1. Experience with the Gravity Model

2. Introduction
There is demand for air travel between every city-pair in the world (can be very small)
We have imperfect data on the actual travel (for many of the larger demands)
The “Gravity Model” is the long-standing traditional formulation
Demand is bigger, the bigger the origin city
Demand is bigger, the bigger the destination
Demand is smaller, the longer the distance
Other things may also matter

3. Prologue We have some experience and prejudices
Doubling the origin city size should double the demand
Doubling the destination size should double the demand
Cost may be a better metric than distance
The “zero” demands should not be left out
Air Demand <200 miles competes with ground
Common Language helps demand
Common “alphabet” helps demand
Different incomes hurts demand
Gravity works better at distributing total outbound demand than at estimating size of total travel
Leisure destinations are origin-specific and arbitrary

4. Act 1: We go Exploring Guidelines (“Pirates’ Code”) : US domestic data
Take the easiest first
Use places you know about
Examine the results in detail
US domestic data
Best reporting quality
One country, one language, one income
Lots of points
Use Seattle (SEA), Boston (BOS), Chicago (CHI)
Disparate types of cities
I’ve lived there

5. SEA, BOS, & CHI Passenger data from US ticket sample
Origin-Destination reporting
Some “breakage” of interline trips
Domestic points only: similar fares, taxes, hassle
Origin gravity weight:
Not population or income or .....
Use outbound departing passengers
Focus results on distributing to destinations
Destination gravity weight:
Arriving passengers to destination
O-D data as used is not directional
Original source has home city for trips
All further data (outside US) will not
So we will use US data in non-directional form

6. Starter Formulation Calibrate gravity model
Pax = WO · WD / Distα
Where Pax = Origin-Destination Demand
WO = Origin weight (size)
WD = Destination weight (size)
Dist = Intercity Distance
α = Distance exponent (calibrated variable)
Use Log formulation, linear least squares fit
Examine forecast to actual Passengers/Demand
Allow origin size WO to be a calibrated variable
One each for SEA, BOS, CHI

7. Early Observations: Some Wild Outliers
Here “Draw” is the ratio of actual to forecast demand, indexed. Distance exponent here is -0.6.

8. “Fitzing” with data Most Dist<200 had low actuals
Demand diverted to surface modes not in data
SEA high actuals were:
Points in Alaska
Had trips interlining in SEA--with “broken’ data
Were dedicated Seattle points—like college towns
BOS high actuals were:
Leisure destinations, for Boston
Characteristics of high actuals
Destinations had small number of origin cities
Destinations had one large demand –to origin
Some were secondary airports in a city
End of Act 1

9. Act 2: Our First Regressions
We eliminate all points<200 miles
Due to ground competition
We eliminate all points with <12 origins
Tend to be captive-to-single-origin points
We did a big side-study on share-of-largest origin
We generate “zeros” by destination (16%)
When 1 or 2 of SEA, BOS, CHI lack demand
Due to log form, zeros don’t work
We try .3, .1, .01, and .001 for zeros
We get rising α with smaller zeros, significantly
We include only 5% zeros, but get same reactions
Smallest value in data is 1 passenger per day. Rounded.

10. Small Demands are a Real Problem
Regression results driven by zero points
Least squares in log form gives equal weight to each demand point
Log form emphasizes percentage error
Actual needs are different:
Forecast big demands with smaller % errors
Forecast the small demands merely as “small”

11. Compromise Ignore small demands and zeros
Require Pax>10 for all three cities
Or drop the destination
Merge multiple airports in a city to a single city destination
(We had been using airports => cities)
We now get same answers, with or without remaining outliers (errors below ½ or above 2)
Errors on large demands more reasonable
Most small markets forecast as small
Exceptions are large for one origin
Could be large for other two, but no online services
Outliers were 24% of data. Before these requirements, answers changed as outliers were removed, interatively.

12. Define “draw” as ratio = actual / forecast
Early Observations
Define “draw” as ratio = actual / forecast

13. Lessons Learned So Far Distance exponent α = -0.66
NOT the same as domestic fares (Fare ~ Dist0.2)
Do not include zero demand points
Destinations with few origins tend to be “captive”
Do not use them in generic calibrations
To improve errors in forecasts of large demands, use only points with large demands
Result will forecast small demands small, mostly
Use Cities, not airports

14. More Lessons City WO fairly consistent with city size
More about this on next slide
Ran against Pax data adjusted to standard fares
Many “under-forecasts” were in discount markets
Ran international destinations
True “O & D” not from US ticketing source
Distance exponent α of -1.5 (much different)
Demands 1/5th of forecasts from domestic
Suggests language, or other barriers count
Goods research found borders act like mi.
Passengers adjusted to “standard” national fare trend formula using price elasticity of -1.2
Research on goods trade between the US and Canada. I read it, but I did not note the reference, as I was not thinking about this research at the time.

15. Play within the Play
Observed different ratios to total outbound travel for SEA, BOS, CHI (Wo).
But not very different
Ran all US domestic pairs (Pax>10)
Using just a single variable (WO · WD), with exponent β
Results:
Distance exponent α = (had been -0.66)
City-Size exponent β = 0.85
Suggests larger cities have smaller demands
Maybe because higher % of demands are >1 and therefore are captured by data base. (Bigger W = ∑ demands.)
Also small cities show more short-haul, which was removed
Otherwise, large cities have more direct services & lower fares !
Interpretation allows β = 1.0 to be “reasonable”
Drive for beta=1.00 is from the “intuition” or “logic” that doubling the city should double the travel (ceteris paribus), and doubling the destination size, ditto.

16. Act 3: European Regional
New set of data points
London, Copenhagen, Istanbul
200 mi < Distance < 2800 mi
All 3 (LON, CPH, IST) have Pax > 10
219 points
Regression Results
Distance exponent α near -0.80
Origin-Specific adjustments not significant
Removing outliers has small effect on answers
Some really big errors in really big markets
Tends to confirm US data experiences

17. Europe: All Points Distance > 200 mi Pax > 10
Least squares regression
Distance exponent α goes to -1.2
Weights (WO · WD) exponent β = 0.4
Gives almost all demands near 40
Results Not Satisfactory
Distance exponent seems “wrong” (beyond -1)
City size (weights) exponent β too far from 1.0
Unsatisfactory forecasts by inspection
Most big markets forecast too small
Most smaller market forecast too big

18. Go Back to Detailed Look
All markets with Pax > 200
Drop 12 high-side outliers
Redefine Error:
Not percentage-error-squared (log least sq.)
Not Diff = (Passengers – Forecast)
Compromise: Diff0.75
Compromise is halfway between size and %
Iterate
Difference (Pax-fore) in absolute value, no - sign

19. Iterative Procedure Start with Distance and Weight Exponents = 1
Adjust scaling so median Forecast/Pax = 1.00
Adjust Weight exponent β to reduce Error
Readjust scaling on each try
Adjust Distance exponent α to reduce Error
Iterate to find min ∑ Diff (min Error)
An “ugly,” unofficial, but practical, process

20. Results from “Procedure”
Distance Exponent α likes to be -1.05
Could be “cultural distance”
City Weights Exponent β likes to be 1.25
Why???
Two effects are independent
Many “too big” forecasts for small demands

21. Poor Fit of Forecast to Data
Forecast is least sq regression resulting in distance exponent of -1.15
Procedure is outlined in slides above.
Results of procedure with exponents of -1.0 for distance and 1.0 for city weights were very similar. And still lousy

22. One Last Regression All Europe – Classic Gravity formula
Pax > 10, Dist > 200
Distance exponent fixed at 1.00
City weight exponent fixed at 1.00
Allowed “factor” for “same country”
Was about 5x, as for US vs International
Nice scatter
Fewer unreasonable forecasts
Huge errors everywhere
Distance fixed because > 1.00 is too big
Weight fixed because it “makes sense” from doubling city= doubling demand

23. Gravity Forecast is Very Poor

24. Obituary on Gravity Model
Forecasts are really bad
Outliers have large effect on answer
Need to be removed
Zeros have large effect on answer
Forecasts more sensible when not included
Results will be misleading
Small markets will be forecast as medium

25. Overall Conclusion Air travel between cities is
Strongly influenced by city-pair specific factors
Not amenable to gravity model approach
If you have to have a forecast
Calibrate from existing larger culturally similar cities to same destinations
Recognize the “same country” effect is large (maybe 5x)

26. More Gravity: Long Haul
All world markets
Distance > 3100 mi (5000 km)
Passengers > 20
No existing nonstop service
Least Squares Regressions
Four Equations (log calibrations):
Traditional: calibrate ratio to gravity term
Distance exponent α only (-1.37)
Whole Gravity term exponent only (0.19)
Separate City Size β and Distance α exponents
(β = 0.18 and α = -0.03)

27. “Best Fit” was not useful measured by either % or value errors
Models 3 & 4 fit best
Fit achieved by low variance
No forecasts at large values
No forecasts at small values
Most forecasts near 40
This is a pretty worthless “forecast”
Model 2 had much worse % misses than 1
Traditional Gravity form had least harmful answers

28. Traditional Gravity was “Best” But not “Good”
This is traditional gravity: Pax = K * (Origin Size * Destin Size)/Distance
Has been rescaled (forecast x 1.5) so averages are about right for Paxx divisions (next slide)

29. Median Forecasts are Weakly Correlated with Actuals