kalman filtering techniques for parameter estimationjaredb/presentations/biomath... · ·...
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Kalman filtering techniques for parameter estimation
Jared BarberDepartment of Mathematics, University of Pittsburgh
Work with Ivan Yotov and Mark TronzoMarch 17, 2011
Outline
• Motivation for Kalman filter
• Details for Kalman filter
• Practical example with linear Kalman filter
• Discussion of other filters
– Extended Kalman filter
– Stochastic Collocation Kalman filter
– Karhunen-Loeve SC Kalman filter
• Results for simplified NEC model
Necrotizing Enterocolitis (NEC) Model
• Ten nonlinear PDEs
• Four layers
• Time consuming simulations for normal computational refinements
• Approximately fifty free parameters– Diffusion rates
– Growth and death rates
– Interaction rates
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Computational Domain-
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Distribution of macrophages
Maximum Likelihood Estimate
• Recall formula (for normal distributions):
• Disadvantages:– Consider all times simultaneously—larger optimization problem
and generally slower– Don’t take into account measurement error and model error
separately– To be more efficient: Want derivative information– Get only one answer…not a distribution (which tells you how
good your answer is)– May be hard to parallelize
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Kalman Filter
• Various versions: Linear KF, Extended KF, Ensemble KF, Stochastic Collocation/Unscented KF, Karhunen-Loeve Kalman Filter
• Advantages of some of these methods (to a lesser or greater extent)– Consider each time separately– Keep track of best estimates for your parameters
(means) and your uncertainties (covariances)– Consider data and measurement error separately– Don’t need derivative information– Easy to parallelize
Kalman Filter Picture: Initial State
Measurement
Model Estimate
Kalman Filter Picture: Analysis/Adjustment Step
Model Estimate
Measurement
True SolutionEstimate
Kalman Filter Picture: Prediction/Forecast Step
Model Estimate
Measurement
True SolutionEstimate
Kalman Filter Picture: Measurement Step
Model Estimate
Measurement
True SolutionEstimate
Kalman Filter: Analysis/Adjustment Step
Model Estimate
Measurement
True SolutionEstimate
Advancing two things: Mean and covariance
Adjusted state vector =Model State Vector + Kk*(Measured Data-Model Data)
Kk = f(Model Covariance, Data Covariance)
Kalman Filter: General Algorithm, Quantities of interest
• Measured data = true data plus measurement noise
• Measurement function
• Optimal “blending factor” Kalman Gain:
• Model/forecast and adjusted state vectors
• Forecast/model function:
• Best/Analyzed model estimate
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Kalman Filter: General Algorithm, Uncertainties/Covariances of interest
• Prescribed: Measured data and application of model covariance
• Forecast state covariances
• Adjusted state covariances
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Kalman Filter: General Algorithm, Kalman Gain
• Recall to adjust the model’s state vector:
• Minimize the sum of the uncertainties associated with the adjusted state to find the right blending factor
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Linear Kalman Filter
• Special Case:
• Covariances given by:
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Example: Exact system
• Physical system modeled exactly by:
• Exact solution for this physical system is then:
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Example: Model
• Pretend we have a model that we think might work for this system:
• We have three unknown variables, y1, y2, and a.
• We wish to find a that makes the model fit the data.
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Example: State vector
• Define the state vector as:
• Note, must make reasonable starting guess for unknown parameter. Here we guessed 1.5 which is close to the actual value of 1.
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Example: Data vector
• Assume we can actually measure y1 and y2 in our system
• Note our measurement function becomes:
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Example: Model/forecast function
• Assuming that the parameter a is a quantity that should not change with time, we can rewrite the system of equations as:
• Forward euler for our system gives us:
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Example: Algorithm step through: Initialization
• Start with state vector data:
• Start with data vector data:
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Example: Algorithm step through: Initialization
• Initial forecast model uncertainty as initial uncertainty in model state vector:
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Example: Algorithm step through: Obtaining other covariances
• Use formulas to find other state vector covariances:
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Example: Algorithm step through:Obtaining other covariances
• Use formulas to find other state vector covariances:
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Example: Algorithm step through: Obtaining Kalman Gain
• Find Kalman Gain:
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Example: Algorithm step through: Obtaining adjusted state vector
• Best estimate, note: halfway in between data and model…this is because the data uncertainties and model uncertainties are the same size
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Example: Algorithm step through: Find adjusted state’s covariance
• Note: Uncertainties are smaller than other covariances…used both data and model to get an estimate with less uncertainty than before
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Example: Algorithm step through
• Predict step, assume dt = 0.1:
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Example: Algorithm step through
• New state covariance matrix:
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Example: Algorithm step through
• “Measure” data
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Extended Kalman Filter:Nonlinear equations
• Special Case:
• Covariances given by:)(;)( 11
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Problems with KF/Ext KF?
• KF only works for linear problems.
• Ext KF
– Works for mildly nonlinear problems
– Must find and store Jacobians
• State vector size >> data vector size
– Reminder: Need to find
– Don’t need any Pxx’s.
– Pxx’s can be big and hard to calculate/keep track of
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Ensemble Kalman Filter: The Ensemble
• Create an ensemble:
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Ensemble Kalman Filter: Ensemble Properties
• The ensemble carries mean and covariance information
system of statemean
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Ensemble Kalman Filter: New algorithm
Linear KF Ensemble KF
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Ensemble KF Advantages
• Handles nonlinearity (better than Ext KF)
• Don’t need Jacobian
• Don’t need Pxx’s
Ensemble KF Disadvantages
• Need many ensemble members to be accurate
– Mean and variances used in algorithm are within O(1/sqrt(q)) of actual (MC integration)
– Accurate representation can affect convergence rate and end error on parameter estimate
• Using many ensemble members requires many model/forecast function evaluations (slow)
Can we use fewer points and obtain more accuracy?
Note: Don’t need a lot of accuracy, just enough to get the job done.
Stochastic Collocation Kalman Filter:Stochastic Collocation
• Consider the expected value of the function g(z) on stochastic space
• ci and zi are collocation weights and locations, respectively
• Collocation exact for linear functions of the components of z and normal pdfs
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Ensemble and Stochastic Collocation Comparison
• En mean (q≈1000 pts) • SC mean (N≈200 pts)
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Stochastic Collocation:Kalman Filter Algorithm
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Stochastic Collocation Advantages and Disadvantages
• Faster than En KF for small numbers
• Slower than En KF for large numbers
• Usually more accurate than En KF
• Can handle nonlinearities
• Curse of dimensionality for pdes
– 20x20x20 grid needs 16001 collocation points
• Is there any way to get around this?
Stochastic Collocation Kalman Filter with Karhunen-Loeve Expansion
• On 3-d grids, above methods assume error is independent of location in computational grid
• Instead, assume error is spatially correlated:
• Hope: Most of the error is captured by the most dominant eigenfunctions
• Idea: Keep only the first twenty-five
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Karhunen-Loeve SC:Kalman Filter Algorithm
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Simplified Necrotizing Enterocolitis Model: The experiment
Epithelial Layer
Create woundwith pipette
Epithelial Layer
Wound
≈150 µm
Simplified Necrotizing Enterocolitis Model: The Model and Equation
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-300
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Simplified Necrotizing Enterocolitis: Perfect Simulated Data
En KF; SC KF; KL SC KF.
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ate
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or
Time (hrs)
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ate
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ate
Time (hrs)
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ate
Simplified Necrotizing Enterocolitis: Imperfect Simulated Data
En KF; SC KF; KL SC KF.
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ate
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Time (hrs)
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Simplified Necrotizing Enterocolitis: Real Data
En KF; SC KF; KL SC KF.
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Are parameter estimates good?
• Produce qualitatively correct results t = 0 h t = 0.5 h t = 1 h t = 1.5 h
t = 2 h t = 2.5 h t = 3 h t = 3.5 h
200 m
Comparisons
• With perfect measurements and a pretty good model, SC does best, then KL, then En
• With imperfect measurements, all are comparable
• With real data, KL fails. Why? Guess: Too much error associated with D
• Additional real data info– Gives temporal information about the parameters
– Gives uncertainty estimates
• All run significantly faster than the direct optimization method used