Monday, January 17, 2011

Angular Deviation for the Covariance Fit

I was able to estimate the deviation in the angle of the line for the method using the largest eigenvector as the direction of the line in two dimensions. One starts with the quadratic equation for the tangent of the angle of the fitted line and using small angle approximations arrives at a formula in terms of the covariance matrix and its estimated deviations.

One then estimates the components of the covariance matrix and their deviations. ξ_max is the maximum distance of the data points from their center along the line.

ξ is assumed to be the distance of a data point along the fitted line and η is its distance perpendicular to the line. In this coordinate system the covariance matrix is diagonalized. One gets a result which is smaller than that obtained for the direction of the center of the inverted distribution of data points and approximatly what was expected.

Note: The components of the covariance matrix are squared "deviations" and one would expect their values to vary by plus or minum the components of δCov' squared for a set of data fits. The sum of squares suggests that the deviations will add vectorially.

Supplemental (Jan 17): In the estimates deviations were substituted for derivatives. Was this procedure valid? I don't seem to have a good handle on this. In the first fundamental form for lengths we relate squares of differentials. We define ds^2 in terms of dx^2, dy^2, etc. What we really want is ds or ds in terms of dx or dx, etc. Also, for the sets of fits we would expect components of the covariance matrix to vary about some mean value. The negative differences would suggest imaginary quantities for some of the deviations. One needs to be careful about confusing deviations and differentials as one would when dealing with vectors and magnitudes. If one is not precise in one's thinking the results are in doubt.

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