Ch11 Curve Fitting
Dr. Deshi Ye
yedeshi@zju.edu.cn
Outline
The method of Least Squares
Inferences based on the Least Squares
Estimators
Curvilinear Regression
Multiple Regression
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11.1 The Method of Least Squares
Study the case where a dependent
variable is to be predicted in terms of a
single independent variable.
The random variable Y depends on a
random variable X.
Regressing curve of Y on x, the relationship
between x and the mean of the
corresponding distribution of Y.
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Linear regression
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Linear regression
Linear regression: for any x, the mean of
the distribution of the Y’s is given by    x
In general, Y will differ from this mean, and we denote
this difference as follows
Y   x 

is a random variable and we can also choose 
so that the mean of the distribution of this random is
equal to zero.
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EX
x
y
1 2 3 4 5 6 7 8 9 10 11 12
16 35 45 64 86 96 106 124 134 156 164 182
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Analysis
ˆ  a  bx
y
ˆi
ei  yi  y
n
as close as possible to zero.
e
i
i 1
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Principle of least squares
Choose a and b so that
n
n
2
e

(
y

(
a

bx
))
  i
i
i 1
2
i
i 1
is minimum. The procedure of finding the equation of the line
which best fits a given set of paired data, called the method
of least squares. Some notations:
n
n
n
( yi )2
i 1
i 1
n
n
n
n
( xi ) 2
i 1
n
S xx   ( xi  x )   xi2 
2
i 1
i 1
S yy   ( yi  y )2   yi2 
n
n
i 1
i 1
n
n
( xi )( yi )
i 1
i 1
n
S xy   ( xi  x )( yi  y )   xi yi 
i 1
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Least squares estimators
a  y  b  x and b 
S xy
S xx
, where x , y are the means of x, y
Fitted (or estimated) regression line
yˆ  a  bx
Residuals: observation – fitted value= y i  (a  bxi )
The minimum value of the sum of squares is called the
residual sum of squares or error sum of squares. We
n
will show that
2
SSE  residual sum of squares=  (yi - a - bxi )
i 1
 S xy  S xy2 / S xx
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EX solution
Y = 14.8 X + 4.35
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X-and-Y
X-axis
independent
predictor
carrier
input
Y-axis
dependent
predicted
response
output
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Example
You’re a marketing analyst for Hasbro
Toys. You gather the following data:
Ad $
Sales (Units)
1
1
2
1
3
2
4
2
5
4
What is the relationship
between sales & advertising?
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Scattergram
Sales vs. Advertising
Sales
4
3
2
1
0
0
1
2
3
4
5
Advertising
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the Least Squares
Estimators
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11.2 Inference based on the Least
Squares Estimators
We assume that the regression is linear in
x and, furthermore, that the n random
variable Yi are independently normally
distribution with the means    xi
Statistical model for straight-line
regression Y     x  
i
i
i
i
are independent normal distributed random
variable having zero means and the common
variance  2
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Standard error of estimate
2

The i-th deviation and the estimate of
is
1 n
2
S 
[
y

(
a

bx
)]

i
i
n  2 i 1
2
e
2

Estimate of
can also be written as follows
S yy 
S 
2
e
( S xy ) 2
S xx
n2
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Statistics for inferences: based on the assumption made
concerning the distribution of the values of Y, the following
theorem holds.
Theorem. The statistics
nS xx
(a   )
(b   )
t
and t 
S xx
2
se
S xx  n( x )
se
are values of random variables having the t distribution
with n-2 degrees of freedom.
Confidence intervals
 : a  t / 2  se
1 ( x )2

n S xx
 : b  t / 2  se
1
S xx
17/30
Example
The following data pertain to number of
computer jobs per day and the central
processing unit (CPU) time required.
Number of jobs
x
1
2
3
4
5
CPU time
y
2
5
4
9
10
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EX
1) Obtain a least squares fit of a line to the
observations on CPU time
b
S xy
S xx
 2, a  y  bx  0
y  2x
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Example
2) Construct a 95% confidence interval for α
s 
2
e
S yy  S xy 2 / S xx
n2
The 95% confidence interval of α,
a  t / 2  se
46  400 /10

2
3
t / 2  t0.025  3.182
1 x2
1 9

 0  3.182 * 2 *

 4.72
n S xx
5 10
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Example
3) Test the null hypothesis
the alternative hypothesis
level of significance.
  1 against
  1 at the 0.05
Solution: the t statistic is given by
(b   )
2 1
t
S xx 
10  2.236
se
2
Criterion:
t  t0.05  2.353
Decision: we cannot reject the null hypothesis
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11.3 Curvilinear Regression
Regression curve is nonlinear.
Polynomial regression:
Y  0  1x  2 x 
2
 px
p
Y on x is exponential, the mean of the distribution of
values of Y is given by y     x
Take logarithms, we have log y  log   x  log 
Thus, we can estimate  ,  by the pairs of value ( xi ,log yi )
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Polynomial regression
If there is no clear indication about the
function form of the regression of Y on x,
we assume it is polynomial regression
Y  a0  a1x  a2 x2   ak xk
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Polynomial Fitting
•Really just a generalization of the
previous case
•Exact solution
•Just big matrices
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11.4 Multiple Regression
The mean of Y on x is given by
b0  b1 x1  b2 x2 
n
Minimize
[ yi  (b0  b1xi1 
 bk xk
 bk xik )]2
i 1
We can solve it when r=2 by the following equations
 y  nb  b  x  b  x
x y  b x b x b x x
 x y b  x  b  x x  b  x
0
1
2
0
0
1
1
1
2
1
1
2
2
2
1
1 2
2
1 2
2
2
2
25/30
Example
P365.
26/30
Multiple Linear Fitting
X1(x), . . .,XM(x) are arbitrary fixed functions of x
(can be nonlinear), called the basis functions
normal equations of the least squares
problem
Can be put in matrix form and solved
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Correlation Models
1. How strong is the linear relationship
between 2 variables?
2. Coefficient of correlation used
Population correlation coefficient denoted

Values range from -1 to +1
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Correlation
Standardized observation
Observation - Sample mean xi  x

Sample standard deviation
sx
The sample correlation coefficient r
1 n xi  x yi  y
r
(
)(
)

n  1 i 1 s x
sy
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Coefficient of Correlation Values
No
Correlation
-1.0
-.5
Increasing degree of
negative correlation
0
+.5
+1.0
Increasing degree of
positive correlation
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