1. Stochastic Process Theory and Spectral Estimation Bijan Pesaran Center for Neural Science New York University

2. Data is modeled as a stochastic process Spikes LFP  Similar considerations for EEG, MEG, ECoG, intracellular membrane potentials, intrinsic and extrinsic optical images, 2-photon line scans and so on

3. Overview  Stochastic process theory  Spectral estimation

4. Stochastic process theory  Defining stochastic processes  Time translation invariance; Ergodicity  Moments (Correlation functions) and spectra  Example Gaussian processes

5. Stochastic processes  Each time series is a realization of a stochastic process  Given a sequence of observations, at times, a stochastic process is characterized by the probability distribution  Akin to rolling a die for each time series Probability distribution for time series  Alternative is deterministic process No stochastic variability

6. Defining stochastic processes  High dimensional random variables Rolling one die picks a point in high dimensional space. Function in N D space.  Indexed families of random variables Roll many dice

7. Challenge of data analysis  We can never know the full probability distribution of the data Curse of dimensionality

8. Parametric methods  Parametric methods infer the PDF by considering a parameterized subspace  Employ relatively strong models of underlying process

9. Non-parametric methods  Non-parametric methods use the observed data to infer statistical properties of the PDF  Employ relatively weak models of the underlying process

10. Stationarity  Stochastic processes don’t exactly repeat themselves  They have statistical regularities: Stationarity

11. Ergodicity  Ensemble averages are equivalent to time averages  Often assumed in experimental work More stringent than stationarity  is not ergodic unless only one constant  Is activity with time-varying constant ergodic?

12. Gaussian processes  Ornstein Uhlenbeck process  Weiner process

13. Fourier Transform  Parseval’s Theorem (Total power is conserved)  Real functions:

14. Examples of Fourier Transforms Time domain Frequency domain

15. Time translation invariance  Leads directly to spectral analysis  Fourier basis is eigenbasis of

16. Implications for second moment  If process is stationary, second moment is time translation invariant  Hence, for  Because

17. Stationarity  Stationarity means neighboring frequencies are uncorrelated  Not true for neighboring times  Also due to stationarity, (In general)

18. Ornstein Uhlenbeck Process  Exponentially decaying correlation function  Obtained by passing passing white noise through a ‘leaky’ integrator  Spectrum is Lorentzian

19. Ornstein Uhlenbeck process

20. Markovian process  “Future depends on the past given the present”  Simplifies joint probability density

21. Wiener process

22. Cross-spectrum and coherence

23. Coherence  Coherence measures the linear association between two time series.  Cross-spectrum is the Fourier transform of the cross-correlation function

24. Coherence  Frequency-dependent time delay

25. Advantages of coherence functions  Neighboring bins are uncorrelated Error bars relatively easy to calculate  Stable statistical estimators  Separate signals together that have different frequencies  Normalized quantities Allow averaging and comparisons

26. Spectral estimation for continuous processes

27.  Spectral estimation: Periodogram Bias Variance  Nonparametric quadratic estimators: Tapering  Multitaper estimates using Slepians Spectrum and coherence

28. Example LFP spectrum Periodogram – Single Trial Multitaper estimate - Single Trial, 2NT=10

29. Spectral estimation problem The Fourier transform requires an infinite sequence of data In reality, we only have finite sequences of data and so we calculate truncated DFT

30. What happens if we have a finite sequence of data? Finite sequence means DFT is convolution of and

31. Fourier transform of a rectangular window  is the Dirichlet kernel: The Fourier transform of a rectangular window  Convolution in frequency = product in time

32. Bias  Bias is the difference between the expected value of an estimator and the true value.  The Dirichlet kernel is not a delta function, therefore the sample estimate is biased and doesn’t equal the true value.

33. Normalized Dirichlet kernel  Narrowband bias: Local bias due to central lobe  Broadband bias: Bias from distant frequencies due to sidelobes 20% height

34. Data tapers  We can do better than multiplying the data by a rectangular kernel.  Choose a function that tapers the data to zero towards the edge of the segment  Many choices of data taper exist: Hanning taper, Hamming taper, triangular taper and so on

35. Triangular taper  Fejer kernel, for triangular taper, compared with Dirichlet kernel, for rectangular taper. Reduces sidelobes Broadens central lobe

36. Spectral concentration problem  Tapering the data reduces sidelobes but broadens the central lobes.  Are there “optimal” tapers?  Find strictly time-localized functions,, whose Fourier transforms are maximally localized on the frequency interval [-W,W]

37. Optimal tapers  The DFT,, of a finite series,  Find series that maximizes energy in a [-W,W] frequency band

38. Discrete Prolate Spheroidal Sequences  Solved by Slepian, Landau and Pollack  Solutions are an orthogonal family of sequences which are solutions to the following eigenvalue functions

39. Slepian functions  Eigenvectors of eigenvalue equation  Orthonormal on [-1/2,1/2]  Orthogonal on [-W,W]  K=2WT-1 eigenvalues are close to 1, the rest are close to 0.  Correspond to 2WT-1 functions within [- W,W]

40.  Power of the kth Slepian function within the bandwidth [-W,W]

42. Comparing Slepian functions  Systematic trade-off between narrowband and broadband bias

43. Advantages of Slepian tapers  Using multiple tapers recovers edge of time window 2WT=6

44. Multitaper spectral estimation  Each data taper provides uncorrelated estimate. Average over them to get spectral estimate.  Treat different trials as additional tapers and average over them as well

45. Cross-spectrum and coherency  Cross-spectrum  Coherency

46. Advantages of multiple tapers  Increasing number of tapers reduces variance of spectral estimators.  Explicitly control trade-off between narrowband bias, broadband bias and variance “Better microscope”  Local frequency basis for analyzing signals

47. Time-frequency resolution  Control resolution in the time-frequency plane using parameters of T and W in Slepians Frequency Time T 2W

48. Example LFP spectrograms Multitaper estimate - T = 0.5s, W = 10Hz Multitaper estimate - T = 0.2s, W = 25Hz

49. Summary  Time series present particular challenges for statistical analysis  Spectral analysis is a valuable form of time series analysis