Predicting ridership?

It's been hard for me to find a good introduction to ridership-prediction models for urban transit and intercity rail lines.

However, I've found
Intercity Rail Ridership Forecasting and the Implementation of High-Speed Rail in California [eScholarship]
Alon Levy's blog entry on Sanity Checks on HSR Ridership | Pedestrian Observations

They mention a "gravity model" that goes something like this:
T(i,j) = P(i)*P(j)/C(i,j)

where T is the number trips, the P's are the origin and destination populations or places to go, and C is a cost function, often modeled as a function of distance or travel time. I've found mentions of gravity models in various other places with the help of Google Scholar.

Alon Levy mentioned that SNCF proposes that the populations be taken to some power of 0.8 to 0.9 for best fit.

I can find riders per station for some systems, and even station-to-station ridership for one system: BART - Monthly Ridership Reports. It would be difficult to test station-to-station ridership models without a lot of detail on urban geography, however. But it may be possible to test hypotheses of preferred destinations.

Also, for urban-rail systems, it may be possible to find ridership statistics over the years for some of the more recent urban-rail systems, since many of them have been tended over the years. So one could test some hypotheses about whether the number of riders is linear (if the lines radiate out from the biggest destination) or more than linear (if many riders do multi-line trips). Places like Portland OR, Sacramento, San Diego, and San Jose.

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On the intercity side, I can't find any station-to-station statistics anywhere in Amtrak's site, though I can find numbers of riders per station and riders for some routes.

So I did something that has a certain apples-and-oranges risk: use airline ridership statistics. I used US ones from the US DoT. The gravity model didn't work very well. Instead, I found a model with a sum instead of a product of populations:
T(i,j) = (P(i)+P(j))/C(i,j)
Curiously, my estimated cost function was nearly flat as a function of distance. I note that the data points have a *lot* of scatter, and that they are bimodal. I found a lot of heavy-traffic routes and a lot of light-traffic ones, with a dividing line of about 6000 trips/direction/year.

But I found a page that suggests something about the rail cost function: Microsoft Word - Final report FINAL.doc - 2006_08_study_air_rail_competition_en.pdf The rail fraction of rail/air ridership can be approximately fit with a line from 100% rail at 1h 30m to 0% rail at 5h 30m. The halfway point is 3h 30m, and that is where the rail cost function likely passes the air one.

The page Microsoft Word - Final report FINAL.doc - 2006_08_study_air_rail_competition_en.pdf
AIR AND RAIL COMPETITION AND COMPLEMENTARITY, August 2006
by Steer Davies Gleave, a London consulting firm

From their rail vs. air data for rail travel time, I fit a logistic function:
F(t0,tw,t) = 100% / (1 + exp((t - t0)/tw))
with
t0 = 3.5 hr
tw = 1.1 hr
using Mathematica's function FindFit

How well does it predict Amtrak's ridership for some of its busier corridor trains? From Amtrak's schedules and Presentation from the US DOT Intercity Passenger Rail Forum | Federal Railroad Administration, I get:

CIty Pair -- Actual % -- Time (hr) -- Predicted %
NYC - DC -- 77% -- 2.75 -- 66%
NYC - Boston -- 54% -- 3.6 -- 48%
LA - San Diego -- 96% -- 2.75 -- 66%
Chicago - Milwaukee -- 89% -- 1.5 -- 86%
St. Louis - Kansas City -- 41% -- 5.67 -- 12%
Seattle - Portland -- 69% -- 2.75 -- 66%
Chicago - St.Louis -- 29% -- 5.5 -- 14%
Chicago - Detroit -- 12% -- 6.3 -- 7%

So if anything, this model is somewhat pessimistic about Amtrak's ridership.

The authors of that paper prepared a more detailed model using typical pricing and a time equivalent for infrequent service, and got very good results. Their frequency-penalty time equivalent is
75 m * (travel time / 180 m)^(0.7)

So from 1 train every 3 hours to 1 train every 1 hour, the time penalty decreases from 75 m to 35 m. At 15 m, it is 13 m. So running trains frequently can be a good alternative to speeding them up.

On short distance corridors it shouldn't be a surprise that the air/rail split would favor high(er)speed trains. What would be more interesting would be comparing the split between air/rail/auto traffic in those same segments. As gas prices rise and urban transit systems provide better connections to intercity train stations, shouldn't train travel increase? The CAHSR project can expect lots of air travelers between SoCal and the Bay Area to jump on the trains, but a lot of the ridership should also come from diverting auto drivers between Fresno and Los Angeles into train passengers. Are there any good ways to model that ridership?

If I find anything on auto vs. train vs. airplane, I'll let you people know. Also including buses if possible.

But in the meantime, I decided to crunch the numbers from BART. At least the March 2013 numbers.

I got the data into Mathematica (some computer-algebra software) and then got to work on it. I did various manipulations, like for all the riders who entered at each station, find the fraction that exited each of the stations, and also the converse of that.

Downtown San Francisco is the most common destination in the system, something that was true everywhere outside of that area. Weekdays: Embarcadero and Montgomery, at the Financial District near the waterfront. Weekends: Powell, a little bit inward. There is a rather curious effect in downtown SF on weekdays. More people exit at Embarcadero than enter, and more people enter at Powell and Civic Center, a bit more inward, than exist. This suggests that many returning East Bay commuters like to try to get seats by entering a bit upstream of the Financial District. Is that something that happens in other transit systems?

It was apparent in the total number of entries and exits at each station, so if one can only get that data, one might be able to find it in other transit systems.

There were some less-important destinations apparent in the data. Downtown Oakland (12th, 19th Sts.), UC Berkeley, SF Airport, and Oakland Coliseum and Airport.

Looking at the ends of the lines, Fremont and East Dublin-Pleasanton have at least twice as many riders as their neighboring stations, but Pittsburg-BayPoint does not. Curiously, Richmond does not have as many riders as the station just inward, El Cerrito del Norte.

To see if this effect happens in similar systems, I checked Washington Metrorail: Metro - About Metro - Public Records. I checked the end stations against the two nearest ones for all 9 outlying line ends, and I found that the outermost station typically had around twice the riders of the next two outermost ones. Separately and not combined. So I think that this end-station effect is real.

BART's yellow line used to terminate at Concord and I think most of the bus connections are still located at Concord, which might explain lower ridership to Pittsburg/Bay Point.

Richmond is kind of a dump city and there isn't much catchment area around the station there. There is a BART O&M base connected to the Richmond BART station, however. Amtrak briefly experimented with stopping the Coast Starlight at Richmond (it's an easy walk between the Richmond Amtrak Station and the BART station), but that experiment didn't last very long.

I crunched the numbers for the Washington Metro. The data that I worked from is the number of station entries for each year from 1977 to 2012, May weekdays of most years.

I found a curious result. Many stations have had approximately constant ridership over the years, though with about 20% - 30% variation. That may mean that once such a station opens, it quickly attracts most of the riders who will use it. That is likely related to how the WM was built. The first stations opened were in downtown DC, and outlying stations followed later. As they opened, they attracted a certain number of commuters, a number when then stayed approximately unchanged.

This page has a nice in-page slideshow of the growth of the WM system: Evolution of Metrorail animation, now with Rush Plus - Greater Greater Washington

lpetrich wrote:How well does it predict Amtrak's ridership for some of its busier corridor trains? From Amtrak's schedules and Presentation from the US DOT Intercity Passenger Rail Forum | Federal Railroad Administration, I get:

CIty Pair -- Actual % -- Time (hr) -- Predicted %
NYC - DC -- 77% -- 2.75 -- 66%
NYC - Boston -- 54% -- 3.6 -- 48%
LA - San Diego -- 96% -- 2.75 -- 66%
Chicago - Milwaukee -- 89% -- 1.5 -- 86%
St. Louis - Kansas City -- 41% -- 5.67 -- 12%
Seattle - Portland -- 69% -- 2.75 -- 66%
Chicago - St.Louis -- 29% -- 5.5 -- 14%
Chicago - Detroit -- 12% -- 6.3 -- 7%

So if anything, this model is somewhat pessimistic about Amtrak's ridership.

Allow me to graph the American data.

electricron wrote:the NYC to BOS market share being the example, is around 50% even though there are dozens of trains every day. I'm not sure how many flights there are between these two cities anymore.

For tomorrow, (a Wednesday), there are 53 flights with approximately 6000 seats, from the NYC area (LGA, JFK, EWR) to Boston Logan, and presumably a similar number going the other way. There's a few other flights to/from other smaller airports "in" both metros, but that's the vast majority.

Keep in mind that planes have a much bigger market than the trains do, as many people on those flights (particularly on the NYC end) are connecting through, and originated from somewhere else, often on a smaller regional flight.

And in practice, you can't really mix transport modes even if relatively easy connections exist, partly because of airline pricing schemes (Ex: Binghamton-EWR-Boston - $175. Want to just do Binghamton-EWR to then catch Amtrak? $150.), and partly because airlines will only honor their own delays for rebooking/refunds. If you miss a flight because the train was late to get you to the airport, the airline will not refund you any money and you're screwed.

So there's a lot of people flying between the two, but many of them are not actually a part of the NYC/BOS market for Amtrak.

millerm277 wrote: So there's a lot of people flying between the two, but many of them are not actually a part of the NYC/BOS market for Amtrak.

A significant fraction of them are doing things like flying from Harrisburg changing planes in one of the NY airports to get to Boston. Or Hartford to Baltimore. Or since the non stops from Albany to DCA are breathtaklingly expense people change planes in New York or Phialdelphia to get from Albany to DC. Or Harrisburg to Hartford. Once the core is faster a lot of the people changing planes disappear too. Not all of them but enough that a flight or two from all the secondary airports to the hub airports, over a day are 20, 30 less. It's not just the shuttle flights that go away. Once the branches are faster even more of them take the train.

... and it doesn't take much-faster on the branches to make significant differences.

Greater metro Albany is 1.1 million and Hartford-Springfield is 1.2. The Connecticut stations had a ridership of 286,479 in fiscal year 2012. Springfield had 143,605. Hartford had the most at 179,536
Albany had 769,413. Tiny little Hudson the next stop south had 167,286.

.... Delaware has 917,092 people. Wilmington had 737,846 Amtrak riders....

It's a virtuous circle. Make it a bit faster and some of the flyers get attracted away which means it's a bit more frequent which attracts away some of the drivers. Which means it makes sense to make it a bit faster which attracts away a few more flyers which makes it once an hour. At once an hour it attracts away more drivers. Which means it makes sense to make it a bit faster....

I'd be more comfortable with those market share numbers if we knew how many were driving themselves. 3, 4 hours is not a long trip and it would not be cheaper than buses. I wonder if downtown to downtown travelers are more inclined to take a train, as opposed to a suburb being either or both origin or destination. I know it's a formula hopefully useful in planning, but I know formulas don't always work. Example- in the c.2000 slump New England had plenty of empty retail space, but developers were building more because a formula showed we had too little retail sf/ person.

As for transit, wouldn't the number of riders rise and fall with the job market and total population? I can understand market share staying constant. And I suggest a generalization that rapid transit terminals have much higher boardings because 1- bus routes are cut back to terminate at them, 2- they have ample parking, and 3- were built with better highway access.