Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-states sequences

In this paper, we propose to mix the approach underlying Bandt-Pompe permutation entropy with Lempel-Ziv complexity, to design what we call Lempel-Ziv permutation complexity. The principle consists of two steps: (i) transformation of a continuous-state series that is intrinsically multivariate or arises from embedding into a sequence of permutation vectors, where the components are the positions of the components of the initial vector when re-arranged; (ii) performing the Lempel-Ziv complexity for this series of `symbols', as part of a discrete finite-size alphabet. On the one hand, the permutation entropy of Bandt-Pompe aims at the study of the entropy of such a sequence; i.e., the entropy of patterns in a sequence (e.g., local increases or decreases). On the other hand, the Lempel-Ziv complexity of a discrete-state sequence aims at the study of the temporal organization of the symbols (i.e., the rate of compressibility of the sequence). Thus, the Lempel-Ziv permutation complexity aims to take advantage of both of these methods. The potential from such a combined approach - of a permutation procedure and a complexity analysis - is evaluated through the illustration of some simulated data and some real data. In both cases, we compare the individual approaches and the combined approach.Comment: 30 pages, 4 figure

Maximum Entropy Production Principle for Stock Returns

In our previous studies we have investigated the structural complexity of time series describing stock returns on New York's and Warsaw's stock exchanges, by employing two estimators of Shannon's entropy rate based on Lempel-Ziv and Context Tree Weighting algorithms, which were originally used for data compression. Such structural complexity of the time series describing logarithmic stock returns can be used as a measure of the inherent (model-free) predictability of the underlying price formation processes, testing the Efficient-Market Hypothesis in practice. We have also correlated the estimated predictability with the profitability of standard trading algorithms, and found that these do not use the structure inherent in the stock returns to any significant degree. To find a way to use the structural complexity of the stock returns for the purpose of predictions we propose the Maximum Entropy Production Principle as applied to stock returns, and test it on the two mentioned markets, inquiring into whether it is possible to enhance prediction of stock returns based on the structural complexity of these and the mentioned principle.Comment: 14 pages, 5 figure

Frequency Effects on Predictability of Stock Returns

We propose that predictability is a prerequisite for profitability on financial markets. We look at ways to measure predictability of price changes using information theoretic approach and employ them on all historical data available for NYSE 100 stocks. This allows us to determine whether frequency of sampling price changes affects the predictability of those. We also relations between price changes predictability and the deviation of the price formation processes from iid as well as the stock's sector. We also briefly comment on the complicated relationship between predictability of price changes and the profitability of algorithmic trading.Comment: 8 pages, 16 figures, submitted for possible publication to Computational Intelligence for Financial Engineering and Economics 2014 conferenc