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Zen, and the Art of Neural Decoding using an EM Algorithm Parameterized Kalman Filter and Gaussian Spatial Smoothing. Michael Prerau, MS. Encoding/Decoding Process. Generate a smoothed Gaussian white noise stimulus - PowerPoint PPT Presentation
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Zen, and the Art of Neural Decoding using an EM Algorithm
Parameterized Kalman Filter and Gaussian Spatial Smoothing
Michael Prerau, MS
Encoding/Decoding Process
Generate a smoothed Gaussian white noise stimulus
Generate a random kernel, D and convolve with the stimulus to generate a spike rate
Drive Poisson spike generator Decode and find K
Use K to decode from new stimuli “real time”
( )( )
( )rsQK
Q
%
%%
rate stim D
Encoding/Decoding Process
Gaussian NoiseStimulus
Random D
Cell Matrix
Poisson SpikeGenerator
Calculate Kernel K
Encode Decode
Kernel K
Cell Matrix
Poisson SpikeGenerator
Stimulus Output
Decoding “Real Time”
Encoding/Decoding
Stimulus
Decoded Estimate
State-Space Modeling
( , )kS x yHidden State: Where sputnik really is ( , )x y
( , )k x yO o oObservations: What the towers see
1( )k k sS PHYSICS S
State equation: How sputnik ideally moves
k k oO S
Observation equation: If we knew where sputnik was, how would that relate to our observations?
{ , , }s o Parameters:
State-Space Modeling
Observations
State estimate
The Kalman Filter
Gaussian state The actual stimulus intensity
Gaussian observations The filtered estimate
1k k kx Ax w State Equation
k k kz Hx Observation Equation
ˆ ˆ ˆ( )k k k kx x K z Hx State Estimate
State Equation: Random Walk AR Model
Observation Equation: Linear Model
Parameters
The Kalman Filter Application to the Intensity Estimate
1k k kx x 2(0, )k N where
k k kz x 2(0, )k N where
2 2( , , , )v
Complete Data Likelihood
Log-likelihood
12
12
2 2 1 2
1
2 2 1 21
1
( ) (2 ) exp{( 2 ) ( ) }
(2 ) exp{( 2 ) ( ) }
K
k kk
K
k kk
p Z x z x
x x
221
0 2 2| 1
( ) ( )1 1log( ( | ))
2 2k k k k
k kk k
x x z xp x Z
The Kalman Filter Application to the Intensity Estimate
Forward Filter Derivation
221
0 2 2| 1
( ) ( )1 1log( ( | ))
2 2k k k k
k kk k
x x z xp x Z
Most likely hidden state will maximize log-likelihood:
0 12 2| 1
log( ( | )) ( )k k k k k k
k k k
p x R x x z x
x
22| 1
12 2 2 2 2 2| 1 | 1
ˆ k kk k k
k k k k
x x z
Maximize for xk and solve:
2| 1
1 12 2 2| 1
( )ˆk k
k k k kk k
x z xx
Arrange Kalman style:
For hidden state variance, first take the 2nd derivative of the log likelihood:
Then take the negative of the inverse for the variance of the hidden state:
2 20
2 2 2| 1
log( ( | )) 1k k
k k k
p x Z
x
2 2| 12
2 2 2| 1
ˆ k kk
k k
Forward Filter Derivation
The EM Algorithm
Suppose we don’t know the parameter values? Use the Expectation Maximization (EM)
Algorithm (Dempster, Laird, and Rubin, 1977) Iterative maximization
E-step: Take the most likely (Expected value) value of the state process given the parameters
M-step: Maximize for the most likely parameters given the estimated state values
E-Step for Intensity Model
( )
22 ( )2
1
22 ( )12
1
log ( ) ||
1 1log(2 ) ( ) ||
2 2
1 1log(2 ) ( ) ||
2 2
K
k kk
K
k kk
E p Z x Z
E K z x Z
E K x x Z
l
l
l
( )
2 ( )
( )11
||
||
||
k K k
kk K
k kk k K
x E x Z
W E x Z
W E x x Z
l
l
l
Take the expected value of the joint likelihood:
We will encounter terms such as:
Can be solved with the state-space covariance algorithm (De Jong and MacKinnon, 1988)
Example :
M-Step for Intensity Model
For the M-Step, maximize with respect to each parameter.
Set equal to zero and solve
2
222 2
1
2 2 ( 1) 2( 1) 22 2
1 1 1
( 1) 2( 12 2|2 2
1 1 1
1 1log(2 ) ( )
2 2
1 1{ [ log(2 ) [ 2 ]}
2 2
1 1{ log(2 ) [ 22 2
K
k kk
K K K
k k k kk k k
K K K
k k K kk k k
E K z x
E K z x z x
K z x z
l l
l l )|
2 ( 1) 2( 1)| |22 2
1 1 1
]}
1[ 2 ]
2 2( )
k K
K K K
k k K k k Kk k k
W
Kz x z W
l l
2 ( 1) 2( 1)| |22 2
1 1 1
2 ( 1) 2( 1)2( 1) 1|
1 1 1
10 [ 2 ]
2 2( )
2
K K K
k k K k k Kk k k
K K K
k k K k k Kk k k
Kz x z W
K z x z W
l l
l ll
M-Step for Intensity Model
1
1 11 1
K K
k k K k Kk k
W W
22 11 1
1
2K
k K k k K k Kk
K W W W
1
( 1)|
11
KK
k K kk Kkk
x zW
l
2 ( 1) 2( 1)2( 1) 1|
1 1 1
2K K K
k k K k k Kk k k
K z x z W
l ll
M-Step Summary:
The EM Algorithm
The EM Algorithm
Kalman Estimate
convolution =
2D Gaussian Spatial Smoothing
Gaussian Spatially Smoothed Estimate
Kalman Filtering the Gaussian Smoothed Estimate
Kalman Filtering the Gaussian Smoothed Estimate
Comparison
Comparison
Comparison
Stimulus
Sest Kalman GaussianSmoothed
SmoothedKalman
fin