Week 15: Multiple Timeseries

Download Report

Transcript Week 15: Multiple Timeseries

Spectral analysis of multiple
Kenneth D. Harris
Continuous processes
β€’ A continuous process defines a probability distribution over the space
of possible signals
π‘₯ 𝑑
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
𝐱 𝑑
Sample space =
all possible multiple signals
Probability density 0.00000343534976
Power spectrum
𝑃 𝑓 =𝐸 π‘₯ 𝑓
𝑃𝑖𝑗 𝑓 = 𝐸 π‘₯π‘–βˆ— 𝑓 π‘₯𝑗 𝑓
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‘₯𝑖 π‘₯π‘—βˆ— has a constant phase if π‘₯𝑖 and π‘₯𝑗 have a constant phase
difference. Absolute phase is irrelevant.
β€’ Cross-spectrum 𝐺𝑖𝑗 𝑓 = 𝐸 π‘₯π‘–βˆ— 𝑓 π‘₯𝑗 𝑓
phase difference, and high power.
is large when constant
Cross spectrum estimation
β€’ Need to average π‘₯ 𝑓
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:
Welch’s method
β€’ Average the squared FFT over multiple windows
β€’ Compute π‘₯π‘–βˆ— 𝑓 π‘₯𝑗 𝑓 of tapered signal in each window. Average over windows
β€’ Arbitrary window start – will change absolute phase but not phase differences.
β€’ 𝐸 π‘₯π‘–βˆ— 𝑓1 π‘₯𝑗 𝑓2 = 0 for stationary signal and 𝑓1 β‰  𝑓2 . (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
β€’ Multiply both signals by taper, then
compute π‘₯π‘–βˆ— 𝑓 π‘₯𝑗 𝑓 .
β€’ NOTE: signals can’t be too short! You need
several cycles of an oscillation to even talk
about a constant phase relationship…
β€’ Maximum value of cross-spectrum occurs with constant phase relationship
Coherence is cross-spectrum divided by RMS of individual spectra:
𝐢𝑖𝑗 𝑓 =
𝐺𝑖𝑗 𝑓
𝐺𝑖𝑖 𝑓
𝐺𝑗𝑗 𝑓
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
𝐺𝑖𝑗 𝑓
𝑇𝑖𝑗 𝑓 =
𝐺𝑖𝑖 𝑓
β€’ Measures how much you should multiply signal 𝑖 to get signal 𝑗.
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 𝐺𝑖𝑗 𝑓 is complex and Hermitian
β€’ Complex version of a symmetric matrix
β€’ It’s transpose is the same as its complex conjugate
β€’ All eigenvalues are real
Eigenvector of cross-spectral matrix
β€’ No need for a seed pixel
β€’ Shows how wave
propagates across cortex
β€’ Computed using SVD first!