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A time-series forecast of average daily traffic volume
Tmnrp. Res.4 Vol. #IA, F’rinud in the U.S.A.
No. I, pp. 51-W.
0191~26a7/86 s3.00+ al986FtrgmmResLtd.
1986
.m
A TIME-SERIES FORECAST OF AVERAGE DAILY TRAFFIC VOLUME Economics kptmcnt
JWAN BENJAMIN andTransportabion hIStiMe, North Carolina A&T State University, Greensboro. NC 27411 U.S.A.
(Received 20 October 1983, in revised form 15 October 1984) AbShet-This paper presents a pmcedwe for forecasting average daily tra!Xc using a time-series analysis. The procedureassumes a logistic function to model traffic volume over a period of years. Model panmetcrs a~ estimated using ordii least-squaresregression. The method was tested empkaliy. Model parameters wm found to be SignifKantfor each of the tluce differentthoroughfares.Further,time-series forrcrrstscompared favorably to observed rraffx and to hwpolated forecasts for the same period. ‘he method is simpler to and moxe economical than the standud demand forecshg procedureand is rccoMllcndcd WkIe land-use pattems are stable and only small modifications to the thoroughfarenetwork are planned.
Econometric analogy
INTRODUCTION
Although many of the travel forecasting models are notbasedoneconometrictechniques,thereisadirect correspondence between econome&ic fomcasting techniques and tbe star&d travel demand forecasting procedure. There are ba&ally two approaches to econometlic forecesting:
a procedure for forecasting average daily trafCc volume. The procedure uses a time-series analysis that models trafk volume as a logistic function.
This paper proposes
The proccdu~ is simpler and less costly than the standard demand forecasting procedure. Background The standard demand forecasting procedule was de-
Structuml mouWs an models that simulate the key pIucesW that lead to the output of interest. These models permit direct analysis of “tihat if?” policy questions. This permits planners and policy makers to make decisions based on an evaluation of their impacts on some criterion variable. Time-series mudels are models that only examine the relationship of the criterion variable to time. Trends in the growth or decline of the variables are modelled without attempting to explain the contributing factors to the trends. These models only permit answers to questions concerning where the criterion variable will be in the future without regard to specific inputs. (The exceptions to this arc multi-variate time-series techniques that take into account intervening variables.)
veloped in the 1960s as an outgrowth of early efforts in the late 1950s to CMte a demand forecasting method that would take into account both land-use growtb patterns and transport decision variables. Today, the standard demand fomca&ng method centers on the well-known UTPS computer package that is maintained by the U.S. Depattment of Transportation. The UTPS package models demand in five basic steps. First, future land uses are forecast, then the trip ends in each area of the city are estimated, the distribution of trips calculated, choice of mode determined and finally route assignments ~IVmade. One important result of this analysis is the estimate of average daily traffic for major segments of the thoroughfare network 20 years in the future. A good review of tbc theory underlying the UTPS models is presented by Kanafani (1983). The UTPS package relies heavily on detailed survey data for parameter estimation, which is typically largescale origin-destination interzonal travel behavior. To reduce these data needs, a generation of disaggregate models was developed that looked at travel behavior in even greater detail. A summary of disaggregate approaches is found in Stopher and Meyburg (1975). These approaches were further extended to examine the basic decision processes of travellers by using attitudinal techniques. A summary of these techniques is presented in Benjamin and Sen (1982). The shortcoming of disaggregate techniques was two-fold. First, data requirements remained high for an entire metropolitan area and second, aggregation of disaggregate data in a multi-stage model was difficult. Most d&aggregate models centered on mode choice.
In travel demand, the criterion variable is average daily traffic volume. Past emphasis has been on developing better structural models with inwing attention to tripmaking decisions. The structural process, however, leads to the development and estimation of parameters for a series of models. In each case, these estimates are subject to assumptions and measurement errors that compound each other progressively. Despite this, only point estimates are produced at the end of the process, giving no range of probable outcomea. Further, no measure of the impact of various assumptions that are made at each level is provided. This is underscored by the need to update model forecasts and the mixed success record of UTPS forecasts. This paper suggests that instead of a structural approach, under proper circumstances, only traffic trends 51
52
J. BENJAMIN
need to be modeled. Such models only use average daily traffic data, which are generally collected for use in traffic engineering studies. The availability and convenience of average traffic information is emphasized by Carey, Hendrickson and Siddharthan (198 1). Nihan and Holmesland (1980) suggest the direct use of traffic counts for traffic forecasting but only for short-term forecasts. The BoxJenkins technique that they use is demonstrated for accurate use only for a l-year forecast and requires mote than 60 periodic observations. Although under stable conditions the Box-Jenkins technique can be used for longer time periods than 1 year, the assumption of a specific functional form when it is known makes forecasts simpler and more accurate. The technique presented in this paper requires substantially fewer observations (which are more typically available) and is demonstrated to be accurate for much longer periods. Despite their advantages, trend techniques, such as the logistic function, cannot estimate sudden shifts in behavior, changes in the transportation network or the introduction of.new modes and they do not allow for input by planners to evaluate alternative plans. However, they doprovideestimatesoffuhm:tnndswhennetworkchangcs ate small changes, such as thoroughfare improvements, or when traffic and land use ate stable. Further, in many cases, the effect of large changes is local, having little effect outside of proximate zones or traffic corridors. Stmctural models are appropriate when large changes are pIaImed. In small urban areas, the vast majority of change is of this small, steady nature. Radical changes, such as the addition of new thoroughfares or modes that necessitate an alternatives analysis based on a structural model, are rare or nonexistent. These patterns are observed in the small urban ateas that are near Greensboro.* For example, in Greensboro, steady growth is observed for all thoroughfares. Further, there is only one new major thoroughfare planned in Greensboro in the next 25 years. In the nearby cities of High Point, Burlington and Jamestown there are no new major thoroughfares under consideration. In Winston-Salem, the only major modification is the interstate highway. In each case, steady growth has been observed over the past decade for virtually all thoroughfares, except when new thoroughfares are introduced or major activity centers arc developed. Planning for these cities instead is frequently a reactive process consisting of a search for troublesome high volume segments and intersections. Upon determination of these trouble spots, transpottation system management techniques are used to remedy the trouble. In these small urban areas, demand determines supply. The role of the planner is more to recognize shifts in demand and to accommodate them than it is to initiate and modify transport systems. The tools needed by these planners are. not tools to investigate the impacts of new networks. Rather, they are the tools to help recognize trends in demand in advance so that system modifications can be made before serious difficulties or even dangerous *Greensborohas a populationof approximately 150,000 and is located in the Piedmont area of North Carolina, U.S.A.
situations arise. The time-series approach that is suggested here is much more suited to this reality of planning today in many small urban areas. A THRORY
OF TRAFFIC GROWTH
The starting point for a theory of traffic growth is land use development. In a residential area, increased development leads to increased trip-making. Commercial and industrial development also results in increased traffic. This basic concept underlies structural models such as the UTPS model (Kanafani, 1983). In a small urban area such as Greensboro, North Carolina, land uses for most areas of the city are stable. Basic industry remains where it has been for decades and residential neighborhoods show little change. Although commercial activities have shifted to suburban malls, these changes were gradual, evolving over decades, as was the gradual development of industrial parks at the city’s periphery. However, although an understanding of the relationship between traffic and land-use development is helpful, because of the gradual change of land patterns, traffic patterns will also change slowly. Traffic volume, therefore, is a function of time that is a surrogate for urban growth. As time passes, more land is developed and traffic increases proportionally. In a particular zone or corridor, land use is initially stable when the land is agriculturally zoned. As land is developed, traffic incteases until all land in the zone or corridor is developed. At this point in time traffic stabilizes. Traf’tjc volume thezeafter remains about the same, inmasing or deecrursing by small percentages based on variations in fuel supply, population density, driving habits and land use. During growth periods. diffetent tracks are developed at different rates that appear as random fluctuations in a smooth growth curve.? Under assumptions of a constant growth rate during growth periods, the curve then rep resents an average growth rate. New development, which may take 1or even several years to complete, will average to a single rate over a period of several years. Under assumptions of constant growth, we find that: dV = A,XV dt
where X = maximum potential new average daily traffic in vehicles per day V = current average daily traffic in vehicles per daY dV the rate of change in volume with respect -dr = to time in average daily traffic per year 4 = the rate of growth in traff5c in average daily traffic per year Equation 1 indicates that changes in traffic volume are proportionate to both the amount of growth to date as tShafer (1977) indicatesthat residentialdevelopmentis a step function in the short term.
Forecast of average daily traffic volume well as potential growth and to a eoustant growth rate. This equation follows from the concept that the proportion of traffic growth is a function of the proportion of land development. One underlying assumption for eq (1) is that the attractiveness of a tract of land for development is enhanced by existing development. This is true if the knowledge of the availability and the at&activeness of residential land is spread primarily by word of mouth. An analogy exists between this form of growth and the spread of contagious diseases. People contact each other and discuss the attractiveness of the tract personally. The more people who know of the tract, the faster it develops. ‘Ibe second assumption is that development is proportionate to available undeveloped land. This is intuitive: the greater the land available, the greater the potential for development. Stated another way, with constraints on available land in a set corridor, once most land is developed, there is little room for further development so growth must be slow. Finally, the constant growth rate with respect to time represents a constant interaction of people cornmunicating the attractiveness of land in a corridor. This rate may change over time as a result of changing economic conditions that affect demand for new homes. In the latter case, growth is a non-linear function of time, which IVsuits in eq (4). Defining X = V, + V, - V where V, is the maximum growth in traffic volume, letting V, equal the minimum traffic while land is undeveloped and solving for V as a function of t, the growth function is the wellknown logistic function: V = v, + v,
exp(A, + A+) 1 + exp(A, + A,t) I
(2)
where A,, is a constant that is introduced to permit nonstaadardizsd measures of time. A0 also serves the purpose of fming the beginning of the growth process in time. The derivation of eq (2) is presented in the Appendix of this paper. The function has a characteristic “s” shape, starting and ending with asymptotically constant rates. The characteristics and estimation of the logistic function parameters am discussed by Kruskal and Tanur (1977). & and A, are found by fmt estimating the asymptotes V, and V, and rearranging terms such that:
v - v, Inv1 -v+v,
= A0 + A,1
Furthermom, the exponent of the logistic function is not aecesJarily a linear function of t but may be some function g(t). If this function of t is a polynomial, the right side of eq (3) becomes:
SW = A,, + A,t + A2t2 + A# + A,t’ + . . .
53
vari&ble&~Gobdness-of-fit;;s available for ordinary least-squares regression analysis such as R2, F statistics and the Durbin-Watson statistic are then applicab1e.S The coefficients of the polynomial terms become important when growth is not monotonic. This permits accurate models when there are temporary fluctuations in traffic volume. Deviations from a linear function of t would occur with changes in economic activity that result from business cycles, with relative cost of gasoline or with proximate changes in the tmnsport network. The logistic growth curve is only one of many curves that have been employed to model new product growth in marketing. Mahajan and Muller (1979) summarize a set of such curves, including exponential, Gompertz and a family of dynamic growth curves. When assumptions differ from logistic function assumptions, other forms should be used. The exponential curve was found best for diffusion of information based on external factors such as various promotional activities. The logistic and Gompettz curves were found to represent situations that were the result of internal interactions between members of potential and current markets. Each of these curves represent situations where the untapped market is fixed at a constant value. Dynamic models wem extensions of these basic models to consider changes in the size of untapped markets over time. Both the logistic and Gompertz curves fit the initial assumptions best in that development of districts within urban amas is primarily the result of personal interactions rather than promotional campaigns and that the untapped market for any area is fixed by the physical and zoning constraints on available potential development. In comparing the logistic and Gompertz curves, one sees that they are similar in shape. One difference between these two curves is that the inflection point is at the point of 50% cumulative growth for the logistic curve and at about 37% for the Gompertz curve. This results in varying growth rates for each. In detailed analysis of both curves, Thomley (1976) points out that the growth of the Gompertz is a direct function of time, whereas the logistic growth rate is constant. A constant rate in growth was initially assumed for this study and the logistic function was adopted. A comparison of constant and polynomial growth rates is also presented. AN EMPIRICALTEST OF LOGKSTKGROWTH The time series forecasting approach was tested in Greensboro. The city has a ring and radial thoroughfare network and has a history of careful planning originating in the early 1950s. Greensboro has stable basic industry as well as a good track record of attracting new industry, as do most cities that are located in the Sunbelt. Thus, there are both stable areas as well as high growth areas of the city.
(4)
The estimatesof A,, and A, (and Al, A,, A,. etc.) are then obtained by using ordinary least-squares regression analysis with In (V - Volvo + V, - V) as the dependent variable and t (and t2, t3, t4, etc.) as the independent
$Thc Durbii-Watsoa statistic tests for auto correlation. Auto correlation occurs in a time-series analysis when the dispersion of the data is r&ted to time. A high auto-correlationusually indicates mis-specifkation of the form of the time-series function.
54
J. BEPJJAMIN
Fig. 1. Locationof traffic count points in Greensboro, NC.
To test the technique, three Greensboro streets were identified that contrasted markedly in theii growth characteristics. Then using data available in 1975, traffic vohunes in 1981 were estimated using both UTP8 and logistic models. These estimates were then compared with observed 1981 traffic measures. UTPS estimation procedwes Because of the expense of developing UTFS estimates, existing year 2008 estimates developed in 1975 were used as the basis for comparison. Estimated volumes for 1981 were linearly interpolated between 1975 measures and 2000 forecasts. The need to interpolate underscores a major disadvantage of UTP!S; that is, its cumbersomeness and expense. Linear interpolation was suggested by local forecasters to be the practical approach that they would use. Although linear interpolation does not necessarily represent variations in the traffic growth rate, these results do provide some idea of how results of the logistic ap preach compare to UTFS, which is the most widely used model.
Test locations The thoroughfare map in Fig. 1 lists the location of the three test sites. The first site was Lee Street, a stable
area of Greensboro with little new development. The sect& site was FSetnUy Avenue, which experjenced rapid development during the decade preceding 1975. The third site was Battleground Avenue, which is in a corridor of rapid continued development. Each route is in a residential-commercial corridor and has available data for at least 25 years. Graphical route-by-route analysis To illustrate these trends, average daily volumes were plotted for the pericxl from 1955 to 1981 for each of the thme routes.8 These graphs are presented in Figs. 2, 3 and 4. By observation, the shape of the curves that best describes the trends on Friendly Avenue and Lee Street is the “s” shape, which is typical of the logistic function. Lee Street completed its growth pattern during the 1950s and 1960s and Friendly Avenue, during the 1960s and 1970s. This is consistent with the land-use growth in Qvolumesweremcas~biann~lyfortheye8rs 1955.1957, 1958, 1960. 1%2, 1964, 1966, KM, 1969, 1971. 1973, 1975, 1977,1979and 1%1.NocuuntswereavailabkonBattleground Avenue for years @or to 1960 and on FriendlyAvenue for 1955 and 1958. Although thae wete inconsisteacics in some periods between okavath. this was a constraintof the available setonday data.
Fomast of avcrsgcdaily trafticvolume
55
20 LEE STREET AVERAGE DAILY TRAFFIC ( x 1000) 10
I
0
I
1
1
,
I
t
1955 LINEAR
POWJO=
I
1
I
1371 ______
I
I
8
1981
YEAR
24 FRIENDLYAVE
AVERAGE DAILY TRAFFIC
( x 1000) 16
8 .
0 1957
1969
1981 YEAR
Fig. 3. Obserwda~~Iestimptedlmffi~~~~~FricDdly
AveauC.
c
56
J. BENJAMIN
BATTLEGROUNDAVE 20 .AVERAGE DAILY TRAFFIC ( x 1000)
10
0 1960
1970
1980
YEAR Fig. 4. Observed and estimated tdic
their respective corridors. In contrast, Battleground Avenue is just beginning to grow. The Battleground Avenue corridor is only partially developed. Functional route-by-route analysis Using a logistic function, a regression analysis was completed using the two-stage procedure described in a prior section of this paper. Fit, the minimum asymptotes were estimated by reviewing the early history of average traffic on the route. Maximum asymptotes were estimated to be proportionate to maximum land utilization in a corridor leading into each traffic count point. The procedure was to first establish corridor boundaries, then to observe which parcels of land within the corridor were developed or suitable for development but not yet developed. The land areas were then aggregated and the proportion of developed land calculated. The growth factor was calculated as the inverse of the proportion of land developed. The most difficult step in the procedure could be setting corridor boundaries. This was not difftcult for the three corridors in the case study because of natural geographical constraints. In each case, the study location was along a radial route leading to the inner beltway and the central business district. This formed the inner side of a pie-slice shaped area leading outward from the CBD. The sides
growth on BattlegroundAvenue.
of each corridor were formed by radial lines biting the augle between the study thoroughfare and the next radial thoroughfate. Natural outer boundaries were formed on Friendly Avenue by regional airport zoning restrictions, on Battleground Avenue by watershed zoning restrictions and on Lee Street by sewage and water restrictions on land development. Detailed information was available from the city planning department on developed tracts as well as long-range plans for future development. These estimates confirmed assumptions that traffic growth would be proportionate to land-use growth hecause planned new development was similar to existing patterns in both use and density. Maximum growth estimation procedures for each corridor are summarixed in Table 1. It should be noted that thii is only one approach to developing maximum growth estimates. More detailed analysis is possible for individual land uses. Another approach is to estimate asymptotes internally from the traffic data. The approach adopted here was in part influenced by the availability of specific land-use information and in part by the resource limitations of the project. Periodic growth for each route was then estimated using all data points in regression analysis. Results are summaked in Table 2.
Forecast of average daily t&k
volume
57
Table.1.
Tl=owfa= Lee stleet Friendly Avenue BattlearoundAve.
1975 Traftk
Petcentage developed land
GKnvtllfactor
16,600 17,500 20.900
97.6 76.1 67.4
1.03 1.31 1.48
For each regression analysis, the constants and coefficients were of the anticipated sign and relative magnitude. Time was coded according to the elapsed time from the start of the study period. Thus Lee Stmet, the first observation, which was for 1955, was coded 1, the second~~~,whichamnredinthethirdyear(1957), was coded 3, etc. The negative constant (A& indicates that growth started before the beginning of the study period. The positive coefficient (A,) indicates a positive growth rate. Each of these adjusted R* values is greater than 30 and each F statistic is significant at the 5% level. For Lee Street and Friendly Avenue, whete substantial growth hasaheadyoccurred,tbepatternissimilartothecharacteristic “s” shape of the logistic function and the R *‘s arc highest. For Battleground Avenue, the function has not yet devleoped and the R2 is substantially lower. The esumated and obsetved functions are graphed in Figs. 2.3 and 4. In Fig. 2. the logistic function accumtely represents the average traffic on Lee Street throughout the period. In Fig. 3, the curve represents the overall tmnd of growth on Friendly Avenue. In Fig. 4, the irregularities of the early stages of growth of the major new msidential area on Battleground Avenue deviate above and below the long-range average growth projected by the logistic function. A comparison of observed and estimated points in each graph illustrates the characteristic randomness of residuals. The Durbin-Watson statistics for Friendly Avenue and Lee Street both indicate that the hypothesis of no
Maximum traffic (V, + V,) 17,ooo 23,CQO 3l.tNm
autocorrelation can be accepted at the 5% significance level. The statistics for Battleground Avenue is inconclusive. The statistics for Friendly Avenue and Lee Street support the appropriateness of the logistic model here. However, there ate too little data available for Battleground Avenue to make such a determination. In comparing fomcasted and observed traffic, the logistic function performs best on Friendly Avenue, which is just completing the growth cycle. Lee Street, which has reached full growth, seems to fluctuate and to actually &crease slightly during the most recent study period. These fluctuations are not simulated by the logistic curve, which assumes constant growth and may be the result of exogenous economic factors, because maximum growth has been attained. Battleground Avenue is at the beginning of its growth period and, as expected, the data have yet to clearly outline the shape of the complete growth data. However, the growth forecast for Battleground Avenue is quite accurate. Compnrison of lo&tic forecasts and observed trafic for 1981
Both point and interval estimates were made for 1981 using logistk fun&m analysis ikim data liited to 1975.11 This informathm is summarked in Table 3. In each case, the observed forecast fell within or proximate to the confidence limits at the .95 confidence interval. (The observed value for Lee Street is slightly below the limits at the .95 level but falls within the limits at the .98 level, which is 12,806 to 17,000.) The forecasts show a high
Table 2.
Parunetefi Assympotes Minimum (V,)
Maximum Growth (V,) Coefficients A0 A’ R’
Adjusted R’ F statistic
Degrees of freedom Durbin-Watson statistic
Lee street
Frkndly Avenue
5400
11.600
6500 16,500
11,300 19,700
-1.73 0.18 0.52 0.48 13.20 (1.12)
-1.56 0.14 0.68 0.65 23.70 (1.11)
- 2.43 0.13 0.37 0.30 5.32 (179)
I.%
1.63
Battleground Avenue
0.83
lparaval estimates were developed under the assmllPtionof known minimum and maximum hits (V, and V,).
J. BENJAMIN
58
Tab’” 3. Comparisonof UTPS and logisticforecastsaad observedII&C in 1981 Traffic Source
Volume Thoroughfare Lee Street
Friendly Avenue
Battleground Avenue
UTPS
18,100
18,800
22,400
Logia tic point estimate interval estimate,
16,900
20,400
22,400
upper limit lover limit
17,0001 14,800
21,300I 10,300
30.9001 11,421
14,600
19,900
22,900
Observed
Traffic
‘Confidence
Interval
calculated
degreeof agrament with observed traffic for each route, even though each of the routes differed sign&a&y in its stage of growth. Contparison of logistic Md VTPS forecasts for 1981 UTF’S fozecasts for 1981 were interpolated from 1975 andyear2OoofonXasts.llTbeseforecasts werecompaled to the logistic forecasts made from the same base year. Forecasts wele made for 1981 using both techniques and then compared to observed tratXc for the same year. This information is also summatizcd in Table 3. In each case, the logistic forecast was as close or closer to the observed traffic than the UTPS forecast. In two of the CM, traBic was overestimated. which may be a result of the depressed growth in development and resulting travel during the onset of a recession. However, in each case, the logistic function performed relatively well despite the lower complexity and cost of the timeseries procedure. Logistic forecasts with polynomial exponent Instead of using a logistic function whose exponent is a linear function oft [as presented in eq (311,a polynomial function [as presented in eqs (5) and (611was used as the exponent. This analysis was particularly useful for Lee Street because of the non-monotonic growth. A fourth degree polynomial was assumed and coefficient estimates are summarized as follows: & 2 4.7, A, = 1.00, A, = 0.046, A3 = 0.00065, and A, was not included by the stepwise procedum The adjusted R* was 0.79, which compares favorably to a base R’ of .27 for a regression with these variables and sample size (McNemar, 1%9, p. 203). The F statistic was 16.9, which is significant at the .OS significance level. A comparison of observed and estimated volumes from both linear and polynomial functions is illustrated in the graph in Fig. 2. The polynomial function successfully (It shouldbe notedthatUTPS forecastsme based on somewhat differentassumptionsof traffic which is Mkctcd by the highervalues reportedfor Lee Stxeet.
at the
.95 confidence
level.
repnsents the reduction in traffic observed after the 1975 peak. Further, traffic level starts to increase again asSympoticallytO maximum capacity toward the year 2000. A DurbiWatson sta!istic of 2.53 indicated no autocorrelation, which fmther confirmed the appropriateness of this model. The shape of the polynomial curve is what would be expected. A&r a temporary reduction in traffic due to hi* gasoline prices and economic recession, final developnreat of the conidor leads to continued growth that reaches a plateau in the future as available land is developed. Bakng unexpected changes in development, the e&mates seem nasonable. Further, they are far more accumte for 1981 than either the linear of UTPS estimates (14,500** as compared to 16,900 for the linear model and 18,000 for the UTPS model; the observed traffic there was 14,600 in 1981). These results underscore the potential of the approach. CONCLUDING REMARKS As compared to the structural approach, the time-series forecasting technique is simpler, easier to understand and less costly. Furthermore. the results are as accurate, or in some cases, more accurate than the UTPS interpolated results. In addition, time-series models have measures of goodness of fit and provide interval estimates at desired confidence levels as well as point estimates. The major disadvantage of these models is that they am not sensitive to large changes in the transportation system. Examples of cases where the time-series ap preach is not best include the introduction of a new mode, the construction of a new thoroughfare, the opening of a new competing thoroughfare or substantial modifications that change the function of an arterial, such as changing traffic to one-way or extension of the arterial into a new cotidor. All of these changes will affect the
**Althoughforecastsnportcd in Table 3 wereestimatedfrom data gathemdon or before 1975,these estimates were derived from analysis of the full data set including all years through 1981.
Forecast of average daily traffic volume
land-use pattems and travel behaviors of urban residents to sufficiently warrant a structural investigation. Further, the time-series approach assumes that travel behavior remains relatively constant. Changes in economic activity, the price of gasoline or merely travel habits cannot be forecast using this approach. However, the results repoited here indicate that historically these factors have had a relatively small linfluence on average
59
of the logistic function and would add the capability to analyze transport policy. Most small urban areas are similar to Greensboro with respect to stability of the thoroughfare system. The use of time-series models should be given careful consideration where appropriate.
RFiFERENCFS
t&fiC.
In light of problems with recent cutbacks in federal highway expenditures and the increasing emphasis on maintenance of existing facilities rather than construction of new highways, the need to build predictive structural models is limited. For example, in Greensboro and SW rounding areas, a few major changes in the highway network am planned in the next 25 years. It is, therefore, reasonable to approach modeling for Greensboro by creating a structural model for the one anticipated major change in the arterial system and to model the remaining network (outside the locally affected area) using the timeseries approach. Estimated cost savings would be substantial and the model would probably be more accurate. Another approach that may reduce the expense of implementing a time-series approach would be to cluster similar regions together and estimate exponent coeff~icients once for many similar routes. Separate values for minimum and maximum levels of traffic would accommodate differences between route capacities. Once computerized, estimates would be produced routinely. Furthetmore, after observing data, diierent growth functions, such as linear functions, may be found best. Completed on a regional basis, growth functions can be assumed that best represent the growth patterns in an area. A second problem with this approach is estimation of maximum volume. The sensitivity of volume estimates to different estimates of maximum traffic volume is proportionate to both the magnitude of the maximum volume variations and the proportion of growth at the time of the estimate. Therefore, the precision of maximum volume estimates will influence answers throughout the growth period, although only to a small extent during early growth. In this paper, quick estimates were developed by projecting future growth and relating it to future traffic. These results seem to be sufficiently accurate for this case, but more precise estimates are possible using more detailed projections of cohorts or even the use of some structural land-use models. In small urban areas, this should not be a serious flaw with the method. However, additional work is needed to investigate the probabilisitic aspect of maximum volume projections and their effect on interval forecasts. Another approach to estimating asymptotes is to use the existing data. In this case, the logistic function, along with average daily trafftc alone would be used to find the maximum daily traffic. This approach should be compared to the outcomes reported in this paper. Finally, the time-series approach is not limited to nonpolicy variables. Policy variables, such as those related
to economic growth or the relative price of fuel to consumers, may be used as additional independent variables. These variables would improve the explanatory ability
A ~Uanual of Procedures to Analyze Am’tudes Towar& Transportation, U.S. Department
Benjamin J. and Sen L. (1982)
of Transportation, Washington, DC. Carey M., Hendrickson C. and SiddharmanK. N. (1981) A method for direct estimation of origin/destination trip matrices, Tmnspn. Sci. IS, 32-49. KanafaniA. (1983) Transportation DemandAnalysis. McGraw-
Hill. New York. Km&al W. H. and Taaur J. M. (1977) lnternarional Encyclopedia of Statistics. ‘Tlw Free Press, New York. MahajanV. and Muller E. (1979) Innovationdiffusion and new pmduct growth models in marketing. J Marketing 43, 5% 68. McNemar Q. (1969) Psychological Srarisrics. 4th ed., p. 203.
Wiley, New York. Nii N. and HoImesknd K. (1980) Use of the Box and Jenkins time saies techniquesin trafficforecasting. Transpn. 9, 125143. Shafer T. W. (1977) [Irban Growth and Economics. Reston
Publishing Company, Reston. VA. Stophg P. R. and MeyburgA. H. (1975) Urban Transportation &xielling and P&n&. Lexington Books. Lexington, MA. Thomky J. (1976) Mathematical Models in Plant Physiology. Academic Press, London.
AFPENDIX (DERIVATlON OF LOGISFICFUNCTION)
The derivation for the logistic function is well known. It is easkst to derive the function for the Inoportionof growth rather than the vdume of growth. The proportion of growth p is p = [(V - V&V,]. whew V is tmfBc volume, V, is minimum traflic volume and V, is maximum traffic volume growth. Potential growth is then [(V, + V, - VW,] = I - p. The rate of change of traffic volmne at any time f is assumed to he s~ to the proportionof growth to date (p), potential - p) and an ovemll growth rate A,. Hence: -
dr
= A&l
- p).
64.1)
The following differential equation thus follows: Q A,dt = -. PC1 -
(A.3 P)
To integratethis function we must set u = (I - p)p-’ and du = -p-%+.K Because the integralof du/u is In V, integrating both sides of (A.2) gives us:
-A,t
I’ =
P
In [(l - p)p-‘1
I
.
(A.3)
PO
b
Applying the boumkry condition that half of all growth will be compkted at r = 0 we get: -A,r - 0 = In [(l - p)p-‘1 - In [(l -
l/2)(1/2)-‘]
-A,r = In [(l - p)p-‘1.
or
(A.4) (A.5)
J. BENJAMIN
60
will appear in (A.4) and the solution for p in tetms of t is:
Taking the antilog of (A.5) gives:
e-4
=
(1
-
p)p-l.
(44.6)
P=
I 1 +
p=p=-
1 + e-4”
+
4’)
64.8)
If it is assumed that the gmvtb rate is a polynomial function of t, the solution for p as a function of t is:
The solution for p as a function of t is:
1
pa
L4.7)
If tbe boundary condition at r = 0 is not p = l/2 but rather mmeotherpmption~,aamstattttenn -& = ln(1 - p)po-I
1 , + e-m’
(A.9
where g(t) is a polynomial function of 1. Solutions for volume V in terms of V,. VI and r are found by appropriate substitutions in eq (A.7). eq (A.8) or eq (A.9). The mults of these substitutions an listed in eq (3). q (4) and eq (5) of the main text ofthepaper.
No. I, pp. 51-W.
0191~26a7/86 s3.00+ al986FtrgmmResLtd.
1986
.m
A TIME-SERIES FORECAST OF AVERAGE DAILY TRAFFIC VOLUME Economics kptmcnt
JWAN BENJAMIN andTransportabion hIStiMe, North Carolina A&T State University, Greensboro. NC 27411 U.S.A.
(Received 20 October 1983, in revised form 15 October 1984) AbShet-This paper presents a pmcedwe for forecasting average daily tra!Xc using a time-series analysis. The procedureassumes a logistic function to model traffic volume over a period of years. Model panmetcrs a~ estimated using ordii least-squaresregression. The method was tested empkaliy. Model parameters wm found to be SignifKantfor each of the tluce differentthoroughfares.Further,time-series forrcrrstscompared favorably to observed rraffx and to hwpolated forecasts for the same period. ‘he method is simpler to and moxe economical than the standud demand forecshg procedureand is rccoMllcndcd WkIe land-use pattems are stable and only small modifications to the thoroughfarenetwork are planned.
Econometric analogy
INTRODUCTION
Although many of the travel forecasting models are notbasedoneconometrictechniques,thereisadirect correspondence between econome&ic fomcasting techniques and tbe star&d travel demand forecasting procedure. There are ba&ally two approaches to econometlic forecesting:
a procedure for forecasting average daily trafCc volume. The procedure uses a time-series analysis that models trafk volume as a logistic function.
This paper proposes
The proccdu~ is simpler and less costly than the standard demand forecasting procedure. Background The standard demand forecasting procedule was de-
Structuml mouWs an models that simulate the key pIucesW that lead to the output of interest. These models permit direct analysis of “tihat if?” policy questions. This permits planners and policy makers to make decisions based on an evaluation of their impacts on some criterion variable. Time-series mudels are models that only examine the relationship of the criterion variable to time. Trends in the growth or decline of the variables are modelled without attempting to explain the contributing factors to the trends. These models only permit answers to questions concerning where the criterion variable will be in the future without regard to specific inputs. (The exceptions to this arc multi-variate time-series techniques that take into account intervening variables.)
veloped in the 1960s as an outgrowth of early efforts in the late 1950s to CMte a demand forecasting method that would take into account both land-use growtb patterns and transport decision variables. Today, the standard demand fomca&ng method centers on the well-known UTPS computer package that is maintained by the U.S. Depattment of Transportation. The UTPS package models demand in five basic steps. First, future land uses are forecast, then the trip ends in each area of the city are estimated, the distribution of trips calculated, choice of mode determined and finally route assignments ~IVmade. One important result of this analysis is the estimate of average daily traffic for major segments of the thoroughfare network 20 years in the future. A good review of tbc theory underlying the UTPS models is presented by Kanafani (1983). The UTPS package relies heavily on detailed survey data for parameter estimation, which is typically largescale origin-destination interzonal travel behavior. To reduce these data needs, a generation of disaggregate models was developed that looked at travel behavior in even greater detail. A summary of disaggregate approaches is found in Stopher and Meyburg (1975). These approaches were further extended to examine the basic decision processes of travellers by using attitudinal techniques. A summary of these techniques is presented in Benjamin and Sen (1982). The shortcoming of disaggregate techniques was two-fold. First, data requirements remained high for an entire metropolitan area and second, aggregation of disaggregate data in a multi-stage model was difficult. Most d&aggregate models centered on mode choice.
In travel demand, the criterion variable is average daily traffic volume. Past emphasis has been on developing better structural models with inwing attention to tripmaking decisions. The structural process, however, leads to the development and estimation of parameters for a series of models. In each case, these estimates are subject to assumptions and measurement errors that compound each other progressively. Despite this, only point estimates are produced at the end of the process, giving no range of probable outcomea. Further, no measure of the impact of various assumptions that are made at each level is provided. This is underscored by the need to update model forecasts and the mixed success record of UTPS forecasts. This paper suggests that instead of a structural approach, under proper circumstances, only traffic trends 51
52
J. BENJAMIN
need to be modeled. Such models only use average daily traffic data, which are generally collected for use in traffic engineering studies. The availability and convenience of average traffic information is emphasized by Carey, Hendrickson and Siddharthan (198 1). Nihan and Holmesland (1980) suggest the direct use of traffic counts for traffic forecasting but only for short-term forecasts. The BoxJenkins technique that they use is demonstrated for accurate use only for a l-year forecast and requires mote than 60 periodic observations. Although under stable conditions the Box-Jenkins technique can be used for longer time periods than 1 year, the assumption of a specific functional form when it is known makes forecasts simpler and more accurate. The technique presented in this paper requires substantially fewer observations (which are more typically available) and is demonstrated to be accurate for much longer periods. Despite their advantages, trend techniques, such as the logistic function, cannot estimate sudden shifts in behavior, changes in the transportation network or the introduction of.new modes and they do not allow for input by planners to evaluate alternative plans. However, they doprovideestimatesoffuhm:tnndswhennetworkchangcs ate small changes, such as thoroughfare improvements, or when traffic and land use ate stable. Further, in many cases, the effect of large changes is local, having little effect outside of proximate zones or traffic corridors. Stmctural models are appropriate when large changes are pIaImed. In small urban areas, the vast majority of change is of this small, steady nature. Radical changes, such as the addition of new thoroughfares or modes that necessitate an alternatives analysis based on a structural model, are rare or nonexistent. These patterns are observed in the small urban ateas that are near Greensboro.* For example, in Greensboro, steady growth is observed for all thoroughfares. Further, there is only one new major thoroughfare planned in Greensboro in the next 25 years. In the nearby cities of High Point, Burlington and Jamestown there are no new major thoroughfares under consideration. In Winston-Salem, the only major modification is the interstate highway. In each case, steady growth has been observed over the past decade for virtually all thoroughfares, except when new thoroughfares are introduced or major activity centers arc developed. Planning for these cities instead is frequently a reactive process consisting of a search for troublesome high volume segments and intersections. Upon determination of these trouble spots, transpottation system management techniques are used to remedy the trouble. In these small urban areas, demand determines supply. The role of the planner is more to recognize shifts in demand and to accommodate them than it is to initiate and modify transport systems. The tools needed by these planners are. not tools to investigate the impacts of new networks. Rather, they are the tools to help recognize trends in demand in advance so that system modifications can be made before serious difficulties or even dangerous *Greensborohas a populationof approximately 150,000 and is located in the Piedmont area of North Carolina, U.S.A.
situations arise. The time-series approach that is suggested here is much more suited to this reality of planning today in many small urban areas. A THRORY
OF TRAFFIC GROWTH
The starting point for a theory of traffic growth is land use development. In a residential area, increased development leads to increased trip-making. Commercial and industrial development also results in increased traffic. This basic concept underlies structural models such as the UTPS model (Kanafani, 1983). In a small urban area such as Greensboro, North Carolina, land uses for most areas of the city are stable. Basic industry remains where it has been for decades and residential neighborhoods show little change. Although commercial activities have shifted to suburban malls, these changes were gradual, evolving over decades, as was the gradual development of industrial parks at the city’s periphery. However, although an understanding of the relationship between traffic and land-use development is helpful, because of the gradual change of land patterns, traffic patterns will also change slowly. Traffic volume, therefore, is a function of time that is a surrogate for urban growth. As time passes, more land is developed and traffic increases proportionally. In a particular zone or corridor, land use is initially stable when the land is agriculturally zoned. As land is developed, traffic incteases until all land in the zone or corridor is developed. At this point in time traffic stabilizes. Traf’tjc volume thezeafter remains about the same, inmasing or deecrursing by small percentages based on variations in fuel supply, population density, driving habits and land use. During growth periods. diffetent tracks are developed at different rates that appear as random fluctuations in a smooth growth curve.? Under assumptions of a constant growth rate during growth periods, the curve then rep resents an average growth rate. New development, which may take 1or even several years to complete, will average to a single rate over a period of several years. Under assumptions of constant growth, we find that: dV = A,XV dt
where X = maximum potential new average daily traffic in vehicles per day V = current average daily traffic in vehicles per daY dV the rate of change in volume with respect -dr = to time in average daily traffic per year 4 = the rate of growth in traff5c in average daily traffic per year Equation 1 indicates that changes in traffic volume are proportionate to both the amount of growth to date as tShafer (1977) indicatesthat residentialdevelopmentis a step function in the short term.
Forecast of average daily traffic volume well as potential growth and to a eoustant growth rate. This equation follows from the concept that the proportion of traffic growth is a function of the proportion of land development. One underlying assumption for eq (1) is that the attractiveness of a tract of land for development is enhanced by existing development. This is true if the knowledge of the availability and the at&activeness of residential land is spread primarily by word of mouth. An analogy exists between this form of growth and the spread of contagious diseases. People contact each other and discuss the attractiveness of the tract personally. The more people who know of the tract, the faster it develops. ‘Ibe second assumption is that development is proportionate to available undeveloped land. This is intuitive: the greater the land available, the greater the potential for development. Stated another way, with constraints on available land in a set corridor, once most land is developed, there is little room for further development so growth must be slow. Finally, the constant growth rate with respect to time represents a constant interaction of people cornmunicating the attractiveness of land in a corridor. This rate may change over time as a result of changing economic conditions that affect demand for new homes. In the latter case, growth is a non-linear function of time, which IVsuits in eq (4). Defining X = V, + V, - V where V, is the maximum growth in traffic volume, letting V, equal the minimum traffic while land is undeveloped and solving for V as a function of t, the growth function is the wellknown logistic function: V = v, + v,
exp(A, + A+) 1 + exp(A, + A,t) I
(2)
where A,, is a constant that is introduced to permit nonstaadardizsd measures of time. A0 also serves the purpose of fming the beginning of the growth process in time. The derivation of eq (2) is presented in the Appendix of this paper. The function has a characteristic “s” shape, starting and ending with asymptotically constant rates. The characteristics and estimation of the logistic function parameters am discussed by Kruskal and Tanur (1977). & and A, are found by fmt estimating the asymptotes V, and V, and rearranging terms such that:
v - v, Inv1 -v+v,
= A0 + A,1
Furthermom, the exponent of the logistic function is not aecesJarily a linear function of t but may be some function g(t). If this function of t is a polynomial, the right side of eq (3) becomes:
SW = A,, + A,t + A2t2 + A# + A,t’ + . . .
53
vari&ble&~Gobdness-of-fit;;s available for ordinary least-squares regression analysis such as R2, F statistics and the Durbin-Watson statistic are then applicab1e.S The coefficients of the polynomial terms become important when growth is not monotonic. This permits accurate models when there are temporary fluctuations in traffic volume. Deviations from a linear function of t would occur with changes in economic activity that result from business cycles, with relative cost of gasoline or with proximate changes in the tmnsport network. The logistic growth curve is only one of many curves that have been employed to model new product growth in marketing. Mahajan and Muller (1979) summarize a set of such curves, including exponential, Gompertz and a family of dynamic growth curves. When assumptions differ from logistic function assumptions, other forms should be used. The exponential curve was found best for diffusion of information based on external factors such as various promotional activities. The logistic and Gompettz curves were found to represent situations that were the result of internal interactions between members of potential and current markets. Each of these curves represent situations where the untapped market is fixed at a constant value. Dynamic models wem extensions of these basic models to consider changes in the size of untapped markets over time. Both the logistic and Gompertz curves fit the initial assumptions best in that development of districts within urban amas is primarily the result of personal interactions rather than promotional campaigns and that the untapped market for any area is fixed by the physical and zoning constraints on available potential development. In comparing the logistic and Gompertz curves, one sees that they are similar in shape. One difference between these two curves is that the inflection point is at the point of 50% cumulative growth for the logistic curve and at about 37% for the Gompertz curve. This results in varying growth rates for each. In detailed analysis of both curves, Thomley (1976) points out that the growth of the Gompertz is a direct function of time, whereas the logistic growth rate is constant. A constant rate in growth was initially assumed for this study and the logistic function was adopted. A comparison of constant and polynomial growth rates is also presented. AN EMPIRICALTEST OF LOGKSTKGROWTH The time series forecasting approach was tested in Greensboro. The city has a ring and radial thoroughfare network and has a history of careful planning originating in the early 1950s. Greensboro has stable basic industry as well as a good track record of attracting new industry, as do most cities that are located in the Sunbelt. Thus, there are both stable areas as well as high growth areas of the city.
(4)
The estimatesof A,, and A, (and Al, A,, A,. etc.) are then obtained by using ordinary least-squares regression analysis with In (V - Volvo + V, - V) as the dependent variable and t (and t2, t3, t4, etc.) as the independent
$Thc Durbii-Watsoa statistic tests for auto correlation. Auto correlation occurs in a time-series analysis when the dispersion of the data is r&ted to time. A high auto-correlationusually indicates mis-specifkation of the form of the time-series function.
54
J. BEPJJAMIN
Fig. 1. Locationof traffic count points in Greensboro, NC.
To test the technique, three Greensboro streets were identified that contrasted markedly in theii growth characteristics. Then using data available in 1975, traffic vohunes in 1981 were estimated using both UTP8 and logistic models. These estimates were then compared with observed 1981 traffic measures. UTPS estimation procedwes Because of the expense of developing UTFS estimates, existing year 2008 estimates developed in 1975 were used as the basis for comparison. Estimated volumes for 1981 were linearly interpolated between 1975 measures and 2000 forecasts. The need to interpolate underscores a major disadvantage of UTP!S; that is, its cumbersomeness and expense. Linear interpolation was suggested by local forecasters to be the practical approach that they would use. Although linear interpolation does not necessarily represent variations in the traffic growth rate, these results do provide some idea of how results of the logistic ap preach compare to UTFS, which is the most widely used model.
Test locations The thoroughfare map in Fig. 1 lists the location of the three test sites. The first site was Lee Street, a stable
area of Greensboro with little new development. The sect& site was FSetnUy Avenue, which experjenced rapid development during the decade preceding 1975. The third site was Battleground Avenue, which is in a corridor of rapid continued development. Each route is in a residential-commercial corridor and has available data for at least 25 years. Graphical route-by-route analysis To illustrate these trends, average daily volumes were plotted for the pericxl from 1955 to 1981 for each of the thme routes.8 These graphs are presented in Figs. 2, 3 and 4. By observation, the shape of the curves that best describes the trends on Friendly Avenue and Lee Street is the “s” shape, which is typical of the logistic function. Lee Street completed its growth pattern during the 1950s and 1960s and Friendly Avenue, during the 1960s and 1970s. This is consistent with the land-use growth in Qvolumesweremcas~biann~lyfortheye8rs 1955.1957, 1958, 1960. 1%2, 1964, 1966, KM, 1969, 1971. 1973, 1975, 1977,1979and 1%1.NocuuntswereavailabkonBattleground Avenue for years @or to 1960 and on FriendlyAvenue for 1955 and 1958. Although thae wete inconsisteacics in some periods between okavath. this was a constraintof the available setonday data.
Fomast of avcrsgcdaily trafticvolume
55
20 LEE STREET AVERAGE DAILY TRAFFIC ( x 1000) 10
I
0
I
1
1
,
I
t
1955 LINEAR
POWJO=
I
1
I
1371 ______
I
I
8
1981
YEAR
24 FRIENDLYAVE
AVERAGE DAILY TRAFFIC
( x 1000) 16
8 .
0 1957
1969
1981 YEAR
Fig. 3. Obserwda~~Iestimptedlmffi~~~~~FricDdly
AveauC.
c
56
J. BENJAMIN
BATTLEGROUNDAVE 20 .AVERAGE DAILY TRAFFIC ( x 1000)
10
0 1960
1970
1980
YEAR Fig. 4. Observed and estimated tdic
their respective corridors. In contrast, Battleground Avenue is just beginning to grow. The Battleground Avenue corridor is only partially developed. Functional route-by-route analysis Using a logistic function, a regression analysis was completed using the two-stage procedure described in a prior section of this paper. Fit, the minimum asymptotes were estimated by reviewing the early history of average traffic on the route. Maximum asymptotes were estimated to be proportionate to maximum land utilization in a corridor leading into each traffic count point. The procedure was to first establish corridor boundaries, then to observe which parcels of land within the corridor were developed or suitable for development but not yet developed. The land areas were then aggregated and the proportion of developed land calculated. The growth factor was calculated as the inverse of the proportion of land developed. The most difficult step in the procedure could be setting corridor boundaries. This was not difftcult for the three corridors in the case study because of natural geographical constraints. In each case, the study location was along a radial route leading to the inner beltway and the central business district. This formed the inner side of a pie-slice shaped area leading outward from the CBD. The sides
growth on BattlegroundAvenue.
of each corridor were formed by radial lines biting the augle between the study thoroughfare and the next radial thoroughfate. Natural outer boundaries were formed on Friendly Avenue by regional airport zoning restrictions, on Battleground Avenue by watershed zoning restrictions and on Lee Street by sewage and water restrictions on land development. Detailed information was available from the city planning department on developed tracts as well as long-range plans for future development. These estimates confirmed assumptions that traffic growth would be proportionate to land-use growth hecause planned new development was similar to existing patterns in both use and density. Maximum growth estimation procedures for each corridor are summarixed in Table 1. It should be noted that thii is only one approach to developing maximum growth estimates. More detailed analysis is possible for individual land uses. Another approach is to estimate asymptotes internally from the traffic data. The approach adopted here was in part influenced by the availability of specific land-use information and in part by the resource limitations of the project. Periodic growth for each route was then estimated using all data points in regression analysis. Results are summaked in Table 2.
Forecast of average daily t&k
volume
57
Table.1.
Tl=owfa= Lee stleet Friendly Avenue BattlearoundAve.
1975 Traftk
Petcentage developed land
GKnvtllfactor
16,600 17,500 20.900
97.6 76.1 67.4
1.03 1.31 1.48
For each regression analysis, the constants and coefficients were of the anticipated sign and relative magnitude. Time was coded according to the elapsed time from the start of the study period. Thus Lee Stmet, the first observation, which was for 1955, was coded 1, the second~~~,whichamnredinthethirdyear(1957), was coded 3, etc. The negative constant (A& indicates that growth started before the beginning of the study period. The positive coefficient (A,) indicates a positive growth rate. Each of these adjusted R* values is greater than 30 and each F statistic is significant at the 5% level. For Lee Street and Friendly Avenue, whete substantial growth hasaheadyoccurred,tbepatternissimilartothecharacteristic “s” shape of the logistic function and the R *‘s arc highest. For Battleground Avenue, the function has not yet devleoped and the R2 is substantially lower. The esumated and obsetved functions are graphed in Figs. 2.3 and 4. In Fig. 2. the logistic function accumtely represents the average traffic on Lee Street throughout the period. In Fig. 3, the curve represents the overall tmnd of growth on Friendly Avenue. In Fig. 4, the irregularities of the early stages of growth of the major new msidential area on Battleground Avenue deviate above and below the long-range average growth projected by the logistic function. A comparison of observed and estimated points in each graph illustrates the characteristic randomness of residuals. The Durbin-Watson statistics for Friendly Avenue and Lee Street both indicate that the hypothesis of no
Maximum traffic (V, + V,) 17,ooo 23,CQO 3l.tNm
autocorrelation can be accepted at the 5% significance level. The statistics for Battleground Avenue is inconclusive. The statistics for Friendly Avenue and Lee Street support the appropriateness of the logistic model here. However, there ate too little data available for Battleground Avenue to make such a determination. In comparing fomcasted and observed traffic, the logistic function performs best on Friendly Avenue, which is just completing the growth cycle. Lee Street, which has reached full growth, seems to fluctuate and to actually &crease slightly during the most recent study period. These fluctuations are not simulated by the logistic curve, which assumes constant growth and may be the result of exogenous economic factors, because maximum growth has been attained. Battleground Avenue is at the beginning of its growth period and, as expected, the data have yet to clearly outline the shape of the complete growth data. However, the growth forecast for Battleground Avenue is quite accurate. Compnrison of lo&tic forecasts and observed trafic for 1981
Both point and interval estimates were made for 1981 using logistk fun&m analysis ikim data liited to 1975.11 This informathm is summarked in Table 3. In each case, the observed forecast fell within or proximate to the confidence limits at the .95 confidence interval. (The observed value for Lee Street is slightly below the limits at the .95 level but falls within the limits at the .98 level, which is 12,806 to 17,000.) The forecasts show a high
Table 2.
Parunetefi Assympotes Minimum (V,)
Maximum Growth (V,) Coefficients A0 A’ R’
Adjusted R’ F statistic
Degrees of freedom Durbin-Watson statistic
Lee street
Frkndly Avenue
5400
11.600
6500 16,500
11,300 19,700
-1.73 0.18 0.52 0.48 13.20 (1.12)
-1.56 0.14 0.68 0.65 23.70 (1.11)
- 2.43 0.13 0.37 0.30 5.32 (179)
I.%
1.63
Battleground Avenue
0.83
lparaval estimates were developed under the assmllPtionof known minimum and maximum hits (V, and V,).
J. BENJAMIN
58
Tab’” 3. Comparisonof UTPS and logisticforecastsaad observedII&C in 1981 Traffic Source
Volume Thoroughfare Lee Street
Friendly Avenue
Battleground Avenue
UTPS
18,100
18,800
22,400
Logia tic point estimate interval estimate,
16,900
20,400
22,400
upper limit lover limit
17,0001 14,800
21,300I 10,300
30.9001 11,421
14,600
19,900
22,900
Observed
Traffic
‘Confidence
Interval
calculated
degreeof agrament with observed traffic for each route, even though each of the routes differed sign&a&y in its stage of growth. Contparison of logistic Md VTPS forecasts for 1981 UTF’S fozecasts for 1981 were interpolated from 1975 andyear2OoofonXasts.llTbeseforecasts werecompaled to the logistic forecasts made from the same base year. Forecasts wele made for 1981 using both techniques and then compared to observed tratXc for the same year. This information is also summatizcd in Table 3. In each case, the logistic forecast was as close or closer to the observed traffic than the UTPS forecast. In two of the CM, traBic was overestimated. which may be a result of the depressed growth in development and resulting travel during the onset of a recession. However, in each case, the logistic function performed relatively well despite the lower complexity and cost of the timeseries procedure. Logistic forecasts with polynomial exponent Instead of using a logistic function whose exponent is a linear function oft [as presented in eq (311,a polynomial function [as presented in eqs (5) and (611was used as the exponent. This analysis was particularly useful for Lee Street because of the non-monotonic growth. A fourth degree polynomial was assumed and coefficient estimates are summarized as follows: & 2 4.7, A, = 1.00, A, = 0.046, A3 = 0.00065, and A, was not included by the stepwise procedum The adjusted R* was 0.79, which compares favorably to a base R’ of .27 for a regression with these variables and sample size (McNemar, 1%9, p. 203). The F statistic was 16.9, which is significant at the .OS significance level. A comparison of observed and estimated volumes from both linear and polynomial functions is illustrated in the graph in Fig. 2. The polynomial function successfully (It shouldbe notedthatUTPS forecastsme based on somewhat differentassumptionsof traffic which is Mkctcd by the highervalues reportedfor Lee Stxeet.
at the
.95 confidence
level.
repnsents the reduction in traffic observed after the 1975 peak. Further, traffic level starts to increase again asSympoticallytO maximum capacity toward the year 2000. A DurbiWatson sta!istic of 2.53 indicated no autocorrelation, which fmther confirmed the appropriateness of this model. The shape of the polynomial curve is what would be expected. A&r a temporary reduction in traffic due to hi* gasoline prices and economic recession, final developnreat of the conidor leads to continued growth that reaches a plateau in the future as available land is developed. Bakng unexpected changes in development, the e&mates seem nasonable. Further, they are far more accumte for 1981 than either the linear of UTPS estimates (14,500** as compared to 16,900 for the linear model and 18,000 for the UTPS model; the observed traffic there was 14,600 in 1981). These results underscore the potential of the approach. CONCLUDING REMARKS As compared to the structural approach, the time-series forecasting technique is simpler, easier to understand and less costly. Furthermore. the results are as accurate, or in some cases, more accurate than the UTPS interpolated results. In addition, time-series models have measures of goodness of fit and provide interval estimates at desired confidence levels as well as point estimates. The major disadvantage of these models is that they am not sensitive to large changes in the transportation system. Examples of cases where the time-series ap preach is not best include the introduction of a new mode, the construction of a new thoroughfare, the opening of a new competing thoroughfare or substantial modifications that change the function of an arterial, such as changing traffic to one-way or extension of the arterial into a new cotidor. All of these changes will affect the
**Althoughforecastsnportcd in Table 3 wereestimatedfrom data gathemdon or before 1975,these estimates were derived from analysis of the full data set including all years through 1981.
Forecast of average daily traffic volume
land-use pattems and travel behaviors of urban residents to sufficiently warrant a structural investigation. Further, the time-series approach assumes that travel behavior remains relatively constant. Changes in economic activity, the price of gasoline or merely travel habits cannot be forecast using this approach. However, the results repoited here indicate that historically these factors have had a relatively small linfluence on average
59
of the logistic function and would add the capability to analyze transport policy. Most small urban areas are similar to Greensboro with respect to stability of the thoroughfare system. The use of time-series models should be given careful consideration where appropriate.
RFiFERENCFS
t&fiC.
In light of problems with recent cutbacks in federal highway expenditures and the increasing emphasis on maintenance of existing facilities rather than construction of new highways, the need to build predictive structural models is limited. For example, in Greensboro and SW rounding areas, a few major changes in the highway network am planned in the next 25 years. It is, therefore, reasonable to approach modeling for Greensboro by creating a structural model for the one anticipated major change in the arterial system and to model the remaining network (outside the locally affected area) using the timeseries approach. Estimated cost savings would be substantial and the model would probably be more accurate. Another approach that may reduce the expense of implementing a time-series approach would be to cluster similar regions together and estimate exponent coeff~icients once for many similar routes. Separate values for minimum and maximum levels of traffic would accommodate differences between route capacities. Once computerized, estimates would be produced routinely. Furthetmore, after observing data, diierent growth functions, such as linear functions, may be found best. Completed on a regional basis, growth functions can be assumed that best represent the growth patterns in an area. A second problem with this approach is estimation of maximum volume. The sensitivity of volume estimates to different estimates of maximum traffic volume is proportionate to both the magnitude of the maximum volume variations and the proportion of growth at the time of the estimate. Therefore, the precision of maximum volume estimates will influence answers throughout the growth period, although only to a small extent during early growth. In this paper, quick estimates were developed by projecting future growth and relating it to future traffic. These results seem to be sufficiently accurate for this case, but more precise estimates are possible using more detailed projections of cohorts or even the use of some structural land-use models. In small urban areas, this should not be a serious flaw with the method. However, additional work is needed to investigate the probabilisitic aspect of maximum volume projections and their effect on interval forecasts. Another approach to estimating asymptotes is to use the existing data. In this case, the logistic function, along with average daily trafftc alone would be used to find the maximum daily traffic. This approach should be compared to the outcomes reported in this paper. Finally, the time-series approach is not limited to nonpolicy variables. Policy variables, such as those related
to economic growth or the relative price of fuel to consumers, may be used as additional independent variables. These variables would improve the explanatory ability
A ~Uanual of Procedures to Analyze Am’tudes Towar& Transportation, U.S. Department
Benjamin J. and Sen L. (1982)
of Transportation, Washington, DC. Carey M., Hendrickson C. and SiddharmanK. N. (1981) A method for direct estimation of origin/destination trip matrices, Tmnspn. Sci. IS, 32-49. KanafaniA. (1983) Transportation DemandAnalysis. McGraw-
Hill. New York. Km&al W. H. and Taaur J. M. (1977) lnternarional Encyclopedia of Statistics. ‘Tlw Free Press, New York. MahajanV. and Muller E. (1979) Innovationdiffusion and new pmduct growth models in marketing. J Marketing 43, 5% 68. McNemar Q. (1969) Psychological Srarisrics. 4th ed., p. 203.
Wiley, New York. Nii N. and HoImesknd K. (1980) Use of the Box and Jenkins time saies techniquesin trafficforecasting. Transpn. 9, 125143. Shafer T. W. (1977) [Irban Growth and Economics. Reston
Publishing Company, Reston. VA. Stophg P. R. and MeyburgA. H. (1975) Urban Transportation &xielling and P&n&. Lexington Books. Lexington, MA. Thomky J. (1976) Mathematical Models in Plant Physiology. Academic Press, London.
AFPENDIX (DERIVATlON OF LOGISFICFUNCTION)
The derivation for the logistic function is well known. It is easkst to derive the function for the Inoportionof growth rather than the vdume of growth. The proportion of growth p is p = [(V - V&V,]. whew V is tmfBc volume, V, is minimum traflic volume and V, is maximum traffic volume growth. Potential growth is then [(V, + V, - VW,] = I - p. The rate of change of traffic volmne at any time f is assumed to he s~ to the proportionof growth to date (p), potential - p) and an ovemll growth rate A,. Hence: -
dr
= A&l
- p).
64.1)
The following differential equation thus follows: Q A,dt = -. PC1 -
(A.3 P)
To integratethis function we must set u = (I - p)p-’ and du = -p-%+.K Because the integralof du/u is In V, integrating both sides of (A.2) gives us:
-A,t
I’ =
P
In [(l - p)p-‘1
I
.
(A.3)
PO
b
Applying the boumkry condition that half of all growth will be compkted at r = 0 we get: -A,r - 0 = In [(l - p)p-‘1 - In [(l -
l/2)(1/2)-‘]
-A,r = In [(l - p)p-‘1.
or
(A.4) (A.5)
J. BENJAMIN
60
will appear in (A.4) and the solution for p in tetms of t is:
Taking the antilog of (A.5) gives:
e-4
=
(1
-
p)p-l.
(44.6)
P=
I 1 +
p=p=-
1 + e-4”
+
4’)
64.8)
If it is assumed that the gmvtb rate is a polynomial function of t, the solution for p as a function of t is:
The solution for p as a function of t is:
1
pa
L4.7)
If tbe boundary condition at r = 0 is not p = l/2 but rather mmeotherpmption~,aamstattttenn -& = ln(1 - p)po-I
1 , + e-m’
(A.9
where g(t) is a polynomial function of 1. Solutions for volume V in terms of V,. VI and r are found by appropriate substitutions in eq (A.7). eq (A.8) or eq (A.9). The mults of these substitutions an listed in eq (3). q (4) and eq (5) of the main text ofthepaper.