Using data the regression approach slide no 51 ence

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Unformatted text preview: st decade – 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.

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