An Independence Test Based on Symbolic Time Series

Table 3

Power test for a time series size of 50.

Generator process	SRS(2,2)	SRS(2,3)	SRS(3,2)	SRS(4,2)	SRS(5,2)	BDS (a)	BDS (b)	BDS (s)	Runs test
Normal	0.32	1.66	0.46	24.72	91.02	20.48	11.14	5.28	3.40
CHI-2(4)	0.46	2.14	0.44	25.36	91.36	15.26	7.56	3.04	4.12
𝑡-Student(4)	0.48	1.86	0.42	24.58	91.44	14.72	7.80	3.32	3.72
Truncated norm.	0.28	1.62	0.50	24.92	91.62	32.94	18.30	13.30	3.06
Beta(1/2,1/2)	0.30	1.80	0.46	24.12	91.06	35.12	18.16	14.38	3.40
Uniform(0,1)	0.34	1.68	0.52	24.42	91.26	32.70	18.10	12.88	3.26
AR(1)	0.10	0.60	0.26	22.34	90.90	21.22	11.16	5.16	0.90
MA(2)	0.02	0.44	0.32	20.54	90.18	12.40	7.30	2.40	0.86
Logistic	10.54	40.20	100.00	100.00	100.00	59.36	68.42	37.94	22.88
Henon	53.94	77.84	100.00	100.00	100.00	99.18	82.34	96.44	22.38
Anosov	0.98	2.14	0.96	24.50	89.82	31.12	17.78	12.60	4.80
Lorenz	90.72	96.84	95.84	99.78	100.00	96.82	95.82	88.08	98.94
TAR	0.24	1.44	0.50	33.80	95.42	24.32	14.28	6.76	5.04
NLSIGN	0.06	0.94	0.40	23.10	90.28	21.50	12.44	5.32	2.40
Bilinear	0.16	5.38	0.48	28.50	93.30	63.08	60.18	44.64	1.54
NLAR	0.02	0.28	0.40	21.28	90.30	21.50	11.44	5.32	0.58
NLMA	0.14	1.18	0.26	20.80	89.56	15.88	7.56	2.82	1.58
BLMA	0.76	2.34	1.44	39.74	96.46	57.24	51.62	35.58	10.56
Modular	0.00	20.18	0.06	55.68	96.02	70.28	66.10	47.84	19.60
GARCH	0.10	0.88	3.12	37.40	93.52	59.14	58.42	38.22	3.24

Note: 5,000 Monte Carlo simulations were conducted applying the pseudorandom numbers from MatLab R2010a. SRS(a,w) indicates the SRS test for a symbolization of a symbols and a length of w. BDS (a), (b), and (s) applied the Kanzler [46] optimal parameters, the Liu et al. [47] best parameters, and the Kanzler [46] simulated critical values for small sample.
Bold values refer to reasonable percentage of rejections considering the process and the critical values.
𝑎 is the alphabet size or number of symbols, 𝑤 is known as the word or a subsequence of symbols.