Summer Camp for Social Scientists! Math Edition

After being here for seven weeks, things have gotten a little strange. Not being a particularly mathematically inclined person, I’ve gradually taken to preferring to reading information in mathematical form, because it reduces the amount of information I have to process. Although it’s fairly clear that I will not be joining the Canadian Mathematical Society anytime soon, almost by magic I’ve started to deeply enjoy the utility and intuition of math.

Today was my last day of seven weeks of math classes, much of it focused on matrix algebra. The second session matrix algebra class turned out to be one of my favourite classes here at the ICPSR summer program, due in large part to the accessibility of the material. Professor Pedro Sanchez spent a week going over the singular value decomposition (SVD) with the class. SVD underlies and is connected to many statistical methods, such as principle components analysis (PCA) and factor analysis (FA). It has been especially interesting for me, because I’m learning about SVD in a math class while also learning about PCA and FA in a stats class. To simplify: singular values are the positive square roots of the eigenvalues of a matrix A. To find the singular values of a matrix A, you multiply it by unitary matrices U’ and V so that you achieve a diagonal matrix D, with singular values on the diagonal.

I thought I’d share a few resources here about SVD. MIT open courseware has an excellent lecture on SVD that I found really helpful. In terms of application, this article by Sirovitch (2003) explains in a very clear and ingenious way how SVD can be used to provide information on the dimensionality in data matrices. The article looks at the decisions of the Renhquist Supreme Court in the USA, and finds that contrary to the conventional thinking that there are nine dimensions of the court, there turns out to be two dimensions that court decisions are aligned on.

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