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stic for the model developed from the estimation  data in Problem 11.2. How well is the model likely to predict? Compare this  indication of predictive performance with the

stic for the model developed from the estimation  data in Problem 11.2. How well is the model likely to predict? Compare this  indication of predictive performance with the

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 31EQ
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Calculate the PRESS statistic for the model developed from the estimation 
data in Problem 11.2. How well is the model likely to predict? Compare this 
indication of predictive performance with the actual performance observed 
in Problem 11.2.

TABLE 11.2 Prediction Data Set for the Delivery Time Example (1) (2) (3) (4) (5) (6) Least-Squares Fit Observed Time, y Observation City Cases, x Distance, X y- ŷ San Diego San Diego Boston 26 22 50.9230 21.1405 905 51.00 0.0770 -4.3405 -4.5957 27 520 16.80 28 15 290 26.16 30.7557 29 Boston 5 500 19.90 17.6207 2.2793 26.4366 -2.4366 3.2734 2.2698 30 Boston 1000 24.00 18,55 31.93 16.95 31 Boston 225 15.2766 32 Boston 10 775 29.6602 33 Boston 4 212 11.8576 5.0924 34 Austin 1 144 7.00 6.0307 0.9693 35 Austin 3 126 14.00 9.0033 4.9967 36 Austin 12 655 37.03 31.1640 5.8660 37 Louisville 10 420 18.62 24.5482 -5.9282 Louisville 15.8125 20.4524 38 7 150 16.10 0.2875 39 Louisville 8 360 24.38 3.9276 40 Louisville 32 1530 64.75 76.0820 -11.3320 this point is quite similar to point 9, which we know to be influential. From an overall perspective, these prediction errors increase our confidence in the usefulness of the model. Note that the prediction errors are generally larger than the residuals from the least-squares fit. This is easily seen by comparing the residual mean square MSRes = 10.6239 from the fitted model and the average squared prediction error 40 332.2809 - 22.1521 i=26 15 15 from the new prediction data. Since MSRes (which may be thought of as the average variance of the residuals from the fit) is smaller than the average squared prediction error, the least-squares regression model does not predict new data as well as it fits the existing data. However, the degradation of performance is not severe, and so we conclude that the least-squares model is likely to be successful as a predictor. Note also that apart from the one extrapolation point the prediction errors from Louisville are not remarkably different from thosc experienced in the citics where the original data were collected. While the sample is small, this is an indication that the model may be portable. More extensive data collection at other sites would be helpful in verifying this conclusion. It is also instructive to compare R from the least-squares fit (0.9596) to the per- centage of variability in the new data explained by the model, say dd 376 vighr 6/27/2017 636:52 P VALIDATION TECHNIQUES 377 40 332.2809 = 1 0.8964 'rediction 40 3206.2338 i=26 Once again, we see that the least-squares model does not predict new obser- vations as well as it fits the original data. However, the "loss" in R for prediction is slight. Collecting new data has indicated that the least-squares fit for the delivery time data results in a reasonably good prediction equation. The interpolation parlor-mance of the model is likely to be better than when the model is used for extrapolation.
Transcribed Image Text:TABLE 11.2 Prediction Data Set for the Delivery Time Example (1) (2) (3) (4) (5) (6) Least-Squares Fit Observed Time, y Observation City Cases, x Distance, X y- ŷ San Diego San Diego Boston 26 22 50.9230 21.1405 905 51.00 0.0770 -4.3405 -4.5957 27 520 16.80 28 15 290 26.16 30.7557 29 Boston 5 500 19.90 17.6207 2.2793 26.4366 -2.4366 3.2734 2.2698 30 Boston 1000 24.00 18,55 31.93 16.95 31 Boston 225 15.2766 32 Boston 10 775 29.6602 33 Boston 4 212 11.8576 5.0924 34 Austin 1 144 7.00 6.0307 0.9693 35 Austin 3 126 14.00 9.0033 4.9967 36 Austin 12 655 37.03 31.1640 5.8660 37 Louisville 10 420 18.62 24.5482 -5.9282 Louisville 15.8125 20.4524 38 7 150 16.10 0.2875 39 Louisville 8 360 24.38 3.9276 40 Louisville 32 1530 64.75 76.0820 -11.3320 this point is quite similar to point 9, which we know to be influential. From an overall perspective, these prediction errors increase our confidence in the usefulness of the model. Note that the prediction errors are generally larger than the residuals from the least-squares fit. This is easily seen by comparing the residual mean square MSRes = 10.6239 from the fitted model and the average squared prediction error 40 332.2809 - 22.1521 i=26 15 15 from the new prediction data. Since MSRes (which may be thought of as the average variance of the residuals from the fit) is smaller than the average squared prediction error, the least-squares regression model does not predict new data as well as it fits the existing data. However, the degradation of performance is not severe, and so we conclude that the least-squares model is likely to be successful as a predictor. Note also that apart from the one extrapolation point the prediction errors from Louisville are not remarkably different from thosc experienced in the citics where the original data were collected. While the sample is small, this is an indication that the model may be portable. More extensive data collection at other sites would be helpful in verifying this conclusion. It is also instructive to compare R from the least-squares fit (0.9596) to the per- centage of variability in the new data explained by the model, say dd 376 vighr 6/27/2017 636:52 P VALIDATION TECHNIQUES 377 40 332.2809 = 1 0.8964 'rediction 40 3206.2338 i=26 Once again, we see that the least-squares model does not predict new obser- vations as well as it fits the original data. However, the "loss" in R for prediction is slight. Collecting new data has indicated that the least-squares fit for the delivery time data results in a reasonably good prediction equation. The interpolation parlor-mance of the model is likely to be better than when the model is used for extrapolation.
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