q-SampEnAve: an adaptive measurement to recognize the patterns for short-term financial time series

Adaptive algorithms are increasingly discussed because of few parameter constraints, which can respond to different inputs and make the results more accurate. In this paper, we propose a self-tuning complexity measurement for short time series, called q-sample entropy average (q-SampEnAve), which is a data-driven approach. This approach avoids the impact of tolerance r in traditional sample entropy (SampEn) by considering all r candidates. Besides, q-sample entropy (q-SampEn), as an intermediate stage, could serve as a good discriminative tool as well. By discussing parameter q, we find that the q plays a decisive role in the results that large positive q filters out the large $p_i$ in a probability distribution, and large negative q filter out the rare events. These two factors make our new method more accurate and complete than existing methods. In simulated experiments, ARFIMA model, uniform distribution and logistic map are employed. It reveals that sequences generated two-component ARFIMA models are distinctly separated from uniform distribution in (0, 1) and logistic mapping ($a=4$). Besides, even for the sequences generated by ARFIMA model with different parameters, four states (mean, std, kurtosis and skewness) show a little different. Most importantly, uniform distribution in (0, 1) and logistic mapping ($a=4$) are relatively disordered. On the contrary, the system generated by the ARFIMA model are relatively regular. For stock indices, the performance of SSE Composite Index (000001.SS) and SZSE Composite Index (399001.SZ) are identical, while Hang Seng Index ($\hat{}$HSI) behaves closer to Dow Jones Industrial Average ( $\hat{}$DJI) and Nasdaq Composite ( $\hat{}$IXIC) than indices in mainland China (000001.SS and 399001.SZ), which may be related to the economic policy and the market environment. It also illustrates that the US stock market is relatively mature and more suitable for investment. Finally, we give a brief discussion about the application in random bit sequence, which illustrates that random bit sequences generated from different parameters can be recognized.