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Regression model A prediction approach
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Prediction Independent variable (input/operating variable) Dependent variable (response variable) Prediction error The prediction model: the linear regression line, the linear regression equation
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Simple regression model The regression line- Expected response Error of simple regression model Actual response ∵
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Graphic explanation
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LSE estimators By definition
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Assumptions of regression model The distribution of residual: e Normality (~N(0, σ2)) Equality of variance (the same variance σ2 for every σi2 ) Independence from X i
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Distribution of the β estimators
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S xx
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Distribution of the β estimators
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Residual analysis Sum of Square of Residuals
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Inference of β ∵ ∴ Therefore,, and
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Confidence interval for β
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Decomposition of Syy, Syy=SSr+SSm X Y Y- Y^ Yi (Yi-Y-) (Yi-Y^) (Y^-Y-) 如果離差越大, 表示 Y^ 不太可能是水平線, 因 為若是水平線, 則差的 平方和將會很小 如果離差越小, 表示 迴歸線越接近真實 值, 預測得越準確 ! Y 的離差, 因給 定 sample 之後 固定不變
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Inference of the mean response α+β x 0
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Confidence interval for the mean response α+β x 0 分母代入 X 2 /(n-2) 之後除去σ2
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Inference of the response at the input level x 0
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Confidence interval for the response at the input level x 0 Normality & equality testing Plotting the random data and regression line Plotting the residuals along the predictive variable
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Determination and sample correlation coefficient By definition,
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More regression models Multiple regression More than one predictive variables Transforming to linearity Log, square Logistic regression For binary response data
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Homework #2 Problem 5,12,26,39,42, optional 48
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