{"id":261,"date":"2017-04-26T15:05:14","date_gmt":"2017-04-26T15:05:14","guid":{"rendered":"http:\/\/ajhassoc.com\/?p=261"},"modified":"2017-05-17T01:35:30","modified_gmt":"2017-05-17T01:35:30","slug":"a-well-choreographed-od-table-dance-part-3-pivot-point-comparison","status":"publish","type":"post","link":"https:\/\/ajhassoc.com\/index.php\/2017\/04\/26\/a-well-choreographed-od-table-dance-part-3-pivot-point-comparison\/","title":{"rendered":"A Well Choreographed OD Table Dance, Part 3: Pivot Point Comparison"},"content":{"rendered":"
\n

Juan Ortuzar\u2019s book indeed provides a pivot point formula that allows two or more choice alternatives to be varied at the same time.\u00a0 So a valid question may be raised:\u00a0 which method works best, pivot point or the gravity model, for OD table forecasts?\u00a0 This blog article extends the example problem to pivot point and compares the answers.\u00a0 As we will see, the selection of the method to be adopted hinges on the amount of error* in the original OD table and how well the gravity assumption matches the reality of the forecast.<\/p>\n

\n

[*I am using \u201cerror\u201d to refer to any inaccuracies in measurement.\u00a0 I am using \u201cmistakes\u201d to refer to any failures or misapplications of theory.]<\/p>\n

Recall the original question posed by Jim Bunch.\u00a0 He has before and after travel time matrices and an observed OD table.\u00a0 He wants to forecast the after OD table without using travel demand modeling software.<\/p>\n

The pivot point formula, attributed to Kumar in 1980 and adapted to this application is:<\/p>\n

Pj<\/sub> = pj<\/sub>o<\/sup>exp(-b\u0394<\/span>tj<\/sub>)\/\u03a3<\/span>k<\/sub>pk<\/sub>o<\/sup>exp(-b\u0394<\/span>tk<\/sub>)\u00a0 \u00a0\u00a0 for choices across rows, and<\/p>\n

Pi<\/sub> = pi<\/sub>o<\/sup>exp(-b\u0394<\/span>ti<\/sub>)\/\u03a3m<\/sub>pm<\/sub>o<\/sup>exp(-b\u0394tm<\/sub>)\u00a0 \u00a0\u00a0 for choices down columns.<\/p>\n

Here \u0394<\/span>t is the difference between the after and before travel times, p is the probability of choosing the the ith or jth OD pair, po<\/sup> is the observed proportion of trips using the ith or jth OD pair, and b is a coefficient on travel time.\u00a0 The subscripts j and k are row indices and i and m are column indices.<\/p>\n

Notice that the pivot point formula is somewhat simpler than the gravity model presented in the first article in this series.<\/p>\n

The setup of this example is identical to the earlier one.\u00a0 QRS II is used to establish \u201ctrue\u201d before and after OD tables, considering a big speed limit increase on the UTown freeway.\u00a0 The analysis is broken out by trip purpose (HBW and HBNW+NHB) and by direction of travel (PtoA and AtoP) exactly as before.<\/p>\n

In this instance the forecasted OD table calculates to these numbers.<\/p>\n\n\n\n\n\n\n\n\n
<\/td>\n\n

3\u00a0\u00a0\u00a0 Northern <\/span><\/b><\/p>\n<\/td>\n

\n

4\u00a0\u00a0\u00a0 Northwestern <\/span><\/b><\/p>\n<\/td>\n

\n

2\u00a0\u00a0\u00a0 Central <\/span><\/b><\/p>\n<\/td>\n

\n

1\u00a0\u00a0\u00a0 Eastern <\/span><\/b><\/p>\n<\/td>\n

\n

5\u00a0\u00a0\u00a0 Southwestern <\/span><\/b><\/p>\n<\/td>\n<\/tr>\n

\n

3\u00a0\u00a0\u00a0 Northern <\/span><\/b><\/p>\n<\/td>\n

\n

44454<\/p>\n<\/td>\n

\n

5379<\/p>\n<\/td>\n

\n

8927<\/p>\n<\/td>\n

\n

16667<\/p>\n<\/td>\n

\n

9389<\/p>\n<\/td>\n<\/tr>\n

\n

4\u00a0\u00a0\u00a0 Northwestern <\/span><\/b><\/p>\n<\/td>\n

\n

4953<\/p>\n<\/td>\n

\n

11228<\/p>\n<\/td>\n

\n

3129<\/p>\n<\/td>\n

\n

5982<\/p>\n<\/td>\n

\n

4429<\/p>\n<\/td>\n<\/tr>\n

\n

2\u00a0\u00a0\u00a0 Central <\/span><\/b><\/p>\n<\/td>\n

\n

8951<\/p>\n<\/td>\n

\n

3100<\/p>\n<\/td>\n

\n

28626<\/p>\n<\/td>\n

\n

20444<\/p>\n<\/td>\n

\n

5686<\/p>\n<\/td>\n<\/tr>\n

\n

1\u00a0\u00a0\u00a0 Eastern <\/span><\/b><\/p>\n<\/td>\n

\n

16610<\/p>\n<\/td>\n

\n

5896<\/p>\n<\/td>\n

\n

20320<\/p>\n<\/td>\n

\n

49329<\/p>\n<\/td>\n

\n

14607<\/p>\n<\/td>\n<\/tr>\n

\n

5\u00a0\u00a0\u00a0 Southwestern <\/span><\/b><\/p>\n<\/td>\n

\n

9445<\/p>\n<\/td>\n

\n

4403<\/p>\n<\/td>\n

\n

5705<\/p>\n<\/td>\n

\n

14705<\/p>\n<\/td>\n

\n

20524<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u00a0And the sizes of the mistakes of this forecast are as so.<\/p>\n\n\n\n\n\n\n\n\n
<\/td>\n\n

3\u00a0\u00a0\u00a0 Northern <\/span><\/b><\/p>\n<\/td>\n

\n

4\u00a0\u00a0\u00a0 Northwestern <\/span><\/b><\/p>\n<\/td>\n

\n

2\u00a0\u00a0\u00a0 Central <\/span><\/b><\/p>\n<\/td>\n

\n

1\u00a0\u00a0\u00a0 Eastern <\/span><\/b><\/p>\n<\/td>\n

\n

5\u00a0\u00a0\u00a0 Southwestern <\/span><\/b><\/p>\n<\/td>\n<\/tr>\n

\n

3\u00a0\u00a0\u00a0 Northern <\/span><\/b><\/p>\n<\/td>\n

\n

-1<\/p>\n<\/td>\n

\n

269<\/p>\n<\/td>\n

\n

-2<\/p>\n<\/td>\n

\n

-97<\/p>\n<\/td>\n

\n

164<\/p>\n<\/td>\n<\/tr>\n

\n

4\u00a0\u00a0\u00a0 Northwestern <\/span><\/b><\/p>\n<\/td>\n

\n

-159<\/p>\n<\/td>\n

\n

51<\/p>\n<\/td>\n

\n

18<\/p>\n<\/td>\n

\n

-353<\/p>\n<\/td>\n

\n

86<\/p>\n<\/td>\n<\/tr>\n

\n

2\u00a0\u00a0\u00a0 Central <\/span><\/b><\/p>\n<\/td>\n

\n

23<\/p>\n<\/td>\n

\n

-8<\/p>\n<\/td>\n

\n

13<\/p>\n<\/td>\n

\n

-81<\/p>\n<\/td>\n

\n

135<\/p>\n<\/td>\n<\/tr>\n

\n

1\u00a0\u00a0\u00a0 Eastern <\/span><\/b><\/p>\n<\/td>\n

\n

-50<\/p>\n<\/td>\n

\n

-399<\/p>\n<\/td>\n

\n

-81<\/p>\n<\/td>\n

\n

188<\/p>\n<\/td>\n

\n

-135<\/p>\n<\/td>\n<\/tr>\n

\n

5\u00a0\u00a0\u00a0 Southwestern <\/span><\/b><\/p>\n<\/td>\n

\n

192<\/p>\n<\/td>\n

\n

51<\/p>\n<\/td>\n

\n

135<\/p>\n<\/td>\n

\n

-136<\/p>\n<\/td>\n

\n

177<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u00a0Which compares with the earlier calculated mistakes from the gravity model, rounded.<\/p>\n\n\n\n\n\n\n\n\n
<\/td>\n\n

3\u00a0\u00a0\u00a0 Northern <\/span><\/b><\/p>\n<\/td>\n

\n

4\u00a0\u00a0\u00a0 Northwestern <\/span><\/b><\/p>\n<\/td>\n

\n

2\u00a0\u00a0\u00a0 Central <\/span><\/b><\/p>\n<\/td>\n

\n

1\u00a0\u00a0\u00a0 Eastern <\/span><\/b><\/p>\n<\/td>\n

\n

5\u00a0\u00a0\u00a0 Southwestern <\/span><\/b><\/p>\n<\/td>\n<\/tr>\n

\n

3\u00a0\u00a0\u00a0 Northern <\/span><\/b><\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n<\/tr>\n

\n

4\u00a0\u00a0\u00a0 Northwestern <\/span><\/b><\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n<\/tr>\n

\n

2\u00a0\u00a0\u00a0 Central <\/span><\/b><\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n<\/tr>\n

\n

1\u00a0\u00a0\u00a0 Eastern <\/span><\/b><\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n<\/tr>\n

\n

5\u00a0\u00a0\u00a0 Southwestern <\/span><\/b><\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n

\n

0<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u00a0All zeros!\u00a0 The mistakes from the application of the gravity model were, as stated earlier, inconsequential.<\/p>\n

Although the pivot point method was OK, the mistakes appear to\u00a0 be far worse than seen earlier with the gravity model.\u00a0 I am dismissing the possibility of a theoretical error in the pivot point equation derivation (which I have not checked) or in the spreadsheet calculations (which I have checked), so what gives?<\/p>\n

To some extent the comparison of the two methods is biased in favor of the gravity model, but I would not have expected the pivot point technique to have done this much worse. A major reason for the differing qualities of the results relates to how each technique treats the errors in the given, before OD table.\u00a0 Remember that splitting the full OD table into four smaller tables introduces a lot of error in each of those smaller tables.\u00a0 The pivot point method thinks these errors are useful, while the gravity model pretty much steamrolls over anything it doesn\u2019t like.<\/p>\n

And the pivot point technique uses the OD proportions from each cell multiple ways.\u00a0 Hence any errors inherent in the OD table are amplified.\u00a0 The effect of any error on the gravity model technique is more muted, since it only utilizes the row and column totals.\u00a0 This by itself is not a reason to abandon the pivot point technique, but it is a good reason for caution.\u00a0 Most empirical OD tables have a lot of error. Do we want a technique that accentuates the error or do we want a technique that quiets the error?<\/p>\n

Notwithstanding the better performance of the gravity model, there are trade-offs.\u00a0 I am sure this is not the last word on this subject.\u00a0 Give me a few weeks to concoct a better test that more fairly demonstrates the effect of errors on these techniques.<\/p>\n

Alan Horowitz, Whitefish Bay, April 26, 2017<\/i><\/p>\n

\n<\/div>\n

 <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Juan Ortuzar\u2019s book indeed provides a pivot point formula that allows two or more choice alternatives to be varied at the same time.\u00a0 So a valid question may be raised:\u00a0 which method works best, pivot point or the gravity model, for OD table forecasts?\u00a0 This blog article extends the example problem to pivot point and …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"enabled":false},"version":2}},"categories":[6],"tags":[],"class_list":["post-261","post","type-post","status-publish","format-standard","category-project-level-forecasting","anons"],"yoast_head":"\nA Well Choreographed OD Table Dance, Part 3: Pivot Point Comparison - 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