#### 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 =πΈ π₯ π
2
Cross-spectrum
πππ π = πΈ π₯πβ 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₯πβ has a constant phase if π₯π and π₯π have a constant phase
difference. Absolute phase is irrelevant.
β’ Cross-spectrum πΊππ π = πΈ π₯πβ 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:
Welchβs method
β’ Average the squared FFT over multiple windows
β’ Compute π₯πβ 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’ πΈ π₯πβ π1 π₯π π2 = 0 for stationary signal and π1 β  π2 . (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₯πβ π π₯π π .
β’ NOTE: signals canβt 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:
πΆππ π =
πΊππ π
πΊππ π
πΊππ 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
πΊππ π
πππ π =
πΊππ 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 πΊππ π is complex and Hermitian
β’ Complex version of a symmetric matrix
β’ Itβs 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!
```