An Independence Test Based on Symbolic Time Series

Table 1

Size of the symbolic random test. Percentage of the null hypothesis rejection.

Length	2 symbols			3 symbols			4 symbols			5 symbols
	𝛼=1%	𝛼=5%	𝛼=10%	𝛼=1%	𝛼=5%	𝛼=10%	𝛼=1%	𝛼=5%	𝛼=10%	𝛼=1%	𝛼=5%	𝛼=10%
	𝑇=50			𝑇=50			𝑇=50			𝑇=50
2	0.01	0.41	1.03	0.28	0.47	0.97	24.68	24.68	24.75	24.68	24.68	24.68
3	0.89	1.95	3.27	98.00	98.00	98.00	100.00	100.00	100.00	100.00	100.00	100.00
4	40.10	40.39	40.68	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00
5	99.80	99.80	99.80	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00
	𝑇=200			𝑇=200			𝑇=200			𝑇=200
2	0.05	0.59	1.14	0.06	0.42	1.06	0.05	0.32	0.82	0.00	0.00	0.00
3	0.54	1.97	3.64	1.18	2.44	3.76	89.67	89.67	89.69	89.67	89.67	89.67
4	1.31	3.74	6.17	99.84	99.84	99.84	100.00	100.00	100.00	100.00	100.00	100.00
5	7.84	10.57	13.06	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00
	𝑇=500			𝑇=500			𝑇=500			𝑇=500
2	0.06	0.54	1.27	0.09	0.47	1.03	0.05	0.31	0.80	0.01	0.01	0.03
3	0.52	1.91	3.49	0.52	1.86	3.69	1.89	3.27	4.73	1.52	1.52	1.52
4	1.28	3.55	6.16	12.23	14.02	16.02	100.00	100.00	100.00	100.00	100.00	100.00
5	2.15	5.65	8.93	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00	100.00
	𝑇=2000			𝑇=2000			𝑇=2000			𝑇=2000
2	0.08	0.44	1.25	0.06	0.33	0.90	0.02	0.42	1.05	0.00	0.00	0.00
3	0.50	2.03	3.45	0.44	1.88	3.79	0.34	1.72	3.51	0.00	0.00	0.00
4	1.32	3.77	6.32	1.14	4.19	6.94	8.78	11.22	13.88	8.01	8.01	8.01
5	2.41	5.79	8.64	8.13	11.63	14.97	100.00	100.00	100.00	100.00	100.00	100.00

Note: 10,000 Monte Carlo simulations were conducted applying the normally distributed pseudorandom numbers from MatLab R2010a.
Bold values refer to reasonable percentage of rejections considering the process and the critical values.