Detecting Unstable Fixed Points Using Kalman Filters ... - IEEE Xplore
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006
Detecting Unstable Fixed Points Using Kalman Filters With Constraints David M. Walker and Michael Small, Member, IEEE
Abstract—We introduce a novel application of the extended Kalman filter for detecting unstable periodic orbits of dynamical systems from experimental time series. The unknown dynamics are approximated using local polynomial functions, and the fixed-point constraint conditions are regarded as additional observation functions in the filter framework. Candidate fixed-point values are considered as parameters to be estimated. We demonstrate the application of this method to test systems and use surrogate time series data to test the statistical significance of the results. Index Terms—Kalman filtering, surrogate time series, unstable fixed points.
I. INTRODUCTION
T
HE unstable periodic orbits of a dynamical system play a crucial role in the theoretical and practical understanding of its properties. In a chaotic system, quantities such as dimension estimates, entropy calculations, and Lyapunov exponents, for example, can be approximated using periodic orbits [1], [2]. There has been a number of techniques proposed by researchers for discovering periodic orbit information from time series data. These methods include studying close returns [3], determining local dynamical models, studying the fixed-point structure of these models [4], [5], and examining the higher order periodic structure of reconstructed models [6]. In this paper, we introduce a new application of nonlinear filtering to extract a periodic structure from experimental time series. Specifically, we use a novel variant of the extended Kalman filter (EKF) subject to constraints [7] to extract the unstable fixed-point (UFP) structure of a dynamical system. We reconstruct the system dynamics locally from time series and incorporate these models into a Kalman filter with constraints (KFC algorithm) satisfied by potential UFPs of the system. The reliability of these estimates is tested using the method of surrogate data analysis. We follow a dynamical systems approach to nonlinear systems identification [8], [9]. The approach is based on Takens’ Embedding Theorem [10], which allows reconstruction of an Manuscript received July 18, 2005; revised March 30, 2006. This work was supported by the Scottish Executive Environment and Rural Affairs Department (SEERAD) and the Hong Kong University Grants Council under Competitive Earmarked Research Grant (CERG) PolyU5235/03E. This paper was recommended by Associate Editor B. Shi. D. M. Walker is with the Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]. edu.hk). M. Small is with the Department of Electrical and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: small@ieee. org). Digital Object Identifier 10.1109/TCSI.2006.883859
equivalent phase space to the original (unknown) system from observations of the system in the form of a scalar time series . The reconstructed state can be represented by
where is the embedding dimension and is the time-delay lag. There are a number of methods that can be used to determine an appropriate , with a good prescription being to choose the first minimum of the average mutual information function [8], [11]. Similarly, a recognized method for selecting a suitable embedding dimension is false nearest neighbors [8], [12]. There are more general time-delay embeddings referred to as nonuniform embedding where the lags are not multiples of a single time delay [13]. These nonuniform embeddings can give better representations of the system state, but, for the purposes of this paper, we will use the standard embedding. The dynamics for a time-delay embedding consist of a simple shift operator together with a (nonlinear) scalar-valued function , i.e.,
There are a number of choices to use as the function . In this paper, we consider local polynomial models, in particular, linear models. These models are reconstructed by identifying nearest embedded data points. The neighbors of from a library of evolution of the selected points is studied to estimate the parameters of a linear approximation of the local dynamics. The (unstable) fixed points of a model in reconstructed phase space can be seen to satisfy (1) where is the fixed-point location. We suggest incorporating (1) as a constraint within a Kalman filter framework to determine . The remainder of this paper is organized as follows. In Section II, we briefly introduce the EKF and describe how it can be modified to incorporate constraints; this is the KFC algorithm. UFPs can be estimated by selecting suitable constraint functions in the KFC algorithm, and we explain how this is done. The statistical significance of candidate UFPs can be investigated using the method of surrogate data and, thus, in Section III, an overview of these techniques is presented. Illustrative examples using data from Chua’s circuit [14], [15], and two other nonlinear dynamical systems are studied to demonstrate the potential of the method in Section IV. A short discussion summarizing the technique and describing the flexible parameters of the method concludes the paper.
1057-7122/$20.00 © 2006 IEEE
WALKER AND SMALL: DETECTING UNSTABLE FIXED POINTS USING KALMAN FILTERS WITH CONSTRAINTS
II. KALMAN FILTER WITH CONSTRAINTS The Kalman filter and its variant to deal with nonlinear systems, the EKF, are statistical state estimators. The algorithms are typically used to achieve noise reduction in signal processing. The system model required by the Kalman filter for this situation is
(2) where represents the reconstructed system state evolving ac; local linear plus cording to the (reconstructed) dynamics time-delay shift in our case. The state is observed using the to obtain a prediction of the time series data scalar function subject to the observational noise term which is assumed . The to be from a Gaussian distribution, i.e., . The filter dynamical noise is represented by , where and is initialized with . The EKF update equations for the system (2) takes the following forms [16]–[18]: time-update equations
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dynamic equations and play a similar role to unknown parameters in Kalman filter estimation [20]. The equation satisfied by a fixed point (1) takes the role of a constraint in the observational system model. The filter update equations are essentially unchanged, but the matrices and vectors in the equations now account for observation functions, where are the number of UFP constraints we try to meet. Explicitly, the update equations become: time-update equations
where
measurement-update equations
measurement-update equations .. .
.. .
.. .
where where
The matrix
is referred to as the Kalman gain. The terms are known as the innovations. The above algorithm has been applied using local polynomial reconstructions for the purposes of noise reduction by Walker [19]. The setup of the Kalman filter is readily amenable to including UFP locations as constraints to be met. We can do this by modifying the observation model of (2) to include the model’s approximation to the UFPs. The system model for applying the Kalman filter with constraints, referred to in the sequel as the KFC algorithm, becomes
where the new terms include , where can reflect the accuracy of the knowledge of the fixed-point location. The UFP locations are essentially appended to the systems
and
where . The above KFC algorithm can be utilized for UFP detection in the following manner. We embed the time series using suitable embedding parameters and possible fixed-point constraints. These possible loconsider cations are initialized by randomly selecting values from an area covering the dynamic range of the time series. The data are presented to the KFC algorithm, and, at each time step, a local linear model of the dynamics is estimated using a portion of the embedded data reserved as a library. The local model is used to perform the time update and measurement update steps of the KFC. estimates of as candidate UFPs We store the evolving and study histograms of the resulting quantities. These could be the final values the reach, or all values as we progress through the time series, or some subset
2820
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006
of these.1 The hope is that true UFPs of the system will reveal themselves as dominant spikes in such histograms. Following Glover and Mees [4], we have found it helpful to calculate the for each and to detect the meanquantity ingfulness of the estimated UFPs. The rationale behind this calculation is that points not near a fixed point will typically yield candidate fixed points far from any real UFP. Conversely, it is expected that points near true fixed points should produce candican date fixed points close to real UFPs. The magnitude of be informative in dismissing spurious fixed-point candidates. The reliability of these estimates are further investigated using surrogate data analysis where the surrogate time series are processed by the KFC algorithm and the corresponding estimates are studied. If the distribution of these estimates is significantly different from the estimates obtained from the original data, we can reliably conclude that the identified UFP estimates are meaningful. A further refinement that we have found to be useful is to apply the KFC algorithm to short sliding windows of length of the scalar time series. For each window, we choose a random and apply the KFC with constraints. The values initial and are used to determine candidate UFPs, the sigof nificance of which are examined using the equivalent quantities obtained from the surrogate time series. III. SURROGATE TIME SERIES The method described in the previous section should be sufficient for the detection of UFPs in a dynamical system from observed time series. We employ surrogate time series methods merely as a form of “sanity check.” The surrogate data methods are used to verify that the UFPs detected by the KFC algorithm are not the result of mere chance. To do this, we employ four distinct surrogate methods: the three standard techniques described by Theiler et al. [21] and a new pseudoperiodic surrogate test introduced in [22]. The original motivation of the method of surrogate data was to provide a scheme to avoid excessive false positives in the search for chaos using correlation dimension estimation techniques. The principle is simple but powerful. For a given observed time series, one estimates a quantity of interest (in the original context, this was usually correlation dimension; we are concerned with UFPs) and then estimate the same quantity from an ensemble of surrogate time series. The surrogate time series are constructed so that they are somehow similar to the original data, but at the same time they are consistent with some trivial (and typically uninteresting) hypothesis [23]. For example, during the search for chaos in experimental data, one may choose to estimate correlation dimension and ascribe a fractional value to the presence of chaos in the underlying system.2 However, given that almost all numbers (represented on a floating point machine) are fractions, what is the chance of getting a fractional correlation dimension through mere happenstance? Surrogate data may be generated from the original time
0
1N is the length of the time series so (N m ) is the length of the embedded time series. 2Even in the ideal situation, a fractional correlation dimension is actually neither necessary nor sufficient, but merely “typical” [24].
series to test this. The surrogate data are similar to the original but contain no nonlinear deterministic dynamics (technically, the surrogates may be consistent with a monotonic nonlinear transformation of linearly filtered noise) and are therefore not chaotic. One then computes the distribution of correlation dimension estimates for the surrogates and the data and compares these with the original. If the two are dissimilar, then the underlying hypothesis may be rejected. Otherwise, it may not. Our rationale is the same. We wish to know whether the candidate UFPs produced by the KFC algorithm reflect a genuine structure in the underlying system, or merely chance. To test this, we generate surrogate data consistent with each of the three linear hypotheses: 0) independent and identically distributed noise; 1) linearly filtered noise; and 2) a monotonic nonlinear transformation of linearly filtered noise. We then compute UFP distributions using the KFC method for both data and surrogates and compare the results. As the surrogates do not contain UFPs, we do not expect the KFC technique to find any. Any evidence of UFPs is purely bad luck. Moreover, we require that evidence of UFPs in the original data must be stronger than that in the surrogates. Conversely, we would also like to determine the typical variability one may expect from our results for data which do exhibit UFPs. For this purpose, we employ the new surrogate technique described in [22] and [25]. The principle of this method is to generate surrogate data which exhibit the same large-scale nonlinear deterministic structure as the observed data, but have noise obscuring the fine-scale structure. The level of noise is an adjustable parameter of the algorithm [25] and may be used to generate surrogates which either have the same nonlinear dynamics as the data [25] or are simply noise-driven periodic orbits of the same general shape as the data. In [22], this method was used to distinguish between chaotic Rössler dynamics with observational noise and the same system in a periodic (period-6) regime with dynamic noise. In Section IV, we apply these surrogates to test for both false identification of UFPs and to develop an estimate of the expected variability of these UFP distributions. IV. EXAMPLES The Chua circuit can be represented by the following equations [14], [15]:
where . The fixed points are given by and with . A double scroll at, tractor can be generated using the parameter set ( , , and ). Fig. 1 shows a time series of the variable together with the values of the three fixed points. (We integrated the Chua circuit equations with initial condition s to s sampling every 0.1 s and ( 1, 0, 0) from then retaining the last 3000 data points.) A calculation of mutual information [11] suggests a time-delay lag of six and false nearest neighbors with the selected lag suggests a three-dimensional embedding [12]. However, we decided to use a four-dimensional embedding.
WALKER AND SMALL: DETECTING UNSTABLE FIXED POINTS USING KALMAN FILTERS WITH CONSTRAINTS
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Fig. 1. x versus t for the Chua circuit. The horizontal lines indicate the location of the (unstable) fixed points.
We reserve the first embedded data points to beused as a library for reconstructing the local linear models and apply the KFC to the next 1000 time-series values. For the given emas the number bedding dimension, we selected a value of of nearest neighbors to use in the local reconstructions. We used and considered a sliding window of length fixed-point constraints. The assumed constraint error covariance . The assumed for each fixed point was set to noise levels for the dynamics was set to , where is the standard deviation of the time-series data. The observational and the initial error covariance for the noise covariance . state and fixed-point estimates were set to We note that all of these parameters are tunable, and so there is great flexibility in applying the KFC for UFP detection. In Fig. 2, we show a plot of against for each candidate fixed point and sliding window. A histogram of the more meaningful estimates are shown in Fig. 3, where only are considered. We can see those points with that there are two large peaks centered about the known UFPs at . The UFP at the origin is not so prominent. The reliability of these estimates can be tested using surrogate data. In Fig. 4, we compare the estimated UFPs for the double scroll data in Fig. 1 to the same calculation of candidate UFPs for linear surrogate time series. In each case, the linear surrogate time series exhibit only a peak in the distribution of UFPs (for algorithms 1 and 2) at 0. There is no indication of false identification of nonzero UFPs. This is to be expected. Fig. 5 repeats the calculation of Fig. 4 for nonlinear surrogates generated using the PPS algorithm described in [22]. In this case, we expect the surrogates to exhibit similar distributions of UFPs as the data. This is precisely what we observe. The three calculations in Fig. 5 are for three distinct levels of dynamic noise in the surrogates. For the lowest level of noise (the top panel of Fig. 5), the distribution of candidate UFPs is very close to that estimated from the data: both data and surrogates exhibit almost the same dynamics and therefore the same UFPs. For increasing levels of dynamic noise, the distinction becomes more pronounced. This is
Fig. 2. Plot of p versus log(v ) for data from Chua’s circuit. There appears to be accumulations about r for small values of v .
6
Fig. 3. Histogram of the candidate UFPs for which log(v are definite peaks about the known UFPs at r .
6
)
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006
Detecting Unstable Fixed Points Using Kalman Filters With Constraints David M. Walker and Michael Small, Member, IEEE
Abstract—We introduce a novel application of the extended Kalman filter for detecting unstable periodic orbits of dynamical systems from experimental time series. The unknown dynamics are approximated using local polynomial functions, and the fixed-point constraint conditions are regarded as additional observation functions in the filter framework. Candidate fixed-point values are considered as parameters to be estimated. We demonstrate the application of this method to test systems and use surrogate time series data to test the statistical significance of the results. Index Terms—Kalman filtering, surrogate time series, unstable fixed points.
I. INTRODUCTION
T
HE unstable periodic orbits of a dynamical system play a crucial role in the theoretical and practical understanding of its properties. In a chaotic system, quantities such as dimension estimates, entropy calculations, and Lyapunov exponents, for example, can be approximated using periodic orbits [1], [2]. There has been a number of techniques proposed by researchers for discovering periodic orbit information from time series data. These methods include studying close returns [3], determining local dynamical models, studying the fixed-point structure of these models [4], [5], and examining the higher order periodic structure of reconstructed models [6]. In this paper, we introduce a new application of nonlinear filtering to extract a periodic structure from experimental time series. Specifically, we use a novel variant of the extended Kalman filter (EKF) subject to constraints [7] to extract the unstable fixed-point (UFP) structure of a dynamical system. We reconstruct the system dynamics locally from time series and incorporate these models into a Kalman filter with constraints (KFC algorithm) satisfied by potential UFPs of the system. The reliability of these estimates is tested using the method of surrogate data analysis. We follow a dynamical systems approach to nonlinear systems identification [8], [9]. The approach is based on Takens’ Embedding Theorem [10], which allows reconstruction of an Manuscript received July 18, 2005; revised March 30, 2006. This work was supported by the Scottish Executive Environment and Rural Affairs Department (SEERAD) and the Hong Kong University Grants Council under Competitive Earmarked Research Grant (CERG) PolyU5235/03E. This paper was recommended by Associate Editor B. Shi. D. M. Walker is with the Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]. edu.hk). M. Small is with the Department of Electrical and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: small@ieee. org). Digital Object Identifier 10.1109/TCSI.2006.883859
equivalent phase space to the original (unknown) system from observations of the system in the form of a scalar time series . The reconstructed state can be represented by
where is the embedding dimension and is the time-delay lag. There are a number of methods that can be used to determine an appropriate , with a good prescription being to choose the first minimum of the average mutual information function [8], [11]. Similarly, a recognized method for selecting a suitable embedding dimension is false nearest neighbors [8], [12]. There are more general time-delay embeddings referred to as nonuniform embedding where the lags are not multiples of a single time delay [13]. These nonuniform embeddings can give better representations of the system state, but, for the purposes of this paper, we will use the standard embedding. The dynamics for a time-delay embedding consist of a simple shift operator together with a (nonlinear) scalar-valued function , i.e.,
There are a number of choices to use as the function . In this paper, we consider local polynomial models, in particular, linear models. These models are reconstructed by identifying nearest embedded data points. The neighbors of from a library of evolution of the selected points is studied to estimate the parameters of a linear approximation of the local dynamics. The (unstable) fixed points of a model in reconstructed phase space can be seen to satisfy (1) where is the fixed-point location. We suggest incorporating (1) as a constraint within a Kalman filter framework to determine . The remainder of this paper is organized as follows. In Section II, we briefly introduce the EKF and describe how it can be modified to incorporate constraints; this is the KFC algorithm. UFPs can be estimated by selecting suitable constraint functions in the KFC algorithm, and we explain how this is done. The statistical significance of candidate UFPs can be investigated using the method of surrogate data and, thus, in Section III, an overview of these techniques is presented. Illustrative examples using data from Chua’s circuit [14], [15], and two other nonlinear dynamical systems are studied to demonstrate the potential of the method in Section IV. A short discussion summarizing the technique and describing the flexible parameters of the method concludes the paper.
1057-7122/$20.00 © 2006 IEEE
WALKER AND SMALL: DETECTING UNSTABLE FIXED POINTS USING KALMAN FILTERS WITH CONSTRAINTS
II. KALMAN FILTER WITH CONSTRAINTS The Kalman filter and its variant to deal with nonlinear systems, the EKF, are statistical state estimators. The algorithms are typically used to achieve noise reduction in signal processing. The system model required by the Kalman filter for this situation is
(2) where represents the reconstructed system state evolving ac; local linear plus cording to the (reconstructed) dynamics time-delay shift in our case. The state is observed using the to obtain a prediction of the time series data scalar function subject to the observational noise term which is assumed . The to be from a Gaussian distribution, i.e., . The filter dynamical noise is represented by , where and is initialized with . The EKF update equations for the system (2) takes the following forms [16]–[18]: time-update equations
2819
dynamic equations and play a similar role to unknown parameters in Kalman filter estimation [20]. The equation satisfied by a fixed point (1) takes the role of a constraint in the observational system model. The filter update equations are essentially unchanged, but the matrices and vectors in the equations now account for observation functions, where are the number of UFP constraints we try to meet. Explicitly, the update equations become: time-update equations
where
measurement-update equations
measurement-update equations .. .
.. .
.. .
where where
The matrix
is referred to as the Kalman gain. The terms are known as the innovations. The above algorithm has been applied using local polynomial reconstructions for the purposes of noise reduction by Walker [19]. The setup of the Kalman filter is readily amenable to including UFP locations as constraints to be met. We can do this by modifying the observation model of (2) to include the model’s approximation to the UFPs. The system model for applying the Kalman filter with constraints, referred to in the sequel as the KFC algorithm, becomes
where the new terms include , where can reflect the accuracy of the knowledge of the fixed-point location. The UFP locations are essentially appended to the systems
and
where . The above KFC algorithm can be utilized for UFP detection in the following manner. We embed the time series using suitable embedding parameters and possible fixed-point constraints. These possible loconsider cations are initialized by randomly selecting values from an area covering the dynamic range of the time series. The data are presented to the KFC algorithm, and, at each time step, a local linear model of the dynamics is estimated using a portion of the embedded data reserved as a library. The local model is used to perform the time update and measurement update steps of the KFC. estimates of as candidate UFPs We store the evolving and study histograms of the resulting quantities. These could be the final values the reach, or all values as we progress through the time series, or some subset
2820
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006
of these.1 The hope is that true UFPs of the system will reveal themselves as dominant spikes in such histograms. Following Glover and Mees [4], we have found it helpful to calculate the for each and to detect the meanquantity ingfulness of the estimated UFPs. The rationale behind this calculation is that points not near a fixed point will typically yield candidate fixed points far from any real UFP. Conversely, it is expected that points near true fixed points should produce candican date fixed points close to real UFPs. The magnitude of be informative in dismissing spurious fixed-point candidates. The reliability of these estimates are further investigated using surrogate data analysis where the surrogate time series are processed by the KFC algorithm and the corresponding estimates are studied. If the distribution of these estimates is significantly different from the estimates obtained from the original data, we can reliably conclude that the identified UFP estimates are meaningful. A further refinement that we have found to be useful is to apply the KFC algorithm to short sliding windows of length of the scalar time series. For each window, we choose a random and apply the KFC with constraints. The values initial and are used to determine candidate UFPs, the sigof nificance of which are examined using the equivalent quantities obtained from the surrogate time series. III. SURROGATE TIME SERIES The method described in the previous section should be sufficient for the detection of UFPs in a dynamical system from observed time series. We employ surrogate time series methods merely as a form of “sanity check.” The surrogate data methods are used to verify that the UFPs detected by the KFC algorithm are not the result of mere chance. To do this, we employ four distinct surrogate methods: the three standard techniques described by Theiler et al. [21] and a new pseudoperiodic surrogate test introduced in [22]. The original motivation of the method of surrogate data was to provide a scheme to avoid excessive false positives in the search for chaos using correlation dimension estimation techniques. The principle is simple but powerful. For a given observed time series, one estimates a quantity of interest (in the original context, this was usually correlation dimension; we are concerned with UFPs) and then estimate the same quantity from an ensemble of surrogate time series. The surrogate time series are constructed so that they are somehow similar to the original data, but at the same time they are consistent with some trivial (and typically uninteresting) hypothesis [23]. For example, during the search for chaos in experimental data, one may choose to estimate correlation dimension and ascribe a fractional value to the presence of chaos in the underlying system.2 However, given that almost all numbers (represented on a floating point machine) are fractions, what is the chance of getting a fractional correlation dimension through mere happenstance? Surrogate data may be generated from the original time
0
1N is the length of the time series so (N m ) is the length of the embedded time series. 2Even in the ideal situation, a fractional correlation dimension is actually neither necessary nor sufficient, but merely “typical” [24].
series to test this. The surrogate data are similar to the original but contain no nonlinear deterministic dynamics (technically, the surrogates may be consistent with a monotonic nonlinear transformation of linearly filtered noise) and are therefore not chaotic. One then computes the distribution of correlation dimension estimates for the surrogates and the data and compares these with the original. If the two are dissimilar, then the underlying hypothesis may be rejected. Otherwise, it may not. Our rationale is the same. We wish to know whether the candidate UFPs produced by the KFC algorithm reflect a genuine structure in the underlying system, or merely chance. To test this, we generate surrogate data consistent with each of the three linear hypotheses: 0) independent and identically distributed noise; 1) linearly filtered noise; and 2) a monotonic nonlinear transformation of linearly filtered noise. We then compute UFP distributions using the KFC method for both data and surrogates and compare the results. As the surrogates do not contain UFPs, we do not expect the KFC technique to find any. Any evidence of UFPs is purely bad luck. Moreover, we require that evidence of UFPs in the original data must be stronger than that in the surrogates. Conversely, we would also like to determine the typical variability one may expect from our results for data which do exhibit UFPs. For this purpose, we employ the new surrogate technique described in [22] and [25]. The principle of this method is to generate surrogate data which exhibit the same large-scale nonlinear deterministic structure as the observed data, but have noise obscuring the fine-scale structure. The level of noise is an adjustable parameter of the algorithm [25] and may be used to generate surrogates which either have the same nonlinear dynamics as the data [25] or are simply noise-driven periodic orbits of the same general shape as the data. In [22], this method was used to distinguish between chaotic Rössler dynamics with observational noise and the same system in a periodic (period-6) regime with dynamic noise. In Section IV, we apply these surrogates to test for both false identification of UFPs and to develop an estimate of the expected variability of these UFP distributions. IV. EXAMPLES The Chua circuit can be represented by the following equations [14], [15]:
where . The fixed points are given by and with . A double scroll at, tractor can be generated using the parameter set ( , , and ). Fig. 1 shows a time series of the variable together with the values of the three fixed points. (We integrated the Chua circuit equations with initial condition s to s sampling every 0.1 s and ( 1, 0, 0) from then retaining the last 3000 data points.) A calculation of mutual information [11] suggests a time-delay lag of six and false nearest neighbors with the selected lag suggests a three-dimensional embedding [12]. However, we decided to use a four-dimensional embedding.
WALKER AND SMALL: DETECTING UNSTABLE FIXED POINTS USING KALMAN FILTERS WITH CONSTRAINTS
2821
Fig. 1. x versus t for the Chua circuit. The horizontal lines indicate the location of the (unstable) fixed points.
We reserve the first embedded data points to beused as a library for reconstructing the local linear models and apply the KFC to the next 1000 time-series values. For the given emas the number bedding dimension, we selected a value of of nearest neighbors to use in the local reconstructions. We used and considered a sliding window of length fixed-point constraints. The assumed constraint error covariance . The assumed for each fixed point was set to noise levels for the dynamics was set to , where is the standard deviation of the time-series data. The observational and the initial error covariance for the noise covariance . state and fixed-point estimates were set to We note that all of these parameters are tunable, and so there is great flexibility in applying the KFC for UFP detection. In Fig. 2, we show a plot of against for each candidate fixed point and sliding window. A histogram of the more meaningful estimates are shown in Fig. 3, where only are considered. We can see those points with that there are two large peaks centered about the known UFPs at . The UFP at the origin is not so prominent. The reliability of these estimates can be tested using surrogate data. In Fig. 4, we compare the estimated UFPs for the double scroll data in Fig. 1 to the same calculation of candidate UFPs for linear surrogate time series. In each case, the linear surrogate time series exhibit only a peak in the distribution of UFPs (for algorithms 1 and 2) at 0. There is no indication of false identification of nonzero UFPs. This is to be expected. Fig. 5 repeats the calculation of Fig. 4 for nonlinear surrogates generated using the PPS algorithm described in [22]. In this case, we expect the surrogates to exhibit similar distributions of UFPs as the data. This is precisely what we observe. The three calculations in Fig. 5 are for three distinct levels of dynamic noise in the surrogates. For the lowest level of noise (the top panel of Fig. 5), the distribution of candidate UFPs is very close to that estimated from the data: both data and surrogates exhibit almost the same dynamics and therefore the same UFPs. For increasing levels of dynamic noise, the distinction becomes more pronounced. This is
Fig. 2. Plot of p versus log(v ) for data from Chua’s circuit. There appears to be accumulations about r for small values of v .
6
Fig. 3. Histogram of the candidate UFPs for which log(v are definite peaks about the known UFPs at r .
6
)
