Transport and society

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Transport and society

• Lecture 10 - 11
• Principles of transport analysis and forecasting

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What is a model?

• An abstract simplified representation of a
  phenomenon, often with the aim to understand or
  change it
• Examples
• An analogy or metaphor
• A map
• A physical model of a building
• But could also be a mathematical model
• A regression model of the relationship between
  car ownership and income
• A simulation model showing how vehicles are
  driving through a signalised intersection
• A comprehensive model of a region that shows how
  road pricing affects transport pattern and land
  use

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Modelling is an art

• What is interesting to model?
• How should the model be formulated?
• What data are available
• Which output variables are of interest?
• How should we interpret the results?
• but also science
• When we calibrate/estimate the model
• When we validate the model
• When we are doing mathematics inside the model

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Why do we need models?

• To be able to answer What if-questions
• To summarise our knowledge about different
  cause-effect relationships, say
• Congestion pricing
• A ring road
• It is most trustworthy to make an experiment
• But expensive, takes a lot of time, the outcome
  is often blurred with effects of other
  uncontrolled changes, and is irreversible
• A model allows many alternatives to be tested,
  where one thing is changed at a time
• Possible to fine-tune the alternatives

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Specifying a model

• You want to model travellers mode choices
• You assume that travellers tend to prefer modes
  with higher utility compared to ones with lower
  utility (utility maximisation)
• You try to describe the utility of each mode
  (car, bus, bike, walking) by specifying relevant
  explaining variables
• For example car utility for trveller n - Cnc
  - (IVTnc X1 OVTnc X2 OPCnc), where Cnc is
  generalised cost
• To calibrate/estimate a model you try to find the
  values of X1 and X2 that best explain (reproduce)
  the choices a sample of travellers have made.
  These values can be interpreted as values of time
• With the calibrated model at hand you can predict
  the behaviour of other travellers

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Classes of model available

• Simple formula (e.g. work trips per household per
  day by employed residents)
• yf(x)
• Time series models (e.g. annual flow of cars on a
  road section)
• Linear
• f (tn)f( t)nx
• where n is the annual increase
• Exponential as an example of non-linear
• f (tn)f(t)exp?n
• where the flow will be doubled for every
• n0log2/? year

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Classes of model available - cont

• Cyclic fluctuation

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Classes of model available - cont

• Growth with saturation

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Regression analysis

• Say to explain car ownership
• Linear regression
• yab1x1 b2x2 b3x3 bnxn
• where y is the dependent variable, x1, x2 ,..xn
  the independent variables, and b1, b2 ,..bn
  coefficients that are calibrated to best explain
  y in terms of x1, x2 ,..xn
• Various software available to do this including
  Excel

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Regression analysis - cont

• Examples

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Elasticity models
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Discrete choice models

• The most common is the logit model
• A traveller has to choice exactly one from a set
  of available choice alternatives (e.g. mode
  choice, destination choice) A 1,2,,I
• Each alternative has a utility ui
• The traveller is assumed to choose the
  alternative i in A that has the highest utility
  ui
• The utilities ui are stochastic of the form
• ui -ci ?i

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Logit models

• where
• ci the deterministically known generalised
  cost of alternative i
• ?i part of the utility that is not known the
  modeller and hence regarded as stochastic
• By assuming that the stochastic parts of the
  utilities, ?i , are independently and identically
  distributed according to a specific distribution
  (Gumbel) one can prove that the probabilities for
  choosing the various alternatives are
• Pi exp-?ci/?k exp-?ck

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Logit models - cont

• where
• ? expresses the sensitivity to the cost
  variable the higher the value of ?, the higher
  the probability of the choosing the alternative
  with the lowest cost (the more deterministic
  is the choice)
• ? is inversely proportional to the standard
  deviation of the random parts of the utilities,
  ?i (? p/(??6)

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Sensitivity to the value of ?

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Data needed an example

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Logit models - cont

• Logit models can be combined into hierarchical
  structures
• Gives a lot of flexibility in model building
• The logit model is now the building block in most
  modern travel demand model systems used all over
  the world
• You will exercise calibration and application of
  logit models in a computer lab
• You will study this further in the course
  Transport modelling

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Matrix estimation models

• Travel demand in an area is often represented in
  form of travel matrix where the element Tijt is
  the number of trips from zone i to zone j at time
  t
• When you want to create a new matrix that you
  believe is not so far from an old matrix
• Updating to year t1 an old matrix for year t0
• Uniform growth
• Tijt1 Tijt0 Gt1t0
• Singly constrained
• Tijt1 Tijt0 Git1t0 New row sums for t1
  given (origin or production constrained)
• Tijt1 Tijt0 Gjt1t0 New column sums for t1
  given (destination or attraction constrained)

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Matrix estimation models - cont

• Doubly constrained
• Both new row sums and new column sums given for
  t1
• You can first multiply each row in the old
  matrix to fit the new row sums. Then multiply
  each column to fit the new column sums and then
  again multiply the rows etc. until the new matrix
  fits both the new row sums and the new column
  sums
• This is called the Furness procedure and
  converges very fast

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Gravity model

• When you have no reasonable old matrix to start
  with
• Singly constrained gravity model (by origin)
• Let Oi be the number of trips that start in zone
  i, cij the generalised travel cost between i and
  j,
• Dj some trip attraction variable for zone j
• Could also be interpreted as a logit model

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Gravity model - cont

• Singly constrained gravity model (by destination)
• Let Dj be the number of trips that end in zone
  j, cij the generalised travel cost between i and
  j,
• Oi some trip attraction variable for zone i
• Could also be interpreted as a logit model

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Gravity model - cont

• Doubly constrained gravity model
• Let Oi be the number of trips that start in zone
  i, Dj be the number of trips that end in zone j,
  and
• cij the generalised travel cost between i and j
• Either or
  equivalently
• Ai and Bj are called balancing factors
• Again, could also be interpreted as a logit model

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Congestion modelling

• In urban road networks, travel time on a link
  typically depends on the flow on the link, e.g.

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Network equilibrium

• Assume 4000 vehicles/hour are going from A to B
  and that they may choose between link 1 and 2
• Link 1
• A
  B
• 
  Link 2

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Network equilibrium

• Under network equilibrium traffic arranges itself
  such that both links will have the same travel
  time (if both are used)
• Link 1
• A
  B
• 
  Link 2
• 
  
  Equilibrium travel
• 
  
  time 27.1 min
• 
  
  Flow link 1is 1522
• 
  
  Flow link 2 is 2478

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Network equilibrium and system optimum

• However network equilibrium is typically not
  system optimal!
• In the previous case, minimum average travel time
  26.8 min is obtained for f11424
• How to achieve system optimum as a network
  equilibrium? Impose optimal congestion charges

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Transport modelling in practice

• How can a traveller react on a change?
• By changing
• Departure time
• Mode
• Route
• Destination
• Trip frequency
• and in the long run also
• Car ownership
• Residential and employment location

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The land-use transport feedback cycle
Wegener (1995)
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Four-stage (sequential) travel demand model
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How can models be used?

• To predict future conditions if nothing is done
• Number of trips, car ownership, mode choice,
  travel times, vehicle kilometres travelled,
  petrol use, CO2-emissions, revenues
• To predict future conditions on the assumption of
  a series of specified policies or investments
• Provides a basis for appraisal of their relative
  costs and benefits
• To test the performance of given policy in a
  series of alternative futures
• To test the robustness given future uncertainty
• To produce short-term forecasts that may be
  needed in sophisticated traffic control systems

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How is a model study carried out?

• A base scenario is compared to one or more
  alternative scenarios
• A description of each scenario is fed into the
  model
• External conditions, usually assumed to be the
  same for all scenarios, are also fed into the
  model
• The results from the model for the different
  scenarios are compared
• Optimisation models
• The model finds the scenario that best fulfils a
  given objective

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The design of a model

• The purpose of model determines
• The geographical coverage and scale
• The level of details
• Output indicators
• Data requirements
• Computing resources

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Desirable features of a model

• Would ideally produce accurate forecast at low
  costs in terms of data and computing resources
• Accuracy and precision
• Economy in data and computing resources
• Ability to produce relevant indicators
• Ability to represent relevant processes and
  interactions (to what should the model be
  sensitive)
• Appropriate geographical spread
• Transparent and user friendly (easy to use
  without making handling mistakes)

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Can we trust the models?

• Practical validation
• What kind of policy issues should be amenable to
  analysis, appropriate system level, which parts
  of the system are modelled, which data are
  available, which mechanisms are modelled, which
  resources and competences are available, how
  transparent and user-friendly is the model, error
  detection, documentation?
• Theoretical validation
• Theoretical foundation, reasonable behavioural
  assumptions, reasonable causal relationships, how
  consistent are different submodels?

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Can we trust the models?

• Internal validation
• How well can the model reproduce the data it was
  estimated on, are all estimated parameters of the
  right sign and significant, does the model
  respond in a reasonable way to external changes?
• External validation
• Can the model reproduce independent data, are
  the elasticities in accordance with elasticities
  in the literature, can the model predict a future
  or historical year, or the situation in another
  town (transferable)?

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The usefulness of models

• Pedagogical tool increases our understanding
• Necessitates more precise formulation of the
  problem
• Identifies lack of knowledge and data
• Checks the consistency of available courses of
  action
• Clarifies goal conflicts
• Clarifies uncertainties
• Provides a common framework for the evaluation of
  different kinds of measures