Download Markov-chain embedded recurrence relations

Learning Markov-chain embedded recurrence
relations for chaotic time series analysis
Jiann-Ming Wu, Ya-Ting Zhou, Chun-Chang Wu
National Dong Hwa University
Department of Applied Mathematics
Hualien, Taiwan
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
Pr( Z[t ] | Z[t  1],..., Z[t   ],..., Z[1])  Pr( Z[t ] | Z[t  1],..., Z[t   ])
 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
y[t ]  Z[t] | z[t - 1], , z[t -  ] 
 G ( z[t  1],..., z[t   ])
equivalently
y[t ]  G (x[t ]),
where
x[t ]  ( z[t  1],..., z[t   ])T
denotes  previous events
y[t ]  z[t ]
denotes current event
Recurrence relation for time series modeling
 5
z[t ]  G( z[t  1],..., z[t   ])
 sin( a1 z[t  1]  a2 z[t  2]    a5 z[t   ]),
t   , ,1000
z[t ]  y[t ]
x[t ]  z[t  1], z[t  2], , z[t   ]
T
predictor
target
y[t ]  G (x[t ])
 sin( a T x[t ])
a  (a1 , a2 ,, a ) T
Mackey-Glass 30 chaotic time series data
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
500
1000
Chaotic time series
Laser data 10000 from the SFI competition
1500
RECURRENCE RELATION APPROXIMATION
• Learning neural networks for approximating underlying
recurrence relation
y[t ]  G(x[t ])  F (x[t ] )
• F denotes a mapping realized by radial or projective basis
functions
•  denotes adaptive network parameters
Recurrence relation approximation
 Form paired predictor and target (x[t ], y[t ]) by assigning
x[t ]  ( z[t  1],..., z[t   ])T and y[t ]  z[t ] for all t
 Define the mean square error of approximating y[t ] by F (x[t ] )
1
E ( ) 
N
2
(
y
[
t
]

F
(
x
[
t
]
|

)

t
 Apply Levenberg-Marquardt learning to resolve unconstrained
optimization
 opt  min  E ( )
 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 (x[t ], y[t ])
Exclusive Memberships
 Each pair is exactly generated by a sub-model
 Let δ[t ] e1 , e2 ,, eM  denote the exclusive membership of x[t ], y[t ]
 where e i denotes a unitary vector with the ith bit active
 By exclusive membership
ei  0,,1,,0
δ[t ]  e k if x[t ], y[t ] is generated by the kth sub - model
 The conditional expectation of y to given x is defined by
y x  δT r
 r denotes local means of the target variable
Overlapping memberships
 A Potts random variable δ is applied to encode overlapping
membership
 The probability of being the kth state is set to
Pr δ  e k   exp(   x[t ]  μ k )
2
where  modulates the overlapping degree and
μ k denotes local mean of the predictor
Normalized radial basis functions ( NRBF )
Since Pr δ  e k   exp(   x[t ]  μ k ) and
2
it follows
vk  Pr δ  e k  

exp   x[t ]  μ k
M

2
 exp   x[t ]  μ h
h 1

M
 Prδ  e   1
k 1
k

2
Let v  v1 ,  , vM 
T
 yx


v r
 exp   x[t ]  μ 
rk exp   x[t ]  μ k
T
2
2
k
h
h
 The conditional expectation exactly sketches a mapping realized by normalized
radial basis functions
Figure 4
Mackey-Glass 17 chaotic time series data
source
1.5
1
0.5
0
0
500
1000
1500
approximation
1.5
1
0.5
0
0
50
100
150
-3
4
200
250
300
350
400
450
500
350
400
450
500
approximating error
x 10
2
0
-2
-4
0
50
100
150
200
Figure 9
250
300
Multiple recurrence relations
 Multiple recurrence relations for modeling more complex
chaotic time series
Chaotic time series
Laser data 10000 from the SFI competition
Markov-chain embedded recurrence relations
 A Markov chain of PGMs (pair-data generative models)
 
 Transition matrix Ti , j
 Ti , j 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
Given an ordered sequence generated by Markov - chain embedded PGMs
The inverse problem for model reconstruc tion can be stated as
1. Find switching points for segmentati on.
2. Estimate transitio n probabilit ies.
Segmentation for phase change
 A time tag is regarded as a switching point if its moving
average error greater than a threshold value
error[t ] 
t 
2
(
y
[
t
]

F
(
x
[
t
];
θ
))


i t 
A simple rule for merging two PGMs
 The goodness of fitting the ith PGM to paired data in Sj is
defined by
Ei, j
1

Si
2
(
y
[
t
]

F
(
x
[
t
];
θ
))

j
x[ t ]S j
 Two PGMs are merged. Si and Sj are regarded from the same
PGM if (Ei,j+Ej,i)/2 is less than a threshold value
1
If Ei , j  E j ,i   
2
 Si and S j are regarded oriented form the same hidden state.
NUMERICAL SIMULATIONS – Synthetic data
K : the number of required hidden states after reduction
 : short time scale.
 : a pre - determined positive threshold
N 0 : the window size of auto - regressive sampling for fixed - length segmentati on.
post - nonlinear functions : sin a1T x , cosaT2 x , exp aT3 x 
( K , ,  , N 0 )  [ 3, 5, 0.1, 200 ]
Translatio n matrix 
Temporal sequence generated by MC-embedded PGMs
Numerical results – original and reconstructed MCembedded PGMs
Translatio n matrix 
Learning
M=60,[K, , , N0 ] = [ 10, 10, 0.001, 500 ]
Chaotic time series
Laser data 10000 from the SFI competition
Markov chain embedded recurrence relations
Generated chaotic time series
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