Saturday, January 15, 2011

Error Intervals and Formulas

A discussion of linear fits would not be complete without actual values for the error intervals. One would expect the center of the data to be within σ/√n of the true line. One can also estimate the expected deviation of the angle of the fitted line from the true line, Δθ. If ctr' is the center of the inverted distribution then cos(Δθ) = |ctr'| and
sin(Δθ) = σ/√n, approximately·

The values shown are the expected deviations for the original linear fit. One would expect most datasets to have a value for the center within 3σ/√n of the line and the direction to be within 3Δθ of the direction of the true line. The direction assumed was that of the center of the inverted distribution of data. The correction using the covariance matrix may be a little better but its formula is much more complicated.

Supplemental (Jan 15): The two centers have the same uncertainty, σ/√n, which would double the uncertainty in the direction of the line to 2Δθ. The correction was about equal to Δθ so the error in the angle of the eigenvector may be approximately Δθ. One could check this numerically. The components of the covariance matrix are estimates which contribute to the uncertainty in the eigenvector. Since the covariance matrix is diagonalized when the horizontal axis is rotated to the line the smallest eigenvalue contributes most to the error. Its uncertainty is σ^2/n so the standard deviation is σ/√n. For small angles the base of a right triangle is approximately equal to its hypotenuse.

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