Experience with the Gravity Model

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 lt200 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)
• 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
• Ive 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 / Dista
• Where Pax Origin-Destination Demand
• WO Origin weight (size)
• WD Destination weight (size)
• Dist Intercity Distance
• a 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

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Early Observations Some Wild Outliers
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Fitzing with data

• Most Distlt200 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 pointslike 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 pointslt200 miles
• Due to ground competition
• We eliminate all points with lt12 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 dont work
• We try .3, .1, .01, and .001 for zeros
• We get rising a with smaller zeros, significantly
• We include only 5 zeros, but get same reactions

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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

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Compromise

• Ignore small demands and zeros
• Require Paxgt10 for all three cities
• Or drop the destination
• Merge multiple airports in a city to a single
  city destination
• (We had been using airports gt 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

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Early Observations

• Define draw as ratio actual / forecast

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Lessons Learned So Far

• Distance exponent a -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

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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 a of -1.5 (much different)
• Demands 1/5th of forecasts from domestic
• Suggests language, or other barriers count
• Goods research found borders act like 3000 mi.

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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 (Paxgt10)
• Using just a single variable (WO WD), with
  exponent ß
• Results
• Distance exponent a -0.55 (had been -0.66)
• City-Size exponent ß 0.85
• Suggests larger cities have smaller demands
• Maybe because higher of demands are gt1 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 ß 1.0 to be reasonable

16
Act 3 European Regional

• New set of data points
• London, Copenhagen, Istanbul
• 200 mi lt Distance lt 2800 mi
• All 3 (LON, CPH, IST) have Pax gt 10
• 219 points
• Regression Results
• Distance exponent a 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

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Europe All Points

• Distance gt 200 mi
• Pax gt 10
• Least squares regression
• Distance exponent a goes to -1.2
• Weights (WO WD) exponent ß 0.4
• Gives almost all demands near 40
• Results Not Satisfactory
• Distance exponent seems wrong (beyond -1)
• City size (weights) exponent ß too far from 1.0
• Unsatisfactory forecasts by inspection
• Most big markets forecast too small
• Most smaller market forecast too big

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Go Back to Detailed Look

• All markets with Pax gt 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

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Iterative Procedure

• Start with Distance and Weight Exponents 1
• Adjust scaling so median Forecast/Pax 1.00
• Adjust Weight exponent ß to reduce Error
• Readjust scaling on each try
• Adjust Distance exponent a to reduce Error
• Readjust scaling on each try
• Iterate to find min ? Diff0.75 (min Error)
• An ugly, unofficial, but practical, process

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Results from Procedure

• Distance Exponent a likes to be -1.05
• Could be cultural distance
• City Weights Exponent ß likes to be 1.25
• Why???
• Two effects are independent
• Many too big forecasts for small demands

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Poor Fit of Forecast to Data
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One Last Regression

• All Europe Classic Gravity formula
• Pax gt 10, Dist gt 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

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Gravity Forecast is Very Poor
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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

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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)

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More Gravity Long Haul

• All world markets
• Distance gt 3100 mi (5000 km)
• Passengers gt 20
• No existing nonstop service
• Least Squares Regressions
• Four Equations (log calibrations)
• Traditional calibrate ratio to gravity term
• Distance exponent a only (-1.37)
• Whole Gravity term exponent only (0.19)
• Separate City Size ß and Distance a exponents
• (ß 0.18 and a -0.03)

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Best Fit was not usefulmeasured 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

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Traditional Gravity was Best But not Good
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Median Forecasts are Weakly Correlated with
Actuals