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 Least Squared Error

 Editorial, notes on derivation

Under Construction

Outline
    fits data points to a user-supplied function.
        eg: quadratic polynomial. coefs unknown.
        data points.
    linear combination.

    wiki refs, history


Example and guidelines ========================================
    The LSE process is often employed when the form of the
    resultant equation is uncertain.  The LSE user has some
    measurements and wishes to know what equation fits the
    data well.  The user can try various forms of equations
    A XX + BX + C, AXXX + C, AXX + C etc.  LSE determines the
    coefficients appearing in the user-chosen equation.

    It's instructive to use a spreadsheet to fabricate data
    for input to LSE and, afterwards, to plot the resultant
    equation and residuals (explained later).  By fabricating
    such data, you can see for yourself the importance of 
    several guidelines, "tips", mentioned in this text.

    Tip 1
    covering the domain of interest. Your data points shouldn't
    favor either end of your graph; areas need to be represented
    for an accurate result.


derivation ====================================================
    The derivation of LSE is elegant, even pretty, and our
    aesthetic response to its elegance somewhat hides some of
    LSE's problems. This section goes through the algebra,
    calculus, and linear algebra behind LSE but the reader should
    feel free to skip this section.

    For now, see the 'derivation section' in this
    wiki link
    



Ramifications, things Gauss didn't tell ya ================

    Data points with big errors have too much effect.
    This means you'll want to check a potential LSE result
    to see if any of your data points have large errors.
    This implies that you'll have to evaluate
    the residuals
    for your data set. You may conclude that one or more of 
    your data points must be wrong. Such rejected points are 
    called outliers.
    You might want to remeasure one or more points or, at 
    least temporarily ignore them.

    In practice, you may have to run the LSE several times
    until you're happy with the results.  All this management,
    I believe, is a consequence of minimizing the square
    of the errors.
        

    A technique called 'weighting' can be used to ignore outliers.
    The concept is apparent if you simply duplicate a measurement entry
    in the list. That duplicated point now has twice the influence
    over the LSE outcome. The summation expressions can be easily
    doctored to include a weighting term which could be 0 or nearly
    any positive number.
    LSE_derivative_weighting.bmp
    If the weight is 0, it's just like that
    point was never measured.  A weight of 1 means the data point
    is counted just as before you considered weighting.