithm for boundary value problems, is limited by the predictability properties of the dynamics, and cannot fill gaps of arbitrary width.

The rest of this Letter is organized as follows: in Section 2 we cast the problem as a variational one, where a functional measures how well a candidate filling trajectory agrees with the vector field defining the observed dynamics. Then an algorithm is proposed for finding a filling trajectory. In Section 3 we give an example of what can be obtained with this method. Finally, we discuss the algorithm and offer some speculations on future works in Section 4.