regressionLectures_4xgs

regressionLectures_4xgs - Part 1 Linear Regression...

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Regression Analysis April 4, 2011 Part 1: Linear Regression - Introduction Example in lab: Design of a casting mold want to reduce the time to finish a batch of castings but need to maintain low percentage of defective castings Casting Mold Applications of regression What regression analysis can do: investigate how variables are related test if there is a relationship between variables predict future values of variables control system output
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Fitting equations to data Two types of mathematical models: Theoretical: Derived from first principles via mathematical reasoning Empirical: Derived by “fitting” an equation to data typically the equation has a simple parametric form parameters are estimated by adjusting them to get the best fit to the data we will study empirical models Linear models (LM’s) E ( Y | X 1 ,..., X p ) = β 0 + β 1 X 1 + β 2 X 2 + · · · + β p X p So Y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β p X p + ǫ where ǫ is variation of Y about E ( Y | X 1 ,..., X p ) In example: Y is the time to finish a cast. X 1 ,..., X p are the design variables of the mold, e.g., cavity width, riser height, .. . Variables From a previous slide: Y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β p X p + ǫ Y : “outcome”, “response”, “dependent variable” X 1 ,..., X p : “predictors”, “independent variables”, “covariates” Y , X 1 ,..., X p : “variables” Parameters From a previous slide: Y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β p X p + ǫ β 0 : “intercept” β 1 ,..., β p : “regression coe ffi cients” = “slopes” = “partial derivatives” β j = X j E ( Y | X 1 ,..., X p ) β 0 , β 1 ,..., β p : “parameters”
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Noise From a previous slide: Y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β p X p + ǫ ǫ is the unpredictable variation in Y ǫ is called the “noise” “error” “residual variation” When is a model linear? From previous page: Y = β 0 + β 1 X 1 + β 2 X 2 + · · · + β p X p + ǫ Definition: A model is linear if Y is linear in the parameters (linear in β 0 ,..., β p ). Key point: X 1 ,..., X p can be anything observable, even indirectly. They can be nonlinear functions of measured quantities. When is a model linear? Which of these are linear models? Y = β 0 + β 1 × ( CavityWidth ) + β 2 × ( CavityWidth ) 2 , Y = β 0 + β 1 × ( CavityWidth ) + ( β 2 × CavityWidth ) 2 , Y = β 0 + β 1 × ( CavityWidth ) + exp ( β 2 × CavityWidth ) , Nonlinear models Nonlinear regression is covered in more advanced courses For now, you only need to know a nonlinear model when you see it Example: Y = β 0 + β 1 exp ( β 2 X ) + ǫ This model is nonlinear because Y is a nonlinear function of β 2
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Data Observe ( Y i , X i , 1 ,..., X i , p ) , for i = 1 ,..., n i = index of “observation” = “case” = “subject” = “row in data spreadsheet” So the linear regression model can be rewritten as: Y i = β 0 + β 1 X i , 1 + β 2 X i , 2 + · · · + β p X i , p + ǫ i Notice that β 0 , β 1 ,..., β p do not depend on i The columns of the data spreadsheet are Y i , X i , 1 ,..., X i , p Least-squares Estimation The predictor of Y i is hatwide Y i = hatwide β 0 + hatwide β 1 X i , 1 + hatwide β 2 X i , 2 + · · · + hatwide β
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