Problem: For a given class of functions
and a set of data
find
a (the?) function
of class
which minimizes
the sum of squares of errors.
Linear regression is the solution to this problem where
is the class of linear functions.
A quadratic function is a function of the form
for fixed constants
,
, and
.
Note that taking we see that a linear function is a
special case of a quadratic function.
Find a quadratic function
minimizing the
sum of squares of errors for the data
.
Setting these equal to zero and performing some algebra, we find
,
, and
.
Problem: Given a set of data
find a function
of two variables (of some specified class) best
fitting the data.
When
is taken to be a linear function, then this is
the problem of multiple linear regression.