Jiann-Ming Wu, Ya-Ting Zhou, Chun-Chang Wu National Dong Hwa University Department of Applied Mathematics Hualien, Taiwan Learning Markov-chain embedded recurrence relations for chaotic time series analysis

Outline Introduction High-order Markov processes for stochastic modeling Nonlinear recurrence relations for deterministic modeling Recurrence relation approximation by supervised learning of radial or projective basis functions Markov-chain embedded recurrence relations Numerical Simulations Conclusions

High-order Markov assumption Let Z[t] denote time series, where t is positive integers High-order Markov assumption- Given chaotic time series are oriented from a generative source well characterized by a high-order Markov process. An order-  Markov process obeys memory-less property Current event only depends on instances of  most recently events instead of all historic events

Recurrence relation Conditional expectation of an upcoming event to  most recently events is expressed by a recurrence relation

Recurrence relation for time series modeling predictor target

Mackey-Glass 30 chaotic time series data Chaotic time series Laser data from the SFI competition

RECURRENCE RELATION APPROXIMATION Learning neural networks for approximating underlying recurrence relation F denotes a mapping realized by radial or projective basis functions denotes adaptive network parameters

Recurrence relation approximation Form paired predictor and target by assigning Define the mean square error of approximating Apply Levenberg-Marquardt learning to resolve unconstrained optimization Apply the proposed pair-data generative model to formulate F

Pair-data generative model (PGM) K sub-models

Mixtures of paired Gaussians A stochastic model for formation emulation of given paired data Each time one of joined pairs is selected according to a set of prior probabilities Apply the selected paired Gaussians to generate paired data

Each pair is exactly generated by a sub-model Let denote the exclusive membership of where denotes a unitary vector with the ith bit active By exclusive membership The conditional expectation of y to given x is defined by r denotes local means of the target variable Exclusive Memberships

Overlapping memberships A Potts random variable is applied to encode overlapping membership The probability of being the kth state is set to where modulates the overlapping degree and denotes local mean of the predictor

Normalized radial basis functions ( NRBF ) The conditional expectation exactly sketches a mapping realized by normalized radial basis functions

Figure 4

Figure 9 Mackey-Glass 17 chaotic time series data

Multiple recurrence relations Multiple recurrence relations for modeling more complex chaotic time series Chaotic time series Laser data from the SFI competition

Markov-chain embedded recurrence relations A Markov chain of PGMs (pair-data generative models) Transition matrix denotes the probability of transition from model i to model j

Data generation Emulate data generation by a stochastic Markov chain of PGMs

Inverse problem of Markov chain embedded PGMs

Segmentation for phase change A time tag is regarded as a switching point if its moving average error greater than a threshold value

A simple rule for merging two PGMs The goodness of fitting the ith PGM to paired data in S j is defined by Two PGMs are merged. S i and S j are regarded from the same PGM if (E i,j +E j,i )/2 is less than a threshold value

NUMERICAL SIMULATIONS – Synthetic data

Temporal sequence generated by MC-embedded PGMs

Numerical results – original and reconstructed MC- embedded PGMs

Chaotic time series Markov chain embedded recurrence relations Generated chaotic time series Laser data from the SFI competition M=60,[K, ,, N 0 ] = [ 10, 10, 0.001, 500 ] Learning

Conclusions This work has presented learning Markov-chain embedded recurrence relations for complex time series analysis. Levenberg-Marquardt supervised learning of neural networks has been shown potential for extracting essential recurrence relation underlying given time series Markov-chain embedded recurrence relations are shown applicable for characterizing complex chaotic time series The proposed systematic approach integrates pattern segmentation, hidden state absorption and transition probability estimation based on supervised learning of neural networks