Measuring information interactions on the ordinal pattern of stock time series

The interactions among time series as individual components of complex systems can be quantified by measuring to what extent they exchange information among each other. In many applications, one focuses not on the original series but on its ordinal pattern. In such cases, trivial noises appear more likely to be filtered and the abrupt influence of extreme values can be weakened. Cross-sample entropy and inner composition alignment have been introduced as prominent methods to estimate the information interactions of complex systems. In this paper, we modify both methods to detect the interactions among the ordinal pattern of stock return and volatility series, and we try to uncover the information exchanges across sectors in Chinese stock markets.

Figure 1

The probability distribution of (a) return and (b) volatility series in 18 sectors.

Figure 2

The linear cross-correlation coefficients $r_{x,y}$ and the dendrogram based on $1-r_{x,y}$ for (a), (c) return and (b), (d) volatility series across Chinese stock sectors. We consider $1-r_{x,y}$ rather than $r_{x,y}$ because $r_{x,y}$ increases with the strength of cross correlation, so we use $1-r_{x,y}$ as the distance to the cluster.

Figure 3

The cross-sample entropy of the ordinal pattern of daily return series across Chinese stock sectors for (a) $d=2$ and (b) $d=3$. (c), (d) Dendrograms based on the values of the cross-sample entropy.

Figure 4

The cross-sample entropy of the ordinal pattern of daily volatility series across Chinese stock sectors for (a) $d=2$ and (b) $d=3$. (c), (d) Dendrograms using the values of the cross-sample entropy as the distance to the cluster.

Figure 5

$h_0=1$ (black cell) if one rejects the null hypothesis that the mean cross-sample entropy equals for two sectors and $h_0=0$ otherwise (white cell) for (a) return series and (b) volatility series.

Figure 6

The entropy derived by inner composition alignment of the ordinal pattern of return series across Chinese stock sectors for (a) $d=2$, (b) $d=3$, (c) $d=4$, and (d) $d=5$. We use the entropy as the distance to the cluster in the dendrograms.

Figure 7

The entropy derived by inner composition alignment of the ordinal pattern of volatility series across Chinese stock sectors for (a) $d=2$, (b) $d=3$, (c) $d=4$, and (d) $d=5$. We use the entropy as the distance to the cluster in the dendrograms.

Figure 8

$h_0=1$ (black cell) if one rejects the null hypothesis that the mean $H_d$ equals for two sectors and $h_0=0$ otherwise (white cell) for (a) return series and (b) volatility series.