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CMU CS 10708 - Kalman Filters Switching Kalman Filter

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Kalman Filters Switching Kalman FilterThe Kalman FilterMultivariate GaussianConditioning a GaussianGaussian is a “Linear Model”Slide 9Conditional Linear Gaussian (CLG) – general caseUnderstanding a linear Gaussian – the 2d caseTracking with a Gaussian 1Tracking with Gaussians 2 – Making observationsOperations in Kalman filterExponential family representation of Gaussian: Canonical FormCanonical formConditioning in canonical formSlide 18Prediction & roll-up in canonical formAnnouncementsWhat if observations are not CLG?Linearization: incorporating non-linear evidenceLinearization as integrationLinearization as numerical integrationOperations in non-linear Kalman filterWhat you need to know about Kalman FiltersWhat if the person chooses different motion models?The moonwalkSlide 29Switching Kalman filterInference in switching KF – one stepMulti-step inferenceVisualizing growth in number of GaussiansComputational complexity of inference in switching Kalman filtersBounding number of GaussiansCollapsing Gaussians – Single Gaussian from a mixtureCollapsing mixture of Gaussians into smaller mixture of GaussiansOperations in non-linear switching Kalman filterAssumed density filteringWhen non-linear KF is not good enoughReparameterized KF for SLATWhen a single Gaussian ain’t good enoughApproximating non-linear KF with mixture of GaussiansWhat you need to know1Kalman FiltersSwitching Kalman FilterGraphical Models – 10708Carlos GuestrinCarnegie Mellon UniversityNovember 20th, 2006Readings:K&F: 4.5, 12.2, 12.3, 12.43 The Kalman FilterAn HMM with Gaussian distributionsHas been around for at least 50 yearsPossibly the most used graphical model everIt’s whatdoes your cruise controltracks missilescontrols robots…And it’s so simple… Possibly explaining why it’s so usedMany interesting models build on it…An example of a Gaussian BN (more on this later)6 Multivariate GaussianMean vector:Covariance matrix:7 Conditioning a GaussianJoint Gaussian:p(X,Y) ~ N(;)Conditional linear Gaussian:p(Y|X) ~ N(Y|X; 2)8 Gaussian is a “Linear Model”Conditional linear Gaussian:p(Y|X) ~ N(0+X; 2)9 Conditioning a GaussianJoint Gaussian:p(X,Y) ~ N(;)Conditional linear Gaussian:p(Y|X) ~ N(Y|X; YY|X)10 Conditional Linear Gaussian (CLG) – general caseConditional linear Gaussian:p(Y|X) ~ N(0+X; YY|X)11 Understanding a linear Gaussian – the 2d caseVariance increases over time (motion noise adds up)Object doesn’t necessarily move in a straight line12 Tracking with a Gaussian 1p(X0) ~ N(0,0)p(Xi+1|Xi) ~ N( Xi +  ; Xi+1|Xi)13 Tracking with Gaussians 2 – Making observationsWe have p(Xi)Detector observes Oi=oiWant to compute p(Xi|Oi=oi)Use Bayes rule:Require a CLG observation modelp(Oi|Xi) ~ N(W Xi + v; Oi|Xi)14 Operations in Kalman filterComputeStart with At each time step t:Condition on observationPrediction (Multiply transition model)Roll-up (marginalize previous time step)I’ll describe one implementation of KF, there are othersInformation filterX1O1 = X5X3X4X2O2 = O3 = O4 = O5 =15 Exponential family representation of Gaussian: Canonical Form16 Canonical formStandard form and canonical forms are related:Conditioning is easy in canonical formMarginalization easy in standard form17 Conditioning in canonical formFirst multiply:Then, condition on value B = y18 Operations in Kalman filterComputeStart with At each time step t:Condition on observationPrediction (Multiply transition model)Roll-up (marginalize previous time step)X1O1 = X5X3X4X2O2 = O3 = O4 = O5 =19 Prediction & roll-up in canonical formFirst multiply:Then, marginalize Xt:10-708 – Carlos Guestrin 200620 AnnouncementsLectures the rest of the semester:Special time: Monday Nov 27 - 5:30-7pm, Wean 4615A: Dynamic BNsWed. 11/30, regular class time: Causality (Richard Scheines)Friday 12/1, regular class time: Finish Dynamic BNs & Overview of Advanced TopicsDeadlines & Presentations:Project Poster Presentations: Dec. 1st 3-6pm (NSH Atrium)popular vote for best posterProject write up: Dec. 8th by 2pm by email 8 pages – limit will be strictly enforcedFinal: Out Dec. 1st, Due Dec. 15th by 2pm (strict deadline)21 What if observations are not CLG?Often observations are not CLGCLG if Oi =  Xi + o + Consider a motion detector Oi = 1 if person is likely to be in the regionPosterior is not Gaussian22 Linearization: incorporating non-linear evidencep(Oi|Xi) not CLG, but…Find a Gaussian approximation of p(Xi,Oi)= p(Xi) p(Oi|Xi)Instantiate evidence Oi=oi and obtain a Gaussian for p(Xi|Oi=oi)Why do we hope this would be any good?Locally, Gaussian may be OK23 Linearization as integrationGaussian approximation of p(Xi,Oi)= p(Xi) p(Oi|Xi)Need to compute momentsE[Oi]E[Oi2]E[Oi Xi]Note: Integral is product of a Gaussian with an arbitrary function24 Linearization as numerical integrationProduct of a Gaussian with arbitrary functionEffective numerical integration with Gaussian quadrature methodApproximate integral as weighted sum over integration pointsGaussian quadrature defines location of points and weightsExact if arbitrary function is polynomial of bounded degreeNumber of integration points exponential in number of dimensions dExact monomials requires exponentially fewer pointsFor 2d+1 points, this method is equivalent to effective Unscented Kalman filterGeneralizes to many more points25 Operations in non-linear Kalman filterComputeStart with At each time step t:Condition on observation (use numerical integration)Prediction (Multiply transition model, use numerical integration)Roll-up (marginalize previous time step)X1O1 = X5X3X4X2O2 = O3 = O4 = O5 =26 What you need to know about Kalman Filters Kalman filterProbably most used BNAssumes Gaussian distributionsEquivalent to linear systemSimple matrix operations for computationsNon-linear Kalman filterUsually, observation or motion model not CLGUse numerical integration to find Gaussian approximation27 What if the person chooses different motion models?With probability , move more or less


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