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

2. Overview
Stochastic process theory – see Appendix
Spectral estimation

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

4. Examples of Fourier Transforms
Time domain
Frequency domain

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

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

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

8. Cross-spectrum and coherence

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

10. Coherence
Frequency-dependent time delay

11. Advantages of spectral estimation
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

12. Estimating spectra Simple spectral estimates: Periodogram
Bias
Variance
Tapering is smoothing your spectrum
Multitaper estimates using Slepians
Spectrum and coherence

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

14. 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

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

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

17. 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.

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

19. 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

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

21. 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]

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

23. 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

24. 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]

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

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

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

29. 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

30. Cross-spectrum and coherency

31. 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

32. Time-frequency resolution2W
Frequency T
Time
Control resolution in the time-frequency plane using parameters of T and W in Slepians

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

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

35. Appendix

36. 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

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

38. 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

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

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

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

42. 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

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

44. 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?

45. Gaussian processes
Ornstein Uhlenbeck process
Weiner process

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

47. Ornstein Uhlenbeck process

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