Identification of Stochastically Perturbed Autonomous Systems from Temporal Sequences of Probability Density Functions
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Abstract
The paper introduces a method for reconstructing onedimensional iterated maps that are driven by an external control input and subjected to an additive stochastic perturbation, from sequences of probability density functions that are generated by the stochastic dynamical systems and observed experimentally.
Keywords
Nonlinear systems Probability density functions Frobenius–Perron operator Stochastic dynamical systemsMathematics Subject Classification
93E12 37H99 65P401 Introduction
There is considerable interest in modeling and analyzing dynamical systems that generate densities of states. Examples of such systems include chaotic systems (Boyarsky and Góra 1997; Lasota and Mackey 1994) and stochastically perturbed dynamical systems (Swishchuk and Islam 2013). Such systems are encountered routinely in physics, biology, engineering and economics (Strogatz 2014; Skinner 1994).
In many practical situations, the system that generates the density of states is unknown and only the densities of states generated by the system or the invariant density associated with the system can be observed, while the individual point trajectories are not measurable. Conventional solutions (Maguire et al. 1998; Han et al. 2004; Príncipe and Kuo 1995; Lai et al. 1999; Lai and Tél 2011; Bollt et al. 2001) rely on time series observations, but for such situations they become unsuitable. The problem of inferring the unknown dynamical system from the observed densities is known as the inverse Frobenius–Perron problem (Boyarsky and Góra 1997; Ershov and Malinetskii 1988). The problem of reconstructing an unknown onedimensional autonomous chaotic map given only knowledge of the invariant density function of the system has been considered by a number of authors (Ershov and Malinetskii 1988; Góra and Boyarsky 1993; Diakonos and Schmelcher 1996; Pingel et al. 1999), while there are special cases in which this problem has a unique solution. Given the invariant symmetric beta density functions, methods were introduced to construct a class of symmetric maps (Diakonos and Schmelcher 1996) and a broader class of continuous unimodal maps whose each brand covers the complete interval (Pingel et al. 1999). Given arbitrary invariant densities other similar approaches were proposed for identifying the maps with specified forms: two types of onedimensional symmetric maps (Koga 1991), smooth chaotic map with closed form (Huang 2006, 2009), multibranches complete chaotic map (Huang 2009). Problems of synthesizing onedimensional maps with prescribed invariant density function or autocorrelation function were tackled in Baranovsky and Daems (1995) and Diakonos et al. (1999). Using positive matrix theory an approach to synthesizing chaotic maps with arbitrary piecewise constant invariant densities and arbitrary mixing properties was developed in Rogers et al. (2004). This method was further extended to synthesizing dynamical systems with desired statistical properties (Rogers et al. 2008a), developing communication networks (Berman et al. 2004) and designing randomly switched chaotic maps and twodimensional chaotic maps used for image generation (Rogers et al. 2008b). In Bollt (2000) and Bollt and Santitissadeekorn (2013), a global and openloop strategy of controlling chaos was presented to solve the inverse problem. The problem was reduced to that of finding a perturbation of the original Frobenius–Perron matrix to achieve the target invariant density function. In general, given only invariant density function, the solution to the inverse problem is not unique, as different maps exhibiting remarkably different dynamics may possess a same invariant density function. Therefore, additional assumptions or constraints are required to ensure the uniqueness of the identification results. A more recent approach (Nie and Coca 2015) addresses the uniqueness issue by considering sequences of density functions generated by the system rather than just the invariant density function of the system. This method allows inferring the map that exhibits the same transient and asymptotic dynamics as the underlying system that generated the data. Although it is shown that the method is robust to noise, the approach does not exploit any a priori knowledge of the noise distribution. In addition, to our knowledge, all existing methods consider only autonomous maps.
In this context, this paper introduces for the first time a method to infer a onedimensional map that is driven by an external control input while being subjected to an additive stochastic perturbation from sequences of observed density functions generated by the unknown system. We formulate the operator transferring the state density function of the stochastic dynamical system in terms of the Frobenius–Perron operator associated with the unperturbed underlying system that we aim to estimate, and derive the matrix representation of the transfer operator in terms of the Frobenius–Perron matrix. Based on this representation the new algorithm is developed to estimate the Frobenius–Perron matrix using temporal sequences of probability density functions generated by the stochastic dynamical system given the density functions of the control input and noise. The approach also determines the monotonicity of general nonlinear transformations over each interval of the partition, which is a crucial step to reconstruct the true dynamical system.
The paper is structured as follows. Section 2 introduces the inverse problem. The stochastic Frobenius–Perron operator associated with stochastically perturbed autonomous systems is derived in Sect. 3. A matrix approximation of the operator is given in Sect. 4. Section 5 introduces the methodology of reconstructing general nonlinear maps from sequences of density functions. Section 6 presents a numerical simulation example to demonstrate the effectiveness of the developed algorithm for the stochastically perturbed autonomous systems. Conclusions are given in Sect. 7.
2 Inverse Problem Formulation
3 The Stochastic Frobenius–Perron Operator Associated with the Stochastically Perturbed Transformation
Remark 1
Remark 2
In the first instance, we assume that S belongs to a special class of nonlinear transformations called piecewise linear semiMarkov transformations and develop the algorithm to reconstruct it. We then show how the reconstruction approach can be applied to approximate more general onedimensional maps.
4 A Matrix Representation of the Transfer Operator \( \bar{P} \)
Let S be a piecewise linear and expanding semiMarkov transformation over the Ninterval partition, \( \Re = \{ R_{1} ,R_{2} , \ldots ,R_{N} \} . \)
Definition 1
A transformation \( S{:}\,R \to R \) is said to be semiMarkov with respect to the partition \( \Re \) (or \( \Re \)semiMarkov) if there exist disjoint intervals \( Q_{j}^{(i)} \) so that \( R_{i} = \cup_{k = 1}^{p(i)} Q_{k}^{(i)} \), \( i = 1, \ldots ,N \), the restriction of S to \( Q_{k}^{(i)} \), denoted \( \left. S \right_{{Q_{k}^{(i)} }} \), is monotonic and \( S(Q_{k}^{(i)} ) \in \Re . \) (Góra and Boyarsky 1993)
Lemma 1
For \( f \in L^{1} \), the sequence \( \bar{P}_{N} f(x) = \sum\nolimits_{i = 1}^{N} {w_{i} \chi_{{R_{i} }} (x)} \) converges in \( L^{1} \) to \( \bar{P}f \) as \( N \to + \infty \).
Let \( Q = MH^{{\prime }} \). The evolution of density functions is formulated as \( w^{{f_{n + 1}^{N} }} = w^{{f_{n}^{N} }} Q \). Q is the matrix representation of the transfer operator \( \bar{P} \). Formula (26) yields the final density function estimated over the Ninterval partition \( \Re \), mapping from the initial piecewise constant density function over the Ninterval partition \( \Re \). This establishes the basis of the new algorithm of reconstructing the unknown transformation S from sequences of probability density functions.
Remark 3
Given the nonsingular transformation \( S{:}\,R \to R \) that induces the Frobenius–Perron matrix M with respect to the partition \( \Re \), input density function \( f_{u} \in L^{1} \) and noise density function \( g \in L^{1} \), from (26) the estimated state density function over \( \Re \) of stochastic dynamical system (1) can be predicted from a piecewise constant initial density function \( f_{0}^{N} \) as \( w^{{f_{n}^{N} }} = w^{{f_{0}^{N} }} Q^{n} . \)
Remark 4
Remark 5
Remark 4 suggests that there exists a stationary density function \( f_{ * }^{N} (x) = \sum\nolimits_{i = 1}^{N} {w_{i}^{ * } \chi_{{R_{i} }} (x)} \) for the transfer operator \( \bar{P}_{N} \). It follows from Lemma 1 that \( f_{ * }^{N} (x) \) converges to \( f_{ * }^{{}} (x) \) of the stochastic dynamical system as \( N \to + \infty \).
5 Solving the Stochastic Inverse Frobenius–Perron Problem for Continuous Nonlinear Transformations
This section introduces a method to reconstruct the underlying map S in Eq. (1) based on a sequence of probability density functions estimated from data, under the general assumption that S is a continuous nonlinear map. Specifically, the method infers a piecewise linear semiMarkov map Ŝ with respect to a uniform partition \( \Re = \{ R_{1} ,R_{2} , \ldots ,R_{N} \} = \{ [0,a_{1} ],(a_{1} ,a_{2} ], \ldots ,\;(a_{N  1} ,a_{N} ]\} ,\,a_{N} = b \), given K random vectors of initial states \( X_{0,i} = \{ x_{j}^{0,i} \}_{j = 1}^{\theta } \), from K initial state densities \( f_{0,i} \), i = 1, …, K, the corresponding final state vectors \( X_{1,i} = \{ x_{j}^{1,i} \}_{j = 1}^{\theta } \), i = 1, …, K under transformation (1) and the density of the noise and of the control input, g and \( f_{u} \), respectively. The matrix M associated with \( P_{S} \) can be approximated arbitrarily well, and thus, Ŝ approximates S to an arbitrary accuracy as \( N \to + \infty \). While g is fixed, \( f_{u} \) can be defined by the user when the experiment is conducted. It is assumed that the correspondence between an initial state measurement \( x_{0,j} \) and its image \( x_{1,j} \) under the transformation is not known and hence the point transformation S in (1) has to be inferred based on the probability density functions \( \left\{ {f_{0,j} } \right\}_{j = 1}^{K} ,\,\left\{ {f_{1,j} } \right\}_{j = 1}^{K} \), g and \( f_{u} \).

Step 1: For K initial, piecewise constant densities \( f_{0,i} \) generate \( X_{0,i} = \{ x_{j}^{0,i} \}_{j = 1}^{\theta } \) and \( X_{t,i} = \{ x_{j}^{t,i} \}_{j = 1}^{\theta } \), i = 1, …, K, t = 1, …, T.

Step 2: Estimate the coefficient vectors \( w^{{f_{t,i}^{N} }} = [w_{1}^{t,i} ,w_{2}^{t,i} , \ldots ,w_{N}^{t,i} ] \) corresponding to the piecewise constant density functions \( f_{t,i}^{N} (x) \) that approximate the new state density functions \( f_{t,i} (x) \) over the regular partition \( \Re \). Compute the matrix H;

Step 3: Identify a trial Frobenius–Perron matrix \( \hat{M} \) firstly to determine the indices of consecutive positive entries of the matrix M that represents the Frobenius–Perron operator \( P_{S} \) associated with the optimal approximate map \( \hat{S} \) and subsequently a refined matrix M;

Step 4: Construct the approximate piecewise linear semiMarkov transformation on \( \Re \), and smooth it to obtain the continuous nonlinear map.
These steps are described below in more detail.
5.1 Step 1: Observe Sets of States to Assemble Sequences of Densities
Let \( X_{0,i} = \{ x_{j}^{0,i} \}_{j = 1}^{\theta } \) be the set of initial conditions obtained by sampling \( f_{0,i} (x) \), and \( X_{t,i} = \{ x_{j}^{t,i} \}_{j = 1}^{\theta } \) be the set of states obtained by applying t times Eq. (1) such that \( x_{j}^{t,i} = S^{t} (x_{j}^{0,i} ) + u_{i} + \xi_{i} \) (mod b) for some \( x_{j}^{0,i} \), where \( U = \{ u_{i} \}_{i = 1}^{\theta } ,\,\varXi = \{ \xi_{i} \}_{i = 1}^{\theta } \) are generated by sampling \( f_{u} \) and g, respectively.
5.2 Step 2: Estimate the Coefficients w and Compute the Matrix H
Given the input and noise density functions \( f_{u} \) and g, the matrix H is computed from (25).
5.3 Step 3: Identify the Frobenius–Perron Matrix M
The approximation to the continuous map may have an infinite number of pieces of monotonicity, and each piece \( \left. S \right_{{R_{i} }} \) can be linearly approximated. Thus, for a piecewise linear semiMarkov approximation \( \hat{S} \), the maximum and minimum column indices of positive entries on two contiguous rows of M are further refined by \( r(i,p(i)) = \left\lfloor {{{[\hat{r}(i,\hat{p}(i)) + \hat{r}(i + 1,1)]} \mathord{\left/ {\vphantom {{[\hat{r}(i,\hat{p}(i)) + \hat{r}(i + 1,1)]} 2}} \right. \kern0pt} 2}} \right\rfloor ,\,r(i + 1,1) = \left\lceil {{{[\hat{r}(i,\hat{p}(i)) + \hat{r}(i + 1,1)]} \mathord{\left/ {\vphantom {{[\hat{r}(i,\hat{p}(i)) + \hat{r}(i + 1,1)]} 2}} \right. \kern0pt} 2}} \right\rceil \), and \( \left. {S^{{\prime }} } \right_{{Q_{p(i)}^{(i)} }} = \left. {S^{{\prime }} } \right_{{Q_{1}^{(i + 1)} }} \) if \( \frac{1}{{\hat{p}(i + 1)}}\sum\nolimits_{k = 1}^{{\hat{p}(i + 1)}} {\hat{r}(i + 1,k)} > \frac{1}{{\hat{p}(i)}}\sum\nolimits_{k = 1}^{{\hat{p}(i)}} {\hat{r}(i,k)} \) and \( \left {\hat{r}(i + 1,1)  \hat{r}(i,\hat{p}(i))} \right > 1 \); \( r(i,1) = \left\lceil {{{[\hat{r}(i,1) + \hat{r}(i + 1,\hat{p}(i + 1))]} \mathord{\left/ {\vphantom {{[\hat{r}(i,1) + \hat{r}(i + 1,\hat{p}(i + 1))]} 2}} \right. \kern0pt} 2}} \right\rceil \), \( r(i + 1,p(i + 1)) = \left\lfloor {{{[\hat{r}(i,1) + \hat{r}(i + 1,\hat{p}(i + 1))]} \mathord{\left/ {\vphantom {{[\hat{r}(i,1) + \hat{r}(i + 1,\hat{p}(i + 1))]} 2}} \right. \kern0pt} 2}} \right\rfloor \) and \( \left. {S^{{\prime }} } \right_{{Q_{1}^{(i)} }} = \left. {S^{{\prime }} } \right_{{Q_{p(i + 1)}^{(i + 1)} }} \) if \( \frac{1}{{\hat{p}(i)}}\sum\nolimits_{k = 1}^{{\hat{p}(i)}} {r(i,k)} > \frac{1}{{\hat{p}(i + 1)}}\sum\nolimits_{k = 1}^{{\hat{p}(i + 1)}} {r(i + 1,k)} \) and \( \left {\hat{r}(i,1)  \hat{r}(i + 1,\hat{p}(i + 1))} \right > 1 \), and that \( \left. {S^{{\prime }} } \right_{{Q_{2}^{(i)} }} = \left. {S^{{\prime }} } \right_{{Q_{j}^{(i)} }} \) for \( j = 3, \ldots ,p(i)  1 \) if \( p(i) \ge 4 \), where \( Q_{k}^{(i)} \) is the newly formed subinterval, and \( \{ r(i,1), \ldots ,r(i,p(i))\} \) are the identified column indices of positive entries in the ith row of the matrix M.
5.4 Step 4: Construct the Nonlinear Map
6 Numerical Simulation Example
7 Conclusions
This paper introduced a new algorithm for reconstructing the underlying onedimensional map for an autonomous dynamical system that is driven by an additive control input and also subjected to an additive stochastic perturbation, given the observed sequences of probability density functions generated by the unknown system, and the input and noise density functions. Evolution of densities was formulated and described by a stochastic Frobenius–Perron operator that has a matrix representation. This forms the basis for the algorithm to identify the Frobenius–Perron matrix associated with a piecewise linear semiMarkov approximation to the underlying nonlinear map. Based on the matrix representation of the stochastic Frobenius–Perron operator the densities generated by the dynamical system and evolving from a given initial condition can be predicted. Convergence of the evolving densities analyzed from the matrix representation reveals a fact that only a limited number of densities characterizing the transient dynamics is observable for arbitrary initial condition, and thus, this requires different initial conditions so as to generate as many as possible temporal sequences of densities to reconstruct the underlying map.
For the situations where only a limited number of initial conditions are available for generating the temporal sequences of densities that converges quickly to the equilibrium distribution, a potential effective solution is to apply multiple linearly independent input density functions to the stochastic dynamical systems so that the densities would diverge to different equilibrium distributions, which will be further explored. From a practical perspective, it is also worthwhile to extend the approach to higherdimensional systems based on sequences of mixture densities generated by the more complex systems.
Furthermore, this paper provides a new insight into identification of stochastic dynamical systems given the density functions of control inputs. It triggers a new scheme to solve the control problem for such systems. Specifically, given the noise density function, the problem aims to determine the optimal input density function so that the dynamical system can have a desired equilibrium distribution that represents the targeted asymptotic dynamics.
Notes
Acknowledgements
The authors gratefully acknowledge the supports from the Department of Automatic Control and Systems Engineering at the University of Sheffield, China Scholarship Council, Innovate UK (RG.GEOG.106459), China Intergovernmental International Scientific and Technological Innovation Cooperation Key Project (2016YFE0128700), MRC (G0802627), BBSRC (BB/M025527/1), Human Frontier Science Program and EPSRC (EP/L014211/1) and thank the anonymous reviewers for their insightful comments that helped improve the paper.
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