Chapter 9 Regression
9.1Introduction
Many engineering and scientific problems are concerned with determining a relationship
between a set of variables. For instance, in a chemical process, we might be interested in
the relationship between the output of the process, the temperature at which it occurs,
and the amount of catalyst employed. Knowledge of such a relationship would enable us
to predict the output for various values of temperature and amount of catalyst.
In many situations, there is a singleresponsevariableY, also called thedependentvari-
able, which depends on the value of a set ofinput, also calledindependent, variables
x 1 ,...,xr. The simplest type of relationship between the dependent variableYand the
input variablesx 1 ,...,xris a linear relationship. That is, for some constantsβ 0 ,β 1 ,...,βr
the equation
Y=β 0 +β 1 x 1 +···+βrxr (9.1.1)would hold. If this was the relationship betweenYand thexi,i=1,...,r, then it would
be possible (once theβiwere learned) to exactly predict the response for any set of input
values. However, in practice, such precision is almost never attainable, and the most that
one can expect is that Equation 9.1.1 would be validsubject to random error. By this we
mean that the explicit relationship is
Y=β 0 +β 1 x 1 +···+βrxr+e (9.1.2)wheree, representing the random error, is assumed to be a random variable having mean
- Indeed, another way of expressing Equation 9.1.2 is as follows:
E[Y|x]=β 0 +β 1 x 1 +···+βrxrwherex=(x 1 ,...,xr) is the set of independent variables, andE[Y|x]is the expected
response given the inputsx.
Equation 9.1.2 is called alinear regression equation. We say that it describes the regression
ofYon the set of independent variablesx 1 ,...,xr. The quantitiesβ 0 ,β 1 ,...,βrare called
351