What is the principle of least squares?
Least squares, also called "regression analysis", is a computational
procedure for fitting an equation to a set of experimental data
points. The criterion of the "best" fit is that the sum of the
squares of the differences between the observed data points,
(Xi,Yi), and the value calculated by the fitting equation,
Ycalc(Xi), i.e.: SUM [Yi - Ycalc(Xi)]^2 is a minimum. The reason for
using the square is to make all of the differences positive numbers.
If the fitting equation
Ycalc(Xi) contains 'j' adjustable parameters: i.e. Ycalc(a1, a2,
..., aj) then the minimum in the sum of difference squared occurs
when: dSUM/da1 = dSUM/da2 = ...= dSUM/daj =0 where:
the quantities dSUM/da1, dSUM/da2, ..., dSUM/daj are the first
derivatives of the sum with respect to the constants a1, a2, ..., aj.
This is the "in principle" part. The actual formulas for
calculating "the best" values of a1, a2, ...,
aj can be messy depending upon the equation chosen for Ycalc(Xi),
but don't let that obscure what is happening "in principle". Most
data analysis programs, even those on hand held calculators,
"crunch" the numbers for you.
There are several things that need to be kept in mind: 1. The
SUM of squares of the differences being a minimum is not the only
criterion of "best fit"; however, it is the most commonly used one.
2. The individual data points (Xi, Yi) can be given different
"weight" if there is reason to expect some of the data to be more
reliable than others. This means "counting" some data points more
than others. 3. Most importantly, inherent in the procedure, is the
condition that there is no error in the Xi. That is, the
independent variable is known with exact precision and accuracy.
This may or may not be true. A consequence of this last condition
is the following: If you reverse the independent and dependent
variables, that is, if you fit (Yi, Xi) instead of (Xi,Yi), you
don't obtain the same fitting equation because in the case, (Yi,Xi)
the condition is that the Yi's are known exactly, instead of the Xi's.
There are a number of statistical tests that can be applied to
the "fit" to determine how good the "fit" is, but that is beyond
what can be done here. One word of caution however. A good "fit"
does not mean that X "caused" Y. It only means that X and Y are
correlated with one another. Correlation is not causation!!
There are many web sites that discuss least squares (or
regression) in varying degrees of sophistication. This one:
gives a pretty good explanation as well as a number of other links.
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Update: June 2012