Least Squared Error
Editorial, notes on derivation
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. 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.