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16 Least Squares Regression

# 16 Least Squares Regression - 16 Least Squares Regression...

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LS Regression - 1 16 - Least Squares Regression The old adage, KISS (keep it simple stupid) forms the basis for regression analysis. In the preceding discussion on Lagrangian interpolation, we started with n points, and obtained a polynomial with exactly n —1 terms that approximated the n points. In regression analysis, we again start with n points, but instead of finding a polynomial with n terms, we fit an arbitrary function (not necessarily a polynomial) with a number of terms that we are free to select, as long as it is less than n —1. The most well-known example of regression is the fitting of a straight line (two-term polynomial) to a series of n data points through a least squares approximation. This is known as least squares linear regression. The following figure shows a series of 100 points. With linear regression we want to fit a straight line of the form y = mx+b to the series of data points. In other words, we want to find m and b . In order to do so, we recognize that each data point represents an equation: ... 2 2 1 1 b mx y b mx y + = + = In matrix form, these relations can be written as:

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LS Regression - 2 = 3 2 1 100 2 1 ... 1 ... ... 1 1 y y y m b x x x Note that it would be trivial to reformulate this problem to fit a parabola or a cubic equation. The formulation for a cubic fit of the form y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 would be = 100 2 1 3 2 1 0 3 100 2 100 100 3 2 2 2 2 3 1 2
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16 Least Squares Regression - 16 Least Squares Regression...

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