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Lecture 13: Polynomial Regression
Stat 102
Textbook
reading
Chapter 5.2.1
Description
Using JMP
Relation to Multiple Regression; tests of coefficients
Choosing the order of the polynomial
CIs
(for the mean of Y and for prediction of an additional observation)
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Polynomial Regression
A Method for Fitting Curvilinear Relationships
•
Reconsider the
simple regression
problem of estimating
Yx
= the conditional mean of
Y
given
x
.
•
For many problems,
is not linear in
x
.
•
We have suggested transformations of
x
(perhaps
accompanying transformations of
Y
) to address this problem.
•
In some situations these yield data in a form suitable for least
squares analysis; but in others they do not work well.
•
Polynomial regression is another leastsquares technique for
fitting curvilinear data.
•
We’ll look at 3 examples, and then explain the theory.
3
Example 1: How does rainfall affect yield of corn?
Data on annual corn yield and average rainfall in six US states
(18901927).
(See
Cornyieldrainfall.jmp)
Bivariate Fit of YIELD By RAINFALL
20
25
30
35
40
YIELD
6
7
8
9
10
11
12
13
14
15
16
17
RAINFALL
Note the generally curved pattern of points, with a max at
about 11.
Such a pattern cannot be well fit by transformations of
x
and/or
Y
.
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Example 2: How do people’s incomes change as they age?
Weekly wages and age of 200 randomly chosen adult males
from the March 1998
C
urrent
P
opulation
S
urvey
3
4
5
6
7
8
log wage
20
30
40
50
60
70
age
We’ll see that there is also a curved pattern here, with max near 45
Note that we’ve already used log
e
(Wage) as the
Y
variable.
5
Arrival Pattern of Calls to a Financial Call Center
Data is for week of 7/15/2002 – 7/19/2002.
Data is from call center of a major US bank.
Here’s what a call center looks like:
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A Call Center (picture is from England, ~1995)
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The number of calls made asking for service by an agent is
automatically recorded (to the nearest second).
The number of calls in each quarter hour of each weekday were
totaled.
For each data point
x
= time (to the quarterhour).
For each data point
y
=
# of calls in that quarter hour
.
− The reason for transforming to the
sq rt
of the # of calls is explained in Brown, Gans,
Mandelbaum, Sakov, Shen, Zeltyn and Zhao, (2005) “Statistical analysis of a telephone
call center: a queueing science perspective”,
Jour. Amer. Statist. Assoc.,
100
, 3650.
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This note was uploaded on 04/04/2012 for the course STAT 102 taught by Professor Shaman during the Spring '08 term at UPenn.
 Spring '08
 SHAMAN
 Statistics

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