1. EVOKED POTENTIALS NOISE REDUCTION BY ENSEMBLE AVERAGING Chapters 4.3.4 (case 2) - 4.3.8 Seppo Mattila (BRU)

2. 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

3. Averaging of Inhomogenious Ensembles Varying noise variance (case 1) Varying signal amplitude but constant noise variance from potential to potential (case 2)

4. 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: w i = a i

5. Varying signal amplitude II ensemble weighted

6. Gaussian noise with varying variance Weights from maximum likelyhood estimation. The joint PDF of the potentials x i (n) at time n: We maximise its logarithm: by setting its derivative wrt s(n) to zero:

7. Gaussian noise with varying variance II weighted average of x i (n) i.e. each potential weighted by identical to the result from SNR maximisation moreover, (ensemble average)

8. 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

9. Ensemble averaging with outlier rejection Consider the generalised Gaussian PDF: where the Gamma function The Laplacian PDF (v = 1) for the noise sample:

10. Ensemble averaging with outlier rejection II ML estimate from again, by setting its derivative wrt s(n) to zero choose s(n) such that exactly half of the sample values greater and half smaller ML estimator of s(n) MEDIAN when Laplacian noise Gaussian noise Laplacian noise

11. Trimmed means ensemble average and ensemble median special cases of where K is the largest integer less than or equal to vM v = 0 for enemble average v = 0.5 for ensemble median

12. Recursive, robust averaging with outlier rejection sign limiterhard limiter closely related to exponential average but has an updated part modified by the influence function influence functions

13. 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

14. Shifts in continuous time signals forthe 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:

15. 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-pass filtering effect due to latency shifts

16. 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

17. Woody method EP affected by latency shift: PDF of observed signal: ML estimate from: best cross-correlation between s(n) and x i (n) update iteratively the ensemble average:

19. Weighting of averaged EPs using Ensemble correlation Weight the individual samples such that the difference between s(n) and the average minimised Sample-by-sample weighted ensemble average: