u niversity of m assachusetts, a mherst department of computer science november 12, 2000memory...

UUNIVERSITY OF NIVERSITY OF MMASSACHUSETTSASSACHUSETTS, A, AMHERST • MHERST • Department of Computer Science Department of Computer Science

November 12, 2000 Memory Management for High-Performance Applications - Ph.D. defense - Emery Berger

Switching Kalman Filters for Prediction and Tracking in an Adaptive Meteorological

Sensing Network

Victoria Manfredi, Sridhar Mahadevan, Jim Kurose

SECON’05September 28, 2005

UUNIVERSITY OF NIVERSITY OF MMASSACHUSETTSASSACHUSETTS, A, AMHERST • MHERST • Department of Computer Science Department of Computer Science 2

Introduction

• CASA– Collaborative Adaptive Sensing of the Atmosphere– Distributed, collaborative, adaptive radar network

Where/what, when, and how to sense?

Configure radars based on predicted locations of meteorological phenomena

Our focus? Storm cells


Problem

• Track storm cells over time• Use predicted storm locations to identify future radar

configurations

• Constraints/Assumptions– Existing meteorological algorithms that identify storms

from raw radar data– Tracking only a single storm cell– Less than 30 seconds for prediction


Outline

• Meteorological vs. Statistical Approaches• Kalman Filter Approaches• Experiments• Conclusions • Future work


Storm Tracking

• Extrapolation– SCIT: linear least-squares over last five points

[JMWMSET98]

– Titan: extrapolation plus cross-correlation [DW93]

– K-means to identify storm clusters, smooth storm movements with Kalman filter [LRD03]

• Knowledge-intensive– Gandolf: model meteorological evolution of each

storm [PHCH00]

– Growth and Decay Storm Tracker: track encompassing storm instead of storm cell [WFHM98]

– Ensemble Kalman Filter: project a set of points forward in time using a meteorological model [E03]

ComputationallyExpensive

Simpler


• Goal:– Good predictions – Satisfy real-time constraints

• Meteorological Approaches– Extrapolation– Knowledge-intensive

Meteorological vs. Statistical

• Other Statistical Approaches– Kalman filter: linear, Gaussian, state– Switching Kalman filter: non-linear, Gaussian, state

SCIT: Linear least-squares regression [JMWMSET98]

Linear, Gaussian, no state(Developed at NSSL, Kurt Hondl)


Kalman Filter (KF)

t=1 t=2 t=3

State

Observation

State transitions : xt+1 = Axt + N[0,Q]Observations : yt+1 = Bxt+1 + N[0,R]

•Model (linear) dynamics of an object•States, Obs: Linear function plus Gaussian noise

X = [lat, long, vlat, vlong

]

Y = [lat, long]


Switching Kalman Filter (SKF)

State transitions : xt+1 = Ai xt + N[0,Qi ]Observations : yt+1 = Bi xt+1 + N[0,Ri ]

•Model object dynamics with set of Kalman filters•Piecewise linear approximation of nonlinear path

t=1 t=2 t=3

State

Observation

X = [lat, long, vlat, vlong

]

Y = [lat, long]

Switch

S = which Kalman filter

Switch

A1 , Q1 , B1 , R1 , 1 , 1

A2 , Q2 , B2 , R2 , 2 , 2

A3 , Q3 , B3 , R3 , 3 , 3


t=4t=1 t=2 t=3

State

Observation

Kalman Filter

X = [ lat, long, vlat, vlong ]

Y = [ lat, long]

Observe Infer Predict

Inference + Prediction

Least-Squares

• Use five most recent observations only


t=4


t=1 t=2 t=3

State

Observation

Switch

Switching Kalman Filter


Y = [ lat, long]

S = which Kalman filter

Switch values unknown inference in SKF is hardt=1: K possible states with K Kalman filters t=2: K2 possible states …t=n: Kn possible states

Solution? Approximate inference: Generalized pseudo-Bayesian– Order 2: Collapse over state, switches two time steps ago

Prediction– Compute most likely sequence of switches– Use corresponding KFs to infer hidden state and predict next

state

Observe Infer Collapse Most Likely Predict


Experiments

• Compare Kalman filter, switching Kalman filter and linear least-squares regression (SCIT [JMWMSET98] ) on tracking and predicting storm locations

• Data– 35 storm tracks courtesy of Kurt Hondl at NSSL– Each track is a sequence of latitude and longitude

coordinates– Range in length from ten to 30 data points– Identified using SCIT [JMWMSET98]


Compare hand-coded parameters with learned parameters

• Kalman filter, switching Kalman filter parameters– What are dynamics of storm cells?– How to obtain model of dynamics?

Parameter Learning

• Expectation-maximization to learn parameters– E-step: Assume parameters are known, compute

expected values of hidden variables (state, switch)– M-step: Assume values of hidden variables are known,

compute maximum likelihood parameters

SKF KF

hand-coded

KF-EM SKF-EM

learned


Results

(Not suprisingly) On nonlinear track, switching Kalman filter performs better


Results

On linear tracks, both methods perform similarly


Results

Method Average 1-Step RMSELatitude Longitude

Average 2-Step RMSE Latitude

Longitude KF 0.2248 0.1923 0.3359 0.2871

KF-EM 0.1967 0.1680 0.2680 0.2389

SKF 0.1914 0.1702 0.2642 0.2421

SKF-EM 0.2577 0.2070 0.3948 0.3110

Least- Squares

0.2114 0.2107 0.3030 0.3081

0.1° lat = 6.9 miles0.1° long 6.9 miles


# of KFs Time Required for 1-Step Prediction (seconds)

avg max min

1 0.000155 0.000864 0.000103

4 0.001689 0.006479 0.001138

8 0.006655 0.028375 0.004553

Timing

Within timing constraints


Conclusions and Future Work

• Although tracks identified with least-squares method (SCIT), KF-EM and SKF have lower prediction error

• Can learn storm dynamics to improve prediction model

• Future work– Obtain more data to improve learned model

• Especially SKF– Incorporate meteorological information– Track multiple targets, other meteorological

phenomena– Combine decision-making with prediction– Add higher layers to the SKF


Thank You.Questions?


References

[JMWMSET98] J. Johnson, P. MacKeen, A. Witt, E. Mitchell, G. Stumpf, M. Eilts, and K.. Thomas. The storm cell identification and tracking algorithm: An enhanced WSR-88D algorithm. Weather and Forecasting, 13:263-276, 1998.

[DW93] M. Dixon and G. Weiner. TITAN: Thunderstorm identification, tracking analysis and nowcasting a radar based methodology. J. Atmos. Ocean. Tech., 10:785-797, 1993.

[LRD03] V. Lakshamanan, R. Rabin, and V. DeBrunner. Multiscale strom identification and forecast. Journal of Atmospheric Research, 367-380, 2003.

[PHCH00] C.Pierce, P. Hardaker, C. Collier, and C. Haggett. GANDOLF: A system for generating automated nowcasts of covective precipitation. Meteorol. Appl., 7:341-360, 2000.

[WFHM98] M. Wolfson, B. Forman, R. Hallowell, and M. Moore. The growth and decay storm tracker. American Meteorological Society 79th Annual Conference, 1999.

[E03] G. Evensen. The ensemble Kalman filter: Theoretical formulation and practical implementatioon. Ocean Dynamics, 53:343-367, 2003.


Generalized Pseudo-Bayesian

• Values of switch variables are unknown inference in SKF is hard– Time step 1: K possible states with K Kalman filters – Time step 2: K2 possible states– …– Time step n: Kn possible states

• Solution? Approximate inference– Generalized pseudo-Bayesian– Variational– Sampling– Viterbi


Generalized Pseudo-Bayesian

• Order two generalized pseudo-Bayesian algorithm– Collapse over everything two time steps ago– x = mean, V = covariance, W = switch probability

(xj, Vj) = Collapse(xij, Vij, Wi)xj = ∑i Wi xij

Vj = ∑i Wi Vij + ∑i Wi (xij -xj)(xij -xj)T

• Covariance depends on observations through x


Linear Least-Squares Regression

• Given a set of points, find best fit line• Assumes constant covariance• Solve Ax=b for coefficient vector x• If too many equations, problem is over-constrained• Error: difference between what model says response

value should be and actual value– Ax - b

• Minimize squared vertical distance to best fit line– ||Ax -b||2

• So instead solve ATAx=ATb for coefficient vector x


Kalman Filter (KF)

• Assume A = Identity and Q = zero matrix– Then for all t, xt+1 = xt

• This can be used to derive the recursive least-squares update equations

• Implies least-squares assumes constant covariance while KF does not

State transitions : xt+1 = Axt + N[0,Q]Observations : yt+1 = Bxt+1 + N[0,R]


Results


t=4 t=5


t=1 t=2 t=3

State

Observation

Kalman Filter

ObserveInferPredict


X = [ lat, long]


t=4 t=5


t=1 t=2 t=3

State

Observation

Switch

Switching Kalman Filter

Least-Squares

• Use five most recent observations only

ObserveInferPredictCollapse


u niversity of m assachusetts, a mherst department of computer science november 12, 2000memory...

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