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Wolpaw Cursor Control Regression Weights

Posted: 22 Jan 2016, 15:55
by MIbciVT

I am currently working on my thesis about motor imagery EEG-based BCI's, and I ran into a bit of a problem trying to understand how the Wadsworth BCI derives the weights for the control signal in cursor control applications.

So in "Emulation of computer mouse control with a noninvasive brain computer interface" they discuss using a linear least-squares criteria and say that feature weights are calculated as b = (X'X)^(-1)X'Y, where X is an m by n matrix formed from the n observations of m predictor variables (i.e., EEG amplitudes at specific frequencies and locations) and Y is the vector of n values (i.e., target predictions) to be predicted.

The actual linear equation controlling (vertical) cursor movement is: delta V = b_V(S_V-a_V) where S_V is just the selected features weighted by the weights obtained from the Least-squares regression, b_V is the gain, and a_V is the mean of the vertical control signal for the user's previous performance.

So my question is whether or not X and Y are augmented with ones to generate an intercept for the regression equation and that value is just tossed out in computing S_V, or if the X and Y are not augmented in which case they force the regression line to pass through the origin of feature space. It just doesn't mathematically make a lot of sense to me if you force the intercept to be 0 when doing regression because your line will not fit the data as well. However, it doesn't make a whole lot of sense to completely throw out the intercept either.

Does anyone have any insight into what I'm not understanding about all of this?

Thank you,

Re: Wolpaw Cursor Control Regression Weights

Posted: 25 Jan 2016, 08:57
by dmcfarland
We didn't mention this in the article, but there was a column of 1s in the X matrix that would represent the intercept. This doesn't matter though as the signal was normalized in real time with respect to the mean and standard deviation.

One other issue- whether the intercept is 0 or not is not a mathematical issue (i.e., either zero or non-zero make mathematical sense). It is an issue of how one wishes to model the data.

Re: Wolpaw Cursor Control Regression Weights

Posted: 26 Jan 2016, 13:57
by MIbciVT
Thank you Dr. McFarland! I just tried it in Matlab, and you do in fact get the same weights whether or not you augment the zero-mean, unit variance matrix X with ones or not.