jakobsen+lins (1)

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Vienna, June 15-17, 2006

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Sequential Monte Carlo Methods

in R

Thomas Jakobsen

Jeffrey Todd Lins

Saxo Bank A/S

[email protected], [email protected]

www.saxobank.com

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Overview

The model

An introduction to sequential Monte Carlo methods

Algorithm & implementation

A factor stochastic volatility model

Example, artificial 2-factor model

Analysis of forex data

Conclusions

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The Model

Markovian, nonlinear, non-Gaussian state-space model:

Described by

Observations arrive sequentially and are noisy.

Problem statement: Estimate recursively in time the posterior distribution p(x|y,). (tracking the state)

Additionally: Estimate.

0x 1x 1tx tx X: Unobserved variables

Y: Observations: Parameters

0y 1y 1ty ty

modelnalObservatio1for),|(

modelspacestateMarkovian1for),|(

ondistributistatePrior)(

1

0

txyp

txxp

xp

tt

tt

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Useful when a (partially observed) state needs to betracked or forecasted:

Tracking problems (robots, vision, radar etc.) Time series analysis (economical/financial data etc.)

General online inference

Sequential Monte Carlo methods are algorithms forinference in hidden state space models.

Also known as particle filters, condensation, samplingimportance resampling etc.


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SMC methods: Basically a nonlinear, non-Gaussianversion of the Kalman filter (but approximate notclosed form)

The posterior at time t-1 is represented by a set ofweighted particles. The particles are drawn i.i.d. andrecursively updated.

Next slide: Illustration of update

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A bootstrap approach (from [1])

Unweighted measure

Compute importance

weights using info at time t-1

Weighted measure

Resampling

Prediction

Apprx. of p(x_t|y_1:t-1)

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Comb. Parameter and State Estimation

Often the parameters are known (or obtained throughseparate analysis).

However: If parameters are unknown, how to carry outcombined estimation of x and ?

Liu & West describe a simple approach.

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Algori thm Liu & West, Combined Parameter and State Estimation:

(auxiliary particle filter with state estimation)

)()(

)((j)

t

)()(11

)(1

)(11)(

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accurate.lysufficientisionapproximatuntil5.-2.Repeat

),|(

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yprobabilitwithN}{1,...,integeranSample2.

),|(:,...,1For1.

.,...,1,weightsand),(sampleCarloMonte:Input

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++


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R Implementation

To describe the model, the user supplies his ownfunctions as arguments to main SMC function (togetherwith Y and [hyper]parameters).

R language very suitable for implementation,

especially because of Vectorization

Built-in statistical functions

The possibility of supplying user-defined functions as arguments

Ease of visualization and interaction

Quite efficient but still computationally heavy. For large datasets, a C/C++ optimization is needed

(we already developed a faster C# version).

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Factor Stochastic Volatility Model

(similar to the model of Liu & West)

),...,diag(

variancesnoiseticIdiosyncra)|(~

matrixvariancesInnovation

sInnovation)|(~

iancesfactor varLog)(

)exp(

iancesFactor var),,(diagFactors),0|(~

matrixloadingsFactor

levelseriesLocal

nsObservatio

1

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titi

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hhN

y

=

++=

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++=

0,

U

U0,

H

Hf

X

Xf

t

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Example, artificial 2-factor model

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Example, artific ial 2-factor model


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Example, FX data

Model exchange rates with a factor stochastic volatility

model.

Per-minute data

EURUSD, GBPUSD, JPYUSD, CHFUSD.

The log return for currencyi on dayt is given by

wheres is the spot rate in US dollars.

)log(,1it

titi

s

sy

=

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FX data, example

Spot rates. 434 bank days of data. Index 1.0 at 2004-10-01.

EURUSD

GBPUSD

JPYUSD

CHFUSD

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FX data, example

Log return, log (s(t)/s(t-1)) (50 data points)

EURUSD

GBPUSD

JPYUSDCHFUSD

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FX data, results

Volatility factor 1

Log returns

Volatility factor 2


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Conclusion

R is flexible and powerful enough for implementingefficient particle filters

For large datasets, however, an optimized C/C++version is really needed (because of the heavycomputational burden).

Combined parameter and state estimation can be usefulbut also unstable when there are too many parameters Alternative: Do separate/offline estimation of parameters (using,

e.g., full MCMC)

Package may be forthcoming

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References

[1] A. Doucet, N. de Freitas and N. Gordon, editors,Sequential Monte Carlo Methods in Practice, Springer,2001.

[2] Liu and West: Combined Parameter and StateEstimation in Simulation-Based Filtering, pp. 197-223, in[1].

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Questions?

Thanks for your attention.

Emails:[email protected]

[email protected]

Web site: www.saxobank.com

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