Multivariate time series analysisBijan PesaranCenter for Neural ScienceNew York University

Overview

Singular value decomposition

Application to space-time data

Application to space-frequency data

Periodic stacking method

Multivariate data, Imaging data types

MEG/EEG/LFP fMRI Optical imaging

Need to find spatial projection to reduce dimensionality

Combine spectral and multivariate tools

Singular value decomposition Eigenvalue decomposition Calculates directions/modes in data space that

contain maximum variance

Singular valuespectrum

Spatialmode

Temporalmode

Application to space-time data

Spatial and temporal correlations modes of spatial correlation matrix

modes of temporal correlation matrix

Truncation defines subspace

Noise tailfor a pxq

matrix

fMRI data set with 1877x500data points, sampled at 5 Hz for 10 s.

2

Spectrum of temporal modes

Reveals physiological features across multiple modes

Application to space-frequency data

A geometric interpretation Project time series into a subspace Use an orthogonal basis set Local-in-frequency projection operator

Ttktk WfPXWfX ;;~,

Advantages of local-in-frequencybasis Combine information across this basis

Ensemble averaging

Choose properties of this basis Select time and frequency

Project onto multiple different subspaces centered on different frequencies

Space-frequency decomposition Local-in-frequency projection

Dimensionality reduction

Multivariate Coherence, Assess degree of low-dimensionality

fMRI data set

Complex-valued spatial modes

Spatial segregation of physiological modes.

1st order modes

fMRI example

Presence or absence of visual stimulus Digitization rate: 5 Hz Duration: 110 s Visual stimulation with red LED patterns (8

Hz).

Visual response in fMRI signal

No stimulus - dashed

Visual stimulus - solid

Visual response in fMRI signal

No stimulus

Visual stimulus

Spatially restricted visual response Coronal slice at the occipital pole

Optical imaging example Isolated procerebral lobe of Limax Presence of voltage sensitive dye Digitization rate: 75 Hz Duration: 23 s 600 um by 200 um

Optical imaging response

Limax procerebral lobe during olfactory stimulation

Principal spatial modes

Quantifying traveling waves Express leading spatial mode

2.5 Hz 1.25 and 2.5 Hz

x

y

LFP example Rubino et al. 2007 Electrode arrays in M1 and PMd of awake

monkey Digitization rate: 1 kHz Duraction Visual instructional cue response

Phase gradients in M1 PMd

M1 M1 PMd

Activity between 10 - 45 Hz

Waves reflect anatomical connections

Periodic stacking

If you have repeated measurements of the response to multiple stimuli, you can order your data to take advantage of the multitaper harmonic analysis methods that we have been shown.

1 2 1 2 1 2

Extraction of the average and differential dynamical response in stimulus-locked experimental data. J Neurosci Methods. 2005 Feb 15;141(2):223-9.

Odd harmonics - differences between responsesEven harmonics - average dynamics

Generalization to N stimuli: N’th harmonics are average dynamics, the rest are differences amongst the stimuli