Meeghat Habibian Analysis of Travel Choice Transportation Demand Analysis Lecture note

Choice context in transportation  Destination choice  Mode choice  Route choice  Also  Travel start time choice  Freight transportation agent choice  … 2 Transportation Demand Analysis - Title

Measurement of Choice Choice can be reflected by:  Number of people  Proportion of population  Therefore, it can be: 3 Independent of the total number of population Transportation Demand Analysis - Title

Choice Process 1.Deterministic  The Decision rule that is used by traveler is consistent and stable 4 The Choice made will be consistently the same Transportation Demand Analysis - Title

Deterministic Choice (Individual Level) V(i)=A i *X i V(i): choice function X i : a vector of demand and supply variables A i : a vector of parameters that represent the effect of each variable 5 Transportation Demand Analysis - Title  The choice rule: V(j)=max[V(i)]

Example  Route choice  The choice function V(i)=-0.2t i -1.0(c i /B) t i : travel time (hours) of alternative i c i : travel cost ($) of alternative i B: annual income of individual (1000$) 6 Transportation Demand Analysis - Title

Example  Marginal rate of substitution between cost and time As: Therefore: 7  Value of time per hour=20% of annual income/1000  The time value of a person with annual income of $20000 would be $4/h Transportation Demand Analysis - Title

Choice Process Due to:  Behavior of choice maker  Absence of rational and consistent decision rule 8 Provide a far superior means for predicting travel behavior than Deterministic one Transportation Demand Analysis - Title 2. Stochastic

Stochastic Choice A stochastic model is preferable because: 1.Idiosyncrasies of traveler behavior isn’t anticipated 2.It is impossible to include all the variables in the choice function 3.Potential traveler don’t have full information about system and alternatives 9 Choice function is considered as a random function Transportation Demand Analysis - Title

Stochastic Choice  Random utility model: U(i)=V(i)+e(i) U(i)=choice function for alternative i V(i)=deterministic function for alternative i e(i)=a stochastic component 10 Statistical assumptions are made regarding to the distributional nature of e(i) Transportation Demand Analysis - Title

Stochastic Choice 11  The probability that the alternative i is chosen: Transportation Demand Analysis - Title

Stochastic Choice 12  Therefore: F: Joint distribution of the random component f i (Φ): Density function of e(i)  Based on f i (Φ), different structures can be derived  Due to lack of knowledge about the error term a number of assumptions could be made (e.g., Normal distribution) Transportation Demand Analysis - Title

The Probit Model Transportation Demand Analysis Lecture note

The Probit Model  Random Utilities [U(i),U(j),…] have a multivariate normal distribution (MVN) 14 n: Number of alternatives б ij : Variance-covariance elements Transportation Demand Analysis - Title

The Probit Model  This is equal to: 15  e i follows:  Multivariate normal distribution  With zero mean  A finite variance-covariance matrix Transportation Demand Analysis - Title

The Probit Model  The resulted model is extensive and expensive  No closed form  An approximated closed form is suggested by Clark 16 Max [U(1),U(2),…,U(n)]~ N( V max, б max 2 ) U(i) multivariate normal variables with means V(i) and covariance elements б ij Transportation Demand Analysis - Title

The Probit Model  Defining for any two normally distributed variable U 1, U 2 17 ρ 12 : correlation coefficient Φ: standard normal distribution Ø: density function  Clark: Transportation Demand Analysis - Title + 12

The Probit Model And correlation between U 3 and max of U 1, U 2 18 Clark: Transportation Demand Analysis - Title

The Probit Model 19 And finally the choice probability is: Transportation Demand Analysis - Title

The Probit Model  For two alternatives from (5-5) 20  For more than 3 alternatives: Transportation Demand Analysis - Title

Example 1 V=[-12,-10,-15] (negative utilities such as cost or travel time) ρ 12 =0.5 (correlation of attribute 1,2) б 12 =0.5*2*2=2 ρ 23 =0 21 Transportation Demand Analysis - Title

Example 1 22 Transportation Demand Analysis - Title

Example 1 from equation (5-12): 23 In a similar manner: p(2)=0.81 p(3)=0.05 p(1)+ p(2)+p(3)= =1.01 is sufficiently close to 1.0 Transportation Demand Analysis - Title

Binary Probit Model U(1),U(2) assumed independent and have normal distribution so: 24 from Eq.(5-12): Transportation Demand Analysis - Title

Binary Probit Model  Even V(1)<V(2) there is a non zero probability of choosing alternative 1  The larger utility function for an alternative the larger probability of its choice 25 All or nothing vs. Probit: Transportation Demand Analysis - Title

The Logit Model Transportation Demand Analysis Lecture note

The Logit Model This model is obtained by assuming that the random component, e(i), of choice utilities are IID:  Independent  Identically distributed through a Gumbel distribution: 27 ɵ= Scale parameter of Gumbel distribution Transportation Demand Analysis - Title

The Logit Model By combining: 28 Transportation Demand Analysis - Title and

The Logit Model Formulation 29 Multinomial Logit Model (MNL) Transportation Demand Analysis - Title

The Logit Model  Advantages: Easier (than the Probit) in terms of  Parameters estimation  Application  Interpretation  Disadvantage: Restricted to the situations where alternatives have independent choices 30 Route choice over a complex network Transportation Demand Analysis - Title

Example Choice vector: V=[-12,-10,-15] Direct application of Eq.(5-20): 31 Transportation Demand Analysis - Title

Comparison of Logit and Probit Models Transportation Demand Analysis Lecture note

Comparison of Logit and Probit Models 33 Logit model P(1)=0.12 P(2)=0.875 P(3)=0.005 Probit model P(1)=0.15 P(2)=0.81 P(3)=0.05 The resulting logit model has the tendency to reduce the choice for low V as P(3) and increase it with high V as P(2) Transportation Demand Analysis - Title

Comparison of Logit and Probit Models  Assuming independence of the utilities There is not much difference between the results 34 Binary logit and probit models of mode choice Transportation Demand Analysis - Title

Comparison of Logit and Probit Models  Alternatives with similar attributes, or with overlapping components that independence can’t be assumed, Probit might be a better model  Alternatives that can be mutually exclusive, a Logit model would be appropriate 35 Alternative routes overlap on a link Intercity travel mode choice Destination choice Transportation Demand Analysis - Title

Example I: can represent a highway mode II: a bus mode with walking access III: a bus mode with walking access 36  Three different routes between points A and B: The overlap part: The alternatives ІІ and ІІІ are not independent Transportation Demand Analysis - Title

The Extent of Error from Independence Assumption x: measures the length of AC in comparison to AB 37 X=0 total overlap between ІІ and ІІІ X=1 independent alternatives  Assuming identical utility function V: Transportation Demand Analysis - Title

The Extent of Error from Independence Assumption  A choice model without dependence between e(ІІ), e(ІІІ) will always predict p(ІІІ)=0.33  A model with dependence term shows a more realistic result 38 Transportation Demand Analysis - Title  Covariance of alternatives II and III is a function of x

The Independence of Irrelevant Alternatives (IIA) Relative odds of choosing one alternative over another 39 Relative odds between any two alternatives are independent of any other alternatives Strength or weakness of model that have this property such as Logit Transportation Demand Analysis - Title

The Independence of Irrelevant Alternatives (IIA) For example in urban mode choice:  Relative odds of taking automobile over taking a bus is independent of there is a train or not.  But presence of a train as a third alternative affects probability of choosing the bus more 40 Transportation Demand Analysis - Title

The Independence of Irrelevant Alternatives (IIA) The binary logit model:  Can be derived as a deterministic choice model  This property leads to make logit model as intrinsically linear: 41 Transportation Demand Analysis - Title

The Nested Logit Model Transportation Demand Analysis Lecture note

The Nested Logit Model Remember the problem:  Logit model: – Private car (0.33) – Red bus (0.33) – Blue bus (0.33)  Expectation: – Private car (0.50) – Red bus (0.25) – Blue bus (0.25) 43 Transportation Demand Analysis - Title

The Nested Logit Model Due to:  Application of dependent alternatives  Difficulty of Probit model The Nested Logit structure is developed: 44 NLBusBlueRedCar MNLBlue BusRed BusCar Transportation Demand Analysis - Title

The Nested Logit Model 45 kmkm Var m ≤ Var k  μ k ≤ μ m  ɵ k =μ k /μ m ≤1 μ=Scale parameter of Gumbel distribution Transportation Demand Analysis - Title

Calibration of Choice Models Transportation Demand Analysis Lecture note

Calibration of Choice Models  Estimate the parameters’ values  Performing the statistical test  Validating the model by comparing with observed data 47 Transportation Demand Analysis - Title

Multinomial Disaggregate Models V(n,m): choice function constructed for each individual n and alternative m x inm : i th variable for alternative m as measured for individual n 48  Assume:  The choice model: V(n,m)= ∑ i β im X inm P n (m)= f [V(n,m)]= g [ ∑ i β im X inm ] Transportation Demand Analysis - Title

Multinomial Disaggregate Models  To estimate the model parameters β im, we define: 1 if individual n chooses alternative m 0 Otherwise N m =Σ n Y nm 49 N m : number of individuals who choose alternative m in the sample Transportation Demand Analysis - Title Y nm =

The Likelihood Function The likelihood of the observed sample is given by: 50 To facilitate the procedure: Transportation Demand Analysis - Title

The Likelihood Function 51 i: origin j: destination Transportation Demand Analysis - Title

Multinomial Disaggregate Models The maximum of LL can be obtained by: 52 The confidence intervals for β im ^ are asymptotically efficient. ƏLL/Əβ im =0 Transportation Demand Analysis - Title

Goodness of Fit Transportation Demand Analysis Lecture note

Definitions  Logarithm of likelihood function is also presented as L(β)  Assuming no variable in model (all β=0)  L(0)  Assuming market share (only constants in utility functions)  L(c)  Note: L(0)<L(c)<L(β)<0 54 Transportation Demand Analysis - Title

Relations  L(0)=Σ n Ln (C n ) – C n : Number of choices for individual (market) n  L(c)=Σ m N m *Ln (N m /N) – N m : Number of individuals adopt alternative m – N: Sampled population 55 Transportation Demand Analysis - Title

Goodness of Fit measures  Market share goodness of fit:  Goodness of fit:  Goodness of fit regarding to market share (Also known as Mc Fadden Goodness of fit):  Adjusted goodness of fit:  Adjusted goodness of fit regarding to market share: 56 Transportation Demand Analysis - Title

Ratio of Likelihood Test  The Chi-square table is adopted: -2[L(C)-L(β )]~χ² k  Generally: -2[L(β r )-L(β u )]~χ² k u -k r 57 Transportation Demand Analysis - Title

Calibration Software Program NLOGIT 5.0 BIOGEME LIMDEP GAUSS Transportation Demand Analysis - Title

Transportation Demand Analysis- Lecture note Finish 59