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Level III

## Is there a good way to score linear dependence

I have an iterative calibration process that is searching for an optimal point in 24-Dimensional space.  The process is well behaved and has guaranteed convergence (in the sense of converging to a random vector with a fixed distribution).  Typically the process achieves strong correlation between successive samples after about 3 iterations.  By strong I mean the correlation quickly reaches values above .98 or so.  I'm looking for a stopping rule that tracks a more sensitive statistic to say when we have reached maximum linear "correlation".  At that point I can stop the process and compare the instrument performance against its requirements with a totally different test (not a practical way to calibrate).  I recall that caution is in order when trying to compare measures of linear association (like correlation) to say one pair of vectors are more linearly dependent than another pair.  This is especially so when all pairs are very close to 1 in correlation.

I did some JMP shopping and looked at the multivariate non-parametric methods.  I found it very interesting that the Hoeffding's D method does a very good job at spreading out the ranks when compared to correlation or any of the other non--parametric methods.

My more general question is what method (statistic) is best at scoring the strength of linear dependence and not simply assessing the significance of the relationship and is using the relative values of this statistic legal to compare different pairs of vectors.  When I read about Hoeffding's D method it does say it is the best at detecting linear dependence against all alternatives.  So maybe it is a good way to know when two successive samples of a random vector are close enough to stop this process.

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