An Independence Test Based on Symbolic Time Series

Abstract

An independence test based on symbolic time series analysis (STSA) is developed. Considering an independent symbolic time series there is a statistic asymptotically distributed as a CHI-2 with − 1 degrees of freedom. Size and power experiments for small samples were conducted applying Monte Carlo simulations and comparing the results with BDS and runs test. The introduced test shows a good performance detecting independence in nonlinear and chaotic systems.

Symbolic Time Series Analysis and Its Application in Social Sciences

“…In [57], the well-known BDS test is introduced, considered as a powerful test to detect nonlinearity. In [58], a simple and powerful test based on STSA is proposed and the results are compared with the BDS and runs test. On one hand, it is found that BDS is not able to detect processes such as the chaotic Anosov and the stochastic processes nonlinear sign model (NLSIGN), nonlinear autoregressive model (NLAR), and nonlinear moving average model (NLMA).…”

“…In [58] is applied the distribution of quadratic forms in normal variables presented in [59]. X=(ε 1 /σ,ε 2 /σ,…,ε n /σ) is distributed multivariate normal N(ø,Σ).…”

“…In [58], it is found that the statistic introduced in (7) is related to the Shannon entropy (H). We can derive the approximation expressed in Eq.…”

“…For instance, consider a string of the first 3000 letters from the book "A Christmas Carol", s 1 = {marleywasdeadtobeginwith…scroogecar} and a random string of 3000 letters from an alphabet of 26, s 2 = {iskynbmhjp…vbbihjfkk}. In [58] is shown that the test is conservative, rejecting the null hypothesis less time than expected. However, it is powerful in detecting nonrandom and nonlinear processes.…”

“…However, Granger test does not detect causality in any case. As asserted by [58] the NLAR process is very difficult to detect. Note that SCN is the only one detecting the causality when T = 5000.…”