TP#24_f08 - CE 3102 Fall, 2008 Thought Problem #24 Fitting...

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Unformatted text preview: CE 3102 Fall, 2008 Thought Problem #24 Fitting Greenshields Model An Introduction to Linear Regression Analysis The following graph shows a plot of traffic speed and traffic density measures from a location on Interstate 5, near Seattle, WA. 100 50 60 50 40 30 20 10 veh/l/hour mph Greenshields traffic flow model hypothesizes that the relationship between mean speed and traffic density is linear, i.e. = j f d d s s 1 where s = speed, (miles/hour) s f = free-flow speed (miles/hour) d = traffic density (vehicles/lane/mile) d j = jam density (vehicles/lane/mile) The capacity flow (i.e. maximum sustainable flows) is then given as V max = (s f d j )/4 For the following data, estimate the free-flow speed and jam density. mph veh/lane/mile 50.9 31.9 37.2 58.7 30.9 59.5 35.8 53.6 38.6 50.3 33.8 58.7 9.1 107.3 23.9 74.6 39.3 52.0 16.1 89.7 38.0 49.5 26.2 68.7 28.1 62.0 26.8 60.3 40.0 52.0 40.0 38.5 40.2 52.8 14.8 92.2 39.9 51.1 32.8 56.1 39.8 45.3 55.2 29.3 50.7 26.8 52.5 25.1 1. Estimate the free-flow speed and the jam density. 2. Compute a 95% confidence interval for the free-flow speed. 3. Test the hypothesis that the free-flow speed equal 60 mph. 4. Compute a point prediction, and 90% confidence interval, for the speed when traffic density equals 20 vehicles/lane/mile. CE 3102 Fall 2008 Thought Problem 24 Instructors Partial Solution 1. With a little algebra, we can express Greenshields model in a standard form s i = s f (s f /d j )d i +e i , or y i = + x i +e i where y i = s i x i = d i =s f = -(s f /d j ) and e i denote the errors between our modeled and measured data. The sum of squared errors is = = + = = n i i i n i i x y e SS 1 2 1 2 )) ( ( The least-squares estimated of and are then found by solving the equations...
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TP#24_f08 - CE 3102 Fall, 2008 Thought Problem #24 Fitting...

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