Evaporation can be a function of other variables such

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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 Y|X 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(Y|X1, . . . , 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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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.

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