Conditional Chow-Liu Tree Structures for Modeling Discrete-Valued Vector Time Series

Conditional Chow-Liu Tree Structures for Modeling Discrete-Valued Vector

Time Series

Sergey Kirshner, UC Irvine

Padhraic Smyth, UC Irvine

Andrew Robertson, IRI

July 10, 2004

2UAI-2004 © Sergey Kirshner, UC Irvine

Overview

• Data and its modeling aspects

• Model description– General Approach

• Hidden Markov models

– Capturing data properties• Chow-Liu trees• Conditional Chow-Liu trees

• Inference and Learning

• Experimental Results

• Summary and Future Extensions


Snapshot of the Data

1

2

3

4

5

1 2 3 4 5 6 7 8 …

…

T

N


Data Aspects

• Correlation– Spatial dependence

• Temporal structure– First order dependence

• Variability of individual series– Interannual variability


Modeling Precipitation Occurrence

Southwestern Australia, 1978-92

Western US, 1952-90


A Bit of Notation

• Vector time series R– R1:T=R1,..,RT

• Vector observation of R at time t– Rt=(At,Bt,…,Zt)

A1

B1

Z1

C1

R1

A2

B2

Z2

C2

R2

AT

BT

ZT

CT

RT


Weather Generator

R1 R2 RT

A1

B1

Z1

C1

A2

B2

Z2

C2

AT

BT

ZT

CT

T

t c

T

t

ttttT ccPcPPPP2 Z,..,A 2

1111:1 )|()()|()()( RRRR

• Does not take correlation into account


Hidden Markov Model

R1 R2 Rt RT-1 RT

S1 S2 St ST-1 ST

T

t

tt

T

t

ttTT SPSSPSPSP12

11:1:1 )|()|()(),( RR


HMM-Conditional-Independence

Rt

St St

At Ct

ZtBt

=

Z,,A

)(

)|Z,,A()|(

c

tt

ttttt

|ScP

SPSP R

R1 R2 Rt RT-1 RT

S1 S2 St ST-1 ST


HMM-CI: Is It Sufficient?

• Simple yet effective

• Requires large number of values for St

• Emissions can be made to capture more spatial dependencies


Chow-Liu Trees

• Approximation of a joint distribution with a tree-structured distribution [Chow and Liu 68]


0.31260.02290.01720.02300.01830.2603

ABACADBCBDCD

(0.56, 0.11, 0.02, 0.31)(0.51, 0.17, 0.17, 0.15)(0.53, 0.15, 0.19, 0.13)(0.44, 0.14, 0.23, 0.19)(0.46, 0.12, 0.26, 0.16)(0.64, 0.04, 0.08, 0.24)

A

C

B

D

A

C

B

D

ABACADBCBDCD

0.31260.02290.01720.02300.01830.2603

(0.56, 0.11, 0.02, 0.31)(0.51, 0.17, 0.17, 0.15)(0.53, 0.15, 0.19, 0.13)(0.44, 0.14, 0.23, 0.19)(0.46, 0.12, 0.26, 0.16)(0.64, 0.04, 0.08, 0.24)

Illustration of CL-Tree Learning

A

C

B

D


Chow-Liu Trees

• Approximation of a joint distribution with a tree-structured distribution [Chow and Liu 68]

• Learning the structure and the probabilities– Compute individual and pairwise marginal distributions for all

pairs of variables – Compute mutual information (MI) for each pair of variables

– Build maximum spanning tree with for a complete graph with variables as nodes and MIs as weights

• Properties– Efficient:

• O(#samples×(#variables)2×(#values per variable)2)

– Optimal

YX YPXP

YXPYXPYX

, )()(

),(log),(),MI(


HMM-Chow-Liu

R1 R2 Rt RT-1 RT

S1 S2 St ST-1 ST

Rt

St

Bt

DtCt

Bt

DtCt

Bt

DtCt

St

St=1 St=2 St=3=

T1(Rt) T2(Rt) T3(Rt)

At AtAt


Improving on Chow-Liu Trees

• Tree edges with low MI add little to the approximation.

• Observations from the previous time point can be more relevant than from the current one.

• Idea: Build Chow-Liu tree allowing to include variables from the current and the previous time point.


Conditional Chow-Liu Forests

• Extension of Chow-Liu trees to conditional distributions– Approximation of conditional multivariate

distribution with a tree-structured distribution– Uses MI to build maximum spanning trees (forest)

• Variables of two consecutive time points as nodes

• All nodes corresponding to the earlier time point considered connected before the tree construction

– Same asymptotic complexity as Chow-Liu trees• O(#samples×(#variables)2×(#values per variable)2)

– Optimal


B’A’

C’

BA

C

0.31260.02290.02300.12070.12530.06230.13920.17000.05590.00330.00300.0625

ABACBCA’AA’BA’CB’AB’BB’CC’AC’BC’C

(0.56, 0.11, 0.02, 0.31)(0.51, 0.17, 0.17, 0.15)(0.44, 0.14, 0.23, 0.19)(0.57, 0.11, 0.11, 0.21)(0.51, 0.17, 0.07, 0.25)(0.54, 0.14, 0.14, 0.18)(0.52, 0.07, 0.16, 0.25)(0.48, 0.10, 0.11, 0.31)(0.47, 0.11, 0.21, 0.21)(0.48, 0.20, 0.20, 0.12)(0.41, 0.26, 0.17, 0.16)(0.53, 0.14, 0.14, 0.19)

ABACBCA’AA’BA’CB’AB’BB’CC’AC’BC’C

0.31260.02290.02300.12070.12530.06230.13920.17000.05590.00330.00300.0625

B’A’

C’

BA

C

Example of CCL-Forest Learning

B’A’

C’

BA

C

B’A’

C’

BA

C


AR-HMM

T

t

ttt

T

t

ttTT SPSSPSPSPSP2

,1

2

1111:1:1 )|()|()|()(),( RRRR

R1 Rt RT

S1 St ST

Rt-1

St-1

R2

S2


HMM-Conditional-Chow-Liu

St

Rt-1 Rt

R1 Rt RT

S1 St ST

Rt-1

St-1

R2

S2

At-1

Bt-1

Ct-1

Dt-1

At Bt

CtDt

Dt-1

Ct-1

Bt-1

At-1

CtDt

At Bt

Dt-1

Ct-1

Bt-1

At-1

Dt Ct

At Bt

St

St=1 St=2 St=3

=


Inference and Learning for HMM-CL and HMM-CCL

• Inference (calculating P(S|R,))– Recursively calculate P(R1:t,St|) and P(Rt+1:T|St,)

(Forward-Backward)

• Learning (Baum-Welch or EM)– E-step: calculate P(S|R,)

• Forward-Backward

• Calculate P(St|R,) and P(St,St+1|R,)

– M-step: • Maximize EP(S|R,)[P(S, R|’)]

• Similar to mixtures of Chow-Liu trees


Chain Chow-Liu Forest (CCLF)

R1 Rt RTRt-1R2

RtRt-1

Bt

CtDt

At

At

Bt

Ct

Dt

=


Complexity Analysis

Model

Criterion

HMM-CI HMM-CL HMM-CCL

# params K2+MK(V-1) K2+K(M-1)(V2-1) K2+KM(V2-1)

Time (per iteration)

O(NTK(K+M)) O(NTK(K+M2V2)) O(NTK(K+

+M2V2))

Space O(NTK(K+M)) O(NTK(K+M)+KM2V2) O(NTK(K+M)+

+KM2V2)

N – number of sequencesT – length of each sequenceK – number of hidden statesM – dimensionality of each vectorV – number of possible values for each vector component


Experimental Setup

• Data– Australia

• 15 seasons, 184 days each, 30 stations

– Western U.S.• 39 seasons, 90 days each, 8 stations

• Measuring predictive performance– Choose K (number of states)– Leave-one-out cross-validation– Log-likelihood– Error for prediction of a single entry given the rest


Australia (log-likelihood)


Australia (predictive error)


Deeper Look at Weather States


Western U.S. (log-likelihood)


Western U.S. (predictive error)


Summary

• Efficient approximation for finite-valued conditional distributions– Conditional Chow-Liu forests

• New models for spatio-temporal finite-valued data– HMM with Chow-Liu trees– HMM with conditional Chow-Liu forests– Chain Chow-Liu forests

• Applied to precipitation modeling


Future Work

• Extension to real-valued data

• Priors on tree structure and parameters [Jaakkola and Meila 00]

– Locations of the stations

• Interannual variability– Atmospheric variables as inputs to non-homogeneous HMM

[Robertson et al 04]

• Other approximations for finite-valued multivariate data– Maximum Entropy– Multivariate probit models (binary)


Acknowledgements

• DOE (DE-FG02-02ER63413)

• NSF (SCI-0225642)

• Dr. Stephen Charles of CSIRO, Australia

• Datalab @ UCI (http://www.datalab.uci.edu)

Documents

Conditional Chow-Liu Tree Structures for Modeling Discrete-Valued Vector Time Series