#### Transcript Week 15: Multiple Timeseries

Spectral analysis of multiple timeseries Kenneth D. Harris 18/2/15 Continuous processes β’ A continuous process defines a probability distribution over the space of possible signals π₯ π‘ Probability density 0.000343534976 Sample space = all possible LFP signals Multivariate continuous processes β’ A continuous process defines a probability distribution over the space of possible signals π± π‘ Sample space = all possible multiple signals Probability density 0.00000343534976 Power spectrum π π =πΈ π₯ π 2 Cross-spectrum πππ π = πΈ π₯πβ π π₯π π Fourier transform: amplitude and phase Constant phase relationship? Complex conjugate β’ Multiplication: phases add β’ Conjugation: flips the phasor upside down (negative of phase) β’ π₯π π₯πβ has a constant phase if π₯π and π₯π have a constant phase difference. Absolute phase is irrelevant. β’ Cross-spectrum πΊππ π = πΈ π₯πβ π π₯π π phase difference, and high power. is large when constant Cross spectrum estimation β’ Need to average π₯ π 2 to reduce estimation error β’ If you observe multiple instantiations of the data, average over them β’ E.g. multiple trials β’ Otherwise, same methods as for power spectrum: Welchβs method β’ Average the squared FFT over multiple windows β’ Compute π₯πβ π π₯π π of tapered signal in each window. Average over windows β’ Arbitrary window start β will change absolute phase but not phase differences. β’ πΈ π₯πβ π1 π₯π π2 = 0 for stationary signal and π1 β π2 . (Why?) Multi-taper method β’ Only one window, but average over different taper shapes β’ Use when you have short signals β’ Taper shapes chosen to have fixed bandwidth β’ Multiply both signals by taper, then compute π₯πβ π π₯π π . β’ NOTE: signals canβt be too short! You need several cycles of an oscillation to even talk about a constant phase relationshipβ¦ Coherence β’ Maximum value of cross-spectrum occurs with constant phase relationship Coherence is cross-spectrum divided by RMS of individual spectra: πΆππ π = πΊππ π πΊππ π πΊππ π A complex number: β’ Magnitude between 0 (independent phases) and 1 (constant phase difference). Note phases do not have to be equal! β’ Argument is mean phase difference. Transfer function β’ Transfer function πΊππ π πππ π = πΊππ π β’ Measures how much you should multiply signal π to get signal π. β’ Can Fourier transform to estimate a linear filter. Seizure over visual cortex Federico Rossi Cross-spectrum with seed pixel Coherence magnitude with seed pixel Cross-spectral matrix β’ Cross-spectral matrix πΊππ π is complex and Hermitian β’ Complex version of a symmetric matrix β’ Itβs transpose is the same as its complex conjugate β’ All eigenvalues are real st 1 Eigenvector of cross-spectral matrix β’ No need for a seed pixel β’ Shows how wave propagates across cortex β’ Computed using SVD first!