A STUDY ABOUT CALIBRATION METHODS OF SOME TRIP DISTRIBUTION MODELS

Mirian Buss Gonçalves Departamento de Matemática – UFSC Florianópolis – SC – Brasil E-mail: [email protected]

Edson Tadeu Bez Pós-Graduação em Engenharia de Produção – UFSC Florianópolis – SC – Brasil E-mail: [email protected]

Abstract We consider the problem of the determination of the parameters involved in some trip distribution models issued from the Transportation Planning. The proceeding to find out the parameters is called calibration. The Statistic Principle of Maximum Likelihood and statistics such as the Mean Sum of Squares Error and Phi-Normalized are the most used calibration criteria. Taking this context, the current research was developed to study and to carry out these criteria, verifying its influence in estimations presented by the models. We have applied the criteria mentioned above to calibrate gravitational entropy and gravitational – opportunities models. The experiments carried out showed, in general, an analogous performance for models independently of the criterion adopted. Thus, based on the facilities and robustness of the computational proceedings related to the maximum likelihood principle, we suggest this criterion as the most suitable to calibrate these models.

keyword: Adjustment measures, Trip Distribution Models, Parameters Identification.

1 INTRODUCTION One of the problems in the scope of transport planning is related to the estimation of the number of trips between the traffic zones, inserted in a determined study area. Some mathematical models, throughout the years, have been developed and improved aiming an improvement in the accuracy of these trips estimates. Among them some models that stand out are the classic gravitational model (WILSON, 1967), the intervening opportunities model (SCHNEIDER, 1960; FINNEY, 1972), and more recently the hybrid, gravitationalopportunity models (WILLS, 1986; GONÇALVES and ULYSSÉA NETO, 1993; DIPLOCK and OPENSHAW, 1996). These models present parameters that have to be determined in such a way that the estimates given by them are the ones that best fit the observed data. This process of parameter determination, aiming the best fitness is called calibration. In the calibration process it is necessary to define which measure will be used to evaluate the fitness of the estimates to the data, i.e., a calibration criterion has to be defined, as well to establish computational procedures in order to determine the parameters of the model leading to the best possible estimate. The most frequently used calibration criterion is based on the statistical principle of maximum likelihood (EVANS, 1971; GONÇALVES and ULYSSÉA-NETO, 1993; YUN and SEN, 1994). Other statistics are also employed, as the Mean Sum of Squares Error (DIPLOCK and OPENSHAW, 1996), the phi-normalized statistic (SMITH and HUTCHINSON, 1981), among others. This paper was elaborated to test these criteria and check their influence in the estimates presented by the models. This work is structured as follows: section 2 presents the trip distribution models used; section 3 describes the fitting measures used as calibration criteria, as well as the numerical procedures used in the models calibration; section 4 presents the data set, the implementation, and a comparative analysis of the results obtained for the different models; and finally section 5 presents the conclusions.

2

TRIP DISTRIBUTION MODELS USED

This study uses versions doubly restricted, simply restricted with origin restriction, and simply restricted with origin restriction, considering the attractivity of the gravitationalentropy model (WILSON, 1967) and gravitational-opportunity model (GONÇALVES and ULYSSÉA NETO, 1993). The following notation is used: i represents a zone of origin; j represents a destination zone; Oi is the total number of trips originated from i; Dj is the total number of trips with destination at j; Tij is the number of trips going from i to j; wij is a measure of the number of intervening opportunities between zones i and j; cij is a measure of spatial separation between zones i and j; n is the number of traffic zones of the study area. In the gravitational-entropy model the following cases are analyzed: a) Gravitational-entropy model doubly restricted:

Tij = Ai B j Oi D j e Ai = [∑ B j D j e

− β ⋅cij

− β ⋅cij

(1)

] −1

(2)

] −1

(3)

j

B j = [∑ Ai Oi e

− β ⋅c ij

i

where Ai and Bj are the balancing factors associated to Oi and Dj , and β is the deterrence parameter associated to the spatial separation between zones i and j. b) Gravitational-entropy model simply restricted with origin restriction: Tij = Ai Oi e Ai = [∑ e

− β ⋅c ij

− β ⋅c ij

(4)

] −1

(5)

j

c) Gravitational-entropy model simply restricted with origin restriction, considering the attractivity: Tij = Ai Oi D j e Ai = [∑ D j e

− β ⋅cij

− β ⋅c ij

(6)

] −1

(7)

j

In the gravitational-opportunity model the following cases are analyzed: d) Gravitational-opportunity model doubly restricted: Tij = Ai B j Oi D j e

− ( β ⋅cij + λwij )

Ai = [∑ B j D j e

− ( β ⋅cij + λwij ) −1

B j = [∑ Ai Oi e

− ( β ⋅cij + λwij ) −1

]

(8) (9)

j

]

(10)

i

where λ is an impedance parameter associated to the intervening opportunities. e) Gravitational-opportunity model simply restricted with origin restriction: Tij = Ai Oi e Ai = [∑ e j

− ( β ⋅cij + λwij )

− ( β ⋅cij + λwij ) −1

]

(11) (12)

f) Gravitational-opportunity model simply restricted with origin restriction, considering the attractivity: Tij = Ai Oi D j e Ai = [∑ D j e

− ( β ⋅cij + λwij )

− ( β ⋅cij + λwij ) −1

]

(13) (14)

j

3 CRITERIA AND NUMERICAL PROCEDURES USED IN THE CALIBRATION The previous section presented the trip distribution models whose parameters have to be determined. The determination of the parameter set, in a way that estimates given by the model are the ones that best fit the observed data is a process called calibration. According to GONÇALVES and SOUZA DE CURSI (1997), the calibration process consists of two steps: “1) Defining a calibration criterion, i.e., defining which measure will be used to evaluate the fitness of the estimates to the data. 2) Establishing computational procedures in order to determine the parameters of the model to obtain the best fitting.” Before presenting the fitting measures that will be used it is necessary to present some notations: Tij = number of trips estimated by cell; Tij* = number of trips observed by cell; *

T = average number of trips observed by cell, given by: T

*

∑T =

* ij

ncel

;

T * = total number of observed trips, where, T * = ∑ Tij* ;

T = total number of estimated trips; ncel = number of matrix cells with estimated flow. This study uses as calibration criteria the statistical principle of maximum likelihood (EVANS, 1971; GONÇALVES and ULYSSÉA-NETO, 1993; YUN and SEN, 1994), the Mean Sum of Squares Error (DIPLOCK and OPENSHAW, 1996), and the phi-normalized statistic (SMITH and HUTCHINSON, 1981), to estimate the parameters of the gravitational and gravitational-opportunity models.

Using the principle of maximum likelihood, EVANS (1971) obtained the following equations for the calibration of the balancing factors and impedance parameter for the gravitational model doubly restricted. Ai = [∑ B j D j e

− β ⋅c ij −1

]

(15)

j

B j = [∑ AiOi e

− β ⋅c ij −1

]

(16)

i

Tij*

∑T

*

ij

cij = ∑

Tij T

ij

cij

(17)

In this study the balancing factors Ai and Bj are determined using the equations (15) and (16) through the method of equilibrium of the matrices of Furness. The impedance parameter β, to the versions from the entropy gravitational model used, was obtained in a way to reproduce the average trip cost observed (equation 17), through the secants method (HYMAN, 1969). GONÇALVES and ULYSSÉA-NETO (1993) applied the statistic principle of maximum likelihood to determine the parameters of the gravitational-opportunity model doubly restricted (equation 8), obtaining results analogous to the ones obtained by EVANS (1971) to the gravitational model. The balancing factors Ai and Bj have to guarantee the restrictions of the consistency of flux, while the impedance parameters are determined in a way to reproduce the average cost and the average number of intervening opportunities observed, respectively. The following equations were obtained by GONÇALVES and ULYSSÉA-NETO (1993): Ai = [∑ B j D j e

− ( β ⋅c ij + λwij ) −1

(18)

]

j

B j = [∑ Ai Oi e

− ( β ⋅ c ij + λwij ) −1

]

(19)

i

∑T c

= ∑ Tij cij

(20)

∑T w = ∑T w

(21)

* ij ij

i j

* ij

i j

i j

ij

ij

ij

i j

In this work, the impedance parameters β and λ, to the gravitational-opportunity models described in the previous section, are determined in a way to reproduce the average trip cost observed and the average number observed of intervening opportunities by trip, through the minimization of the function 2

 Tij ( β , λ )   Tij ( β , λ )  F ( β , λ ) =  c − ∑ ⋅ cij  +  w − ∑ ⋅ wij  T T ij ij    

where,

2

(22)

∑ Tij*cij c=

∑T

* ij

ij

e

T*

w=

wij

ij

T*

(23)

The function F (β , λ ) was minimized through the random perturbation of the gradient method POGU and SOUZA DE CURSI (1994). The statistics Mean Sum of Squares Error was used by DIPLOCK, OPENSHAW (1996) in a study about the use of genetic algorithms to the calibration of trip distribution models. The equation that represents this statistic is given by the following expression: f (error ) = ∑

(Tij* − Tij )2

ij

ncel

(24)

The smaller the value determined by f (error) , the better is the fitting between the matrices of observed and estimated trips. In this study the expression (24) is applied both to the gravitational as to the gravitationalopportunity models. The third fitting measure used as a calibration criterion was the phi-normalized statistic, presented by SMITH, HUTCHINSON (1981). The expression that represents this statistic is given by:  Tij*  f = ∑ * ln   Tij  ij T   Tij*

(25)

The smaller the value of f, the better is the fitting between the matrices of observed and estimated trips. The phi-normalized statistic is based on the information theory, and assumes value equal to zero, when the matrices of observed and estimated trips coincide. Both to the Mean Sum of Squares Error as to the phi-normalized statistic, the Fibonacci Search (NOVAES, 1978) was used to determine the parameter β, in the three cases of the gravitational model. To the determination of the parameters β and λ, in the three cases of the gravitational-opportunity model, the random perturbation of the gradient method (GONÇALVES and SOUZA DE CURSI, 1997; GONÇALVES and SOUZA DE CURSI, 2001) was used. This method was chosen having in view the numerical difficulties presented in the calibration of these models, as the no convexity of the function to be minimized, as verified in Figure 1. The balancing factors Ai and Bj of the versions doubly restricted from the mentioned models were determined through the Furness method.

Function value (average sum of the squared errors criterium)

Lambda parameter

Beta parameter

Figure 1 – Gravitational-opportunity model simply restricted with origin restriction, considering the attractivity ( rectangle [0;2]x[0;1]; mesh interval 0.02)

4

DATA SET, IMPLEMENTATION, AND RESULT ANALYSIS

The data set used in this work was collected by the researcher Lourdes Maria Werle de Almeida, and used in the developing of a practical application of the presented methodology in her PhD thesis: “Desenvolvimento de uma metodologia para análise locacional de sistemas educacionais usando modelos de interação espacial e indicadores de acessibilidade.” This data set was collected in the urban area of Londrina, a city located in the north of the state of Parana, south of Brazil, with an area of 116,80 Km2. This area was divided in 12 traffic zones , based on data obtained from IPUL (Urban Planning Institute of Londrina). The centroid of each zone was determined taking into account the biggest populational concentration of the zone. Each zone can be seen as a trip origin and destination zone. The study was centered in the public (state) net of high school, that was composed by 29 units of public schools in 1997, with 18701 students enrolled. Once chosen the school segment with its several schools spatially distributed, it was possible to obtain an observed matrix of students trips to the schools through a research in the 29 schools about the origin zone of their students. It was also possible to define the demand and the attractivity of the traffic zones, taking into account these schools.

As attractivity measure of the destination zone j, Dj , was considered the number of vacancy of all the schools from zone j, obtained from the Regional Nucleus of Education from Londrina (NRE). The demand Oi of students from each zone i was obtained from IBGE (Brazilian Institute of Geography and Statistics) (GONÇALVES, ALMEIDA and BEZ, 2000). The spatial separation matrix between each pair of zones (i, j ) was determined through the time taken by the student to go from their house to the school, including the waiting and walking time, using the bus as a mean of transportation. This transportation was chosen by ALMEIDA (1999), since it is the only public transportation available in the city and also because it is used by the majority of the students interviewed. To determine the intervening opportunities matrix to each unit of the school net in study was associated the utility that it provides to the students, according to the methodology presented by ALMEIDA and GONÇALVES (1998). Having in view the objective of this study, that is testing the several calibration criteria used to determine the parameters involved in the models in study, it was necessary the development of several computational programs. These programs were implemented with the assistance of the software MATLAB for Windows version 5.1.0.421. Once calibrated the models, the analysis of the criteria adequacy is done based on two basic aspects: the numerical difficulties and the models performance. The models performance is carried out by means of trip-length frequency distributions and the followed goodness-of-fit statistics: Dissimilarity Index (DI) (THOMAS, 1977) is defined by the equation:

DI =

50 T * − Tij * ∑ ij T ij

(26)

Normalized Absolute Average Error (NAAE) (SMITH, HUTCHINSON, 1981) is defined by:

NAAE = ∑

Tij* − Tij

(27)

−

ij

T

*

Root Mean Square Error (RMSE) (WILSON, 1976) is defined by:  (Tij* − Tij ) 2  RMSE = ∑  ncel   ij

1

2

(28)

Chi-square Error ( ℵ2 ) (WILSON, 1976) is defined by: ℵ2 = ∑ ij

(Tij* − Tij ) 2 Tij

(29)

Besides these statistics, the ones used as calibration criteria were also used, being them: the Mean Sum of Squares Error (DIPLOCK and OPENSHAW, 1996), and the phi-normalized statistic (SMITH and HUTCHINSON, 1981). In a fitting evaluation a smaller value determined by these statistics represents a better fitting between the observed and estimated trip matrices. Table 1 presents the results obtained for the three versions of the gravitational model (Equations (1), (4) e (6)). The difference between the observed average cost (CMO) and estimated average cost (CME) and the values obtained to the impedance parameter β, are also presented.

TABLE 1 – Values obtained in the statistics Criteria à Statistics ↓

Criterion of the maximum likelihood

(1) Dissimilarity index Normalized absolute average error Mean Sum of Square Error Root Mean Square Error Chi-square Phi-normalized

| CMO – CME | Parameter β value

(1)

(4)

(6)

(1)

(4)

(6)

38.323 38.301 110.37 110.30

25.001 72.00

38.678 111.39

38.523 110.94

25.361 73.04

38.260 110.19

34.819 100.28

17022 130.46 14531 0.505

31817 38004 178.37 194.94 28432 28182 0.768 0.852

17048 130.57 15056 0.503

32279 38470 179.66 196.13 26412 26773 0.749 0.847

17022 130.46 14571 0.504

31768 32039 178.23 178.99 29643 276994 0.780 1.174

0

0

(6)

Criterion of the Mean Sum of Squares Error

25.395 73.13

0.0889

(4)

Criterion of the Phinormalized statistic

0

0.0808 0.0629

0.30

1.02

1.29

0.02

0.44

5.46

0.0921

0.0752

0.0535

0.0892

0.0835

0.1535

Observing the columns from Table 1, a global evaluation shows that the accuracy level of the estimates obtained using the maximum likelihood and phi-normalized criteria is very close, being superior to the one obtained using the Mean Sum of Squares Error statistic. The version simply restricted with origin restriction, when considering the attractivity, presented interesting results to the Mean Sum of Squares Error criterion. Analyzing the columns related to the model given by equation (6), it is possible to see that the first four statistics show a superior performance of the model using the Mean Sum of Squares Error criterion. However, when observing the values obtained in the Chi-square and Phi-normalized statistics, it is verified that it behaved in an absurd way, obtaining inconsistent values for an analysis. In the frequency distribution presented in Figure 2, it is possible to notice this fact. The other distributions to the doubly restricted and simply restricted with origin restriction versions are set aside due to space restrictions. It is also interesting to observe that, from the versions tested for the gravitational model, a better performance was noticed from the model in its version doubly restricted.

Number of trips ( in percentage)

70

60

50

Observed distribution M a x i m u m likelihood criterion Phi-normalized criterion M S S E criterion

40

30

20

10

0 10

15

20

25

30

35

40

45

50

55

60

Trips cost (6 classes)

Figure 2 – Frequency distribution – Gravitational model simply restricted with origin restriction, considering the attractivity

Table 2 presents the results obtained for the three versions of the gravitational-opportunity model. It presents the difference between the observed average cost (CMO) and estimated average cost (CME) and the average number of estimated intervening opportunities (WME) and observed intervening opportunities (WMO), that are also relevant to verify the models performance. The obtained values for the impedance parameters are also presented.

TABLE 2 – Values obtained in the statistics Criteria à Statistics ↓

Criterion of the maximum likelihood

(8)

(11)

(13)

Criterion of the Phinormalized statistic

(8)

(11)

(13)

Dissimilarity rate Normalized absolute average error Mean Sum of Square Error Root Mean Square Error Chi-square Phi-normalized

22.431 64.60

27.576 38.287 22.319 79.42 110.26 64.27

27.586 38.476 79.44 110.81

19.638 56.55

26.517 76.37

36.542 105.24

12037 109.71 16015 0.467

17206 39601 11888 131.17 199.00 109.03 23130 29138 16361 0.583 0.861 0.467

17216 38895 131.21 197.21 23089 27123 0.583 0.850

10898 104.39 33300 0.502

15959 36457 126.33 190.93 30129 822408 0.607 1.380

| CMO – CME | | WMO – WME | Parameter β value Parameter λ value

0 0

0.06 0.10

0.05 0.11

1.29 0.73

1.78 1.69

1.75 1.18

5.65 5.31

0.0512 0.0025

0.0129 0.1272

0.0096 0.1024

0.1379 0.0507

0.15 0.43

(8)

0.01 0.04

(11)

0.0230 0.00009 0.0547 0.0190 0.00008 0.0831 0.0994 0.0083 0.0881 0.0994

(13)

Criterion of the Mean Sum of Squares Error

Analysing Table 2, the results obtained show in a global evaluation, once more, that the accuracy level of the estimates obtained using the maximum likelihood and phi-normalized criteria is very close, being superior to the one obtained using the Mean Sum of Squares Error statistic. Once more the Mean Sum of Squares Error criterion presented inconsistent results in the Chisquare and Phi-normalized statistics, in the version simply restricted with origin restriction when considering the attractivity. Another interesting fact in this version of the gravitational–opportunity model is that its graphic representation of the frequency distributions is similar to the one presented in Figure 2. Analyzing Tables 1 and 2 it is also noticed that when using the Mean Sum of Squares Error criterion, the gravitational model simply restricted considering the attractivity (Equation (6)) presents a better performance than the correspondent version of the gravitational opportunity model (Equation (13)). Finally, analyzing the set of all statistics used, we can conclude that model (8) (doubly restricted gravitational opportunity model) presents the best performance between the studied models.

5

CONCLUSIONS

From the criteria applied to the entropy gravitational models, it was noticed in some cases an analogous performance of the models. However, in general, the application of the maximum likelihood criterion determined a better performance of these models. Besides, the computational implementation did not present any problem. In this way, it was empirically noticed that this is the most appropriate criterion for the calibration of these models. In the cases described for the gravitational–opportunity models it was noticed in some cases, once more, an analogous performance of the models. However, since the maximum likelihood criterion did not present in any case any problem on its computational implementation it was empirically noticed that this is the most appropriate criterion for the calibration of these models. In a general analysis it was noticed that the models obtained a better performance when applying the maximum likelihood criterion. Besides, taking into account the facility of its computational implementation and the fact that no numerical difficulties arose during the calibration process, this criterion is suggested as the best alternative for the calibration of the studied models.

6

REFERENCES

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