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– Annual population growth in 19th century France
and annual cancer deaths rate in the U.S. in the
20th century CHAPTER 10. USING DATA The Regression Approach Slide No. 51
ENCE 627 ©Assakkaf Correlation
Separation of Variation TV = EV + UV
TV = total variation
EV = explained variation
UV = unexplained variation 26 Slide No. 52 CHAPTER 10. USING DATA ENCE 627 ©Assakkaf The Regression Approach
Correlation – Separation of Variation: A Set of
Observations on a Random Variable Y
TV = EV + UV
1= ˆ
ˆ
∑ (y − Y ) = ∑ (y − Y ) + ∑ ( y − y )
n EV UV
+
TV TV EV
R=
TV i =1 ˆ
∑ (y
n R2 = i =1
n ∑ (y
i =1 i i n 2 i −Y ) −Y ) i =1 2 i i =1 n ∑x y 2 = n 2 i =1 i i 2 i i 1 n n − ∑ xi ∑ yi n i =1 i =1 1 n xi2 − ∑ xi ∑ n i =1 i =1
n CHAPTER 10. USING DATA The Regression Approach 2 1 n yi2 − ∑ yi ∑ n i =1 i =1
n 2 Slide No. 53
ENCE 627 ©Assakkaf Correlation
Separation of variation: (a) total variation;
(b) explained variation; (c) unexplained
variation. 27 CHAPTER 10. USING DATA The Regression Approach Slide No. 54
ENCE 627 ©Assakkaf Need for Regression
– When dealing with two or more variables,
the functional relationship between the
variables is often of interest.
– However, if one or two variables (in twovariable case) are random, there is no
unique relationship between the values of
the two variables. CHAPTER 10. USING DATA The Regression Approach Slide No. 55
ENCE 627 ©Assakkaf Need for Regression
– Given a value of one variable (the
controlled or independent variable), there
is a range of possible values of the other.
– Thus, a probabilistic description is
required. 28 CHAPTER 10. USING DATA The Regression Approach Slide No. 56
ENCE 627 ©Assakkaf Regression Analysis
Regression analysis is the probabilistic
relationship between random variables
when this relationship is described in terms
of the mean and variance of one random
variable as a function of the value of the
other random variables. CHAPTER 10. USING DATA The Regression Approach Slide No. 57
ENCE 627 ©Assakkaf Optimization
– The process of deriving a relationship
between a random variable and measured
values of other variables is called
“optimization” or model “calibration”
– The objective of optimization is to find the
values of vectors of unknowns that
provides the minimum or maximum value
of some function. 29 CHAPTER 10. USING DATA The Regression Approach Slide No. 58
ENCE 627 ©Assakkaf Correlation Versus Regression
– Correlation analysis provides a measure of
goodness of fit.
– Regression analysis is a means of
calibrating the unknown coefficients of a
prediction equation.
– Correlation has its usefulness in model
formulation and verification. CHAPTER 10. USING DATA The Regression Approach Slide No. 59
ENCE 627 ©Assakkaf Elements of Statistical Optimization
1. An objective function, which defines what is
meant by the best fit.
2. A mathematical model, which is a n explicit
function relating a criterion variable (i.e., Y) to
vectors of unknowns and predictor (i.e., X)
variable(s)
3. A matrix of measured data 30 Slide No. 60 CHAPTE...
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This note was uploaded on 10/24/2012 for the course ESI 6385 taught by Professor Mansoorehmollaghasemi during the Fall '12 term at University of Central Florida.
 Fall '12
 MansoorehMollaghasemi

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