Detecting nonlinear stochastic systems using two independent hypothesis tests

Various systems in the real world can be nonlinear and stochastic, but because nonlinear time series analysis has been developed to distinguish nonlinear deterministic systems from linear stochastic systems, there have been no appropriate methods developed so far for testing the nonlinear stochasticity for a given system. Thus, here we propose a set of two hypothesis tests, one for the nonlinearity and one for the stochasticity, independent of each other. The test for the linearity is based on Fourier-transform-based surrogate data with a nonlinear test statistic, while the test for determinism depends on the theory of ordinal patterns or permutations recently developed intensively. We demonstrate the proposed set of tests with time series generated from toy models. In addition, we show that both a foreign exchange market and a temperature series in Tokyo could be nonlinear and stochastic, as well as sometimes with determinism beyond pseudoperiodicity.

Figure 1

The classifications of models depending on stationarity- nonstationarity, determinism-stochasticity, and linearity-nonlinearity. Associated methods for time series analysis are also annotated in the corresponding regions of these panels.

Figure 2

Classification of dynamic systems. Region 1 corresponds to the class of nonlinear deterministic systems. Region 2 corresponds to the class of linear stochastic systems. Region 3 corresponds to the class of nonlinear stochastic systems. The blue region is the class of systems the proposed approach can identify. In addition, we show the class of systems that pseudoperiodic surrogates [11] can identify.

Figure 3

The increases of the number of appearing permutations $N\left(l\right)$ given the length $l$ of permutations. In this figure, the values $lnN\left(l\right)$ for each model are aligned so that the values for $l=4$ become the same point. To evaluate the straightness, concavity, or convexity of these lines, use a straight edge.

Figure 4

Parts of recurrence plots for five models.

Figure 5

The results for surrogate data analysis on testing linearity. In each panel, the histogram shows the distribution for the surrogate data of Ref. [4] and the red dash-dotted vertical line corresponds to the value of the test statistic for the original data.

Figure 6

The results for surrogate data analysis on testing determinism beyond pseudoperiodicity. In each panel, the test statistics are normalized so that the minimum and the maximum for the surrogate data become 0 and 1, respectively.

Figure 7

Bifurcation diagram for the AR model converted into the logistic map.

Figure 8

Results for the AR model converted into the logistic map. Panel (a) shows the results for the test of linearity. Panel (b) shows the combined results for surrogate data analysis on determinism beyond pseudoperiodicity and the test of exponential growth for the number of appearing permutations. Panel (c) shows the results on the test of expansiveness. In panel (a), the test statistic for each $a$ is scaled so that the minimum and the maximum for the surrogate data become 0 and 1, respectively.