Google Patentsopen access

Unified nonlinear modeling approach for machine learning and artificial … — Nxgen Partners Ip, Llc (US12488266B2)

ABSTRACT

Abstract

A method for predicting future behavior for a dynamic system using an artificial intelligence system implemented within a computer hardware system. A predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system are received at the artificial intelligence system. An attractor is constructed from the time series group of data that defines the previous behavior of the dynamic system using the artificial intelligence system. The attractor models the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system. A prediction horizon for the predetermined amount of the time series group of data is determined with the artificial intelligence system using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor. The prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly. Prediction values of future behavior of the dynamic system are generated with the artificial intelligence system using the constructed attractor and the determined prediction horizon.

Description

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 16/161,840, filed Oct. 16, 2018, entitled UNIFIED NONLINEAR MODELING APPROACH FOR MACHINE LEARNING AND ARTIFICIAL INTELLIGENCE (ATTRACTOR ASSISTED AI), now U.S. Pat. No. 10,755,188, will issue on Aug. 25, 2020, which claims benefit of U.S. Provisional Application No. 62/574,039, filed Oct. 18, 2017, entitled NONLINEAR MODELING AND FORECASTING AND ITS APPLICATION TO ARTIFICIAL INTELLIGENCE, the specification of which is incorporated by reference herein in its entirety.

TECHNICAL FIELD

The present invention relates to artificial intelligence systems, and more particularly, to a method for providing an improved artificial intelligence system using nonlinear modeling and forecasting.

BACKGROUND

Artificial intelligence (AI) systems be used for processing time series data in order to provide predictions of future events. Various techniques have been utilized for predicting future events using AI systems which have varying degrees of effectiveness. Some manner for improving the predictive capabilities of AI systems with respect to time-series data in dynamical and chaotic systems would be of great benefit in the AI system arena.

SUMMARY

The present invention, as disclosed and described herein, in one aspect thereof comprises a method for predicting future behavior for a dynamic system using an artificial intelligence system implemented within a computer hardware system. A predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system are received at the artificial intelligence system. An attractor is constructed from the time series group of data that defines the previous behavior of the dynamic system using the artificial intelligence system. The attractor models the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system. A prediction horizon for the predetermined amount of the time series group of data is determined with the artificial intelligence system using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor. The prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly. Prediction values of future behavior of the dynamic system are generated with the artificial intelligence system using the constructed attractor and the determined prediction horizon.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to the following description taken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates a combination of nonlinear modeling and forecasting with an artificial intelligence system;

FIG. 2 illustrates a flow diagram of the process for nonlinear modeling and forecasting;

FIG. 3 illustrates a forecast horizon comprising a function of attractor dimensions and Lyapunov exponents;

FIG. 4 illustrates a flow diagram of a process enabling AI systems to forecast future values based upon past time-series data;

FIG. 5 illustrates a flow diagram of a process for predicting future data from historical data records;

FIG. 6 illustrates a flow diagram of a process for merging predictions from nearest neighbor approximations;

FIG. 7 illustrates a flow diagram of a process for establishing and embedding dimension;

FIG. 8 illustrates a flow diagram for computing mutual information;

FIG. 9 illustrates a Rossler attractor;

FIG. 10 illustrates mutual information for Rossler 4096 and 8192 points;

FIG. 11 illustrates a first embodiment of a reconstructed Rossler attractor;

FIG. 12 illustrates a second embodiment of a reconstructed Rossler attractor;

FIG. 13 illustrates a third embodiment of a reconstructed Rossler attractor;

FIG. 14 illustrates a fourth embodiment of a reconstructed Rossler attractor;

FIG. 15 illustrates mutual information of a Rossler attractor with noise;

FIG. 16 illustrates a daily solar flux time series;

FIG. 17 illustrates filtered and unfiltered daily solar flux;

FIG. 18 illustrates variations of the period between minimas;

FIG. 19 illustrates the power spectrum of variations between minimas;

FIG. 20 illustrates solar flux averaged over solar rotations;

FIG. 21 illustrates mutual information of averaged flux;

FIG. 22 illustrates a preliminary solar flux forecast;

FIG. 23 illustrates an artificial neural network/AI implemented within a field programmable gate array (FPGA);

FIG. 24 illustrates an artificial neural network;

FIG. 25 illustrates the feedforward and recurrent topologies that may be associated with an artificial neural network;

FIG. 26 illustrates a flow diagram of the process for training an artificial neural network;

FIG. 27 illustrates various schools of machine learning;

FIG. 28 illustrates a process for modeling system behavior from an attractor;

FIG. 29 illustrates a system for detecting glucose nonlinear dynamics;

FIG. 30 illustrates a flow diagram of the process for detecting glucose nonlinear dynamics;

FIGS. 31 - 34 illustrates various intensity images taken using the system of FIG. 29 ;

FIG. 35 illustrates a flow diagram of a process for processing intensity images for glucose and sucrose;

FIG. 36 illustrates a photonic reservoir system with output feedback;

FIG. 37 illustrates a flow diagram of a process for providing an improved extreme learning machine;

FIG. 38 illustrates various types of quantum logic circuits;

FIG. 39 illustrates a single bit rotation gate;

FIG. 40 illustrates a two bit controlled NOT gate; and

FIG. 41 illustrates a Qudit neural network.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are used herein to designate like elements throughout, the various views and embodiments of the unified nonlinear modeling approach for machine learning and artificial intelligence are illustrated and described, and other possible embodiments are described. The figures are not necessarily drawn to scale, and in some instances the drawings have been exaggerated and/or simplified in places for illustrative purposes only. One of ordinary skill in the art will appreciate the many possible applications and variations based on the following examples of possible embodiments.

Referring now to the drawings, and more particularly to FIG. 1 , there is illustrated the manner in which an artificial intelligence system 102 may be combined with a process for nonlinear modeling and forecasting 104 as described herein to provide an improved artificial intelligence system 106 . FIG. 2 is a flow diagram illustrating the general process for nonlinear modeling and forecasting 104 that can be implemented within an artificial intelligence system 102 . Initially, with respect to a group of data being analyzed, a time series group of the data is generated at step 202 . A time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus, it is a sequence of discrete-time data. Time series data has a natural temporal progression. Next, a delay value for the time series is created at step 204 . Based upon the time series group of data and the created the attractor is reconstructed at step 206 . A prediction horizon for the data set is determined at step 208 .

An attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that come close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space in physical systems, the n-dimensions may be, for example, two or three positional coordinates for each one or more physical entity. If the evolving variable is two or three dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions. An attractor can be a point, a finite set of points, a curve, a manifold or even a complicated set with a fractal structure known as a strange attractor. Describing the attractors of chaotic dynamical systems may be done in Chaos Theory.

A dynamical system is a system in which a function describes the time dependence of a point in a geometric space. Examples include the mathematical models describing the swinging of a clock pendulum, the flow of water in a pipe and the number of fish each spring in a lake. At any given time, a dynamical system has a state even by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables. A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forwarding time. This enables the attractor to be utilized for predictions. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, with the flow in the neighborhood is away from the set, the set is not an attractor but instead called a repeller. From the attractor various predictions may be generated at step 210 in order to predict future behavior based upon the provided time series data. The process for nonlinear modeling and forecasting 104 utilized herein is more particularly described hereinbelow.

In the past, the modeling and forecasting of time series data that had nonlinear structure involved attempts to model the systems underlying physics. For example highly entangled dynamics in solar activity data was uncovered as disclosed in S. Ashrafi and L. Roszman, Detecting and Disentangling Nonlinear Structure from Solar Flux Time Series, 43rd Congress of the International Astronautical Federation, August 1992, which is incorporated herein by reference. This disclosure discovered that the general lack of predictability in solar activity data arises from its nonlinear nature as more fully discussed in S. Ashrafi and L. Roszman, “Evidence of Chaotic Pattern in SolarFlux Through a Reproducible Sequence of Period-Doubling-Type Bifurcations,” Proceedings of Flight ]Mechanics/Estimation Theory Symposium , National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, May 1991 and S. Ashrafi, Combining Schatten's Solar Activity Prediction Model With a Chaotic Prediction Model , National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, 554-FDD-911125, November 1991, each of which are incorporated herein by reference in their entirety.

Nonlinear dynamics allows the prediction of time series data more accurately than is possible using stochastic methods for time scales shorter than a characteristic horizon, and with about the same accuracy as using stochastic techniques when the forecasted data exceed this horizon. In some embodiments the horizon may be an Ashrafi Horizon as described in S. Ashrafi and L. Roszman, “Lyapunov Exponent of Solar Flux Time Series,” Proceedings of First Experimental Chaos Conference , June 1991 and S. Ashrafi and L. Roszman, Solar Flux Forecasting Using Mutual Information with Optimal Delay, AAS 93-311, Spaceflight Dynamics 1993, American Astronautical Society Publication, Advances in the Astronautical Sciences, Volume 84 Part II pages 901-913, each of which are incorporated herein by reference. As shown in FIG. 3 , the forecast horizon 302 is a function of two dynamical invariants: the attractor dimension 304 and the Lyapunov exponent 306 as shown in S. Ashrafi and L. Roszman “Nonlinear Techniques for Forecasting Solar Activity Directly From its Time Series,” Proceedings of Flight Mechanics/Estimation Theory Symposium . National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland May 21-23, 1992 which is incorporated herein by reference. The techniques introduced herein are applicable to any time series of data generated from any physical, social or economic system.

Estimation of th

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 16/161,840, filed Oct. 16, 2018, entitled UNIFIED NONLINEAR MODELING APPROACH FOR MACHINE LEARNING AND ARTIFICIAL INTELLIGENCE (ATTRACTOR ASSISTED AI), now U.S. Pat. No. 10,755,188, will issue on Aug. 25, 2020, which claims benefit of U.S. Provisional Application No. 62/574,039, filed Oct. 18, 2017, entitled NONLINEAR MODELING AND FORECASTING AND ITS APPLICATION TO ARTIFICIAL INTELLIGENCE, the specification of which is incorporated by reference herein in its entirety.

TECHNICAL FIELD

The present invention relates to artificial intelligence systems, and more particularly, to a method for providing an improved artificial intelligence system using nonlinear modeling and forecasting.

BACKGROUND

Artificial intelligence (AI) systems be used for processing time series data in order to provide predictions of future events. Various techniques have been utilized for predicting future events using AI systems which have varying degrees of effectiveness. Some manner for improving the predictive capabilities of AI systems with respect to time-series data in dynamical and chaotic systems would be of great benefit in the AI system arena.

SUMMARY

The present invention, as disclosed and described herein, in one aspect thereof comprises a method for predicting future behavior for a dynamic system using an artificial intelligence system implemented within a computer hardware system. A predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system are received at the artificial intelligence system. An attractor is constructed from the time series group of data that defines the previous behavior of the dynamic system using the artificial intelligence system. The attractor models the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system. A prediction horizon for the predetermined amount of the time series group of data is determined with the artificial intelligence system using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor. The prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly. Prediction values of future behavior of the dynamic system are generated with the artificial intelligence system using the constructed attractor and the determined prediction horizon.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to the following description taken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates a combination of nonlinear modeling and forecasting with an artificial intelligence system;

FIG. 2 illustrates a flow diagram of the process for nonlinear modeling and forecasting;

FIG. 3 illustrates a forecast horizon comprising a function of attractor dimensions and Lyapunov exponents;

FIG. 4 illustrates a flow diagram of a process enabling AI systems to forecast future values based upon past time-series data;

FIG. 5 illustrates a flow diagram of a process for predicting future data from historical data records;

FIG. 6 illustrates a flow diagram of a process for merging predictions from nearest neighbor approximations;

FIG. 7 illustrates a flow diagram of a process for establishing and embedding dimension;

FIG. 8 illustrates a flow diagram for computing mutual information;

FIG. 9 illustrates a Rossler attractor;

FIG. 10 illustrates mutual information for Rossler 4096 and 8192 points;

FIG. 11 illustrates a first embodiment of a reconstructed Rossler attractor;

FIG. 12 illustrates a second embodiment of a reconstructed Rossler attractor;

FIG. 13 illustrates a third embodiment of a reconstructed Rossler attractor;

FIG. 14 illustrates a fourth embodiment of a reconstructed Rossler attractor;

FIG. 15 illustrates mutual information of a Rossler attractor with noise;

FIG. 16 illustrates a daily solar flux time series;

FIG. 17 illustrates filtered and unfiltered daily solar flux;

FIG. 18 illustrates variations of the period between minimas;

FIG. 19 illustrates the power spectrum of variations between minimas;

FIG. 20 illustrates solar flux averaged over solar rotations;

FIG. 21 illustrates mutual information of averaged flux;

FIG. 22 illustrates a preliminary solar flux forecast;

FIG. 23 illustrates an artificial neural network/AI implemented within a field programmable gate array (FPGA);

FIG. 24 illustrates an artificial neural network;

FIG. 25 illustrates the feedforward and recurrent topologies that may be associated with an artificial neural network;

FIG. 26 illustrates a flow diagram of the process for training an artificial neural network;

FIG. 27 illustrates various schools of machine learning;

FIG. 28 illustrates a process for modeling system behavior from an attractor;

FIG. 29 illustrates a system for detecting glucose nonlinear dynamics;

FIG. 30 illustrates a flow diagram of the process for detecting glucose nonlinear dynamics;

FIGS. 31 - 34 illustrates various intensity images taken using the system of FIG. 29 ;

FIG. 35 illustrates a flow diagram of a process for processing intensity images for glucose and sucrose;

FIG. 36 illustrates a photonic reservoir system with output feedback;

FIG. 37 illustrates a flow diagram of a process for providing an improved extreme learning machine;

FIG. 38 illustrates various types of quantum logic circuits;

FIG. 39 illustrates a single bit rotation gate;

FIG. 40 illustrates a two bit controlled NOT gate; and

FIG. 41 illustrates a Qudit neural network.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are used herein to designate like elements throughout, the various views and embodiments of the unified nonlinear modeling approach for machine learning and artificial intelligence are illustrated and described, and other possible embodiments are described. The figures are not necessarily drawn to scale, and in some instances the drawings have been exaggerated and/or simplified in places for illustrative purposes only. One of ordinary skill in the art will appreciate the many possible applications and variations based on the following examples of possible embodiments.

Referring now to the drawings, and more particularly to FIG. 1 , there is illustrated the manner in which an artificial intelligence system 102 may be combined with a process for nonlinear modeling and forecasting 104 as described herein to provide an improved artificial intelligence system 106 . FIG. 2 is a flow diagram illustrating the general process for nonlinear modeling and forecasting 104 that can be implemented within an artificial intelligence system 102 . Initially, with respect to a group of data being analyzed, a time series group of the data is generated at step 202 . A time series is a series of data points indexed in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus, it is a sequence of discrete-time data. Time series data has a natural temporal progression. Next, a delay value for the time series is created at step 204 . Based upon the time series group of data and the created the attractor is reconstructed at step 206 . A prediction horizon for the data set is determined at step 208 .

An attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that come close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space in physical systems, the n-dimensions may be, for example, two or three positional coordinates for each one or more physical entity. If the evolving variable is two or three dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions. An attractor can be a point, a finite set of points, a curve, a manifold or even a complicated set with a fractal structure known as a strange attractor. Describing the attractors of chaotic dynamical systems may be done in Chaos Theory.

A dynamical system is a system in which a function describes the time dependence of a point in a geometric space. Examples include the mathematical models describing the swinging of a clock pendulum, the flow of water in a pipe and the number of fish each spring in a lake. At any given time, a dynamical system has a state even by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables. A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forwarding time. This enables the attractor to be utilized for predictions. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, with the flow in the neighborhood is away from the set, the set is not an attractor but instead called a repeller. From the attractor various predictions may be generated at step 210 in order to predict future behavior based upon the provided time series data. The process for nonlinear modeling and forecasting 104 utilized herein is more particularly described hereinbelow.

In the past, the modeling and forecasting of time series data that had nonlinear structure involved attempts to model the systems underlying physics. For example highly entangled dynamics in solar activity data was uncovered as disclosed in S. Ashrafi and L. Roszman, Detecting and Disentangling Nonlinear Structure from Solar Flux Time Series, 43rd Congress of the International Astronautical Federation, August 1992, which is incorporated herein by reference. This disclosure discovered that the general lack of predictability in solar activity data arises from its nonlinear nature as more fully discussed in S. Ashrafi and L. Roszman, “Evidence of Chaotic Pattern in SolarFlux Through a Reproducible Sequence of Period-Doubling-Type Bifurcations,” Proceedings of Flight ]Mechanics/Estimation Theory Symposium , National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, May 1991 and S. Ashrafi, Combining Schatten's Solar Activity Prediction Model With a Chaotic Prediction Model , National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, 554-FDD-911125, November 1991, each of which are incorporated herein by reference in their entirety.

Nonlinear dynamics allows the prediction of time series data more accurately than is possible using stochastic methods for time scales shorter than a characteristic horizon, and with about the same accuracy as using stochastic techniques when the forecasted data exceed this horizon. In some embodiments the horizon may be an Ashrafi Horizon as described in S. Ashrafi and L. Roszman, “Lyapunov Exponent of Solar Flux Time Series,” Proceedings of First Experimental Chaos Conference , June 1991 and S. Ashrafi and L. Roszman, Solar Flux Forecasting Using Mutual Information with Optimal Delay, AAS 93-311, Spaceflight Dynamics 1993, American Astronautical Society Publication, Advances in the Astronautical Sciences, Volume 84 Part II pages 901-913, each of which are incorporated herein by reference. As shown in FIG. 3 , the forecast horizon 302 is a function of two dynamical invariants: the attractor dimension 304 and the Lyapunov exponent 306 as shown in S. Ashrafi and L. Roszman “Nonlinear Techniques for Forecasting Solar Activity Directly From its Time Series,” Proceedings of Flight Mechanics/Estimation Theory Symposium . National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland May 21-23, 1992 which is incorporated herein by reference. The techniques introduced herein are applicable to any time series of data generated from any physical, social or economic system.

Estimation of the attractor dimension 304 reconstructed from a time series of data has become an important tool in data analysis. Many possible characterizations of dimension have been introduced. Grassberger and Procaccia have introduced the notion of correlation dimension, which has become a popular method for estimating the attractor dimension for low-dimensional systems. In calculating the invariants of the system, the first necessary step is the reconstruction of the attractor for the system from the time-delayed values of the time series as shown in F. Takens, “Detecting Strange Attractors in Turbulence,” In Lecture Notes in Mathematics , Vol. 898, 366 Berlin: Springer-Verlag, 1981, which is incorporated herein by reference. The choice of the time delay is critical for this reconstruction.

For an infinite amount of noise-free data, the time delay can, in principle, be chosen almost arbitrarily. However, the quality of the phase portraits produced using the time-delay technique is determined by the value chosen for the delay time. Studies by Fraser and Swinney such as those described in A. M. Fraser and H. L. Swinney, Phys. Rev . A33, 1134, 1986, which is incorporated herein by reference, have shown that a good choice for this time delay is the one suggested by Shaw, in R. S. Shaw, The Dripping Faucet as a Model Chaotic System, Aerial Press, C A, 1985, which is incorporated herein by reference, which uses the first local minimum of the mutual information rather than the autocorrelation function to determine the time delay. A refinement of this criterion is described hereinbelow and applies the refined technique to a sample time series data (solar flux data) to produce a forecast of the solar activity. However, the technique may be applied to any time series data.

Referring now to FIG. 4 there is illustrated a process for enabling an artificial intelligence system to forecast future values of data based upon a past time series group of data. The time series data is received at 402 and using this various dynamical invariants may be extracted at step 404 . Using the dynamical invariants phase-space representations of the time series data are constructed at step 406 . Phase-space representations comprise a space representation in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase-space. The phase-space representations are used for forecasting future values of the data at step 408 . Numerical techniques enable construction of the phase-space representations of the time series data so that the future values of the time series data can be forecasted directly from its past time series. This approach makes it possible to model the behavior of a system using an attractor created in terms of extracted dynamical invariants from the system's dynamics without reference to any underlying physics or dynamics of the data. The dynamical evolution of time series in this reconstructed phase-space that captures the properties of the attractor, give a procedure for extracting an optimal time delay using mutual information, and present preliminary predictions made for the magnitude and phase of a sample data (next maximum).

Nonlinear Structure in Systems

Until recently, there was little reason to doubt that weather in principle is predictable, given enough data. Recently, a striking discovery changed this view. Simple deterministic systems with only a few degrees of freedom can generate seemingly random behavior. When a system exhibits apparent random behavior that is fundamental to its dynamics such that no amount of information gathering will make the system predictable, the system is considered chaotic. For example, much evidence supports the assertion that solar flux falls in this category. Perhaps paradoxically, chaos is generated by fixed rules that do not themselves involve any element of chance. The future of a dynamic system is completely determined by present and past conditions. In practice, amplification of small initial uncertainties makes a system with short-term predictability unpredictable in the long term.

New developments in chaos and nonlinear dynamics allow the modeling of the behavior of a system in terms of some invariants directly extractable from the system's dynamics, without reference to any underlying physics. Using chaos theory, short-term activity can be predicted more accurately than with statistical methods. In addition, chaos theory imposes a fundamental limit on accurate long-term predictions.

Review of Chaotic Dynamics

Self-Organization and Attractors

To gain an understanding of some of the concepts underlying nonlinear dynamics, consider a simple pendulum. The pendulum exhibits two fundamental degrees of freedom: position and momentum. However, in its stable periodic state (referred to as a limit cycle), the pendulum exhibits only one degree of freedom, the phase angle. The dynamics of the system are attracted to a lower dimensional phase-space, and the dimension of this reduced phase-space is equal to the number of active degrees of freedom in the system. The trajectory of the pendulum in phase-space is called the attractor for the system. Attractors are not limited to zero dimension (fixed points in phase-space) or one dimension (limit cycles, like the simple pendulum). For nonlinear systems, attractors can have higher dimensions and, in some cases, even fractional dimensions. When the attractor for a system has a fractional dimension, the attractor is a fractal and is referred to as a strange attractor. For non-linear systems, a generated attractor has a dimension equal to the number of active degrees of freedom in the system and these attractors can have multiple dimensions and in some cases fractional dimensions.

Phase-Space Construction Directly from a Time Series

When confronted with a complicated physical system, an experimenter normally measures at regular and discrete intervals of time the value of some state variable (for example, records the time series s(t 0 ), s(t 1 ), s(t 2 ), . . . , with s(t i )∈

and t i =t 0 +i

t. This comprises the creation of a time series sequence of data as mention above. From the observed time series, the experimenter attempts to infer something about the dynamics (for example, the physics) of the system. Because the time series is one-dimensional, it is an incomplete description of the system during its time evolution. Nevertheless, many features of the dynamics of the system can be inferred from the time series. Takens (F. Takens, “Detecting Strange Attractors in Turbulence,” In Lecture Notes in Mathematics , Vol. 898, 366 Berlin: Springer-Verlag, 1981 which is incorporated herein by reference) and Packard et al. (N. Packard et al., “Geometry From a Time Series,” Phys. Rev. Lett., 45, 1980 which is incorporated herein by reference) have shown that for chaotic systems, the time series can be embedded into a higher dimensional space using time-delayed values of the series and thus recover the dynamics of the system. Vectors with components as:

X ( t )=[ s ( t ), s ( t +τ), s ( t+ 2τ), . . . , s ( t +( m− 1)τ)] T

where τ (time delay) and m (the embedding dimension) are parameters of the embedding procedure. Here X(t) represents a more complete description of dynamics than s(t).

An embedding dimension of m>2D+1, where D is the fractal dimension of the attractor, almost always ensures the construction of the topology of the attractor (Takens' theorem). If unlimited, infinitely precise data are available, almost any delay time τ and embedding dimension m>D will work, at least in principle. However, choosing the optimal parameters for real data is a nontrivial process.

If τ is too large, then the components s(t) and s(t+(m−1)τ) of the reconstructed vector will be effectively uncorrelated, which will cause inaccurate reconstruction of the attractor and thus inflate the estimated dimension of the system. On the other hand, if (m−1)τ is too small, then the components s(t), . . . , s(t+(m−1)τ) will all be very nearly equal, and the reconstructed attractor will fall near a long diagonal line. Thus, if the time delay is too large then it will cause an inaccurate reconstructions of the attractor and inflate the estimated dimension of the system, but if the time delay is too small then the reconstructed attractor will fall near a diagonal line. Generally, τ (time delay) must not be less than some characteristic decorrelation time, and (m−1)τ must not be too much greater than this decorrelation time. One such characteristic of time is the local minima of the autocorrelation function. This criterion gives only the linear dependence of the function s(t) with s(t+τ). The system here is nonlinear. Therefore, the mutual information of the system is a better property to use to select the optimal delay because the mutual information is (roughly) a nonlinear analogue of the autocorrelation function. That is, the mutual information is a property that shows the general dependence of s(t+τ) on s(t) rather than its linear dependence.

As an example, after embedding the time series in a state space using the Ruelle-Takens-Packard delay coordinate technique, the induced nonlinear mapping is learned using a local approximation. This procedure enables short-term forecasting of the future behavior of the time series using information based only on past values. Farmer and Sidorowich (J. Fanner and J. Sidorowich, “Predicting Chaotic Time Series,” Phys. Rev. Letts., 59, 1987 which is incorporated herein by reference) have already developed the error estimate of such a technique:

E≈Ce (m+1)KT N (m+1)/D)

where E=normalized error of prediction (0≤E≥1, where 0 is perfect prediction and 1 is a prediction no better than average)

m=order of local approximation K=Kolmogorov entropy T=forecasting window N=number of data points D=dimension of the attractor C=normalization constant

Using the Fanner-Sidorowich relation, the prediction horizon T for the zeroth order of local approximation can be found at step 206 . Any prediction above Tmax is no better than an average constant prediction, where T max is found using:

E ( T max )=1

For m=0, K is the largest Lyapunov exponent λ. Therefore, T max can be calculated from:

e KT

max

N −1/D ≈1

which can be written as:

T

max

≈

In

⁡

(

N

)

K

⁢

D

≈

In

⁡

(

N

)

λ

⁢

⁢

D

Any prediction beyond the indicated horizon (Ashrafi-Conway Horizon) is no better than an average value. The connection between the local and the global Lyapunov exponents has recently been found by Abarbanel et al. (H. Abarbanel et al., “Lyapunov Exponents in Chaotic Systems,” Rev. Modern Phys. Letts . (BJ, (in press) which is incorporated herein by reference) to be a power law of the form:

λ( l )=λ G +C/l v

where λ(l)=local Lyapunov exponent

l=length of observed data (observation window) v=a constant dependent to the dynamical system (0.5≤v≤1.0) c=a constant dependent to initial conditions of the system λ G =well-known global Lyapunov exponent ω=frequency of data points

Because any data set has a finite length, the Abarbanel-Kennel power law and Farmer-Sidorowich relation can be used to show that Tmax must have the form:

T max

≈

ln ⁡

(

l ⁢

⁢ ω

)

(

λ g

+

c ⁢

/

⁢

l v

)

⁢ D

This equation shows that as l increases linearly, Tmax increases logarithmically (point of diminishing return).

Structure of the Forecasting Algorithm

Once the state space representation is known, the next goal is to fit a model to the data to generate predictions at step 210 . Several methods can be used. The simplest method is to assume that the dynamics can be written as a map in the form:

X n+1 =M ( X n )

where the current state is Xn, and Xn+1 is a future state. Methods such as the polynomial method, rational approximation, radial basis functions, neural networks, and local approximations have all proven to be successful approaches to modeling. The local approximation technique is described here because it is the method used to structure the forecasting algorithm presented in this patent. A description of some of these modeling techniques is described below.

Local Approximation

The basic idea of this approach is to break up the domain of M into local neighborhoods and fit different parameters into each neighborhood. This fit is generally better than global approximation, especially for large data sets. Most global representations reach a point of diminishing return. At this point, adding more parameters or data gives only an incremental improvement in accuracy. After a certain point, adding more local neighborhoods is usually more efficient than adding more parameters and going to a higher order. With local approximation, it is possible to use a given functional representation efficiently. The key is to choose the local neighborhood size correctly, so that each neighborhood has just enough points to make the local parameter fit stable. Thus, adding more parameters or given data gives only an incremental improvement in accuracy so it is important to choose the local neighborhood size correctly when doing local approximation.

An example of local approximation is the first order, or nearest neighbor, approximation. This approach involves looking through the data set for the nearest neighbor to the current point and predicting the evolution of the state based on what the neighbor did at later times. This may also be described as establishing a sphere around a present point, where the sphere encompasses the nearest neighborhood points and the present point comprises the current point. Thus as illustrated in FIG. 5 , to predict future occurrences such as tomorrow's solar flux or another recurring data pattern using local approximation to first order, it is necessary to search the historical record at step 502 and find the solar flux or other data pattern most similar to that of today at step 504 . Tomorrow's solar flux or other data pattern should be the same as the neighboring pattern one day later may be determined at 506 . This determined future pattern may be used to determine future data values at step 508 using an AI system. Referring now also to FIG. 6 , first order approximation can sometimes be improved by finding more neighbors at step 602 , generating predictions for each of the additional neighbors (spheres) at step 604 and merging their predictions at step 606 , for example, by weighting according to distance from the current state. When the data is noisy, it is better to use a larger number of neighbors (spheres). This procedure can be improved by weighting the contributions of neighbors (spheres) according to their distance from the current state. The advantage of linear approximation is that the neighborhood size grows slowly with the embedding dimension. The order of approximation may depend on factors such as the choice of neighborhoods, the dimension, or peculiarities of the data set. For low-dimensional problems, a third order approximation is good enough.

Finding neighbors (spheres) in a multidimensional data set is time consuming when considering many points. A straightforward method is to compute the distance to each point when finding neighbors in a multidimensional data set, which takes approximately N steps for N points. This method can be reduced to roughly log N steps by organizing the data with a decision tree, such as a k-d tree. This approach has been taken for the models presented here.

To implement this approach, the parameters required for the reconstruction of the attractor (step 208 ) must be determined accurately. This determination will be discussed in the remainder of this section.

Choice of the Embedding Dimension d

In this section, the technique used to determine the correct value of the embedding dimension d from the scalar time series x(n), n=1, 2, . . . , N D will be covered. There must be enough data in the time series to remove concerns with statistical issues about numerical accuracy. Extrinsic noise in the data will be ignored for this analysis. By following Takens' phase-space attractor reconstruction technique for the time series, the dynamics of the system will be captured and embedded in phase-space. This procedure requires a correct choice of τ (time shift), which will be discussed in the next section.

Referring now to FIG. 7 , there is illustrated a flow diagram of the process to establish an embedding dimension. To establish the embedding dimension d, a characteristic of the attractor that becomes unchanging as d becomes large is needed. This invariant characteristic of the attractor is the attractor dimension d A . An initial value of d is selected at step 702 and an attractor dimension d A is determined at step 704 . Inquiry step 706 determines if the attractor dimension is constant. If not, one increases d at step 708 until inquiry step 706 determines d A becomes constant and identifies the minimum d where d A “saturates” as the embedding dimension at step 710 . Computation of d A is difficult, so the correlation function D(r) proposed by Grassberger and Procaccia in P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346, 1983, which is incorporated herein by reference is used for computing d A . The correlation function is given by:

D ⁡

(

r , N , d

)

=

2

N ⁡

(

N - 1

)

⁢

∑

∑

U ⁡

(

r -

ï˜

X ⁡

( j )

-

X ⁡

( i )



)

,

i ≠ j

where U(r) is the unit step function. For large values of N, the behavior of D(r, N, d) for r becomes independent of N and takes the form:

D ( r,N,d )=Φ( r,d ) r v(d)

By plotting D(r, N, d) versus r, the correct value of the dimension d can be singled out.

Choice of the Time Shift τ

The optimal choice of r has been discussed extensively in the literature such as J. H. Bentley, “Multidimensional Binary Search Trees in Data Base Applications,” IEEE Transactions on Software Engineering, 5 (4), 1979, which is incorporated herein by reference. Previous studies of the solar activity set r equal to the first local minimum of the autocorrelation function. The work of Fraser and Swinney suggests that a better criterion for the choice of the time delay is given by the first local minimum of the mutual information for the system. In this work, a modification of that criterion is used to forecast the sample time series.

Mutual Information

The autocorrelation function is given by:

R (τ)=∫ −∞ ∞ f ( t ) f ( t +τ) dt

This function measures the linear dependence of the function ƒ(t) with the time-shifted function ƒ(t+τ). Because the system under study here is nonlinear, the mutual information of the system is a better choice of functions if the optimal time delay is to be determined. The mutual information is a measure of the general dependence of the function with its time-shifted value, rather than a measure of the linear dependence. The mutual information between a time series Q and the time-shifted series S is given by:

I ( S,Q )= I ( Q,S )= H ( Q )+ H ( S )− H ( S,Q )

where H(Q) is the entropy of system Q, given by:

H ( Q )=−Σ i P q ( q i )log P q ( q i )

and H(S, Q) is the joint entropy of systems Sand Q, given by:

H ( S,Q )=Σ ij P sq ( S i ,q j )log P sq ( S p ,q i )

Pq and Psq in these equations are the probability densities for the corresponding states. The mutual information calculated in this manner gives a general measure of the independence of the time series S (the original time series shifted by the amount r) relative to the time series Q.

Computation of Mutual Information.

Because the mutual information is basically a sum of entropies, the probability density P for S, Q, and the joint system must be calculated. For a time series, this calculation cannot be performed analytically. An approximation to the calculation is possible following the prescription given in Fraser and Swinney. The procedure can be summarized as follows as shown in FIG. 8 . Let Q be the original time series at step 802 , and let S n be the series with the first n points removed at step 804 . Plotting Q against S n at step 806 produces a curve in a two-dimensional space that is used in the calculation of I(S, Q). Because the time series has a finite number of elements, a finite number of points occurs in the space of the system. Now subdivide each coordinate in the space at step 808 so that the same number of points falls in each of the subdivided regions. Thus, if the time series has i elements, the Q-axis is divided at location q j , chosen so that there are i/2 elements with q component greater than q j , and i/2 elements with q component less than q j . The S n -axis is divided in a similar manner. When the system is divided in this manner, the probabilities P(Q) and P(S n ) for the components q and S n of any point are the same for each region. This process simplifies the calculation of the mutual information because the only nontrivial probability density in the equation for I(S n , Q) is the mutual probability density, Psq.

Once the division of the coordinate axes has been accomplished as described above, the space of the two-dimensional system is divided into four regions. The probability that a randomly selected point in the system (but not in the original time series) will fall in region m of the space can be approximated at step 810 by:

P m

=

n m

N

where n m is the number of points from the time series that lie in region m, and N is the total number of points in the time series. This procedure can be repeated, dividing the regions into subregions, dividing the resulting subregions into sub-subregions, and so on, until the approximate probability density calculated using this technique becomes an accurate representation of the true probability density of the system.

Two cautions are in order here. First, the depth of the subdivisions must not be too shallow. If it is, then the resulting probability density will not accurately reflect the details of the true density. Second, if the subdivisions are made too often, the resulting probability density will pick up fluctuations arising from the discrete nature of the time series and will be bumpier than the true probability density. Thus, the criteria for halting subdivision of the regions must be handled carefully. The procedure taken here is to halt subdivision if the resulting subregions are equiprobable to within a specified tolerance. This tolerance is taken to be 20 percent as measured using a x-square test in this patent.

A Simple Model System

To test the effectiveness of the mutual information algorithm, several model systems have been analyzed. To illustrate the results, the Rossler system will be examined. The equations that describe a Rossler system are:

d

⁢

x

d

⁢

t

=

-

y

-

z

d

⁢

y

d

⁢

t

=

x

+

0

.

2

⁢

y

d

⁢

z

d

⁢

t

=

0

.

2

+

x

⁢

z

-

5

.

7

⁢

z

These equations have been integrated numerically using a fourth order Runge-Kutta algorithm with a fixed step size of Δt=0.05. The system produces the chaotic attractor 902 shown in FIG. 9 that is a Rossler Attractor. A Rossler system is a system of three non-linear ordinary differential equations originally studied by Otto Rossler. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. An orbit within the attractor follows and out spirals close to the X, Y plane around an unstable fixed point. Once the graph spirals out enough, a second fixed point influences the graph, causing a rise in twist in the Z dimension. In the time domain, it becomes apparent that although each variable is oscillating within a fixed range of values, the oscillations are chaotic. This attractor has some similarities to the Lorenz attractor, but is simpler and has only one manifold. The data generated for the x, y, and z coordinates of the system have been treated as independent time series, and the resulting mutual information for these time series has been calculated using the algorithm described above. The mutual information for the x-coordinate as a function of the time shift is shown in FIG. 10 . It illustrates mutual information for Rossler 4096 points 1002 and 8192 points 1004 .

The Rossler system illustrates the features of good and bad choices of the time delay. The goal of the analysis of the mutual information is the selection of the time shift which given a time series, best reconstructs the attractor of the system using the Ruelle-Takens-Packard technique. The suggestion that the first local minimum of the mutual information produces the optimal time shift can be examined for this system. For convenience, consider the time series constructed from the x coordinate data for the system. As can be seen in FIG. 10 , the first local minimum occurs at about t=32 steps−1.6 time units. (Note that minima that occur due to fluctuations arising from the discrete nature of the time series are omitted.) The time shift suggested is therefore τ=1.6. Because the Rossler system is three-dimensional, this choice gives the location of the points on the reconstructed attractor as:

r {right arrow over (i)} =[x ( t ;), x ( t i +1.6), x ( t i +3.2)] T

When the resulting vectors are plotted in phase-space, the attractor does resemble the original system (see FIG. 11 ). Adjacent sections along the trajectory are spaced in the same way as on the original attractor. The time series does not contain information about the range of the y or z variables, so the actual orientation and scaling of the reconstructed attractor are different than for the actual system.

For the purposes of prediction, the reconstructed attractor shown in FIG. 11 has one disturbing feature. For one specific region 1102 through which many trajectories pass (the cusp on the left side of the phase portrait), any prediction made across this region will be inaccurate. However, a better choice exists for the time delay than that made above. This better choice is seen by first examining a much worse choice. Suppose that the first local maximum of the mutual information is chosen for the time shift, so that τ=3.15. The reconstructed attractor 1202 for this choice is shown in FIG. 12 . This reconstructed attractor 1202 does not preserve even the appearance of the original attractor 902 . This difference is explained by considering what the mutual information tells about the system. The mutual information is a measure of the dependence of the system with the time delayed version of itself. For reconstruction, the reconstructed axes should be as orthogonal to one another as possible. This pseudo-orthogonality produces the optimal spread in the trajectory of the attractor and, therefore, the best reconstruction for forecasting the system. The problem with the reconstruction 1202 based on τ=1.6 is that the (t+2τ)-axis hits near the first local maximum of the mutual information and is, therefore, far from satisfying the pseudo-orthogonality criterion. In this work, the axes are made pseudo-orthogonal by finding the time shift τ that minimizes the murual information at each of the time-shifted values of τ. That is, the optimal choice of the time shift for a d-dimension system is found by minimizing:

I

t

=

∑

j

=

1

d

-

1

⁢

I

⁡

(

τ

j

)

For the Rossler system, this minimization indicates that a time delay of τ≈0.85 should be used when reconstructing the attractor. The resulting system is shown in FIG. 13 . For comparison, the reconstruction 1402 using a time delay of τ=0.40 is shown in FIG. 14 . The optimal reconstruction 1302 shows the best spread of the trajectories possible given the limitations of the time series data. Therefore, it is this optimization criterion that must be applied to the time series for the solar flux data to produce the best reconstructed attractor on which to forecast.

Preliminary Forecasts of the Solar Flux or Other Data

Solar flux or other data is not noise free. The goal of this work is to determine the optimal time shift for the reconstruction of the attractor for the solar flux or other data. Because the criterion chosen for the determination of this time shift depends on the mutual information, the effects of noise on the mutual information should be examined. FIG. 15 shows the effect of additive noise on the mutual information for the Rossler system. The extremas of th

CLAIMS

Claims ( 20 )

What is claimed is:

1 . A method for predicting future behavior for a dynamic system using an artificial intelligence system, comprising:

implementing the artificial intelligence system within a programmable processing system; configuring a chaotic oscillator implemented within the artificial intelligence system within the programmable processing system for providing a chaotic time series; receiving a predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system at the artificial intelligence system implemented within the programmable processing system; constructing an attractor from the time series group of data defining the previous behavior of the dynamic system using the artificial intelligence system implemented within the programmable processing system, the attractor modeling the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system; determining a prediction horizon for the predetermined amount of the time series group of data with the artificial intelligence system implemented within the programmable processing system using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor, wherein the prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly; implementing the constructed attractor, the determined prediction horizon and the chaotic time series provided by the chaotic oscillator within the artificial intelligence system implemented within the programmable processing system, the artificial intelligence system providing nonlinear modeling and forecasting for predictive capabilities of the artificial intelligence system; generating predicted values of future behavior of data generated by the dynamic system with the artificial intelligence system implementing the programmable processing system using the constructed attractor, the determined prediction horizon and the chaotic time series provided by the chaotic oscillator; and predicting a future behavior of the dynamic system responsive to the generated predicted values using the artificial intelligence system.

2 . The method of claim 1 , wherein the step of constructing further comprises:

creating a time delay value for the predetermined amount of the time series group of data using the artificial intelligence system implemented within the programmable processing system; and constructing the attractor responsive to the predetermined amount of the time series group of data and the time delay value using the artificial intelligence system implemented within the programmable processing system.

3 . The method of claim 2 , wherein the step of creating the time delay value further comprises:

determining mutual information having a first local minimum for the time series group of data; and determining the time delay value using the first local minimum of the mutual information.

4 . The method of claim 1 , wherein the step of constructing further comprises:

determining drivers of the dynamic system using the artificial intelligence system implemented within the programmable processing system; and constructing the attractor responsive to determined drivers using the artificial intelligence system implemented within the programmable processing system.

5 . The method of claim 1 , wherein the step of constructing further comprises:

extracting invariants from the dynamic system using the artificial intelligence system implemented within the programmable processing system implemented without reference to underlying physics of the dynamic system; and constructing the attractor responsive to the invariants using the artificial intelligence system implemented within the programmable processing system.

6 . The method of claim 1 , wherein the step of determining the prediction horizon further comprises:

determining the attractor dimension of the attractor using the artificial intelligence system implemented within the programmable processing system; determining the Lyapunov exponent of the attractor using the artificial intelligence system implemented within the programmable processing system; and calculating the prediction horizon responsive to the attractor dimension and the Lyapunov exponent, wherein the prediction horizon increases logarithmically as the length of the predetermined amount of the time series group of data from the dynamic system increases linearly using the artificial intelligence system implemented within the programmable processing system.

7 . The method of claim 1 , wherein the step of generating the predicted values of future behavior further comprises:

selecting a plurality of neighbors within the time series group of data; examining data sets of the attractor for the plurality of neighbors; and generating the predicted values of the future behavior responsive to values of the data sets of the attractor for the plurality of neighbors.

8 . The method of claim 1 , wherein the step of predicting further comprises:

looking through a data set of the attractor for a nearest neighbor to a current point for which the predicted value is being generated; and predicting an evolution of a state of the current point based on what the nearest neighbor did at later times.

9 . The method of claim 8 further including the step of computing a distance to each point in a multidimensional data set when finding neighbors in the multidimensional data set.

10 . A method for predicting future behavior for a dynamic system using an artificial intelligence system, comprising:

implementing the artificial intelligence system within a graphics processing unit; receiving a predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system at the artificial intelligence system implemented within the graphics processing unit; constructing an attractor from the time series group of data defining the previous behavior of the dynamic system using the artificial intelligence system implemented within the graphics processing unit, the attractor modeling the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system, wherein the attractor predicts future behaviors of the dynamic system more accurately than a statistical model; determining a prediction horizon for the predetermined amount of the time series group of data with the artificial intelligence system implemented within the graphics processing unit using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor, wherein the prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly; implementing the constructed attractor and the determined prediction horizon within the artificial intelligence system implemented within the graphics processing unit, the artificial intelligence system providing nonlinear modeling and forecasting for predictive capabilities of the artificial intelligence system; generating predicted values of future behavior of data generated by the dynamic system with the artificial intelligence system implemented within the graphics processing unit and implementing the constructed attractor and the determined prediction horizon; and predicting a future behavior of the dynamic system responsive to the generated predicted values using the artificial intelligence system.

11 . The method of claim 10 , wherein the step of constructing further comprises:

creating a time delay value for the predetermined amount of the time series group of data using the artificial intelligence system implemented within the graphics processing unit; and constructing the attractor responsive to the predetermined amount of the time series group of data and the time delay value using the artificial intelligence system implemented within the graphics processing unit.

12 . The method of claim 11 , wherein the step of creating the time delay value further comprises:

determining mutual information having a first local minimum for the time series group of data; and determining the time delay value using the first local minimum of the mutual information.

13 . The method of claim 10 , wherein the step of constructing further comprises:

determining drivers of the dynamic system using the artificial intelligence system implemented within the graphics processing unit; and constructing the attractor responsive to determined drivers using the artificial intelligence system implemented within the graphics processing unit.

14 . The method of claim 10 , wherein the step of constructing further comprises:

extracting invariants from the dynamic system using the artificial intelligence system implemented within the graphics processing unit implemented without reference to underlying physics of the dynamic system; and constructing the attractor responsive to the invariants using the artificial intelligence system implemented within the graphics processing unit.

15 . The method of claim 10 , wherein the step of determining the prediction horizon further comprises:

determining the attractor dimension of the attractor using the artificial intelligence system implemented within the graphics processing unit; determining the Lyapunov exponent of the attractor using the artificial intelligence system implemented within the graphics processing unit; and calculating the prediction horizon responsive to the attractor dimension and the Lyapunov exponent, wherein the prediction horizon increases logarithmically as the length of the predetermined amount of the time series group of data from the dynamic system increases linearly using the artificial intelligence system implemented within the graphics processing unit.

16 . The method of claim 10 , wherein the step of generating the predicted values of future behavior further comprises:

selecting a plurality of neighbors within the time series group of data; examining data sets of the attractor for the plurality of neighbors; and generating the predicted values of the future behavior responsive to values of the data sets of the attractor for the plurality of neighbors.

17 . A method for predicting future behavior for a dynamic system using an artificial intelligence system, comprising:

implementing the artificial intelligence system within programmable processing system; configuring a chaotic oscillator implemented within the artificial intelligence system within the programmable processing system for providing a chaotic time series; receiving a predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system at the artificial intelligence system implemented within programmable processing system, wherein at least a portion of the time series group of data comprises drivers of the dynamic system; constructing an attractor from the time series group of data defining the previous behavior of the dynamic system using the artificial intelligence system implemented within the programmable processing system, the attractor modeling the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system; determining a prediction horizon for the predetermined amount of the time series group of data with the artificial intelligence system implemented within the programmable processing system using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor, wherein the prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly; implementing the constructed attractor, the determined prediction horizon and the chaotic time series provided by the chaotic oscillator within the artificial intelligence system implemented within the programmable processing system, the artificial intelligence system providing nonlinear modeling and forecasting for predictive capabilities of the artificial intelligence system; generating predicted values of future behavior of data generated by the dynamic system with the artificial intelligence system implemented within the programmable processing system implementing the constructed attractor and the determined prediction horizon and the chaotic time series provided by the chaotic oscillator; and predicting a future behavior of the dynamic system responsive to the generated predicted values using the artificial intelligence system.

18 . The method of claim 17 , wherein the attractor dimension equals a number of active degrees of freedom of the dynamic system, further wherein the attractor has multiple dimensions including fractional dimensions.

19 . A method for predicting future behavior for a dynamic system using an artificial intelligence system, comprising:

implementing the artificial intelligence system within a programmable processing system; configuring a chaotic oscillator implemented within the artificial intelligence system within the programmable processing system for providing a chaotic time series; receiving a predetermined amount of a time series group of data from the dynamic system defining previous behavior of the dynamic system at the artificial intelligence system; extracting invariants from the dynamic system using the artificial intelligence system implemented without reference to underlying physics of the dynamic system; constructing an attractor from the time series group of data defining the previous behavior of the dynamic system and the invariants using the artificial intelligence system, the attractor modeling the previous behavior of the dynamic system based on the predetermined amount of the time series group of data of the dynamic system and the invariants; determining a prediction horizon for the predetermined amount of the time series group of data with the artificial intelligence system using an attractor dimension of the constructed attractor and a Lyapunov exponent of the constructed attractor, wherein the prediction horizon increases logarithmically as a length of the predetermined amount of the time series group of data from the dynamic system increases linearly; implementing the constructed attractor, the determined prediction horizon and the chaotic time series provided by the chaotic oscillator within the artificial intelligence system implemented within the programmable processing system, the artificial intelligence system providing nonlinear modeling and forecasting for predictive capabilities of the artificial intelligence system; generating prediction values of future behavior of the dynamic system with the artificial intelligence system implementing the constructed attractor, the determined prediction horizon and the chaotic time series provided by the chaotic oscillator; and predicting a future behavior of the dynamic system responsive to the generated predicted values using the artificial intelligence system.

20 . The method of claim 19 , wherein the prediction horizon defines a limit to which the prediction values are accurately generated, the prediction values beyond the prediction horizon are no more accurate than the prediction values generated from a statistical model.

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S. Ashrafi, Combining Schatten's Solar Activity Prediction Model With a Chaotic Prediction Model, National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, 554-FDD-911125, Nov. 1991.

Sadoudi, S., Azzaz, M. S., Tanougast, C., & Dandache, A. (Dec. 2009). Real time hardware implementation of a new Duffing's chaotic attractor. In 2009 16th IEEE International Conference on Electronics, Circuits and Systems—(ICECS 2009) (pp. 559-562). IEEE. (Year: 2009).

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