EVOKED POTENTIALS

NOISE REDUCTION BY ENSEMBLE AVERAGING

Chapters 4.3.4 (case 2) - 4.3.8

Seppo Mattila (BRU)

Overview

Averaging of Inhomogenious Ensembles

Spike Artifacts and Robust Averaging

The effect of Latency Shifts Estimation of Latency Shifts Weighting of Averaged EPs Using

Ensemble Correlation

Averaging of Inhomogenious Ensembles

Varying noise variance (case 1) Varying signal amplitude but

constant noise variance from potential to potential (case 2)

Varying signal amplitude

assume that signal amplitude differs from potential to potential:

optimal weights from the eigenvalue problem:

where

all eigenvalues equal to zero, except:

and optimla weight vector proportional to corresponding eigenvector:

normalise: wi = ai

Varying signal amplitude IIensemble

weighted

Gaussian noise with varying variance

Weights from maximum likelyhood estimation.The joint PDF of the potentials xi(n) at time n:

We maximise its logarithm:

by setting its derivative wrt s(n) to zero:

Gaussian noise with varying variance II

weighted average of xi(n)

i.e. each potential weighted by

identical to the result from SNR maximisation

moreover,

(ensemble average)

Spike artifacts & robust averaging

ensemble & exponential averaging perform well when Gaussian noise

spike (outlier) artifacts degrade performance

need more robust methods: ensemble averaging with outlier rejection recursive, robust averaging with outlier

rejection

Ensemble averaging with outlier rejectionConsider the generalised Gaussian PDF:

where

the Gamma function

The Laplacian PDF (v = 1) for the noise sample:

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

EastWestNorth

Ensemble averaging with outlier rejection II

ML estimate from

again, by setting its derivative wrt s(n) to zero

choose s(n) such that exactlyhalf of the sample values greaterand half smaller

ML estimator of s(n)MEDIAN when Laplacian noise

Gaussian noise

Laplacian noise

Trimmed means• ensemble average and ensemble median special cases of

where K is the largest integer less than or equal to vMv = 0 for enemble averagev = 0.5 for ensemble median

Recursive, robust averaging with outlier rejection

sign limiter hard limiter

closely related to exponential average but has an updated partmodified by the influence function

influence functions

Effect of latency shifts• variations in latency distortion in ensemble average• shifts in continuous-time signals

• caused by biological mechanisms• not constrained to sampling time grid

• shifts in discrete-time signals• variations taking place in sampled signal

ensemble average

Shifts in continuous time signals

for the expected value of ensemble average:

The convolution integral can be expressed as a product in the frequency domain:

Zero-mean Gaussian PDF is an example of the characteristic function:

Shifts in continuous time signals

Latency shifts can act as a lowpass filter on s(t)

sampling intervalGaussian sigma

-3 dB cut-off frequency associated with the low-passfiltering effect due to latency shifts

Estimation of latency shifts

need to find the shift in each individual potential

compute latency corrected ensamble average

Woody method most well-known estimates individual shifts estimates latency corrected ensemble average interative procedure for improving the estimates

Woody method

EP affected by latency shift:

PDF of observed signal:

ML estimate from:

best cross-correlation betweens(n) and xi(n)

update iteratively the ensemble average:

Weighting of averaged EPs using Ensemble correlation

Weight the individual samplessuch that the difference betweens(n) and the average minimised

Sample-by-sample weightedensemble average: