# Multivariate Auto-Regressive Modeling

## Modeling strategy: vector auto-regressive analysis

Unlike
the pure data-driven approach adopted in 3dGC.R in which the
analysis focuses only on the connection of a target voxel with a seed region
in a bivariate fashion similar to simple correlation or
context-dependent correlation (aka PPI) analysis, we take the approach
of vector (or multivariate) auto-regressive (VAR or MAR) analysis,
similar to structural equation modeling (SEM), by considering a number
of pre-selected regions. But unlike SEM, the primary causality analysis
is done with all the regions' time series at individual subject level.

The modeling approach is a mixture. It is model-based in the sense that we start with a number of pre-selected regions and the between-regions relationship is assumed to be linear. But the analysis is also exploratory in the sense all the path connectivities statistically tested in a data-driven fashion. The pitfall of this approach is that any missing regions that should be in the network but aren't included in the model could ruin the whole analysis and end up with spurious connectivities.

For more discussion, see*G. Chen, et al., Vector autoregression, structural equation modeling, and their synthesis in neuroimaging data analysis, Comput. Biol. Med. (2011), doi:10.1016/j.compbiomed.2011.09.004*

## Program 1dGC

1dGC, written in R, can be run on all major platforms such as unix-based systems and Windows, and requires R installation:

Choose a mirror site
geographically close to you, and download the appropriate binary for
your platform (or the source code and then compile yourself). Set your
path appropriately. For example, my R executable is under
/Applications/R.app/Contents/MacOS on my Mac OS X, so I add
/Applications/R.app/Contents/MacOS as one of the search paths in my C
shell startup configuration file .cshrc

If the installation is successful, start the R interface with the following command on the prompt

R

You can also work with the GUI version of R on Mac OS and Windows.

1dGC.R should be already in the most recent version of AFNI package (say, under ~/abin/). You can also download it from here, and place it wherever you prefer.

If you know the directory (e.g., ~/abin/) 1dGC.R is in, launch it inside R by typing/copying, for example,

source("~/abin/1dGC.R")

If you are not sure about the location of 1dGC.R, copy the following into R (assuming 1dGC.R is under the same directory as the AFNI graphic viewer):

source(paste(strsplit(system('which afni', intern=TRUE), "afni")[[1]], "1dGC.R", sep=""))

or (assuming 1dGC.R is on one of the search paths)

source(Find(file.exists, file.path(strsplit(system("echo $PATH", intern=TRUE), "\\:")[[1]], "1dGC.R")))

1dGC.R works in a procedural or streamlined fashion with a string of inputs about parameters and options, and can run analysis at both individual subject and group levels. Hopefully anything else should be self-evident from there.

In case the program chokes because of failure on installing packages such as vars, network, tcltk, etc. for some reason, run the following commands in R:

install.packages("vars",dependencies=TRUE)

install.packages("network",dependencies=TRUE)

install.packages("gsl",dependencies=TRUE)

install.packages("car",dependencies=TRUE)

To quit R, type

q()

(or hit letter "d" while holding down CTRL key on UNIX-based systems).

## Input data

Some
suggestions about input files below might be too specific for FMRI
data. Make proper adjustment if the program is used under other
circumstances.

(1) All input data are assumed having a suffix of
.1D in the AFNI convention, and they are typically time series at
regions of interest (ROIs) and from covariates such as conditions/tasks
of no interest, head motion and physiological noises). They should be
structured in a pure text format of either multiple one-column files or
one multi-column file.

(2) Header is NOT allowed in a
one-column input file, but is optional for a multi-column input file:
If provided in the first row, it has to be the labels of those
ROIs/nodes, as the format of data frame in R. ROI time series as input
are required, but covariates are optional. All ROI time series can be
stored in one data frame, or multiple one-column files, but not a
mixture.

(3) The minimum pre-processing steps for the ROI time
series include slice timing correction and volume registration. Spatial
smoothing is typically recommended for noise reduction, but not
mandatory. It is the change relative to the baseline that is comparable
across blocks/runs, regions, and subjects, therefore signal
normalization through scaling in terms of the loose concept of percent
signal change is very important, and it can be done during the
pre-processing, or you can leave it for 1dGC.R to handle.

(4)
Any confounding effects are better entered as covariates in the
causality model unless they are orthogonal to the autocorrelation
present in the network, which is rarely true.

(5) Since the low
frequencies (drifting effect including baseline) in the signal can be
modeled with polynomials embedded in the causality model, it's NOT
recommended to remove the trend (including baseline) during the
pre-processing because of argument (4) above. You don't have to include
the polynomial time series from the design matrix as covariates for
input, but if you do decide to include them, disable the embedded
option in the program by specifying an order of -1 for polynomials in
the program.

(6) Covariates should be in separate file(s) from
ROI time series files, and can be stripped from the design matrix of
the individual subject regression analysis. All the ROI time series can
be multiple one-column files or one multi-column file, but not a
mixture of both formats. The same is true for covariates, but they
don't have to be of the same format as the ROI files. All time series
from ROI and covariates must have the same length and match up in
temporal sequence. They can be data from multiple blocks and/or runs
stitched together.

(7) If you want to censor out a few time points in the time series, create one covariate for each censored time point with the same length of the time series and with all 0's except putting a 1 at the censored time point.

(8) Input files for group analysis are supposed to be path coefficients (plus corresponding *t*-statistics) saved from analysis at individual subject level.

## Features of 1dGC

Compared to other generic Granger analysis tools, 1dGC is used in time domain with the following features:

(1) written in an open source language and executable on all computer platforms;

(2) allowing breaks in the data;

(3)
extending VAR model with all possible covariates included as part of
the analysis instead of being regressed out prior to the analysis, and
fine-tuning the model by removing covariates deemed insignificant;

(4) providing 4 criteria for VAR order selection: AIC, FPE, HQ, and SC;

(5) outputting one network per lag (based on path coefficients and their t-statistics) instead of lumping all lags into one network (based on overall F-statistics across lags);

(6) diagnosing the model from various perspectives: order selection criteria, stationarity (testing roots of characteristic polynomial), residual normality tests (Gaussian process, skewness, and kurtosis), residual autocorrelation (portmanteau, Breusch-Godfrey, and Edgerton-Shukur tests), autoregressive conditional heteroskedasticity (ARCH) test for time-varying volatility, and structure stability (or stationarity detection);

(7) running group analysis on path coefficients and t-statistics per lag from individual subjects instead of overall F values across all lags.

## Useful links

1. Vector auto-regressive modeling

## Acknowledgements

I'd like to thank Patrick Brandt and Chris Sims for theoretical consultation, Bernhard Pfaff for programming support, Jim Bjork,** **Paul
Hamilton, Jonathan Omuircheartaigh, Kai Hwang, and Jeremy I. Skipper
for help in testing the program and for providing feedback.

Last modified 2011-10-08 17:29