graphical models for time series - spsc · 2019-05-07 · spsc – advanced signal processing...

SPSC – Advanced Signal Processing Seminar 2

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Walch Daniel Graz, 17. 5. 2011 Graphical Models for Time Series

Graphical Models for Time Series

Gaining insight into their computational implementation


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Contents

• Time Series – short Introduction

• Developing a graphical representation

• Latent Markov models

• Switching Linear Dynamical Systems (SLDS)

• Gaußian Sum Filtering

• Noisy Signal Reconstruction

• Example: Traffic flow

• Reset models

• Conclusions


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Time Series

• Realizations (samples) from a process • Process itself is non-random, but

• Random influences (noise)

• TS analysis is central to many problems • Signal Processing

• Finance


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Time Series

• Examples


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Developing a Graphical Representation

• Notation: 𝑦1:𝑇 = 𝑦1, … , 𝑦𝑇 … time series

• A probabilistic model of a time series is a

specification of a joint distribution 𝑝 𝑦1:𝑇

• Bayes‘ rule:

𝑝 𝐴 𝐵 =𝑝(𝐴, 𝐵)

𝑝(𝐵)

𝑝 𝑦1:𝑇 = 𝑝 𝑦𝑇 𝑦1:𝑇−1 𝑝(𝑦1:𝑇−1)


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Developing a Graphical Representation

𝑝 𝑦1:𝑇 = 𝑝 𝑦𝑇 𝑦1:𝑇−1 𝑝(𝑦1:𝑇−1)

• Recursively applying Bayes‘ rule, any distribution can

be written in a causal form:

𝑝 𝑦1:𝑇 = 𝑝(𝑦𝑡|𝑦1:𝑡−1)

𝑇

𝑖=1

• For each factor, the present depends only on the

past!

𝑝 𝑦1, … , 𝑦𝑁 = 𝑝(𝑦𝑖|pa(

𝑁

𝑖=1

𝑦𝑖))


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Graphical Model

• Each node represents a variable 𝑦𝑖.

• Variables that point to 𝑦𝑖 are parents of this variable.

• Applied to belief networks, each node corresponds to

a factor in the joint distribution over all variables

• Example (blackboard)


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Some examples

𝑝 𝑦1, … , 𝑦𝑁 = 𝑝(𝑦𝑖|pa(

𝑁

𝑖=1

𝑦𝑖))

• First order Markov model: pa 𝑦𝑖 = 𝑦𝑖−1

• Second order Markov model: pa 𝑦𝑖 = 𝑦𝑖−1, 𝑦𝑖−2

• Lth order auto-regressive model 𝑦𝑡 = 𝑎𝑙𝑦𝑡−𝑙 + η𝑡𝐿𝑖=1

with η𝑡~𝑁(η𝑡|0, 𝜎2) corresponds to the transition

𝑝 𝑦𝑡 𝑦𝑡−𝐿:𝑡−1 = 𝑁 𝑦𝑡| 𝑎𝑙𝑦𝑡−𝑙

𝐿

𝑙=1

, 𝜎2


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Parameter Learning

• Parameter θ of a model in 𝑝 𝑦1:𝑇 𝜃

• Bayes: 𝑝 𝑦1:𝑇 , 𝜃 = 𝑝 𝑦1:𝑇 𝜃 𝑝 𝜃

• All questions relating to parameter estimation are

computed from the parameter posteriori density

𝑝 𝜃 𝑦1:𝑇 =𝑝 𝑦1:𝑇 𝜃 𝑝(𝜃)

𝑝(𝑦1:𝑇)

• First order Markov model:

𝑝 𝑦1:𝑇 , 𝜃 = 𝑝(𝜃) 𝑝(𝑦𝑡|𝑦𝑡−1, 𝜃)

𝑡

Sketch on blackboard!


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Latent Markov Models

• Unobserved variable 𝑥𝑡

• Observations 𝑦𝑡

• Example: Tracking an object

𝑥𝑡 is the position of the object that is assumed to move

according a transition dynamics 𝑝 𝑥𝑡 𝑥𝑡−1 and 𝑦𝑡 is a

noisy function of it

(noisy radar reading 𝑦𝑡 of the approximate distance to

the object)


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HMM (hidden Markov model)

• „Discrete Latent State Markov Models“

• The latent variables 𝑥𝑡 are discrete (square nodes)

• The observations 𝑦𝑡 can be continuous or discrete

• Able to model discrete changes in the underlying

state


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Continuous State Latent Markov Models

• Continuous variable distributions analytically

seldom tractable

• LDS (Linear Dynamical Systems) play a special role

(LDS ≜ discrete-state HMM)

𝑥𝑡 = 𝐴𝑥𝑡−1 + 𝜂𝑡, 𝑦𝑡 = 𝐶𝑥𝑡 + 𝜐𝑡

𝜂𝑡, 𝜐𝑡 … Gaussian noise terms

≜ Kalman filtering

• As a probabilistic model, the LDS corresponds to

𝑝 𝑥𝑡 𝑥𝑡−1 = 𝑁 𝑥𝑡 𝐴𝑥𝑡−1, 𝑄 , 𝑝 𝑦𝑡 𝑥𝑡 = 𝑁(𝑦𝑡|𝐶𝑥𝑡 , 𝑅)


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Inference in Latent Markov Models

• We often want to infer the distribution of the latent

state 𝑥𝑡 based on noisy observations.

• Notation: 𝑑𝑋, that either integrates or sums over the

domain of X.

• Conclusions that will be drawn: • Same procedure applies in all models consistent with the belief

network representation

• Can numerically only be implemented in a restricted class of

transition and observation distributions (discrete latent variables

(HMM) and linear Gaußian transition and observations (LDS) –

Kalman filter)


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Latent Markov Models:

Filtering/Smoothing • Probably skip this section


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Inference in Linear Dynamical Systems

• Probably skip this section

• Well-known Kalman filtering and smoothing

recursions.


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Switching Linear Dynamical System

• „Marrying“ HMM and LDS: breaking the time series

into segments, each modelled by a (different) LDS

• Useful when the underlying model may change from

one parameter setting to another

• Application in econometrics and machine learning


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Exact inference?

𝛼 𝑠𝑡 , 𝑥𝑡 = 𝑝(𝑦𝑡|𝑥𝑡 , 𝑠𝑡)

∙ 𝑝(𝑠𝑡 , 𝑥𝑡|𝑠𝑡−1, 𝑥𝑡−1, 𝑦𝑡)𝛼(𝑠𝑡−1, 𝑥𝑡−1)𝑥𝑡−1𝑠𝑡−1

• Computationally intractable

• Summation over the states 𝑠𝑡 exponentially many

Gaußians 𝑆𝑡−1, at time 𝑡.

• Approximate inference necessary (Monte Carlo

methods, deterministic variational techniques, and …


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Gaußian Sum Filtering

• Keep the exponential explosion in check

• S² Gaußian mixture is collapsed back to an S

component Gaußian mixture.

• Example: Two states (S = 2), three mixture

components (I = 3)


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Gaußian Sum Filtering

Two states: Red and Blue

Three mixture states I = 3

Area of ellipse

corresponds to the weight

of each component

One method: ignore the

lowest weight components


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Noisy Signal Reconstruction

• Example: SAR model (essentially an HMM)

• 𝑥𝑡 indicate which of a set of AR models is active


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Noisy Signal Reconstruction (cont‘d)


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Traffic Flow

• Traffic flows into junction a

• Goes to d via different routes

• a and b with traffic lights

• Routing dependent on

state


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Traffic flow (cont‘d)

• Time evolution of the traffic flow (into the network and

out of the network)


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(a) Correct latent flows and switch variables

(b) Filtered flow based on I = 2 Gaußian sum forward

pass algorithm

(c) Smoothed flows and traffic light states


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Reset models

• Switching LDS powerful, but computationally difficult

to implement

• Reset model: Switching models where the switch can

reset the latent 𝑥𝑡 (isolating present from the past)


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Example: Poisson reset model

• Intensity constant, but may „jump“

• 𝑐𝑡 indicates wheter there is a jump or not

• Deadly coal mining desasters in England


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Conclusions

• Graphical models provide a compact description of

the basic independence assumptions behind a model

• A useful way of communicating ideas

• Makes it easy to envisage new models tailored for a

particular environment

Thank you for your attention!


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References

[1] „Graphical Models for Time Series“, D. Barber, A. T.

Cemgil, IEEE Signal Processing Magazine,

November 2010

[2]

http://upload.wikimedia.org/wikipedia/commons/9/94/I

ntVerglArblos.PNG, visited 2011-05-16

[3] http://blog.iwenzo.de/wp-

content/uploads/2008/02/boersenkurs.gif, visited

2011-05-16

http://upload.wikimedia.org/wikipedia/commons/9/94/IntVerglArblos.PNG




http://blog.iwenzo.de/wp-content/uploads/2008/02/boersenkurs.gif





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References (cont‘d)

[4]

http://read.pudn.com/downloads187/sourcecode/mat

h/877147/kalman%2520simulink/html/runkalmanfilter

_01.png, visited 2011-05-16

http://read.pudn.com/downloads187/sourcecode/math/877147/kalman%20simulink/html/runkalmanfilter_01.png



graphical models for time series - spsc · 2019-05-07 · spsc – advanced signal processing...

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