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Stochastic Process Theory and Spectral Estimation Bijan Pesaran Center for Neural Science New York University
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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
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Overview Stochastic process theory Spectral estimation
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Stochastic process theory Defining stochastic processes Time translation invariance; Ergodicity Moments (Correlation functions) and spectra Example Gaussian processes
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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
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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
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Challenge of data analysis We can never know the full probability distribution of the data Curse of dimensionality
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Parametric methods Parametric methods infer the PDF by considering a parameterized subspace Employ relatively strong models of underlying process
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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
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Stationarity Stochastic processes don’t exactly repeat themselves They have statistical regularities: Stationarity
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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?
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Gaussian processes Ornstein Uhlenbeck process Weiner process
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Fourier Transform Parseval’s Theorem (Total power is conserved) Real functions:
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Examples of Fourier Transforms Time domain Frequency domain
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Time translation invariance Leads directly to spectral analysis Fourier basis is eigenbasis of
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Implications for second moment If process is stationary, second moment is time translation invariant Hence, for Because
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Stationarity Stationarity means neighboring frequencies are uncorrelated Not true for neighboring times Also due to stationarity, (In general)
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Ornstein Uhlenbeck Process Exponentially decaying correlation function Obtained by passing passing white noise through a ‘leaky’ integrator Spectrum is Lorentzian
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Ornstein Uhlenbeck process
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Markovian process “Future depends on the past given the present” Simplifies joint probability density
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Wiener process
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Cross-spectrum and coherence
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Coherence Coherence measures the linear association between two time series. Cross-spectrum is the Fourier transform of the cross-correlation function
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Coherence Frequency-dependent time delay
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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
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Spectral estimation for continuous processes
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Spectral estimation: Periodogram Bias Variance Nonparametric quadratic estimators: Tapering Multitaper estimates using Slepians Spectrum and coherence
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Example LFP spectrum Periodogram – Single Trial Multitaper estimate - Single Trial, 2NT=10
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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
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What happens if we have a finite sequence of data? Finite sequence means DFT is convolution of and
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Fourier transform of a rectangular window is the Dirichlet kernel: The Fourier transform of a rectangular window Convolution in frequency = product in time
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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.
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Normalized Dirichlet kernel Narrowband bias: Local bias due to central lobe Broadband bias: Bias from distant frequencies due to sidelobes 20% height
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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
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Triangular taper Fejer kernel, for triangular taper, compared with Dirichlet kernel, for rectangular taper. Reduces sidelobes Broadens central lobe
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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]
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Optimal tapers The DFT,, of a finite series, Find series that maximizes energy in a [-W,W] frequency band
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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
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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]
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Power of the kth Slepian function within the bandwidth [-W,W]
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Comparing Slepian functions Systematic trade-off between narrowband and broadband bias
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Advantages of Slepian tapers Using multiple tapers recovers edge of time window 2WT=6
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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
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Cross-spectrum and coherency Cross-spectrum Coherency
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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
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Time-frequency resolution Control resolution in the time-frequency plane using parameters of T and W in Slepians Frequency Time T 2W
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Example LFP spectrograms Multitaper estimate - T = 0.5s, W = 10Hz Multitaper estimate - T = 0.2s, W = 25Hz
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Summary Time series present particular challenges for statistical analysis Spectral analysis is a valuable form of time series analysis
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