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Simple Linear Regression-Coumputional Aspects-ECO6416

Simple Linear Regression-Coumputional Aspects-ECO6416 -...

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Simple Linear Regression: Computational Aspects The regression analysis has three goals: predicting, modeling, and characterization. What would be the logical order in which to tackle these three goals such that one task leads to and /or and justifies the other tasks? Clearly, it depends on what the prime objective is. Sometimes you wish to model in order to get better prediction. Then the order is obvious. Sometimes, you just want to understand and explain what is going on. Then modeling is again the key, though out-of-sample predicting may be used to test any model. Often modeling and predicting proceed in an iterative way and there is no 'logical order' in the broadest sense. You may model to get predictions, which enable better control, but iteration is again likely to be present and there are sometimes special approaches to control problems. The following contains the main essential steps during modeling and analysis of regression model building, presented in the context of an applied numerical example. Formulas and Notations: = Σ x /n This is just the mean of the x values. = Σ y /n This is just the mean of the y values. S xx = SS xx = Σ (x(i) - ) 2 = Σ x 2 - ( Σ x) 2 / n S yy = SS yy = Σ (y(i) - ) 2 = Σ y 2 - ( Σ y) 2 / n S xy = SS xy = Σ (x(i) - )(y(i) - ) = Σ (x y) – ( Σ x) ( Σ y) / n Slope m = SS xy / SS xx Intercept, b = - m . y-predicted = yhat(i) = m x(i) + b Residual(i) = Error(i) = y – yhat(i) SSE = S res = SS res = SS errors = Σ [y(i) – yhat(i)] 2 = SS yy – m SS xy Standard deviation of residuals = s = S res = S errors = [SS res / (n-2)] 1/2 Standard error of the slope (m) = S res / SS xx 1/2 Standard error of the intercept (b) = S res [(SS xx + n. 2 ) /(n SS xx ] 1/2 R 2 = (SS yy - SSE) / SS yy A computational Example: A taxicab company manager believes that the monthly repair costs (Y) of cabs are related to age (X) of the cabs. Five cabs are selected randomly and from their records we obtained the following data: (x, y) = {(2, 2), (3, 5), (4, 7), (5, 10), (6, 11)}. The first step in constructing a simple linear regression model is to draw a scattered diagram, as shown in the following figure for our numerical example:
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Click on the image to enlarge it and THEN print it. A Visual Procedure as an Assessment Tool and Decision Process for Linearity of the Best Fit Based on the Scattered Diagram The linear dependency of Y variable with variable X can be checked graphically by carefully examining all the points in the scatter diagram, and see if it is possible to bound all the points within two parallel lines, shown in green in the
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