Nonparametric testing for serial independence using the NRL statistic

ABSTRACT

Serial independence is tested using two measures of the effects of noise reduction in chaotic data, proposed by Orzeszko (2005). The extensive Monte Carlo simulations on the size and power of the new permutation-based tests are performed. Four popular nonparametric tests for serial independence are employed as a benchmark. The conducted simulations show that the new tests may be effective tools for detecting different kinds of dependencies. Moreover, they can distinguish between nonlinearity in the mean and nonlinearity in the variance.

KEYWORDS:

• Nonlinearity
• Nonparametric tests
• NRL statistic
• Serial independence

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G10
• 91B84

Funding

This work was supported by the National Science Centre under Grant 2013/11/B/HS4/00578.

Notes

1 Orzeszko (2008) proposed the modified quantity NRL, where a diameter (i.e., the maximal distance between delay vectors) is used to measure the distortion caused by noise reduction. In this version, the quantity NRL is defined by the following formula: where d_min and diam relate to a series after noise reduction, and d⁰_min, diam⁰ are calculated for the raw noisy data. Since the modified quantity NRL consists in comparing two series—the original and cleaned ones, it measures a relative decrease in d_min. In contrast to the quantity NRL defined by (3) it makes this variant of the quantity NRL useless in the context of testing for serial independence.

2 In this context they are applied directly to investigated time series so no minimization is performed.

3 In fact, samples of 1,000 and 1,700 series were generated, respectively, but, in order to attenuate the effect of initial values, the first 700 observations were truncated.

4 As argued in Section 2, for chaotic series it is expected that the values of the d_min and NRL statistics should be lower than for independent data. In such a case, the left-sided tests are justified. For non-chaotic data there are no such clear reasons for applying left-sided tests, so the two-sided tests were considered in simulations, too. However, such an approach resulted in the decrease in the power of both the d_min and NRL tests.

5 In the simulations, the values m = 2, 3,…, 8 were considered (note that m = 1 is excluded from the analysis, since in this case both test statistics are constant for each permutation sample). The results of the study on the size of the test were similar for each m. A different situation took place when the power of the test was examined—it increased as m increased. This relation was evident for small values of m, especially in the case of the statistic d_min. For larger m the power of the test was similar regardless of m. Thus, the value m = 5 seems reasonable considering both the power of the test and the speed of calculations.

6 In each of the following cases ϵ_t∼ iid N(0,1).

7 To calculate the BDS statistic and the mutual information measure, the scripts created by L. Kanzler and A. Leontitsis, respectively, were used.

8 A symbol σ_T, m(ϵ) denotes the standard deviation of the statistic (see, e.g., Brock et al., 1991).

9 In our calculations p = 1 is considered.