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Urban Start-up Agglomeration and Venture Capital Investment (view source)
Revision as of 17:55, 29 October 2020
, 17:55, 29 October 2020→Standard Estimators
Thus the trace is the square of the distance between points (for H, between a point within a cluster and the cluster's centroid). The trace of a product is the product of the traces only under specific circumstances and not in general, though we likely meet those circumstances (no zero elements on the diagonal, or no points exactly on the centroids).
=====Standard EstimatorsTest Statistics=====
We'd like to use some measure of variance explained, but variance is now a matrix. So, instead, we should probably turn to a The standard estimator test statistics for MANOVA, which is are the closest equivalents to the direct equivalent. These ANOVA F-statistic from earlier, are:
Wilk's Lambda: <math>\Lambda^* = \frac{\det \mathbf{E}}{\det(\mathbf{H}+\mathbf{E})}</math>
Pillai Trace: <math>V = Tr(H(H+E)^{-1})</math>
So, ... the two trace statistics are very close to what we would get if we calculate used scalar distancesand used either scalar definition of variance explained.The main difference is the lack of correction for degrees of freedom. =====An Opportunity?===== We can't find a decent, let alone seminal, reference for using the elbow method to select the number of clusters. Our problem, which uses geographic coordinates, is also a special case anyway. So, we could implement a method using scalar distance and put a description of it, and its relationship to other measures, in the appendix.It might be a good value-added for the paper.
====The Heuristic Method Justification====
