Measuring statistical dependences in a time series. (English) Zbl 1102.62340
Summary: We propose two methods to measure all (linear and nonlinear) statistical dependences in a stationary time series. Presuming ergodicity, the measures can be obtained from efficient numerical algorithms.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 60G18 | Self-similar stochastic processes |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37M10 | Time series analysis of dynamical systems |
References:
| [1] | St. Bingham and M. Kot, Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity,Phys. Lett. 140A:327-330 (1989). |
| [2] | G. E. P. Box and G. M. Jenkins,Time Series Analysis?Forecasting and Control (Prentice Hall, Englewood Cliffs, New Jersey, 1976). · Zbl 0363.62069 |
| [3] | W. A. Brock, Causality, chaos, explanation and prediction in economics and finance, inBeyond Belief: Randomness, Prediction and Explanation in Science, J. L. Casti and A. Karlqvist, eds. (CRC Press, Boca Raton, Florida, 1991). |
| [4] | L. L. Campbell, A coding theorem and Rényi’s entropy,Information Control 8:423-429 (1965); Definition of entropy by means of a coding problem,Z. Wahrsch. Verw. Geb. 6:113-118 (1966). · Zbl 0138.15103 · doi:10.1016/S0019-9958(65)90332-3 |
| [5] | P. Collet and J.-P. Eckman,Iterated Maps on an Interval as Dynamical Systems (Birkhäuser, Boston, 1980). |
| [6] | J.-P. Eckman and D. Ruelle, Ergodic theory of chaos and strange attractors,Rev. Mod. Phys. 57:617-656 (1985). · Zbl 0989.37516 · doi:10.1103/RevModPhys.57.617 |
| [7] | A. M. Fraser and H. L. Swinney, Using mutual information to find independent coordinates for strange attractors,Phys. Rev. A 33:1134-1140 (1986); A. M Fraser, Information and entropy in strange attractors,IEEE Trans. Information Theory 35: 245-262 (1989). · Zbl 1184.37027 · doi:10.1103/PhysRevA.33.1134 |
| [8] | P. Grassberger, Finite sample corrections to entropy and dimension estimates,Phys. Lett. A 128:369-373 (1988); An optimized box-assisted algorithm for fractal dimensions,Phys. Lett. A 148:63-68 (1990). · doi:10.1016/0375-9601(88)90193-4 |
| [9] | P. Grassberger and I. Procaccia, On the characterization of strange attractors,Phys. Rev. Lett. 50:346 (1983); Measuring the strangeness of strange attractors,Physica 9D:189-208; P. Grassberger, Generalized dimensions of strange attractors,Phys. Lett. 97A:227-230 (1983). · Zbl 0593.58024 · doi:10.1103/PhysRevLett.50.346 |
| [10] | P. Grassberger, Th. Schreiber, and C. Schaffrath, Nonlinear time sequence analysis,Int. J. Bifurcation Chaos 1:521-547 (1991). · Zbl 0874.58029 · doi:10.1142/S0218127491000403 |
| [11] | S. Grossmann and S. Thomae, Invariant distributions and stationary correlation functions of one-dimensional discrete processes,Z. Naturforsch. 32a:1353-1363 (1977). |
| [12] | M. Heilfort and B. Pompe, in preparation. |
| [13] | N. S. Jayant and P. Noll,Digital Coding of Waveforms-Principles and Applications to Speech and Video (Prentice-Hall, Englewood Cliffs, New Jersey, 1984). |
| [14] | R. W. Leven and B. Pompe, Über die Möglichkeit der Zustandsvorhersage chaotischer Systeme,Ann. Phys. (Leipzig)43:259-278 (1986). |
| [15] | K. Pawelzik and H. G. Schuster, Generalized dimensions and entropies from a measured time series,Phys. Rev. A 35:481-484 (1987); K. Pawelzik,Nichtlineare Dynamik und Hirnaktivität (Verlag Harri Deutsch, Thun, Frankfurt am Main, 1991), pp. 84-85. · doi:10.1103/PhysRevA.35.481 |
| [16] | L. R. Rabiner and R. W. Schafer,Digital Processing of Speech Signals (Prentice-Hall, Englewood Cliffs, New Jersey, 1978). · Zbl 1162.94003 |
| [17] | A. Rényi,Probability Theory (North-Holland, Amsterdam, 1970). |
| [18] | J. A. Scheinkman and B. LeBaron, Nonlinear dynamics and stock returns,J. Business 62:311-337 (1989). · doi:10.1086/296465 |
| [19] | B. W. Silverman,Density Estimation for Statistics and Data Analysis (Chapman and Hall, London, 1986). · Zbl 0617.62042 |
| [20] | F. Takens, Invariants related to dimension and entropy, inAlas do 13{\(\deg\)} Coloquio Brasileiro de Mathematica (1983). · Zbl 0572.58008 |
| [21] | J. Theiler, Efficient algorithm for estimating the correlation dimension from a set of discrete points,Phys. Rev. A 36:4456-4462 (1987); Statistical precision of dimension estimators,Phys. Rev. A 41:3038-3051 (1990). · doi:10.1103/PhysRevA.36.4456 |
| [22] | H. Tong,Non-linear Time Series?A Dynamical System Approach (Clarendon Press, Oxford, 1990). · Zbl 0716.62085 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
