cognitive dynamic systems - mcmaster...
TRANSCRIPT
1
ems
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Cognitive Dynamic Syst
Simon HaykinCognitive Systems Laboratory
McMaster, UniversityHamilton, Ontario, Canada
email: [email protected] site: http://soma.mcmaster.ca
2
e Dynamic Sys-
orynvironment
the Environmentsemonstrating the
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Outline of the Lecture1. Background Behind the Emergence of Cognitiv
tems (CDS)2. Cognition3. Highlights of Research into CDS in my Laborat4. Bayesian Filtering for State Estimation of the E5. Cubature Kalman Filters6. Feedback Information7. Dynamic Programming and Optimal Control of8. Summarizing Remarks on the Cognitive Proces9. Experiment on Cognitive Tracking Radar for D
Power of Cognition10. Final Remarks
Last Note
3
rgence ofCDS)
for my current
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
1. Background Behind the EmeCognitive Dynamic Systems (
Broad Array of Subjects that have prepared meresearch passion: CDS
Signal Processing;
Control Theory;
Adaptive Systems;
Communications;
Radar; and
Neural Information Processing
4
rain-empoweredal on Selectede on Cognitive
of the Future”,1, January 2006.
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Background (continued)
Two Seminal Journal Papers
(1) Simon Haykin, “Cognitive Radio: BWireless Communications”, IEEE JournAreas in Communications, Special IssuNetworks, pp. 201-220, February, 2005.
(2) Simon Haykin, “Cognitive Radar: A WayIEEE Signal Processing Magazine, pp. 30-4
5
d Cognitivetatistical signalon theory, andnes drawn from game theory.r the design anddynamic sys-itive radar withe hallmarks of
, Point of View
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Background (continued)
Predictive Article3
“I see the emergence of a new discipline, calleDynamic Systems, which builds on ideas in sprocessing, stochastic control, and informatiweaves those well-developed ideas into new oneuroscience, statistical learning theory, andThe discipline will provide principled tools fodevelopment of a new generation of wireless tems exemplified by cognitive radio and cognefficiency, effectiveness, and robustness as thperformance”.
3.Simon Haykin, “Cognitive Dynamic Systems”, Proc. IEEEarticle, November 2006.
6
NIPS Workshops, Whistler, BC, December 2009 (Haykin)2. Cognition
Figure 1: Information-processing Cycle in Cognitive Radio1
Radioenvironment
(Outside world)
Radio-scene
analysis
Receiver
Channel-stateestimation, and predictive modeling
Transmit-power control, and spectrum management
Action:
Transmitter
Spectrum holes
transmittedsignal
Interferencetemperature
Quantized channel capacity
RF
Noise-floor statisticsTraffic statistics
stimuli
7
ic closed-loop
Receiver
esianet-ker
Priorknowledge
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Figure 2: Block diagram of cognitive radar viewed as a dynam
feedback system2
Environment
Radar-sceneanalyzer
IntelligentIlluminatorof theenvironment
Transmittedradar signal Radar returns
Environmentalmodel parameters
Transmitter
Statistical parameter estimates and probabilistic decisions on the environment
Information-bearing signalson theenvironment Bay
targtrac
Othersensors
8
n the Visual Brain,
tionng)n
Observations received from the environment
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Figure 3: Graphical representation of perception-action cycle i
(D.A. Milner and M.A. Goodale, 2006; J.M. Fuster, 2005)
he n ironment
Percep(Sensi
ction(Control)
Feedback Channel F
Feedback Link
Action Perceptio
TThe Environment
Action taken on the environment
9
DS in
t
ating this novelbeen completed,
motivated by
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
3. Highlights of Research into Cmy Laboratory
(i) Cognitive Radio
Self-organizing dynamic spectrum managemen
for cognitive radio networks
The design of a software testbed for demonstrDSM strategy (using 5,000 lines of codes) hasready for experimentation; the strategy isHebbian learning.
10
edical
mory-
elated
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
(ii) Cognitive Mobile Assistants
New generation of hand-held biom
wireless devices used as aids for me
impaired patients, and other r
applications.
11
dar, revisited in light
rceptionensing)
eption:-of-theronment
ator
Radar returns
Receiver
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
(iii) Cognitive Tracking Radar
Figure 4: Cognitive information-processing cycle of tracking raof the perception-action cycle in the visual brain
he n ironment
Pe(S
ction(Control)
Feedback Channel F Entropycomputer
Action: Transmit- waveform controller
Perc State envi estim
The Radar Environment
Information onpredicted state-estimation error vector
TransmittedWaveform
Transmitter
12
tion964)
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
4. Bayesian Filtering for State Estima of the Environment (Ho and Lee, 1
State-space Model
1. System (state) sub-model
2. Measurement sub-model
where k = discrete timexk = state at time kyk = observable at time kωk = process noise
= measurement noise
xk 1+ a xk( ) ωk+=
yk b xk( ) νk+=
νk
13
n, withe dynamic
he digitalts.
are statisticallyan and known
a .( )
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Prior Assumptions:
• Nonlinear functions and are knowbeing derived from underlying physics of th
system under study and derived from tinstrumentation used to obtain measuremen
• Process noise and measurement noise independent Gaussian processes of zero mecovariance matrices.
• Sequence of observations
a .( ) b .( )
b .( )
ωk νk
Yk yi{ } i=1k
=
14
NIPS Workshops, Whistler, BC, December 2009 (Haykin)Up-date Equations:
1. Time-update:
where Rn denotes the n-dimensional state space.
2. Measurement-update:
where Ck is the normalizing constant defined by the integral
p xk Yk -1( ) p xk xk -1( ) p xk -1 Yk -1( ) xk -1dR
n∫={ {
Prior Old
distribution posterior distribution
{Predictivedistribution
p xk Yk( ) 1Ck------ . p xk Y
k -1( )l yk xk( )=
Updated Predictive Likelihoodposterior distribution functiondistribution
{ { {Ck p xk Yk -1( )l yk xk( ) xkd
Rn
∫=
15
n about the state ofs.
bodying time and state-space model:
posteriori (MAP)
plicable to a linearspecial case of the
nvironment is non-osed-form solutionsates, in which case:
e Bayesian filter
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Notes on the Bayesian Filter
(i) The posterior fully defines the available informatiothe environment, given the sequence of observation
(ii) The Bayesian filter propagates the posteriori (emmeasurement updates for each iteration) across the
The Bayesian filter is therefore the maximum aestimator of the state
(iii) The celebrated Kalman filter (Kalman, 1960), apdynamic system in a Gaussian environment, is aBayesian filter.
(iv) If the dynamic system is nonlinear and/or the eGaussian, then it is no longer feasible to obtain clfor the integrals in the time- and measurement-upd
We have to be content with approximate forms of th
16
linearter so as toabout the stateions Yk
is made directly
utomatic Control, pp.
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
5. Cubature Kalman Filters4
Objective
Using numerically rigorous mathematics in nonestimation theory, approximate the Bayesian filcompletely preserve second-order information xk that is contained in the sequence of observat
In cubature Kalman filters, this approximationand in a local manner.
4. I. Arasaratnam and S. Haykin, “Cubature Kalman filters”, IEEE Trans. A
1254-1269, June 2009.
17
e Bayesian filterm:
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Steps involved in deriving the CKF:
In a Gaussian environment, approximating thinvolves computing moment integrals of the for
where x is the state.
h f( ) f x( ) xT x–( )exp xd∫= { {Arbitrarynonlinearfunction
Gaussianfunction
18
ing the cubature
nsformed into a
the spherical surface
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
CKF Steps (continued)
(i) Cubature rule, which is constructed by forcpoints to obey symmetry:
Let x = rz with zTz = 1 for 0 < r < ∞
The Cartesian coordinate system is thus traspherical-radial coordinate system, yielding
where
, dσ(z) is an elemental measure of
and n is the dimension of vector x (state).
h S r( )rn-1
r2
–( )exp rd0
∞∫=
S r( ) f rz( ) σ z( )dR
n∫=
19
NIPS Workshops, Whistler, BC, December 2009 (Haykin)CKF Steps (continued)
(ii) Spherical rule of third-degree:
where w is a scaling factor.
f rz( ) σ z( )dR
n∫ w f u[ ] i
i=1
2n
∑≈ {2n cubature points resultingfrom the generator [u]
20
nown to bengle dimension,
own generalized
i w xi( )=
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
CKF Steps (continued)
(iii) Radial rule, using Gaussian quadrature kefficient for computing integrals in a siwhich yields
where
, and
and the integral is in the form of the well-knGauss-Laguerre formula.
f x( )w x( ) x wi f xi( )i=1
n
∑≈d∫ {
Weightingfunction
w x( ) xn-1
x2
–( )exp= 0 x< ∞< w
21
erivative-free on-lineration to the next forility.
Bayesian filter are all
an filter grows as n3,
reserves second-orderons; in this sense, it isayesian filter.
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Properties of the Cubature Kalman Filter
• Property 1: The cubature Kalman filter (CKF) is a dsequential-state estimator, relying on integration from one iteits operation; hence, the CKF has a built-in smoothing capab
• Property 2: Approximations of the moment integrals in thelinear in the number of function evaluations.
• Property 3: Computational complexity of the cubature Kalmwhere n is the dimensionality of the state.
• Property 4: The cubature Kalman filter completely pinformation about the state that is contained in the observatithe best known information-theoretic approximation to the B
22
inued)
ure Kalman filter bynown to play a role
wn properties of theproved accuracy and
em, depending on how
feasible
representation of theional cost of the CKFropagate the cubature
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Properties of the Cubature Kalman Filter (cont
• Property 5: Regularization is naturally built into the cubatvirtue of the fact that the prior in Bayesian filtering is kequivalent to regularization.
• Property 6: The cubature Kalman filter inherits well-knoclassical Kalman filter, including square-root filtering for imreliability.
• Property 7: The CKF eases the curse-of-dimensionality problnonlinear the filter is:
The less nonlinear the filter is, the higher is thestate-space dimensionality of the filter.
• Property 8: The equally weighted cubature points provide aestimator’s statistics (i.e., mean and covariance); computatmay therefore be reduced by modifying the time-update to ppoints.
23
ated Turns5
to fairly small noiseation
l state of the aircraft
ons and andely; ωdenotes the turn
; the noise term
ally independent
errors due to turbu-
-Discrete Nonlinear. Signal Processing.
ε̇ ,η̇ ζ̇
0]T
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Hybrid CKF: Application to Tracking Coordin
For a (nearly) coordinated turn in three-dimensional space subjectmodeled by independent Brownian motions, we write the state equ
where, in an air-traffic-control environment, the seven-dimensiona
with and denoting positidenoting velocities in the x, y and z Cartesian coordinates, respectiv
rate; the drift function
, involving seven mutu
standard Brownian motions, accounts for unpredictable modeling lence, wind force, etc.
5. I. Arasaratnam, s. Haykin, and T. Hurd, Cubature Filtering for Continuous Systems: Theory with an Application to Tracking, submitted to IEEE Trans
dx t( ) f x t( )( )dt Qdβ t( )+=
x ε ε̇ η η̇ ζ ,ζ̇ ω, , , , ,[ ]T
= ε,η ζ
f x( ) ε̇ ωη̇–( ) η̇ ω ε̇ ζ̇ 0, , , , , ,[=
β t( ) β1 t( ) β2 t( ) …,β7 t( ), ,[ ] T=
24
. For the experiment
d to measure the
erval of T. Hence, we
.
0m, 0ms-1, ω deg/s]T.
σr2,σθ
2] )
θ 0.1deg;=
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
The gain matrix
at hand, a radar was located at the origin and digitally equippe
range, r, and azimuth angle, , at a measurement sampling intwrite the measurement equation:
where the measurement noise wk ~ N (0,R) with
Data.
and the true initial state x0 = [1000m, 0ms-1, 2650m, 150ms-1, 20
Q diag 0 σ12,0 σ1
2,0 σ12,0,σ2
2,,,[ ]( )=
θ
rk
θk εk
2 ηk2 ζk
2+ +
ηk
εk------
1–tan
wk+=
R diag [(=
σ1 0.2; σ2 7 10 3–;× σr 50m; σ= = =
25
F)
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
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 CK
26
of the CKF are
a
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Matlab Codes
The Matlab codes for the discrete-time versionavailable on the
Website: http://soma.mcmaster.c
27
Kalman filteran-like filteringisual cortex.
ts computation,
ered in neuralevisit the Rao- EKF.
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
The Visual Cortex Revisited:
• In Rao and Ballard (1997), the extended(EKF) was used to demonstrate that Kalm(i.e., predictive coding) is performed in the v
• The EKF relies on differentiation for iwhereas the CKF relies on integration.
• Since integration is commonly encountcomputations, it would be revealing to rBallard model using the CKF in place of the
28
less of its kind, weservables about theble, and exploit the
te-estimation error
ocessing, entropy ofmation delivered to
alf the logarithm ofconstant term.
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
6. Feedback Information
(i) Principle of Information PreservationIn designing an information-processing system, regardshould strive to preserve the information content of obstate of the environment as far as computationally feasiavailable information in the most cost-effective manner.
(ii) ComputationAt the receiver, the CKF computes the predicted stavector.
With information preservation as the goal of cognitive prthis error vector is the natural measure of feedback inforthe transmitter by the receiver.
For Gaussian error vectors, the entropy is equal to one-hdeterminent of the error covariance matrix, except for a
29
Optimal6
tion.forms with varying
environment that isramming algorithmransmitter updatesthe entropy of the
005); and vol. 2(2007),
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
7. Dynamic Programming and
Control of the Environment
Design of the transmitter builds on two basic ideas:
(i) Bellman’s dynamic programming and its approxima(ii) Library of linear frequency-modulated (LFM) wave
slopes, both positive and negative.
Given the feedback information about the state of thedelivered by the receiver, an approximate dynamic prog(e.g., Q-learning, least squares policy iteration) in the tselection of the current LFM waveform so as to reducepredicted state-error vector.
6. Dimitri Bertsekas, “Dynamic Programming and Optimal Control”, Vol. 1 (2Athena Scientific.
30
e
o updates:
e the radar environ-itter by adapting it.
he receiver so as toed as the cost-to-go
rth, until the radar
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
8. Summarizing Remarks on thCognitive Process
Each cycle of the cognitive process in radar consists of tw
(i) Transmitted waveform-update.By delivering feedback information about state pf thment, the receiver reinforces the action of the transmto update selection of the transmitted LFM waveform
(ii) Feedback information-update.The transmitter, in turn, reinforces the action of tupdate the entropy of the feedback information, viewfunction of the dynamic programming algorithm.
This cycle of joint-reinforcement continues, back and foachieves its ultimate target objective.
31
Radar fortive Process
s change as the
E of velocity
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
9. Experiment on Cognitive TrackingDemonstrating the Power of Cogni
Object Falling in Space, where the dynamicobject reenters the atmosphere
Figure 5: RMSFigure 6: RMSE of altitude
32
tive Radio for thes presented in thisin designing newnition exemplified
tilization of the
f biomedical and
proved accuracy,
n and integration
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
10. Final RemarksEmboldened by my extensive work done on Cognipast four years and the exciting experimental resultlecture on Cognitive Radar, I see breakthroughsgeneration of engineering systems that exploit cogby
• Cognitive radio networks for improved uelectromagnetic spectrum
• Cognitive mobile assistants for a multitude osocial-networking applications
• Cognitive radar systems with significantly imresolution, reliability, and fast response
• Cognitive energy systems for improved utilizatioof different sources of energy
33
ology, which is systems,
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Final Remarks (continued)
Simply stated:
Cognition is a transformative software technapplicable to a multitude of engineering
old and new.
34
tivated by ideasarly, the visual
n brain, it is myms (particularly,ve may well help
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Cognitive Dynamic Systems and Neuroscience
The study of cognitive dynamic systems is modrawn from cognitive neuroscience, particulbrain.
Just as we learn from ideas basic to the humabelief that the study of cognitive dynamic systecognitive radar) from an engineering perspectius understand some aspects of the brain.
35
perceptual taskspter 29 inth Edition, 2009).
Responseorticalotor
rea
}
NIPS Workshops, Whistler, BC, December 2009 (Haykin)Figure 7: Block-diagram representation of processing stages in(Adapted from C. Speidemann, Y. Chen, and W.S. Geisler, chaM.S. Gassaniga, editor-in-chief, The Cognitive Neurosciences, 4
Stimulus
Feedback information
Corticalsensoryarea
Corticalassociationareas
Cma
Encoder Decoder}
36
radar system
eiverResponse
coder}
NIPS Workshops, Whistler, BC, December 2009 (Haykin)Figure 8: Block-diagram representation of processing stages in
TransmitterConvolutionwith targetimpulseresponse
Rec
Unknowntarget
Transmittedwaveform
Radarreturns
De
}Encoder
Feedback Information
37
ownloadable
NIPS Workshops, Whistler, BC, December 2009 (Haykin)
Last Note
The complete set of slides for this lecture is d
from our website:
http://soma.mcmaster.ca