Zen, and the Art of Neural Decoding using an EM Algorithm Parameterized Kalman Filter and Gaussian...
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Transcript of Zen, and the Art of Neural Decoding using an EM Algorithm Parameterized Kalman Filter and Gaussian...
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”
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:
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: