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Unformatted text preview: Linear Regression OEM 2009 ESI 6321 –Applied Probability Methods in Engineering 2 Regression Analysis A technique to model and investigate the relationship between two or more variables. Regression analysis can be used to describe a relationship, as well as to forecast the value of one variable as a function of the others. 3 Example 1 The number of pounds of steam used per month by a chemical plant is thought to be related to the average ambient temperature for that month. We have data on the past year’s usage and temperatures. This information can be summarized in a socalled scatter diagram . 4 Scatter Diagram 100 200 300 400 500 600 700 800 20 40 60 80 Temperature Usage (1000's) 5 Scatter Diagram A scatter diagram plots the corresponding values of two variables. For regression analysis, we usually place the independent variable on the horizontal axis, and the dependent variable on the vertical axis. The independent variable is sometimes also called the explanatory variable or regressor , and is usually denoted by x . The dependent variable , also called response or endogenous variable, is usually denoted by Y . 6 Theoretical, Empirical Models Sometimes, we will have a theoretical model describing the relationship between the variables. In the absence of a theory, the model choice can be based on visual inspection of a scatter diagram, yielding an empirical model . 7 Linear Regression Model The scatter diagram suggests that the relationship between temperature and steam usage is approximately linear . More formally, we may expect that for some constants β and β 1 . The term ε is an error term that captures the deviation from the linear relationship. Y = ! + ! 1 x + " 8 Error Term The error term ε can be caused by: Inaccuracies in observing Y . Inappropriateness of the linear form of the relationship. The presence of other variables that influence the value of Y , but are absent from the model. We will assume that the error term is a random variable with mean E ( ε ) = 0 , and variance V ( ε ) = σ 2 ; Independent between observations. 9 Linear Regression Model In terms of the linear regression model, this actually means that Y is a random variable! Given the value of x , we have that the random variable Y has mean and variance E ( Y ; x ) = E ( ! + ! 1 x + " ) V ( Y ; x ) = V ( ! + ! 1 x + " ) = ! + ! 1 x = ! + ! 1 x + E ( " ) = ! 2 = V ( ! ) 10 Linear Regression Model The linear model thus describes the behavior of the mean of Y as a function of x . The variability of Y for a given value of x is measured by the error variance σ 2 . The smaller the error variance, the closer we can expect observed values of Y for different values of x to be to the line describing the mean of Y . 11 Example 2 The amount of property taxes to pay can be expected to depend (partly) on the price of a house....
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This note was uploaded on 05/12/2010 for the course ESI 6321 taught by Professor Josephgeunes during the Spring '07 term at University of Florida.
 Spring '07
 JosephGeunes

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