Multifractals embedded in short time series: An unbiased estimation of probability moment

An exact estimation of probability moments is the base for several essential concepts, such as the multifractals, the Tsallis entropy, and the transfer entropy. By means of approximation theory we propose a new method called factorial-moment-based estimation of probability moments. Theoretical prediction and computational results show that it can provide us an unbiased estimation of the probability moments of continuous order. Calculations on probability redistribution model verify that it can extract exactly multifractal behaviors from several hundred recordings. Its powerfulness in monitoring evolution of scaling behaviors is exemplified by two empirical cases, i.e., the gait time series for fast, normal, and slow trials of a healthy volunteer, and the closing price series for Shanghai stock market. By using short time series with several hundred lengths, a comparison with the well-established tools displays significant advantages of its performance over the other methods. The factorial-moment-based estimation can evaluate correctly the scaling behaviors in a scale range about three generations wider than the multifractal detrended fluctuation analysis and the basic estimation. The estimation of partition function given by the wavelet transform modulus maxima has unacceptable fluctuations. Besides the scaling invariance focused in the present paper, the proposed factorial moment of continuous order can find its various uses, such as finding nonextensive behaviors of a complex system and reconstructing the causality relationship network between elements of a complex system.