Wednesday, January 12, 2011

Estimating the Angle of the Fitted Line for the Original Dataset

The moment function is fairly linear near its zero so one can use Newton's Method to estimate the angle of the fitted line. If we let θ represent the angle from the horizontal axis to the line, ε'(θ) the direction of the line and ε(θ) the normal direction for increasing θ then we can compute the moment function and its derivative to find improved estimates of the angle of the line. We started by finding the direction to a point farthest from the center of the distribution of datapoints, then computed the projections of the Δx onto this direction and inverted through the center those with negative projections. This resulted in a new distribution, Δx', whose direction from the original center was e_0. The angle for this direction is the first approximation of the angle of the fitted line, θ_0.

Using Newton's Method gives improved estimates of the angle which rapidly approach the angle of the eigenvector obtained using Mathcad 11's eigenvector function. Since we divide the moment, M, by its derivative, M', we can ignore any multiplicative constants. If we define the moment as the derivative of the variance it will have an additional factor of 2. Using expected values adds a factor of 1/n to both the variance and the moment.

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