Fisher information and Rényi dimensions: A thermodynamical formalism

Abstract

The relation between the Fisher information and Rényi dimensions is established: the Fisher information can be expressed as a linear combination of the first and second derivatives of the Rényi dimensions with respect to the Rényi parameter β. The Rényi parameter β is the parameter of the Fisher information. A thermodynamical description based on the Fisher information with β being the inverse temperature is introduced for chaotic systems. The link between the Fisher information and the heat capacity is emphasized, and the Fisher heat capacity is introduced.

Symbolic dynamics techniques for complex systems: Application to share price dynamics

“…We will investigate the correlations and complex behavior associated with discrete symbol sequences generated from observed share price returns on various time scales. We will quantify this by the calculation of the corresponding Rényi entropies [1,[11][12][13] for symbol sequences based on historical data sets of share price returns for various sectors, both on long (daily) time scales and on short time scales (minutes). It turns out that the stochastic process of symbol sequences observed for real share price data exhibit non-Markovian character.…”

“…2, it is meaningful to find a proper way to measure multifractal features. For this purpose we use the well-known concept of Rényi information [1,[11][12][13] defined as…”

“…It has been applied to many low-dimensional maps exhibiting chaotic or critical behavior. Nontrivial correlations in the dynamical system manifest themselves in a nontrivial spectrum of Rényi entropies [1,4,5,[11][12][13] and other observables associated with the set of allowed symbol sequences and their corresponding probabilities. This technique, which was very popular in the 80s and 90s when a lot of research on 1-dimensional maps was done [2][3][4][5]10], has been revived in more recent work and applied in a more general context [14][15][16][17].…”

Rényi’s divergence as a chemical similarity criterion

“…Considering a historical point of view, perhaps Rényi's work can be considered the first attempt to generalize Shannon's entropy. Since the 1990s, Rényi's entropy has been profusely used in the field of atomic and molecular physics [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], and these works were focused on the study of the electron correlation phenomenon.…”

Fisher information and Rényi entropies in dynamical systems

“…In our previous paper [7] this new formalism was generalized to chaotic systems. The thermodynamics of chaotic systems [8][9][10][11][12] is a useful technique for studying multifractals.…”