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Progress and problems in the development of recreational trip generation and trip distribution modelsGordon O. Ewing aa Associate Professor of Geography , McGill University , Montreal, CanadaPublished online: 13 Jul 2009.
To cite this article: Gordon O. Ewing (1980) Progress and problems in the development of recreational trip generation and trip distribution models, Leisure Sciences: An Interdisciplinary Journal, 3:1, 1-24, DOI: 10.1080/01490408009512924
To link to this article: http://dx.doi.org/10.1080/01490408009512924
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Progress and Problems in theDevelopment of Recreational TripGeneration and TripDistribution Models
Gordon O. EwingAssociate Professor of GeographyMcGill UniversityMontreal, Canada
Continuing problems in the development of recreational spatialinteraction models are discussed. The paper begins by consider-ing the relative behavioral relevance to recreational travel be-havior of the unconstrained gravity model, the origin-constrainedgravity model, and a simultaneous trip generation and trip dis-tribution model. It elaborates on the behavioral rationale for thesupply-generated participation effect hypothesized in the lattermodel. A detailed discussion ensues of modeling problems re-lated to the travel cost/distance and the attractiveness compon-ents of recreational trip distribution models. Particular topicscovered are the continued inability of models to estimate theseparate effects of various elements of travel cost and the evi-dence that, in addition to a distance deterrence effect, there is anearest opportunity effect, an intervening opportunities effect,and perceptual barrier effects on recreational travel flows. A re-view of recent developments in estimating destination attractive-ness leads to a discussion of a multinomial logit regression modelof trip distribution and of the effect of crowding on the interpre-
Leisure Sciences, Volume 3, Number 10149-0400/80/0101-0001 $02.00/0Copyright © 1980 by Crane, Russak & Company, Inc.
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2 Gordon O. Ewing
tation of revealed attractiveness estimates. The paper concludeswith a discussion of the effect of spatial aggregation on param-eter estimates and the implications for planning applications.KEY WORDS: spatial interaction, trip generation, trip distribution,travel cost, attractiveness.
This paper attempts a synthesis of some recent developments in themodeling of recreational trip generation and trip distribution1, and,in addition to indicating what progress has been made, it highlightssome of the continuing unresolved problems facing researchers inthis field and their relevance to policy issues. The emphasis is pri-marily on behavioral issues rather than on statistical ones or thoserelated to calibration techniques on which there is a substantialliterature. (See, for example, Batty, 1976; Batty and Mackie, 1972;Evans, 1971;Hyman, 1969;Kirby, 1974.)
Three Basic Spatial Interaction ModelsIn the development of spatial interaction modeling in general, it ispossible to identify three levels of model of varying levels of be-havioral realism and difficulty of calibration. The simplest andearliest to be used is the unconstrained gravity model, which takesthe general form.
tij = kPiAjfCCy) (1)where tij = the number of trips from origin i to destination j ;
Pi = the population of origin i, or some other measureof origin mass;
Aj = a measure of the quality or attractiveness of des-tination j , or some surrogate measure, such assize, and
Cij = a measure of the cost of travel from i to j , or somesurrogate of cost such as distance.
1 "Trip generation" refers to the volume of trips emitted by an origin, and"trip distribution" refers to the proportionate allocation of these trips to dif-ferent destinations. The term "spatial interaction" is often used as an umbrellaphrase to describe the joint resultant of trip generation and distribution,namely trip flows between all pairs of origins and destinations in a system.
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Recreational Trip Generation and Trip Distribution Models 3
An implicit assumption of Equation (1) is that there is no theoreti-cal upper limit on tu, or more specifically that there is no upperlimit on demand as supply is increased. Thus, the structure ofEquation (1) implies that the addition of new destinations to thesystem evokes no reduction in tu values relating to the originaldestinations, but simply generates new trips from each origin to thenew destinations. In principle, if an infinitely large number of newdestinations were created the new trips generated from each origincould exceed its population. Although the example is unrealistic, itnevertheless points out a fundamental logical flaw in the model.In essence the debatable assumption of the model, as far as leisuretrips are concerned, is that the number of trips from any origin toexisting destinations is wholly unaffected by the addition of newdestinations; hence, by implication it assumes that the volume oftrips from any origin to one destination is unaffected by the locationand attractiveness of other alternative destinations. If the researcheror planner finds these assumptions unacceptable, but still uses sucha model, it is necessary to recalibrate it whenever there is anychange in the system under study, be it the addition of facilities orhighway improvement. As such, it means the model is both descrip-tive and ideographic.
In a move away from the facile assumptions of the unconstrainedgravity model, the origin-constrained gravity model was proposedinitially by Huff (1962) with the general form:
(2)
where py = Arf (C y ) /S Ak f (Cik) (3)
This has two important related properties. Firstly, pu is by defini-tion a proportion, such that for any origin i, ? p^ equals one. If newdestinations are added, the proportions allocated to the existingdestinations are reduced, and the total allocated from i to all des-tinations remains kPi. Thus the model assumes that a fixed numberof trips emanate from i irrespective of the number, attractiveness,and accessibility of the destinations. This is possibly an appropriateassumption to make for non-discretionary trips such as most cate-
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4 Gordon O. Ewing
gories of shopping trips, where the quantity of goods consumed isrelatively unaffected by the density, accessibility, and attractivenessof shopping areas. A second important property of the model whichsets it apart from Equation (1) is that it assumes that the propor-tion of trips from i terminating in j is a function of the attractive-ness and accessibility of j in relation to the attractiveness and ac-cessibility of all alternate destinations in the system. This gives it amore plausible behavioral basis than the unconstrained gravitymodel, which assumes tij is immutable irrespective of changes inthe alternatives. Significantly the major application of Equation (2)has been in connection with shopping trips (Huff, 1962; Lakshma-nan and Hansen, 1965).
To allow for the possibility that the number of trips emanatingfrom i is a variable, depending on the number, attractiveness, andaccessibility of an origin's potential destinations and on intrinsiccharacteristics of i's population, k in Equation (2) can be replacedso that Equation (2) becomes:
tij = ( t i . ) ( p u ) ( 4 )= F i d fi [2Ak f2(Gk)] Ajf2(Ci3)/E Akf2(Cik) (5)
where ti. = the number of trips emanating from origin i, irre-spective of destination;
= the proportion of trips originating from i that gotoj;
i = a measure of the intrinsic per capita trip generat-ing potential of origin i.
This formulation is more appropriate to discretionary trips wherethe participation rate is thought to be a function of the supply offacilities and of certain characteristics of a population. Effectively,Equations (2) and (3) are a special case of Equation (5) and,therefore, discussion will focus on the more general equation.
The parameters to be estimated in Equation (5) can include thevalues of Aj (j = 1,2,. . .,n), Gi (i = 1,2,. . .,m), the parametersof the so-called "supply-generated participation" effect (Beamanet al., 1977) contained in fi, and the parameters of the distancedeterrence function, f2. Although estimation of such an array ofparameters in a non-linear expression is in itself a considerable
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Recreational Trip Generation and Trip Distribution Models 5
problem—which is discussed in a later section—it is important notto be side-tracked by essentially technical calibration problems be-fore having considered the degree of the behavioral realism ofEquation (5) .
Behavioral Arguments about the Supply-GeneratedParticipation EffectIt is implicit in fi in Equation (5) that any increase in the number,attractiveness, or accessibility of destinations will result in some in-crease in the number of trips generated by origin i. It is helpful tobe specific about the potential sources of this expected increase intrips.
Improvements in existing facilities and the addition of new onescan result in increased trips directed at these facilities alone, or itis conceivable that such improvements can stimulate an across-the-board increase in trips to all facilities in general. These additionaltrips can either be in the form of additional trips by existing patronsof the improved facilities, or in the form of patrons of unim-proved facilities switching to improved or new facilities and usingthem more often, or in the form of previous non-participantsusing the new or improved facilities due to their "resistance thresh-old" being overcome by these facilities.
Turning to the question of why such increases in trips are likelyto occur, it is useful to consider separately the impact of improve-ments to or additions of single facilities and more widespread fa-cility improvements or additions. If a single new or improvedfacility does not have a higher attractiveness/cost ratio for residentsof origin i than any other facility, increased trips to that destinationcan nevertheless occur either because: (1) it provides a uniquecombination of attractiveness and accessibility not previously avail-able to residents of i and fulfilling a previously unmet demand; or(2) because incomplete knowledge about available alternativesmay make it appear a more preferable alternative than it actuallyis; or (3) because it is the most preferred alternative for a segmentof i's population; or (4) because some if not all recreational tripmakers prefer to vary their destinations from time to time or as amatter of course.
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6 Gordon O. Ewing
There are at least three possible reasons why more widespreadfacility improvement or addition might evince higher volumes oftrips. One effect of the increased spatial supply of facilities is to in-crease the probability that knowledge about these facilities in gen-eral will be diffused more widely in the population. In the literatureon innovation diffusion, the adoption process through time hascommonly been described by the logistic curve model (Griliches,1957; Casetti, 1969). It is possible to suggest an analogy in therelationship of ti./Pi and ^ Ak£>(Gk) in Equation (4) (see Figure1) in that initial increments in trip generation due to informationdiffusion are likely to be slow as £ Ak fa(Cik) increases at lowlevels, but that progressively trip generation increases as spatialsupply increases, only to level off as some saturation level is reached.Algebraically this relationship can be described as
ti./Pt = l -exp{-£[£Akf2(Cik)]) (6)where p = a parameter describing the steepness of the S-
shaped logistic curve; andexp = the base of natural logarithms, e = 2.72.
This form of expression enables a more complex relationship be-tween trip generation and supply to be described than the powerfunction, which has typically been used to describe fi in empiricalstudies (see, for example, Cesario, 1975b; Edwards and Dennis,1976; McAllister and Klett, 1976). With the power function, tripgeneration increases either at a marginally declining rate (P < 1),at a constant rate (p = 1), or at a marginally increasing rate (p > 1)with respect to an increase in spatial supply. And since the latter isinconsistent with an upper bound on the number of trips generated,it is usual to assume p ^ 1 (see Cesario, 1975b: 215-216). Theredo not appear to be any studies which have compared the perfor-mance of the two functions in describing trip generation as afunction of supply.
A second possible explanation of increased trip generation asgeneral supply increases is that crowding at facilities can be re-duced by an increase in supply, which in turn has a positive feed-back on the perceived attractiveness of facilities in general, leading
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Recreational Trip Generation and Trip Distribution Models
y
r
Figure 1: The * 'Supply-Gene rated-Participation" Effect.
to increased usage of the then less crowded facilities. This explana-tion depends, of course, on the initial existence of crowding atsome facilities.
A third possible reason for increased trip generation is that theincreased aggregate attractiveness or accessibility of the facilitiesmay lead to them being used as substitutes for other leisure activi-ties, assuming there is no commensurate change in the provision offacilities for these other leisure pursuits.
Thus, there seem to be good behavioral grounds for hypothesiz-ing a "supply-generated participation" effect. However, if we con-sider four recent studies of the effect (Vickerman, 1974; Cesario,1975b; Edwards and Dennis, 1976; McAllister and Klett, 1976)no single conclusion can be drawn. In Vickerman's study of urbanleisure, only one out of five different regressions of mode-specificleisure trip rates against a supply measure similar to fi in Equation(5) explains any statistically significant amount of variance. Theother three studies are all concerned with longer distance day tripsby car to visit either rural parks, ski resorts, or rural beauty spotsand involve calibrating models with the general form of Equation
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8 Gordon O. Ewing
(5). In all three fi, the "supply-generated participation" expressionin Equation (5), is defined as (EAkCik*)/3. Unfortunately, none ofthe three papers indicates the partial correlation between tu andthis term. However, the following values for p are obtained: 0.6(McAllister and Klett, 1976), 0.425 (Cesario, 1975b), no signifi-cant difference from zero (Edwards and Dennis, 1976). It shouldbe added that only Edwards and Dennis perform significance testson ft. One reason for the Edwards and Dennis result may be thattheir supply measure is improperly defined as the sum of attractive-ness/cost ratios for only the destination zones of day trips. Sincetheirs is a study of day trips to the countryside by car, it is arguablethat a significant part of the utility of such trips derives from thepleasure experienced on part or all of the trip route and not justthat experienced at the zone most distant from home (see Coppockand Duffield, 1975: 202). If the latter is a valid assumption, itwould necessitate a redefinition of spatial supply for such trips.
One potentially serious source of error in the trip generationcomponent of Equation (5), as it stands now, is that no allowanceis made for the possible competitive effect of substitutable recrea-tion activities. Just as in the trip distribution component of Equa-tion (5) the proportion of trips from i to j is assumed to be affectedby the nature of alternatives to j , so it is equally plausible to assumethat the number of trips relating to a certain kind of activity thatare generated by i will be affected by the kind and extent of alterna-tives to that activity that are available to residents of origin i.2 Butexactly how such a set of alternative activities and their competitivestrength could be operationally defined within a trip generationmodel is a vexed question. But the absence of such a term shouldevoke some doubt about the validity of the parameters estimatedin existing generation models.
2 This presumes some variation between origins in their supply of facilitiesfor substitutable activities. Thus, for example, the larger number of substitut-able leisure activities in a metropolis than in a small town may mean for equallevels of supply of a particular recreation facility, a higher number of trips isgenerated in the small center with fewer substitutes.
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Recreational Trip Generation and Trip Distribution Models 9
Travel CostA behaviorally valid definition of travel cost, Gj, in Equation (5)is another problematic area in trip distribution modeling. In prin-ciple, travel cost (Gj) can be seen as composed of three majorelements—the out-of-pocket expense of travel, which is primarily afunction of the distance from i to j (Dy); the opportunity cost oftravel time, which reflects the fact that the time spent travelingcould be devoted to alternative activities of some worth to the in-dividual; and the physiological and psychological effort of travel,which for car-borne leisure trips is possibly a function of traveltime from i to j (Tij). If the values of units of distance, time, andeffort could all be defined in a commensurable unit of measurement,say money, it would be possible to define generalized travel costas a linear sum of the form:
(7)where ai = a parameter defining the average perceived vehicle
operating cost per unit of distance;a2 = a parameter defining the perceived value of a unit
of time; andas — a parameter defining the value of the effort ex-
pended per unit of travel time.
Given exogenously defined estimates of ai, a2, and as, the value ofthe cost matrix, C, could be obtained and the deterrence effect ofcost estimated in terms of the parameters of f2 in Equation (5).However, it is an open question whether valid estimates of a*, andmore particularly as, can be obtained in monetary terms, and there-fore questionable whether a valid measure of generalized cost, Gj,can be obtained.
The value of a2 has been variously estimated for commuter tripsas somewhere between one-fourth and one-half of the hourly wagerate (Cesario, 1976b: 37), or according to Edwards and Dennis(1976: 244) in the range between 20 percent and 30 percent ofhourly incomes. Goodwin (1976), however, contends that there iswide variation in the results of studies trying to estimate values of
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10 Gordon O. Ewing
in-vehicle, waiting, and walking time on commuter trips. In addi-tion it is questionable whether the value of travel time on the wayto and from work is the same as, for example, weekend leisuretravel time. There is also the added complication that the leisuretraveler may derive some pleasure from the trip itself as well asfrom the destination activity.
But if reliable estimates of au are in doubt, it is even more dubi-ous whether it is feasible to obtain monetary estimates of the valueof travel effort. Indeed, one of the reasons why an effort term isusually not included in a generalized cost equation is the difficultyin putting a monetary value on effort as well as the problem ofmeasuring effort involved in a trip. Goodwin (1976) provides avery useful discussion of these problems and of the fallacious con-clusions that can be reached about changes in trip volumes if effortis a relevant but excluded element of travel cost.
The consequence of being unable to determine commensurableweights for ai, a2, and a.s, and therefore being unable to measureCij, is that the cost deterrence function, fo, cannot be defined simplyin terms of Cij. Rather it becomes some more complicated expres-sion involving separate weighting parameters to be estimated for theeffects of distance, time, and effort. The author knows of no attemptto include such an expression in calibrating a trip distributionmodel.
The common simplifying assumption made is either to ignore a3in Equation (7) and assume some value for a2 in line with Cesario'sor Edwards and Dennis' suggestion in order to calculate Cij, or toassume that travel distance or time is a surrogate for generalizedtravel cost. The latter is a particularly appealing assumption tomake when, as in the case of rural recreation trips, there is a highcorrelation between travel distance and travel time, because in sit-uations with high correlations between "independent" variablessuch as Du and Ty, no reliance could be placed on the estimatedvalues of their associated parameters. However, the policy-relatedvalue of a model that uses such a surrogate is limited. For example,if distance is used as a surrogate of generalized cost and there is anincrease in the per mile cost of vehicle operation, there is likely tobe a decrease in the volume and some change in trip destinations
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Recreational Trip Generation and Trip Distribution Models 11
of recreation trips, which the calibrated trip generation/distributionmodel will not predict since the actual Dy values will not havechanged.
Typically, distance in spatial interaction models is defined interms of distance, or in some cases the number of intervening op-portunities between i and j . However, consider the case where twodestinations j and j ' are equidistant from origin i and alike in everyother respect, except that there is an intervening opportunity k onthe route from i to j . Arguably this will tend to diminish the numberof trips from i to j more than the number from i to j ' . This inter-vening opportunity effect occurs in addition to the fact that k,regardless of its position, competes with j and j ' for trips from i, andcan be regarded as if it "stretched" Dy relative to Di/. Baxter andEwing (1979) have estimated this effect by redefining Dy as
D i j D i k D k j ( 8 )
where k = an intervening opportunity between i and j ; and$ = a "stretching" coefficient which is estimated empiri-
cally in a trip distribution model.
In applications to two separate sets of recreation trip data, esti-mates of S in excess of the theoretical minimum of one were ob-tained, and in both instances the inclusion of the parameter resultedin sizeable improvements of fit of the model's predictions.
Facility AttractivenessIn the majority of recreation and shopping travel models it hasbeen common to treat the "destination effect" as exogenously de-fined, and treat only travel cost effects as parameters to be endo-genously estimated in some form of production constrained gravitymodel such as Equations (2) and (3) . For example, in shoppingmodels, the destination effect, sometimes referred to as destinationattractiveness, has commonly been defined as some measure of re-tailing space in a shopping center (Huff, 1962; Lakshmanan andHansen, 1965; Wilson, 1971). Similarly, in recreation studies com-mon surrogates for facility attractiveness have been variables suchas park acreage (Grubb and Goodwin, 1968) and number of camp-
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12 Gordon O. Ewing
sites (Wennergren and Nielsen, 1970).:l Initially, attempts to movebeyond using a single surrogate variable for facility attractiveness(Aj) involved merely defining Aj in terms of a composite scorereflecting the jth destination's possession of various attributes andtheir relative importance as judged by the researcher (Cheung,1972; Ellis and Van Doren, 1966). But as Vickerman (1974) hassuggested, "many of the attraction factors will be subjective, al-though there is no reason why future research into consumer be-havior should not be able to identify more objective criteria bywhich to classify destinations." One step in this direction has beento obtain behaviorally based estimates of the relative importance ofvarious variables in contributing to facility attractiveness. Thus, forexample, McAllister and Klett (1976) replaced Aj in a trip distri-bution model with the expression TT Xkjyu, where Xkj is the amountof variable k possessed by destination j , and yk is a weighting esti-mated by calibrating the model using observed travel behavior data.
An alternative and more flexible approach is to let Aj values betreated as parameters to be estimated (together with cost functionparameter(s)) in the process of calibrating a production con-strained gravity model such as Equation (3) and the trip distribu-tion component of Equation (4). There are, after all, few a priorigrounds on which to exclude particular destination characteristicsfrom consideration in a trip distribution model. Although it mightbe equally possible to estimate the parameters in the expressions
7T Xkjyk or s yk Xkj for a large set of k characteristics, nothing is lostk "
and calibration problems are simplified if instead Aj (j = 1,2,... ,n)is estimated in calibrating the model.
If endogenous estimates of the Aj values for a particular set offacilities are to be an end in themselves, no further analysis of the
j values is required. If, however, the attractiveness of other facili-
3 It should be noted that retail turnover has been used as a surrogate of at-
tractiveness (Aj) in some retailing studies. Its equivalent in a study of park or
other recreation facility usage would be the number of users of a facility in a
given period (Σi t i j ) . However, neither measure is a valid surrogate of attrac-
tiveness if it is assumed that tij and therefore Σi t i j, is affected not only by Aj
but by the travel cost from the set of origins to facility j .
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Recreational Trip Generation and Trip Distribution Models 13
ties is to be inferred from the results of the model calibration, thenit is first necessary to treat the A., estimates as dependent variableswhose variance is to be explained by a set of destination attributes.One advantage of this approach is that alternative methods of ex-plaining variance in Aj such as linear and non-linear regression, anANOVA model, and Automatic Interaction Detection (AID) canbe used depending on the researcher's assumptions about the dis-tributional properties of both the dependent and independent vari-ables (see, for example, Cesario, 1975a).
There have been a variety of methods used to estimate Aj endo-genously. Ross (1973) employed a variation of the Method ofPaired Comparison from psychology, although it has been arguedelsewhere (Ewing, 1977; Ewing and Kulka, 1979) that the spatialdistribution of origins can bias the Aj estimates. Cesario (1973,1974, 1975a), in a more mathematical vein, has estimated Aj(j =1,2,. . .,n) in a model of the form:
(9)where Ei = emissiveness of origin i; and
Ei, i = 1,2. . .,m; Aj, j = 1,2,. . .,n; and /? are parameters to beestimated.
However, as already argued this form of unconstrained gravitymodel is behaviorally unrealistic as it implicitly assumes that thereis no upper bound on tij or 2 tij as destinations are added or mademore attractive. It can, however, be argued that Ei, being the onlyterm unique to origin i in Equation (9), is the equivalent ofPiGifi[2Akf2(Cik)]/2Ak£»(Cik) in Equation (5), since that expres-sion is also unique to origin i.4
4 In fact, Baxter and Ewing (1979) have shown that the Ej parametersestimated by Cesario (1976a) in a trip distribution model of the form:
P i j = kE i A j f(C i j )
are almost identical to the values they obtain for the "competition factor,"1/ΣkAkf(Cik), in Equation (3) when calibrating it using the same data as
Cesario. Jn both studies the A1 estimates were very similar. The main disad-
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14 Gordon O. Ewing
Recently there have been two attempts to calibrate joint tripgeneration and constrained trip distribution models similar to Equa-tion (5), with Aj(j = 1,2,. . . ,n) treated as parameters to be esti-mated endogenously. Significantly, both are in the field of recrea-tional travel. In an unpublished paper, Goodchild and Booth(1975) solve for the attractiveness of urban swimming pools andother parameters using Powell's iterative method for finding theminimum of a function of several variables without calculatingderivatives (Powell, 1965). And Edwards and Dennis (1976) ina study of day trips in England use their own ad hoc iterative re-gression procedure to estimate day trip destination attractivenessand other parameters. However, these iterative optimization meth-ods suffer several weaknesses. One is that there is inevitably somedoubt as to whether or not the final solution represents a global orlocal optimum, and, therefore, some doubt about whether theparameter estimates obtained depend on the parameter values usedto initiate the iterative procedure. Another weakness is that theseprocedures typically require large amounts of computer time rela-tive to analytical methods, and that the requirement increases ex-ponentially with the number of parameters to be estimated. Andfinally, they are not suited to testing hypotheses about parameterestimates and putting confidence intervals on them.
Recently, Baxter (1979) has shown that all the above problemscan be avoided by application of the multinomial extension of thelogit regression model. Building on the work of Thiel (1969) andMcFadden (1974), he has shown how weighted linear least-squaresregression may be used to obtain estimates of the parameters a,p,yand Aj(j = 1,2,. . . ,n) in the trip distribution model
pij = Aj Dij ~7 exp| -aDijP] /2 Ak Dik " 7 exp [-aDlk0] (10)k
vantage of Cesario's model is that the Ei estimates are unique to the geometryof the specific origin-destination system under study. And by being largely afunction of the Aj and cost function parameters, the Et values can in nosense be considered as independent parameters containing additional informa-tion about the trip generating propensity of each origin, as Cesario suggests.
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Recreational Trip Generation and Trip Distribution Models 15
To linearize Equation (10) it is necessary to remove the unwieldydenominator, all of the parameters of which are to be found in thenumerator. This is achieved for each origin i by dividing each p^ byone of the pik(k = 1,2,. . .,n) and logarithmically transformingthe result to give
^ l = l o g A . i - l o g A k - 7 l o g | g i i | - « [ D , j ^ - D , ^ ] (11)Pik D
It is assumed that p is either equal to one or estimated by using asearch procedure to find that value of p , which minimizes theweighted sum of squares objective function.
The assumption also has to be made that a large number ofobservations are available for each origin, and that the observed pijare samples drawn from a multinomial distribution in whichvar(pu) is not constant but can be defined and used to weightobservations accordingly. The procedure is rapid and, being ana-lytical, does not have the problem of local optima that remains alingering doubt when using non-linear optimization techniques. Inaddition least-squares statistical theory can be used to test hypoth-eses about parameter values. Although more experience with themethod is clearly needed before its value can be properly assessed,initial investigation (Baxter and Ewing, 1979), using data setspreviously analyzed by Cesario (1973, 1974, 1975a), suggests itperforms as well as, or better than, previous methods for estimatingAj values in a production constrained trip distribution model.
A further advantage of the logit regression model is that it canhandle the highly generalized form of distance decay functionevident in Equation (10) . Special cases of that function for par-ticular values of «, p , and y include the exponential function y = 0,P = 1), the power function (« = 0 ) , and Tanner's function (P= 1). Thus, if for theoretical or comparative purposes a specifictype of decay function is required, the researcher can specify itwithout affecting the method of solution.
The above discussion indicates the progress that has been madein attempts to obtain reliable estimates of facility attractiveness by
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using observed trip distribution data to calibrate a trip distributionmodel in which facility attractiveness (Aj) is a parameter.
The Effect of Crowding on AttractivenessSince all destination-specific effects on tu are encapsulated in theestimate Aj, it should occasion no surprise if one of the factorsinfluencing a facility's attractiveness is the level of use it experi-ences. However, unlike the one-way relationship between a facility'sphysical characteristics and its attractiveness, a two-way relation-ship exists between attractiveness and level of use. Thus, whileattractiveness influences use, certain levels of use are liable todiminish a facility's attractiveness. In essence, "a feedback loopwill operate to make the site less attractive when use is heavy, andperhaps also when it is excessively light" (Goodchild, 1977). Thismeans that while tij is in part a function of Aj, Aj may also be inpart a function of 2 tij/Q, where Q is some measure of facility j'scapacity. Insofar as the previous discussion of methods of estimatingAj concentrated on its being estimated endogenously, the effect ofuse level as well as physical characteristics of the site are, there-fore, implicit in the Aj estimate.
This creates problems if Aj estimates are to be used to predicttrip flows after changes have been made to the set of destinations.Any change resulting in a diversion or increase in trips to destina-tion j(?tij) may change Aj, insofar as Aj is partly dependent on^tij/Cj. So, strictly speaking, the same Aj estimates could not beused to predict trip flows after any change in the system had oc-curred which materially affected Stij values. To re-predict trip flowsafter such a change, it would be necessary to define the form ofthe function
AJ = f(2ti j/Q) (12)i
Then, using a joint trip generation and distribution model such asEquation (5) predictively, it is possible, given the changed system,to estimate an initial changed tij matrix. Calculating the resultantEtij/Cj for each facility, the A/s can be re-estimated using Equa-tion (12). This in turn would necessitate re-estimating the tij matrix
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using Equation (5), and again the Aj values using Equation (12).This recursive process would be repeated until the Aj values did notchange between two successive iterations.
Parameter Reliability and ZoningThe importance of the reliability of parameter estimates in a spatialinteraction model cannot be underestimated if the model is to beused predictively or to measure various origins' spatial supply offacilities. If we consider a simple example of two origins, each withonly one feasible destination with attractiveness and distance valuesas described in Table 1, it is clear that small changes in the value
TABLE 1The Effect of a Distance Parameter on a Measure of Spatial Supply.
Origin 1 Origin 2"values A 1 = 1 , D 1 1 = 2 A2 = 4, D22 = 4
Value of A /D..« for different a values-1.5 0.35 ! " 0.50-2.0 0.25 0.25-2.5 0.18 0.12
of the distance parameter a in the expression Aj/Dija can makesizeable differences to the relative supply measures of the two ori-gins as defined by that ratio.5 Likewise, it can make sizeable differ-ences to the predicted pattern of trip flows.
One source of bias in distance parameter estimates is the zoningsystem used to define origins and/or destinations. Several authorshave discussed how changes in zone size and zone centroid canaffect parameter estimates (Batty, 1976; Baxter, 1978; Broadbent,1969; Dalvi and Martin, 1975; Judge, 1974; Thomas, 1969). Theimplication of these findings in recreational trip modeling whereorigins may be spatially diffused is that the smaller the origin zoneschosen and the higher the density of origins in the vicinity of azone centroid, the more accurate are distance measures and the
5 Beaman and Smith (1977) give a more detailed, formal illustration ofthis point.
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more reliable are parameter estimates. If, however, these conditionscannot be met, it is important to test the sensitivity of a model'scalibrated parameters to changes in the zonal aggregation beforeusing the model for purposes of prediction or the measurement oforigins' spatial supply of facilities.
Approaches to Model ImprovementGiven the conceptual simplicity of a joint trip generation/tripdistribution model such as Equation (5), it should occasion littlesurprise that the goodness of fit of such models, when calibratedusing real data, leaves a lot to be desired. In part, this can be attrib-uted to sample error in observed trip flows, but even with large tripflow samples there is evidence that models do not fit the data aswell as is necessary to place faith in the validity of a model. Whatis more, the patterns of residuals in such models, i.e., the patternof differences between observed and expected trip flows, is oftendecidedly non-random, which suggests there are undetected factorsat work which account for some of this residual unexplained vari-ance in trip flows. Indeed, it is probably wise to think of a modelsuch as Equation (5) as one that shows the degree to which threegeneral factors, namely origin-specific effects, embodied in Gi,destination-specific effects, embodied in Aj, and origin-destinationcost/distance effects, embodied in f2, do not account for trip flowvariance. Having removed these general factors, the pattern ofresidual flows may contain more interesting and challenging infor-mation as far as theory development is concerned. For example,some recent work by Baxter and Ewing in which residuals of a tripdistribution model similar to Equation (3) were examined revealedevidence not only for an intervening opportunity effect, overlaidon the distance effect, as discussed earlier, but also evidence thatcertain distances were perceptually "stretched" by the presence of aperceived rather than a real barrier to flows. The fact that themodel used only objective distance, as defined by road mileage,resulted in a significant pattern of overpredictions of trips thatcrossed that perceptual barrier. There is considerable evidence nowfrom the fields of environmental psychology and behavioral geog-raphy that people perceive certain distances as greater than objec-
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tive distances due to a variety of measurable factors such as routedirectness, number of corners on a route, familiarity with a route,monotony of a route, etc. While we prefer simple models to com-plex ones, if the evidence from even aggregate spatial interactionmodels is that individual perceptual distortions are sufficientlypervasive to affect the performance of aggregate models, it is nec-essary to include such effects, where possible, in our aggregatemodels.
In the same work, the pattern of residuals also suggested that thenearest of several particular types of destination, such as beachresorts and mountainous areas, drew more visitors than would bepredicted from its relative distance alone. This suggests an exten-sion of the intervening opportunity effect to cover nearest oppor-tunities whether or not they intervene between an origin and amore distant destination.
Inevitably, even if a model is modified to account for such fac-tors, it is quite possible that the effect may be peculiar to one dataset and of no general interest. For this reason, and failing anymajor departures that significantly improved upon the conventionalproduction-constrained gravity model such as Equation (3) , thetime is ripe for comparative calibration of the same model, usingmany different data sets to see if there are any common sources ofresiduals that might suggest general improvements that could bemade to the model.
Sensitivity AnalysisAt present, then, there are sufficient sources of possible misspecifi-cation of spatial interaction models to be wary about placing strongreliance on exact parameter estimates. Failing any dramatic im-provement in model performance, it would seem imperative thatsome form of sensitivity testing procedure be adopted as a matterof course before using such models in recreational planning. Totake a simple example, if a forecast of patronage of a new recrea-tion facility is sought, it is necessary to determine what differencewill be made to the model's predictions if certain of its parametervalues are changed. For example, the range of values for eachattractiveness estimate and each distance function parameter esti-
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mate obtained, using a variety of zonal aggregations and of distancemeasures, could be used as a basis for generating random valuesfor each parameter and calculating the variation in trip predictionsthat would result. If this process was repeated numerous times, thedistribution of predictions could be examined to determine wherethe worst prediction lay relative to the minimum desirable patron-age level. Alternatively, if a minimum desirable level of use waspredetermined, it might be possible to calculate what the range ofindividual parameter estimates would have to be in order for thatminimum level to be generated on a particular percentage of pre-dictions based on randomly generated parameter estimates. Suchsensitivity analysis is now a necessary part of any traffic forecastingprocedure.
Having couched the above comments in terms of the trip distri-bution model in which alternatives are defined as other destinations,it bears re-emphasis that in trip generation prediction it is probablyas important to include a component that describes alternative ac-tivities to the one being predicted. Having said that, it seems clearthat the problem of defining and measuring alternative activities isimmensely more complex than the problem of defining and measur-ing alternative destinations in a trip distribution model. For thisreason it would seem prudent for the time being to tackle thedevelopment of trip generation and trip distribution models sepa-rately, so that obstacles in one do not impede progress in the other.In any case, it makes sense in the early stages of model develop-ment to reduce data to their fundamental "building blocks" so thatthe decision of which activity to participate in is treated separatelyfrom the decision where to go, given that a particular activity is tobe engaged in.
ConclusionThis paper has reviewed three common trip generation and tripdistribution models and has discussed the relative behavioral meritsof each with respect to recreational travel. It has sought to providea rationale for the "supply-generating-participation" component ofa trip generation model, as well as discussing the serious problems
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of estimating the effects of various elements of travel "cost." Areview of recent developments in estimating destination attractive-ness is provided, leading to a discussion of the multinomial logitregression model in trip distribution modeling. Attention was thenfocused on the dynamic nature of revealed attractiveness estimatesas affected by destination "crowding" and on the reduced reliabilityof parameter estimates when trip flow data are spatially aggregated.Finally, some possible avenues of future research in trip distribu-tion modeling were suggested, and in light of known sources ofvariability in parameter estimates, suggestions were made for test-ing the sensitivity of model predictions to parameter variability.
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