Topics in Brain Computer Interfaces CS295-7

Transcription

1 Topics in Bain Compute Intefaces CS295-7 ofesso: MICHAEL BLACK TA: FRANK WOOD Sping 25 Automated Spie Soting Fan Wood - fwood@cs.bown.edu

2 Today aticle Filte Homewo Discussion and Review Kalman Filte Review CA Intoduction EM Review Spie Soting Fan Wood - fwood@cs.bown.edu

3 aticle Filteing Movies Fan Wood - fwood@cs.bown.edu

4 Homewo Results? Bette than CC X.5, CC Y.8? How? What state estimato did you use (ML/E[])? Why? When did you estimate the state? aticle e-sampling schedule? Remaining questions? Initial state estimate? How did the homewo synthesie with the lectue notes and eadings? Fan Wood - fwood@cs.bown.edu

5 Fan Wood - fwood@cs.bown.edu Viewing the Bayesian Recusion afte implementing aticle Filteing ( ) ( ) ( ) ( ), and the model and given ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) 2 2,, ( ) ( ) ( ) ( ) ,,,, M Stat hee with paticles epesenting the posteio. Stat hee with paticles epesenting the posteio. System model System model Obsevation model Obsevation model

6 Net Homewo: The Kalman Filte Closed fom solution to ecusive Bayesian estimation whee the obsevation and state models ae linea + Gaussian noise. Seminal pape published in 96: R.E. Kalman, A New Appoach to Linea Filteing and ediction oblems Obsevation model H + q q ~ N(, Q) State model + A w w ~ N(, W ) Fan Wood - fwood@cs.bown.edu

7 Fan Wood - fwood@cs.bown.edu and ˆ estimate of Initial - Time Update Measuement Update Welch and Bishop 22 io estimate Eo covaiance osteio estimate Kalman gain Eo covaiance W A A A T + ˆ ˆ ) ( ) ( ) ˆ ( ˆ ˆ + + Q H H H K H K I H K T T v The Kalman Filte Algoithm

8 Whee do these equations come fom? Find an unbiased minimum vaiance estimato of the state at time + of the fom ˆ ˆ + K + + K + + We ll We ll loo loo at at this this today. today. ˆ + Fo to be unbiased means: E [ ] ˆ + + This This bit bit is is much much ticie. ticie. A lin lin to to a full full deivation is is on on the the web. web. Ecepted and modified fom aticouses.com Fan Wood - fwood@cs.bown.edu

9 ½ Unbiased Estimate E E Remembe fom the pevious slide [ K ˆ + + K ] Tic alet! [ K ˆ ( ) + + K + H + + q + + K + + K + ] [ ( ˆ ) ( ( ) ) ( ) ] + + K+ H A + w + + q + A + w + + K+ [ ( ) + ( K HA A+ K ) + ( K H I) w + K q ] E K E K ˆ ( K HA A+ K ) E[ ] + + K+ HA A+ K+ o K ( + I K+ H)A E ˆ + + ˆ ˆ + K + + K + + [ ] Fan Wood - fwood@cs.bown.edu Ecepted and modified fom aticouses.com

10 ulling it togethe (a bit) ˆ ˆ + K + + K + + ( I K ) + H A K + ˆ ˆ ˆ Aˆ + + K + + Can Can get get the the Kalman gain gain by by minimiing the the vaiance of of the the estimation eo. eo. ( HAˆ ) Fan Wood - fwood@cs.bown.edu Remembe fom the pevious slide K osteio estimate ˆ ˆ + Eo covaiance ( I Kalman gain K H ( I K H)A + + K K T v ( H ) ( H Hˆ H T ) + Q) Ecepted and modified fom aticouses.com

11 Fan Wood - fwood@cs.bown.edu and ˆ estimate of Initial - Time Update Measuement Update Welch and Bishop 22 io estimate Eo covaiance osteio estimate Kalman gain Eo covaiance W A A A T + ˆ ˆ ) ( ) ( ) ˆ ( ˆ ˆ + + Q H H H K H K I H K T T v The Kalman Filte Algoithm

12 Good time fo a Bea Changing geas to CA/EM/Mitue Modeling Fan Wood - fwood@cs.bown.edu

13 incipal Component Analysis (CA) The cental idea of [CA] is to educe the dimensionality of a data set consisting of a lage numbe of inteelated vaiables, while etaining as much as possible of the vaiation pesent in the pincipal components (Cs), which ae uncoelated, and which ae odeed so that the fist few etain most of the vaiation pesent in all of the oiginal vaiables., I.T. Joliffe Eample applications Compession Noise Reduction Dimensionality Reduction Eigenfaces, etc. Fan Wood - fwood@cs.bown.edu

14 The Gist of CA 8 Gaussian cloud num_points 5; angle pi/4; vaiances [5 ;.5]' otation [cos(angle) -sin(angle); sin(angle) cos(angle)] data otation*(vaiances*andn(2,)); [pcadata,eigenvectos,eigenvalues] pca(data,2); ecoveed_otation eigenvectos ecoveed_vaiances sqt(eigenvalues) vaiances 5..5 otation ecoveed_otation ecoveed_vaiances Histogam of data pojected onto fist C Fan Wood - fwood@cs.bown.edu

15 The Math of CA Fist step: Find a linea function (a pojection) of a R.V. that has maimum vaiance. i.e. p T α α + α + L+ α th Second though step: Find the subsequent uncoelated pojection with maimum vaiance etc. i.e. α 2, α3, K, α Continue until enough vaiance is accounted fo o up to, the dimensionality of. 2 2 p p α j j j Fan Wood - fwood@cs.bown.edu

16 Finding a pincipal component (C) Maimie the vaiance of the pojection: ag ma E [ ( ) T ] T T α α Easy to do! Set α Solution: constain T α α α ag ma E α T ag maα Σα α [ ] T T α α Fan Wood - fwood@cs.bown.edu

17 Constained Optimiation Use Lagange multiplie and diffeentiate: α ( T ( T α Σα λ α α ) Σα λα ( Σ Ι ) α λ o Σ α λ α α Σ λ So is an eigenvecto of and is the coesponding eigenvalue. Fan Wood - fwood@cs.bown.edu

18 Optimal opeties of C s The second, thid, etc. C s can be found using a simila deivation subject of couse to additional constaints. It can be shown that a choosing B to be the fist q eigenvectos of the covaiance mati Σ of that the othonomal linea tansfomation y B maimies the covaiance of y. Fan Wood - fwood@cs.bown.edu

19 The Gist of CA 8 Gaussian cloud num_points 5; angle pi/4; vaiances [5 ;.5]' otation [cos(angle) -sin(angle); sin(angle) cos(angle)] data otation*(vaiances*andn(2,)); [pcadata,eigenvectos,eigenvalues] pca(data,2); ecoveed_otation eigenvectos ecoveed_vaiances sqt(eigenvalues) vaiances 5..5 otation ecoveed_otation ecoveed_vaiances Histogam of data pojected onto fist C Fan Wood - fwood@cs.bown.edu

20 EM fo Gaussian Mitue Models Epectation Maimiation is a ecusive method fo estimating the paametes of data distibutions with missing o unobseved data. In ou case, the missing data is data class membeships. obability obability of of i assuming i assuming log( L( θ Χ, Υ)) log( ( Χ, Υ θ )) Shothand Shothand fo fo the the pio pio pobability pobability of of the the class class labeled labeled y y i i ( y ) p( θ ) This epesents a geneative view with latent stuctue. that that it it came came fom fom that that class. class. n log i p i i y i Fan Wood - fwood@cs.bown.edu

21 Fan Wood - fwood@cs.bown.edu Closed fom E & M steps fo GMM ( ) Θ N i g i new l l p N, α ( ) ( ) Θ Θ N i g i N i g i i new l l p l p,, µ ( )( ) ( ) ( ) Θ Θ Σ N i g i N i new l i new l i g i new l l p l p T,, µ µ Fom A Gentle Tutoial of the EM Algoithm and its Application to aamete Estimation fo Gaussian Mitue and Hidden Maov Models, Jeff A. Bilmes

22 Eample Application Spie Soting evious Solutions Non-paametic template matching Vaious clusteing's of pinciple components (Lewii) EM on mitues of multivaiate t- distibutions (Shoham, et al) Wavelet pacets (Hulata, et al) oblems Manually detemine wavefom templates Manually detemine numbe of clustes Manually identify noise Wavefom vaiability Inte-spie inteval Off-line vs. on-line Fan Wood - fwood@cs.bown.edu

23 Neual Decoding / ostheses Signal Decode (Kalman filte, linea filte, etc) Detection Spie Soting Rate Estimation multi-unit activity single unit activity EEG, LF, etc. Voluntay contol signal Compute cuso and eyboad enty Robotic am Stimulation of Muscles, Spinal Cod, and Bain Fan Wood - fwood@cs.bown.edu

24 A Slight Untuth has been Implied The data you have is soted hoibly. Most of the 42 channels you have ae multiunits not actually single neuons. It is vitually impossible to isolate and ecod a single neuon with pefect cetainty with any ecoding technology and is even moe difficult to do with an aay due to its andom insetion. Fan Wood - fwood@cs.bown.edu

25 Spie Soting. Ou definition: wavefoms captued at theshold cossings ae soted by deciding : which ae spies how many neuons thee ae which neuons each came fom. Not detection! Results fom Bionic micoelectode aay. Fan Wood - fwood@cs.bown.edu

26 Fan Wood - fwood@cs.bown.edu

27 Off-line Sote Sceen-Captue Fan Wood - fwood@cs.bown.edu

28 Spie Soting s dity little secet. Inspied by Hais et al (2) we conducted a study of spie soting subjectivity. Real data 5 Epet sotes 2 Repesentative channels Subject A B C D E Spies Units Fan Wood - fwood@cs.bown.edu

29 Two people soting the same channel. Subject D Noise: 9484 Unit A: 3474 Subject E Noise: 574 Unit A: 43 Unit B: 3539 Unit C: µ sec. Subject D µ sec. Subject E Seconds Seconds Fan Wood - fwood@cs.bown.edu

30 Ou Goal Bette decoding accuacy by way of impoved spie soting. Bette spie soting fo neuoscience would be geat to achieve as well but is a slightly diffeent goal. Fan Wood - fwood@cs.bown.edu

31 A Geedy Automatic Spie Soting Algoithm Fo evey channel in the ecoding opose,2,3 units on cuent channel. Assign evey channel to have unit and decode using Kalman filte. Recod the MSE of the econstuction. Initialie mitue model with n mitue components by spectal clusteing ala Ng and Jodan (2) Optimie model paametes (EM) Decode using all channels using Kalman filte ala Wu et al (23) Yes Best decoding MSE? No Fan Wood - fwood@cs.bown.edu Fo full details see the pape.

32 Why did we tal about CA? The wavefoms ae lagely simila (even between two diffeent neuons). The intinsic dimensionality of a wavefom is pobably much lowe than the 48 samples we had fo each. Speeds computation consideably and maes estimates of mitue model paametes moe obust. Fan Wood - fwood@cs.bown.edu

33 Automatic Spie Soting Visual Results Wavefoms Noise : 42 Unit A: Unit B: Unit C: Noise : 7 Unit A: 544 Unit B: Unit C: Coesponding 2 lagest CA coefficients. Noise : 75 Unit A: 556 Unit B: 578 Unit C: Noise : 6 Unit A: 387 Unit B: 84 Unit C: Fan Wood - fwood@cs.bown.edu

34 Decoding Results Subject A B C D Ave. Human Random None Auto Ma Auto Weighted Neuons Spies MSE (cm^2).45 +/ / / / / / / / /-.5 Neual Reconstuction Actual Money hand position Ran: Kalman Auto Soted filte, tained No Soting on 3 minutes Randomly of pinball Soted data, Human aveage Soted! esults fo 5 one minute decoding segments. Fan Wood - fwood@cs.bown.edu

35 Conclusions and Discussion This automatic soting algoithm poduces bette spie tains fo neual decoding. Maybe spie soting isn t necessay fo good decoding? Hints at using a diffeent signal instead? Lining decoding to soting may not identify physiological neuons. Net Steps Fully leveage pobabilistic intepetation fo enhanced ate estimation. Diffeent cost function. Etend to continuous signal. Fan Wood - fwood@cs.bown.edu

36 Net Wee Fun! Cay papes thought povoing ethics aticle. Get the Kalman filte assignment out of the way quicly. Given what you now now it should be quite easy! Fan Wood - fwood@cs.bown.edu