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    Orthogonal Regression

    In ordinary linear regression, the goal is to minimize the sum of the squared vertical distances between the y data values and the corresponding y values on the fitted line. In orthogonal regression the goal is to minimize the orthogonal (perpendicular) distances from the data points to the fitted line.

    The slope-intercept equation for a line is:

    Y = m*X + b

    where m is the slope and b is the intercept.

    A line perpendicular to this line will have -(1/m) slope, so the equation will be:

    Y' = -X/m + b'

    If this line passes through some data point (X0,Y0), its equation will be:

    Y' = -X/m + (X0/m + Y0)

    The perpendicular line will intersect the fitted line at a point (Xi,Yi) where Xi and Yi are defined by:

    Xi = (X0 + m*Y0 - m*b) / (m^2 + 1)
    Yi = m*Xi + b

    So the orthogonal distance from (X0,X0) to the fitted line is the distance between (X0,Y0) and (Xi,Yi) which is computed as:

    distance = sqrt((X0-Xi)^2 + (Y0-Yi)^2)

    So the goal of the NLREG program is to minimize the sum of these orthogonal distances. Here is a NLREG program that does this:

    Title "Fit a line to data points minimizing orthogonal distances";
    Variables X0, Y0;
    Parameters m, b;
    Double Xi, Yi, distance;
    Xi = (X0 + m*Y0 - m*b) / (m^2 + 1);
    Yi = m*Xi + b;
    distance = sqrt((X0-Xi)^2 + (Y0-Yi)^2);
    Function distance;

    For an example of a NLREG program that performs orthogonal regression to a 3D plane, please click here.

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