Unformatted text preview: R 10. USING DATA The Regression Approach ENCE 627 ©Assakkaf Example: Evaporation
– In irrigation projects, it is necessary to
provide estimates of evaporation.
– Evaporation can be a function of other
variables such as air temperature,
humidity, and air mass.
– If measurement of air temperature are
available, a relationship or a model can be
developed. Slide No. 61 CHAPTER 10. USING DATA The Regression Approach ENCE 627 ©Assakkaf Example (cont’d): Evaporation ˆ
E = β 0 + β1T
b0 and b1 = the unknown coefficients ˆ
E = the predicted value of E
T = air temperature 31 CHAPTER 10. USING DATA The Regression Approach Slide No. 62
ENCE 627 ©Assakkaf Example (cont’d): Evaporation
– If we are interested in daily evaporation
rates, we may measure both the total
evaporation for each day in a year and the
corresponding mean daily temperature.
– An objective function should be established
to evaluate the unknowns.
– Regression minimizes the sum of the
squares of the differences between the
predicted and measured values. CHAPTER 10. USING DATA The Regression Approach Slide No. 63
ENCE 627 ©Assakkaf Regression Definitions
– The objective of regression is to evaluate
the coefficients of an equation relating the
criterion variable to one or more variables,
which are called the predictor variables.
– The predictor variables are variables in
which their variation is deemed to cause or
agree with variation in criterion variable 32 Slide No. 64 CHAPTER 10. USING DATA The Regression Approach ENCE 627 ©Assakkaf Linear Regression
– The conditional expected value of Y is
linear in the X’s
– In symbols: E( 1,...,Xk ) = β0 + β1X1 +...+ βk Xk
YX
The β ’s are coefficients, and they serve the purpose of combining the X
values to obtain a conditional expected value for Y. CHAPTER 10. USING DATA The Regression Approach Slide No. 65
ENCE 627 ©Assakkaf Note:
1. It is important to remember that the
equation defines a relationship between
the explanatory variables and the
expected Y. The actual Y value will be
above or below this expected value to
some extend; this is where the
uncertainty and the conditional probability
distribution of Y come into play. 33 Slide No. 66 CHAPTER 10. USING DATA The Regression Approach ENCE 627 ©Assakkaf 2. The distribution around the conditional
expected value has the same shape
regardless of the particular X values
ˆ
Y = E (Y  X 1 ,..., X k ) + ε The conditional distribution (and the corresponding density) of Y, given
the X’s, ahs the same shape as the distribution (or density) of the errors,
but it is just shifted so that the distribution is centered on the expected
value E(YX1, . . . , Xk) CHAPTER 10. USING DATA The Regression Approach Slide No. 67
ENCE 627 ©Assakkaf Bivariate Model ˆ
Y = β 0 + β1 X
Multivariate model ˆ
Y = β 0 + β1 X 1 + β 2 X 2 + .... + β k X k 34 Slide No. 68 CHAPTER 10. USING DATA The Regression Approach ENCE 627 ©Assakkaf Principle of Least Squares
– The principle of least squares is the
process of obtaining the best estimates of
the coefficients (β0, β1,.., βk).
– This principle is referred to as the
regression method.
– To express the principle of least squares, it
is important to define the error e Slide No. 69 CHAPTER 10. USING DATA The Regression Approach ENCE 627 ©Assakkaf Principle of Least Squares
– Objective Function
n n i =1 i =1 ˆ
F = min ∑ ei2 = mi...
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 Fall '12
 MansoorehMollaghasemi
 Regression Analysis, Probability theory, Yi, SLIDE NO.

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