Dynamically Measuring Statistical Dependencies in Multivariate Financial Time Series Using Independent Component Analysis

Table 2

Table showing accuracy of four measures of dependence, that is, information coupling (𝜂), linear correlation (𝜌), rank correlation (𝜌_𝑅), and normalised mutual information (𝐼_𝑁), when used to estimate the level of dependence in a coupled system with varying levels of true correlation (𝜌_TC). 𝜌_TC is induced in an independent bivariate, randomly distributed, system using the Iman-Conover method as described in the text. The dependence estimates, together with their standard deviation values, shown in the table are obtained using 1000 independent simulations using 1000 data points long data sets for each simulation. The last row of the table gives values for the mean absolute error (MAE).

𝜌TC	𝜂	𝜌	𝜌𝑅	𝐼𝑁	|𝜌TC−𝜂|	|𝜌TC−𝜌|	|𝜌TC−𝜌𝑅|	|𝜌TC−𝐼𝑁|
0	0.0046±0.0026	0.0038±0.0022	0.0147±0.0115	0.1851±0.0401	0.0046	0.0038	0.0147	0.1851
0.1	0.1079±0.0073	0.0917±0.0058	0.0942±0.0184	0.1687±0.0412	0.0079	0.0083	0.0058	0.0687
0.2	0.2145±0.0102	0.1862±0.0093	0.1899±0.0177	0.1131±0.0464	0.0145	0.0138	0.0111	0.0869
0.3	0.3176±0.0138	0.2812±0.0130	0.2863±0.0167	0.1644±0.0538	0.0176	0.0188	0.0137	0.1356
0.4	0.4166±0.0210	0.3764±0.0165	0.3835±0.0153	0.2787±0.0500	0.0166	0.0236	0.0165	0.1213
0.5	0.5135±0.0196	0.4720±0.0197	0.4818±0.0136	0.3819±0.0475	0.0135	0.0280	0.0182	0.1181
0.6	0.6070±0.0347	0.5688±0.0226	0.5815±0.0117	0.4788±0.0458	0.0070	0.0312	0.0185	0.1212
0.7	0.7011±0.0232	0.6670±0.0242	0.6830±0.0093	0.5728±0.0483	0.0011	0.0330	0.0170	0.1272
0.8	0.7936±0.0239	0.7676±0.0249	0.7864±0.0066	0.6652±0.0490	0.0064	0.0324	0.0136	0.1348
0.9	0.8864±0.0252	0.8713±0.0249	0.8919±0.0034	0.7595±0.0501	0.0136	0.0287	0.0081	0.1405
1.0	1.0000±0.0000	1.0000±0.0000	1.0000±0.0000	0.9601±0.0185	0.0000	0.0000	0.0000	0.0399
MAE					0.0093	0.0201	0.0125	0.1163