Page 233 - Matrix Analysis & Applied Linear Algebra

228 Chapter 4 Vector Spaces

To help seat these ideas, consider the problem of predicting the amount of
weight that a pint of ice cream loses when it is stored at very low temperatures.
There are many factors that may contribute to weight loss—e.g., storage tem-
perature, storage time, humidity, atmospheric pressure, butterfat content, the
amount of corn syrup, the amounts of various gums (guar gum, carob bean gum,
locust bean gum, cellulose gum), and the never-ending list of other additives and
preservatives. It is reasonable to believe that storage time and temperature are
the primary factors, so to predict weight loss we will make a linear hypothesis of
the form
y = α 0 + α 1 t 1 + α 2 t 2 + ε,
where y = weight loss (grams), t 1 = storage time (weeks), t 2 = storage tem-
o
perature ( F ), and ε is a random function to account for all other factors. The
assumption is that all other factors “average out” to zero, so the expected (or
mean) weight loss at each point (t 1 ,t 2 )is

E(y)= α 0 + α 1 t 1 + α 2 t 2 . (4.6.4)
Suppose that we conduct an experiment in which values for weight loss are
measured for various values of storage time and temperature as shown below.
Time (weeks) 1 1 1 2 2 2 3 3 3

o
Temp ( F) −10 −5 0 −10 −5 0 −10 −5 0
Loss (grams) .15 .18 .20 .17 .19 .22 .20 .23 .25
If
11 −10 .15
   
 11 −5   .18 
 11 0   


 .20 
   
 12 −10  α 0  .17 


A =  12 −5  , x =  α 1  , and b =  .19  ,




 12 0  α 2  


 .22 
 13 −10   


 .20 
13 −5 .23
   
13 0 .25
and if we were lucky enough to exactly observe the mean weight loss each time
(i.e., if b i = E(y i ) ), then equation (4.6.4) would insure that Ax = b is a
consistent system, so we could solve for the unknown parameters α 0 ,α 1 , and
α 2 . However, it is virtually impossible to observe the exact value of the mean
weight loss for a given storage time and temperature, and almost certainly the
system defined by Ax = b will be inconsistent—especially when the number
of observations greatly exceeds the number of parameters. Since we can’t solve
Ax = b to find exact values for the α i ’s, the best we can hope for is a set of
“good estimates” for these parameters.