ENMA 420-520 Lecture 8 Slides

ENMA 420-520 Lecture 8 Slides - Statistical Concepts for...

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Click to edit Master subtitle style 10/17/09 Statistical Concepts for Engineering Management ENMA 420 / 520 Lecture #8 Simple Linear Regression 11
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10/17/09 Regression Models Regression analysis: Focus is on the relationship between a dependent variable and one or more independent variables. How the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. 22
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10/17/09 Regression Models (Cont’d) The variable to be predicted, y, is called the dependent (or response) variable The variables used to predict y are called independent variables (typically x i ) 33
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10/17/09 Model Assumptions For a perfect relationship, a model equation would exactly match all data points Typically does not Measurement errors Model errors Linear Regression: Assumes the relationship between the 44
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10/17/09 Linear Relationships 55 0 2 4 6 8 10 12 14 16 18 20 0 50 100 150 200 250 300 350 400 Constant Acceleration x = ½ at2 Constant Velocity x = vt Tim e Positio n Line ar Nonline ar
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10/17/09 Simple Linear Regression Model y = ˜ 0 + &1x + & where: y = independent variable x = dependent variable E(y) = Deterministic component, &0 + &1x l = Random error component 66
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10/17/09 Assumptions The mean of the probability distribution of ½ is 0. The variance of the probability distribution for ˜ is constant for all x The probability distribution for ' is normal The errors associated with any two different observations are independent 77
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10/17/09 The Method of Least Squares Given a set of data points (x, y) and a hypothetical “best” line fitting the data The deviations are the errors in y given x The method of least squares minimizes the sum of the squares of the deviations 88
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10/17/09 The Method of Least Squares (Cont’d) 99 : StraightLineModel : = : 0 + : 1 l + : : EstimateofStraightLineModel l = : : 0 + : : 1 l ( For agivendatapoint x i , y ) theestimateof : is llll l : 0 + : : 1 l l : Theerror intheestimateisthedeviation l l − :: l : − : : 0 − : : 1 l l ( : SotheSumofSquaresoffor Errors SSE is o Q % = σ ll l − : : 0 − : : 1 l l l 2 l l = 1 . Thegoal istominimizetheseerrors Com : putingthepartial derivatives : −Q % : : : 0 2l l l − : : : 0 + : : 1 l l ll l 1l l l = 1 : −Q % : : : 1 l − : : : 0 + : : 1 l l −: : : : := 1
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10/17/09 The Method of Least Squares (Cont’d) 1010 l o «. l l l 0 = : 2l l l − : : : 0 + : : 1 l l ll l 1l l l = 1 : − : : : 1 l − : : : 0 + : : 1 l l −: : : : := 1 : Settingthesederivativesequal tozerotofindtheminimum : : : : 1 − : : : 0 l l = 1 − : : 1 l l l l l = 1 = : : : : 1 − : : : 0 − : : 1 l l l l l = 1 = 0 : : : : : : 1 − : : 0 l l l l l = 1 : 1 l l l 2 l l = 1 = 0
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10/17/09 The Method of Least Squares (Cont’d) 1111 : Settingthesederivativesequal tozerotofindtheminimum : : : : := 1 − : : : 0 l l = 1 − : : 1 l l l l l = 1 = : : : : 1 − : : : 0 − : : 1 l l l l l = 1 = 0 : : : : : : 1 − : : 0 l l l l l = 1 : 1 l l l 2 l l = 1 = 0 Simplifying : : : : 0 + : : 1 l l l l l = 1 = : : : : 1 : : 0 l l l l l = 1 : 1 l l l 2 l l = 1 = : : : : : : 1
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10/17/09 The Method of Least Squares (Cont’d) 1212 : : : 0 + : : 1 l l l l l = 1 = : : : : := 1 : : 0 l l l l l = 1 : 1
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This note was uploaded on 10/17/2009 for the course MET 387 taught by Professor Dean during the Spring '09 term at Old Dominion.

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ENMA 420-520 Lecture 8 Slides - Statistical Concepts for...

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