1. Discrete Choice Models for Modal Split Overview

2. Outline n General procedure for model application n Basic assumptions in Random Utility Model n Uncertainty in choice n Utility & Logit model n Numerical example n Application issues in four step model n Summary

3. Choice Model Formulation Predict Exogenous Explanatory & Policy Variables Estimate Dissagregate Choice Model(s) Individual & Travel Data Apply Prediction Procedure Aggregate (TAZ) Travel Prediction Insert Predicted Proportions for Each Mode in the Four Step Sequence

4. Theory from microeconomics n We will skip the more theoretical description of principles, theorems, lemas n Emphasize practical aspects n Look at examples n Note: Dan McFadden is Professor of Economics and Nobel Laureate in Economics http://emlab.berkeley.edu/users/mcfadden/ Nobel Laureate in Economics http://emlab.berkeley.edu/users/mcfadden/Nobel Laureate in Economics http://emlab.berkeley.edu/users/mcfadden/ n A site that contains a very good bibliography on Random Utility Models

5. Basic Assumptions (1) n n Suppose a trip maker i faces J options (choices or alternatives) with index j=1,2,3…J. n n Assume that each trip maker associates with each choice j=1,2,...,J a function called UTILITY representing the "convenience" of choosing mode j. n n j=1,2,..., J is called the choice set. This is the set from which a decision maker chooses an option. n n Note: Let’s assume that choice and consideration sets are the same.

6. Basic Assumptions Example –A person, i, needs to go to work from home to the downtown area. –Suppose this person has three possible modes to choose from: Car (j=1), Bus (j=2), and her Bike (j=3). Total number of options (J=3). –One possible form of the person’s convenience function (called utility) is: U car =F (car attributes, person characteristics, trip attributes) U bus =F (bus attributes, person characteristics, trip attributes) U bike =F (bike attributes, person characteristics, trip attributes)

7. Utility components n n Variables describing the individual --> this is an attempt to represent "taste variation" from person to person. In our example if young persons have systematically differing preferences from the older individuals, then, age would be one of the variables. n n Variables describing the choice characteristics (called choice attributes) in the choice set. For example, some travel modes are less expensive than others. Cost of the trip for each available mode would be another variable in the utility. Travel time is another key variable. n n Variables describing the context such as the trip type, time of day, budget constraints.

8. Key Assumption (maximum utility) n Travelers (decision makers) formulate for each option a utility and they calculate its value. n Then, they choose the option with the most advantageous utility (maximum utility). n Example: U (car,bus,bike) =-0.5*cost-2*waiting time n Cost by bus=$1, Waiting time=5 minutes n Cost by car=$2.5, Waiting time=1 minute n Cost by bike=$0.2, Waiting time=0 minutes Which mode is the most desirable, second less desirable, etc? Utility is actually an Indirect Conditional Utility

9. Uncertainty in utility (1) n We (analysts) do not know all the factors that influence choice behavior n Travelers (decision makers) do not always make choices consistently n We are not interested in including all possible variables that affect behavior in our models n We are interested in policy variables (taxes, fares, gasoline costs, waiting times) that we can “manipulate” to find travelers reaction n We are also interested in social, demographic, and economic traveler characteristics because these variables allow us to link models to TAZs

10. Incorporating uncertainty and traveler/trip characteristics n The example becomes: U (bus,car,bike) =-0.5*cost- 2*waiting time + SOMETHING ELSE n The “something else” is an indicator of “general mode attractiveness” AND a random component n Let’s look at the details:

11. Utility elements U ij =  j -0.5*cost j -2*waiting U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij time j +  j * age i +  ij Utility of person i for mode j

12. U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij A constant for each mode j. Captures desirability of j for unknown reasons Utility elements

13. U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij Waiting time is different for each mode j Cost is different for each mode j Utility elements

14. U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij The effect of the age variable is different for each alternate mode (Class: Let’s talk about behavioral meaning - bikes?) Utility elements

15. U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij The key indicator of uncertainty = our ignorance & traveler variability for unknown reasons Utility elements

16. n In a similar way as for age we can include other traveler and trip characteristics (explanatory) U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij In applications: These are parameters we estimate from data using regression methods Utility elements

17. U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij U ij =  j -0.5*cost j -2*waiting time j +  j * age i +  ij Can write as: U ij = V jj +  ij Can write as: U ij = V jj +  ij Utility elements Systematic & measurable part Random

18. Numerical example (trip from home to work/school) U icar =  - 0.5*cost - 2*waiting time +  * age i +  icar U icar =  - 0.5*cost - 2*waiting time +  * age i +  icar U ibus = 5 - 0.5*cost - 2*waiting time + 0.25 * age i +  ibus U ibus = 5 - 0.5*cost - 2*waiting time + 0.25 * age i +  ibus U ibike = 12 - 0.5*cost - 2*waiting time - 0.3 * age i +  ibike U ibike = 12 - 0.5*cost - 2*waiting time - 0.3 * age i +  ibike Note: Different age coefficients - why?

19. Compare systematic part (V) n Compute for each person the systematic part of utility for each mode n Plot all V (syst. utilities) for all persons n Horizontal = age n Vertical = V the systematic part of utility of each mode

21. Probability of Choice n We need to convert utilities to an estimate of the chance to choose a mode The specific equation to use depends on the probability distribution of the random component (  ) in the utility function (U=V+  ) The specific equation to use depends on the probability distribution of the random component (  ) in the utility function (U=V+  ) n Ease of calculations should be considered in selecting a probability function

22. LOGIT Model Assume the random components (  i ) of the utility are independent identically Gumbel distributed random variables then: Assume the random components (  i ) of the utility are independent identically Gumbel distributed random variables then:

25. Applications n Modal split (type of mode) n Route choice (link by link or entire path) n Car ownership (type of car) n Destination choice (shopping place) n Activity types (type of activity) n Residential unit (size and type of home)

26. Practical issues n Choice set - consideration set n Variables to include in utility n Measurement of mode attributes (e.g.,in- vehicle-travel-time) n Need survey data and mode by mode attributes! n Next: TAZ application and “complete” enumeration

27. Choice Model Formulation Predict Exogenous Explanatory & Policy Variables Estimate Dissagregate Choice Model(s) Individual & Travel Data Apply Prediction Procedure Aggregate (TAZ) Travel Prediction Insert in the Four Step Sequence

28. For the four step modal split n We need aggregate TAZ proportions by each mode (% of trips by car, % trips by bus, % trips by bike) n We have a disaggregate (individual) model which tells us the likelihood (chance) of a person to choose each mode n We need a procedure to go from disaggregate predictions of chance to aggregate predictions of proportions

29. Taking Average TAZ Characteristics Does Not Work n (Pa+Pb)/2 is not the same as P ([Va+Vb]/2) - a and b are value points for V n When the two are equated we have the Naïve method of aggregation n Bias depends on how close the probability function is to a linear function n Following is an example from Probability to choose bus as an option

30. V=2V=12 P(V=2)=0.034 P(V=12)=0.679 Consider a TAZ with two persons with V=2 &V=12

31. What is the correct TAZ Proportion of Choosing the Bus? n (P(V=2)+P(V=12))/2 n or n P((2+12)/2)=P(V=7)

32. V=2V=7V=12 P(V=2)=0.034 P(V=12)=0.679 P(V=7)=0.223 The correct value is: [P(V=2)+P(V=12)]/2=0.357

33. V=2V=7V=12 P(V=7)=0.223 [P(V=2)+P(V=12)]/2=0.357 Bias (see page 310 OW)

34. Naïve Aggregation n For each TAZ take the average value of explanatory variables n Compute average value for each utility function for each mode n Compute the corresponding probability and use it as the TAZ proportion choosing each mode

35. Market Segmenation n Divide the residents in each TAZ into relatively homogeneous segments n Apply Naïve aggregation to each segment and get proportions for each mode n Compute the TAZ proportion either as average segment-specific proportion or weighted segment-specific proportion

36. Complete Enumeration n Compute for each person and for each mode the probability to choose a mode n Compute the proportion for each mode as an average of the individual probabilities n Stochastic microsimulation is a method derived from this - see also Chapter 12 of Goulias, 2003 (red book)

37. Example (TAZ with four persons) V icar =  - 0.5*cost - 2*waiting time +  * age i V ibus = 5 - 0.5*cost - 2*waiting time + 0.25 * age i V ibike = 12 - 0.5*cost - 2*waiting time - 0.3 * age i

38. Compare values of the three methods

39. Theoretical issues n Gumbel IID convenient but is it realistic? n IID components imply unrelated options in the unobserved components - new models account for relations n Trips are related - different formulations n See CE 523

40. Additional sources n http://www.bts.gov/ntl/DOCS/SICM.html (Spear’s report on how to apply models) n http://www.bts.gov/ntl/DOCS/UT.html (self-instructional overview with examples) n http://www.tfhrc.gov///////safety/pedbike/vol 2/sec2.5.htm (simple description of most of the key issues) n All sites accessed September 22, 2003

41. Summary n Rational economic behavior n Utility linear in systematic and random components n Choice probability is function of utilities – non linear function! n Application by enumeration is best - weighted average by market segments may be good - depends on application! n Aggregate models are also available – approximate! n Surveys must be used for this step

42. Additional reading suggestions (for future reference) n Ortuzar Willumsen - Chapter 8 (8.1, 8.2, 8.3) n Ortuzar Willumsen - Chapter 9 (9.1, 9.2, 9.3)