COS424SML 302 Linear Regression 18 46 Example predicting

Cos424sml 302 linear regression 18 46 example

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COS424/SML 302 Linear Regression February 25, 2019 18 / 46
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Example: predicting height using shoe size We fit a linear model to D , removing the one outlier point, and find that ˆ β 0 = 24 . 1 and ˆ β = 1 . 7 versus ˆ β 0 = 25 . 39, ˆ β = 1 . 65 60 65 70 75 80 85 22.5 25.0 27.5 Foot size (cm) Height (inches) How can one point have such a big impact on the fitted regression line? COS424/SML 302 Linear Regression February 25, 2019 19 / 46
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Example: predicting height using shoe size Now, say I am a forensics expert, and I find a suspect’s shoe print at a crime scene; the shoe is 26 cm long. What do I know about the suspect? 60 65 70 75 80 85 22.5 25.0 27.5 Foot size (cm) Height (inches) y * = ˆ β 0 + ˆ β x * 68 . 3 = 24 . 1 + 1 . 7 × 26 . COS424/SML 302 Linear Regression February 25, 2019 20 / 46
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Definition: residuals We define the residual as follows (dropping β 0 ): r i = y i - ˆ β x i = y i - ˆ y i ˆ β = our estimated β ˆ y i = our predicted value These r i are called the residuals . For each sample, the error is the same as the residual: i = y i - ˆ β x i = r i . From our definition of the error term E [ r i ] = E [ i ] = E [ y i - ˆ β x i ] = 0 I.e., the expected residual is zero for the Gaussian model. COS424/SML 302 Linear Regression February 25, 2019 21 / 46
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Linear regression: residuals For a fitted linear regression model and a data set D = { ( x 1 , y 1 ) , . . . , ( x n , y n ) } , I can compute the residuals: r i = y i - ˆ β x i = y i - ˆ y i X Y COS424/SML 302 Linear Regression February 25, 2019 22 / 46
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Example: predicting height using shoe size We can plot the residuals of the fitted model: -5 0 5 10 15 20 60 65 70 Height (inches) Residual error Are these Gaussian like the model assumes they are? COS424/SML 302 Linear Regression February 25, 2019 23 / 46
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Example: predicting height using shoe size We can examine the residuals of the fitted model: 0.00 0.05 0.10 0.15 0.20 0 10 20 Residual error density These look approximately Gaussian with zero mean, with one outlier. COS424/SML 302 Linear Regression February 25, 2019 24 / 46
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Evaluating the fit of the regression model to data I How do we quantify the quality of the predicted response values? There are a number of metrics, all of which are a function of the residuals: Residual Sum of Squares (RSS): RSS( ˆ β, D ) = n X i =1 ( y i - ˆ β T x i ) 2 RSS: total squared difference between predicted and true response values; lower values indicate better fit. COS424/SML 302 Linear Regression February 25, 2019 25 / 46
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Evaluating the fit of the regression model to data II How do we quantify the quality of the predicted response values? There are a number of metrics, all of which are a function of the residuals: Mean Squared Error (MSE): MSE = 1 n RSS( ˆ β, D ) Root Mean Squared Error (RMSE): RMSE = MSE MSE, RMSE: (root) average squared difference between predicted and true response values; lower values indicate better fit. COS424/SML 302 Linear Regression February 25, 2019 26 / 46
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Evaluating the fit of the regression model to data III How do we quantify the quality of the predicted response values? There are a number of metrics, all of which are a function of the residuals: Coefficient of determination ( r 2 ) : r 2 = 1 - RSS ( ˆ β, D ) n i =1 ( y i - 1 n n i =1 y i ) 2 r 2 : correlation between predicted and true response values; higher values indicate better predictions.
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