Cognitive Dynamic Systems - Cognitive Systems Laboratory

Cognitive Dynamic SystemsSimon HaykinCognitive Systems LaboratoryMcMaster, UniversityHamilton, Ontario, Canadaemail: haykin@mcmaster.caWeb site: http://soma.mcmaster.caNIPS Workshops, Whistler, BC, December 2009 (Haykin)1

Outline of the Lecture1. Background Behind the Emergence of Cognitive Dynamic Systems (CDS)2. Cognition3. Highlights of Research into CDS in my Laboratory4. Bayesian Filtering for State Estimation of the Environment5. Cubature Kalman Filters6. Feedback Information7. Dynamic Programming and Optimal Control of the Environment8. Summarizing Remarks on the Cognitive Process9. Experiment on Cognitive Tracking Radar for Demonstrating thePower of Cognition10. Final RemarksLast NoteNIPS Workshops, Whistler, BC, December 2009 (Haykin)2

1. Background Behind the Emergence ofCognitive Dynamic Systems (CDS)Broad Array of Subjects that have prepared me for my currentresearch passion: CDSSignal Processing;Control Theory;Adaptive Systems;Communications;Radar; andNeural Information ProcessingNIPS Workshops, Whistler, BC, December 2009 (Haykin)3

Background (continued)Two Seminal Journal Papers(1) Simon Haykin, “Cognitive Radio: Brain-empoweredWireless Communications”, IEEE Journal on SelectedAreas in Communications, Special Issue on CognitiveNetworks, pp. 201-220, February, 2005.(2) Simon Haykin, “Cognitive Radar: A Way of the Future”,IEEE Signal Processing Magazine, pp. 30-41, January 2006.NIPS Workshops, Whistler, BC, December 2009 (Haykin)4

Background (continued)Predictive Article3“Isee the emergence of a new discipline, called CognitiveDynamic Systems, which builds on ideas in statistical signalprocessing, stochastic control, and information theory, andweaves those well-developed ideas into new ones drawn fromneuroscience, statistical learning theory, and game theory.The discipline will provide principled tools for the design anddevelopment of a new generation of wireless dynamic systems exemplified by cognitive radio and cognitive radar withefficiency, effectiveness, and robustness as the hallmarks ofperformance”.3.Simon Haykin, “Cognitive Dynamic Systems”, Proc. IEEE, Point of Viewarticle, November 2006.NIPS Workshops, Whistler, BC, December 2009 (Haykin)5

2. CognitionRadioenvironment(Outside l, andspectrummanagementRFstimuliSpectrum holesNoise-floor statisticsTraffic statisticsInterferencetemperatureQuantizedchannel estimation, andpredictivemodelingReceiverFigure 1: Information-processing Cycle in Cognitive Radio1NIPS Workshops, Whistler, BC, December 2009 (Haykin)6

Transmittedradar signalRadar luminatorof theenvironmentOthersensorsInformationbearing signalson theenvironmentRadarBayesianscenetargetanalyzer Environmental trackermodel parametersPriorknowledgeStatistical parameter estimates and probabilisticdecisions on the environmentFigure 2: Block diagram of cognitive radar viewed as a dynamic closed-loopfeedback system2NIPS Workshops, Whistler, BC, December 2009 (Haykin)7

Action taken on theenvironmentObservationsreceived from theenvironmentThe Environmenthe n Control)FeedbackChannelLinkFeedbackFFigure 3: Graphical representation of perception-action cycle in the Visual Brain,(D.A. Milner and M.A. Goodale, 2006; J.M. Fuster, 2005)NIPS Workshops, Whistler, BC, December 2009 (Haykin)8

3. Highlights of Research into CDS inmy Laboratory(i) Cognitive RadioSelf-organizing dynamic spectrum managementfor cognitive radio networksThe design of a software testbed for demonstrating this novelDSM strategy (using 5,000 lines of codes) has been completed,ready for experimentation; the strategy is motivated byHebbian learning.NIPS Workshops, Whistler, BC, December 2009 (Haykin)9

(ii)Cognitive Mobile AssistantsNew generation of hand-held biomedicalwireless devices used as aids for memoryimpaired patients, and other relatedapplications.NIPS Workshops, Whistler, BC, December 2009 (Haykin)10

(iii) Cognitive Tracking RadarTransmittedWaveformTransmitterThe Radar Environmenthe n elFeedbackcomputerInformation onpredicted stateestimation error vectorFigure 4: Cognitive information-processing cycle of tracking radar, revisited in lightof the perception-action cycle in the visual brainNIPS Workshops, Whistler, BC, December 2009 (Haykin)11

4. Bayesian Filtering for State Estimationof the Environment (Ho and Lee, 1964)State-space Model1. System (state) sub-modelxk 1 a ( xk ) ωk2. Measurement sub-modelyk b ( xk ) νkwhere k discrete timexk state at time kyk observable at time kωk process noiseν measurement noisekNIPS Workshops, Whistler, BC, December 2009 (Haykin)12

Prior Assumptions: Nonlinear functions a ( . ) and b ( . ) are known, with a ( . )being derived from underlying physics of the dynamicsystem under study and b ( . ) derived from the digitalinstrumentation used to obtain measurements. Process noise ωk and measurement noise νk are statisticallyindependent Gaussian processes of zero mean and knowncovariance matrices. Sequence of observationsYk k{ y i } i 1NIPS Workshops, Whistler, BC, December 2009 (Haykin)13

Up-date Equations:Time-update:{p ( x k Y k -1 ) Predictivedistribution n p( xk xk -1 ) p( xk -1 Yk -1 )dxk ere Rn denotes the n-dimensional state ution{{{1 .p ( x k Y k -1 )l ( y k x k )p ( x k Y k ) re Ck is the normalizing constant defined by the integralCk n p( xk Yk -1 )l ( yk xk )dxkRNIPS Workshops, Whistler, BC, December 2009 (Haykin)14

Notes on the Bayesian Filter(i)The posterior fully defines the available information about the state ofthe environment, given the sequence of observations.(ii) The Bayesian filter propagates the posteriori (embodying time andmeasurement updates for each iteration) across the state-space model:The Bayesian filter is therefore the maximum a posteriori (MAP)estimator of the state(iii) The celebrated Kalman filter (Kalman, 1960), applicable to a lineardynamic system in a Gaussian environment, is a special case of theBayesian filter.(iv) If the dynamic system is nonlinear and/or the environment is nonGaussian, then it is no longer feasible to obtain closed-form solutionsfor the integrals in the time- and measurement-updates, in which case:We have to be content with approximate forms of the Bayesian filterNIPS Workshops, Whistler, BC, December 2009 (Haykin)15

5. Cubature Kalman4FiltersObjectiveUsing numerically rigorous mathematics in nonlinearestimation theory, approximate the Bayesian filter so as tocompletely preserve second-order information about the statexk that is contained in the sequence of observations YkIn cubature Kalman filters, this approximation is made directlyand in a local manner.4.I. Arasaratnam and S. Haykin, “Cubature Kalman filters”, IEEE Trans. Automatic Control, pp.1254-1269, June2009.NIPS Workshops, Whistler, BC, December 2009 (Haykin)16

Steps involved in deriving the CKF:In a Gaussian environment, approximating the Bayesian filterinvolves computing moment integrals of the form: f ( x ) exp ( –x x ) dxArbitrarynonlinearfunction{T{h(f ) Gaussianfunctionwhere x is the state.NIPS Workshops, Whistler, BC, December 2009 (Haykin)17

CKF Steps (continued)(i) Cubature rule, which is constructed by forcing the cubaturepoints to obey symmetry:Let x rz with zTz 1 for 0 r The Cartesian coordinate system is thus transformed into aspherical-radial coordinate system, yieldingh 0 S ( r )rn-12exp ( – r ) drwhereS (r) f ( rz ) dσ ( z ),Rdσ(z) is an elemental measure of the spherical surfacenand n is the dimension of vector x (state).NIPS Workshops, Whistler, BC, December 2009 (Haykin)18

CKF Steps (continued)(ii) Spherical rule of third-degree:2n f [ u ]i f ( rz ) dσ ( z ) w i 1n{R2n cubature points resultingfrom the generator [u]where w is a scaling factor.NIPS Workshops, Whistler, BC, December 2009 (Haykin)19

CKF Steps (continued)(iii) Radial rule, using Gaussian quadrature known to beefficient for computing integrals in a single dimension,which yieldsn{ wi f ( xi ) f ( x )w ( x ) dx i 1Weightingfunctionwherew( x) xn-12exp ( – x ) ,0 x and w i w ( x i )and the integral is in the form of the well-known generalizedGauss-Laguerre formula.NIPS Workshops, Whistler, BC, December 2009 (Haykin)20

Properties of the Cubature Kalman Filter Property 1: The cubature Kalman filter (CKF) is a derivative-free on-linesequential-state estimator, relying on integration from one iteration to the next forits operation; hence, the CKF has a built-in smoothing capability. Property 2: Approximations of the moment integrals in the Bayesian filter are alllinear in the number of function evaluations. Property 3: Computational complexity of the cubature Kalman filter grows as n3,where n is the dimensionality of the state. Property 4: The cubature Kalman filter completely preserves second-orderinformation about the state that is contained in the observations; in this sense, it isthe best known information-theoretic approximation to the Bayesian filter.NIPS Workshops, Whistler, BC, December 2009 (Haykin)21

Properties of the Cubature Kalman Filter (continued) Property 5: Regularization is naturally built into the cubature Kalman filter byvirtue of the fact that the prior in Bayesian filtering is known to play a roleequivalent to regularization. Property 6: The cubature Kalman filter inherits well-known properties of theclassical Kalman filter, including square-root filtering for improved accuracy andreliability. Property 7: The CKF eases the curse-of-dimensionality problem, depending on hownonlinear the filter is:The less nonlinear the filter is, the higher is the feasiblestate-space dimensionality of the filter. Property 8: The equally weighted cubature points provide a representation of theestimator’s statistics (i.e., mean and covariance); computational cost of the CKFmay therefore be reduced by modifying the time-update to propagate the cubaturepoints.NIPS Workshops, Whistler, BC, December 2009 (Haykin)22

Hybrid CKF: Application to Tracking Coordinated Turns5For a (nearly) coordinated turn in three-dimensional space subject to fairly small noisemodeled by independent Brownian motions, we write the state equationdx ( t ) f ( x ( t ) )dt Qdβ ( t )where, in an air-traffic-control environment, the seven-dimensional state of the aircraftx [ ε, ε̇, η, η̇, ζ,ζ̇, ω ]Twith ε,η and ζ denoting positions and ε̇,η̇ and ζ̇denoting velocities in the x, y and z Cartesian coordinates, respectively; ω denotes the turnTf ( x ) [ ε̇ ( – ωη̇ ), η̇, ω, ε̇, ζ̇, 0, 0 ] ; the noise termTβ ( t ) [ β 1 ( t ), β 2 ( t ), ,β 7 ( t ) ] , involving seven mutually independentrate; the drift functionstandard Brownian motions, accounts for unpredictable modeling errors due to turbulence, wind force, etc.5.I. Arasaratnam, s. Haykin, and T. Hurd, Cubature Filtering for Continuous-Discrete NonlinearSystems: Theory with an Application to Tracking, submitted to IEEE Trans. Signal Processing.NIPS Workshops, Whistler, BC, December 2009 (Haykin)23

The gain matrix Q diag ( [ 0, σ 1 ,0, σ 1 ,0, σ 1 ,0,σ 2 ] ) . For the experimentat hand, a radar was located at the origin and digitally equipped to measure the2222range, r, and azimuth angle, θ , at a measurement sampling interval of T. Hence, wewrite the measurement equation: ε 2 η 2 ζ 2 kk k rk wk –1 η tan -----k- θ k εk where the measurement noise wk N (0,R) withData.σ1 22R diag ( [ σ r ,σ θ ] ) .–30.2; σ 2 7 10 ; σ r 50m; σ θ 0.1deg;and the true initial state x0 [1000m, 0ms-1, 2650m, 150ms-1, 200m, 0ms-1, ω deg/s]T.NIPS Workshops, Whistler, BC, December 2009 (Haykin)24

Figure 5: Accumulative RMSE Plots for a fixed sampling interval T 6s and varying turn rates:first row, ω 3 deg/s; second row, ω 4.5 deg/s; third row, ω 6 deg/s(Solid thin with empty circles-EKF, dashed thin with filled squares-UKF, dashed thick-hybrid CKF)NIPS Workshops, Whistler, BC, December 2009 (Haykin)25

Matlab CodesThe Matlab codes for the discrete-time version of the CKF areavailable on theWebsite: http://soma.mcmaster.caNIPS Workshops, Whistler, BC, December 2009 (Haykin)26