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A note on the balancing factors of gravity models
0038-0121/81/050249-05102.00/0 Pergamon Press Ltd.
Socio-Econ. Plan. Sci. Vol. IS, No. 5, pp. 249-253, 1981 Printed in Great Main
A NOTE ON THE BALANCING FACTORS OF GRAVITY MODELS ERIC I. PAS Duke University, Durham, NC 27706,U.S.A.
MARK A. TODES Technion-Israel Institute of Technology, Haifa, Israelt (Received
1 September
1980; in
revised form 19 March 1981)
Abstract-Application of the production constrained gravity-type spatial interaction model generally incorporates a “balancing of attractions” procedure to yield the fully constrained condition. Two techniques for balancing a singly constrained gravity model are compared in this note. These are the Federal Highways Administration adjusted attraction factor method and the Furness column and row balancing procedure. This comparison shows firstly that the two techniques are identical, and secondly that the balancing procedure results in an arbitrary distortion of the calibrated distribution function. Furthermore, some empirical results show that the balancing procedure does not necessarily improve the model’spredictionson a cell by cell basis. These results indicate that balancing may be an unnecessary as well as undesirable step in the application of the singly constrained gravity model. 1.
INTRODUCTION
constrained model is derived by arbitrarily setting Ci = 1 for all j,S which yields the following formulation for &:
In recent years the approach to analytical transport
planning has shifted largely towards the development of disaggregate behavioral models. Nevertheless, the conventional aggregate models are still being widely applied and elucidation of their appropriate use is therefore important. Perhaps the most widely applied of the conventional models is the well known gravity-type spatial interaction model for trip distribution. In an area with N origins and M destinations this model may be stated as follows: tii = B;C,PA,f(cij)
i = 172,. . . . ,N j=1,2 , . . . . . (M
Bi=[k$,Arf(cik)]-’
i=1,2 ,....,
N.
(4)
Substituting for Bi in eqn (1) yields the production con-
strained formulation of the gravity model: tij _
MP,Af(c,) AJ(Cik)
z,
(1)
(5)
Setting the factor Cj = 1 for all j is equivalent to relaxing the “attractions” constraint (3) shown above. Thus the gravity model shown in (5) will be production constrained-i.e. the model-predicted row totals will be equal to the input productions (Pi), but the predicted column totals will not necessarily be equal to the input attractions (Ai). Two alternative methods have been developed to “balance the attractions”. These are the Furness row and column balancing method and the Federal Highways Administration (FHWA) attraction factor method. This note presents a comparison of these two methods in which it is shown that they are, in fact, identical. The comparison also yields a formulation of the balanced (doubly constrained) model which enables further interpretation of the effect of the balancing factors.
where tii is the estimated number of trips made between origin i and destination j; Pi is the number of trips produced at origin i; Ai is the number of trips attracted to destination j; f(cij) is a function of the cost (generalized) of travel between zones i and j; and Bi and Ci are “balancing factors” which ensure that the origin and destination trip-totals constraints are satisfied. These constraints are:
ztij =
Pi
$+tij=Ap This note is concerned with the derivation, use and interpretation of these balancing factors. The most commonly applied form of the gravity model is the production constrained formulation. This singly
XMETHODS
OF BALANCING A SINGLY-CONSTRAINEDGRAVITYTYPEMODEL
The FHWA balancing method
Kurrent address: Murray, Biesenbach and Badenhorst Ing/Inc. Consulting Agricultural Engineers, P.O. Box 111571, Brooklyn 00111Pretoria. SOf course, if we set Bi = I for all i, we obtain the attraction constrained formulation, which is also singly constrained.
The computer programs for urban transportation planning distributed by the FHWA [ l] use the production constrained gravity model described above for the analysis of trip distribution. The distribution function [f(Cij)] is calibrated by an iterative procedure which aims to replicate the observed trip “length” frequency distribution. Once this is completed, the balancing of
249
E. I. PAS
250
and M. A. TODES
attractions is carried out by a second iterative process, using an adjusted attraction which can be expressed generally as follows: Aj
a,Ol’= afn-U x
I
2
I=1
(G[n-II) t1j
I,
PiaF’f
=
$,
The FHWA attraction factor, shown in eqn (6) can be reformulated as follows:
(6)
a,+l’
aj(“‘=Ajx F
where: a?) is the adjusted attraction in zone j at the nth balancing iteration; and tj~[“-” is the trips predicted by the gravity model at the (n - 1)th balancing iteration. Then the number of trips from i to j predicted at the nth balancing iteration are given by: pnl)
Comparison of the FHWA and Fumess balancing procedures
&ll,
but, a!“-‘) = Aj x I
and therefore,
(Cij)
(7) a (n-2)
a:“‘f(Gd
q@“=,J,x F
In this case, A, is the observed total trips attracted to zone j. When the model is used to predict future trip distribution, an initial predicted matrix is calculated using the calibrated distribution function and the attractions are balanced using the procedure described above. In this case, Aj is the predicted total trips attracted to zone j.
t;“-ll~
x
F
:,k21~~
Similarly, substituting for a/“-‘),. . . . , a?‘, and noting that aJo’ I = A.1,we obtain ay’=Ajx
The Fumess balancing method
The Furness method consists of alternate column and row balancing iterations which can be expressed in general terms as follows: and thus: (8)
a!“’ = AiRi I
(13)
where:
That is, Rj”) is an adjustment factor for the nth balancing iteration of the FHWA attraction factor method. Now, substituting eqn (13) into eqn (7) we obtain: t{x”l) I,
=
P.A. (cij)Ri(“) 1
(19
x AJ(Cik)R*‘“” k p+‘l’= t,
$x””
j
x T
(11)
;yl)
where: t!F[“” is the estimated trips between i and j after the nth “Furness” column balancing iteration; t::‘“” is the estimated trips between i and j after the nth “Furness” row balancing iteration; and t~~[ol)is the trips between i and j estimated originally by the gravity model. That is, ~~F[oI) _
_P,M(Cij)
3, AJ(cit)
.
(12)
Equation (15) is a general expression for the simulated trips from i to j after the nth balancing iteration of the FHWA attraction factor method. For n = 1, eqn (14) yields:
Rro) = 7 2&o,Y
Therefore; eqn (8) may be rewritten as: t
(16)
A note on the balancingfactors of gravity models and substituting from (16) and (5) into (9); we obtain:
251
Thus, in general: @‘“‘I = $rnl).
(24)
That is, the two alternative balancing procedures compared in this note lead to identical results. 3,DISCUSSIONAND INTERPRETATIONOF ANALYTIC RESULTS
Two significant points arise from the above analysis. First, the numerous properties of the Furness procedure that have been developed in the literature are equally Simplifying (17) yields: applicable to the FHWA adjusted attraction factor method. For example, Evans[Z] shows that if all rij > 0, pw, = PiAif( C,)R/” (18) then the Furness process will converge to a unique solution that has all rows and columns “balanced”. The I’ T AJ(Cik)R,‘“’ above analysis therefore implies that the FHWA technique will converge to the same unique, balanced soluEquation (18) represents the simulated trip distribution tion. It should be noted that the condition that all tii >O after 1 column and 1 row balancing iteration using the is a sufficient, but not a necessary condition for conFurness method. Now, substituting n = 1 in the general vergence. Cesario[3] shows that a matrix containing expression for the FHWA method feqn 15), we find some zero elements will still converge to the solution, as long as the sum of attractions in all columns other than the column containing a zero, is not less than the production in the row containing that same zero. Thus in large study areas, with large trip productions and attractions, the “zero elements” condition will not generally However, the r.h.s. of eqns (18) and (19) are identical. prevent convergence. In fact, the zero elements problem Thus, we have: does not arise with the FHWA method, since the gravity model can only produce zero trips between any zone pair r
m
J
=i PiAjf( Cij)Ri[2’ T AJ(cidRk”’
4. E~I~CALANALYS~ Gravity models were calibrated using three different sets of data and two different calibration techniques. The first technique was the FHWA iterative procedure based (23) on matching the observed and simulated trip “length”
E. I. PASand M. A. TODES
252
frequency distributions, and the second calibration method was a linear regression technique which is designed to derive zone-to-zone specific factors that minimize the sum of squares between the observed and simulated origin-destination matrix[5]. The data were obtained from origin-destination studies conducted in Cape Town, South Africa and Skokie, Illinois. The Skokie data consists of trips within a 22 zone area (data set A). The Cape Town data consists of trips to work by car in a 105 zone area, made by White and Coloured (mixed race) commuters (data sets B and C).t The origin-destination matrices predicted by the gravity models before and after balancing were compared with the respective observed matrices using the error measures described below. The error measures employed for comparing the results are those used by Koppelman[6]. The root mean square error (RMSE) is given by: RMSE = [i?
7 (REii)Z]I’*
(25)
where N is the total numper of cells in the distribution matrix (m X n); RE, is [(tii - fij)/tij]; tij is the observed trips between i and j; and tij is the model estimated trips between i and j. Koppelman shows that the RMSE can be separated into components representing the average error (AE) and the standard deviation of the error (SDE) as follows: RMSE* = AE* + SDE* where AE=$~RE, 1 I SDE = [ $F
F (REU- AE)2]“Y
(28)
tThe technical analyst finds it hard to avoid the reality of racially separated residential areas in South African cities.
This formulation enables broader interpretation of the errors. It can be seen from eqn (27) that AE is sensitive to the sign of RE. Thus if AE is positive, it indicates that the model has generally overpredicted the trjp interchange values since it implies that on average,(!,j - tij) is greater than zero. Similarly, if AE is negative, It Indicates that the model has generally underpredicted. The SDE provides a measure of the deviation of the error about AE. If for example, the AE is small but the SDE is large, this indicates that the model is generally providing good estimates, but that it is not well specified in certain cases. It can be seen in Table 1 that in all cases the average error (AE) was reduced by balancing the attractions. However, this does not necessarily imply an improved fit with the observed matrix, since it is possible that the balancing of attractions may also have resulted in a better balancing of overpredictions and underpredictions. In some cases the standard deviation of the error (SDE) was also reduced, thus indicating that there was in fact an overall improvement. However, in other cases the standard deviation of the error increased, indicating that the errors were generally larger after balancing. In most cases these changes were small and therefore not significant. In case B.2 however, the increase in the standard deviation of the error was significantly large. Thus the balancing procedure did not consistently produce an improved fit to the observed origin-destination matrix. Because of the disaggregation of the root mean square error into its two components, we can see that on the average the balancing procedure does improve the fit between the observed ans simulated trip distribution. However, we see also that the standard deviation of the error increases, indicating that the balancing procedure has the effect of producing large differences between observed and simulated trips for some origin-destination pairs. Furthermore, the results in Table 1 indicate that in those cases where the root mean square error is reduced by the balancing process, the change is relatively small. 5. CONCLUSIONS It has been shown that the balancing of attractions factor procedure commonly used in applications of the
Table I. Base year prediction errors of six gravity models Before
Balancing
After
MODEL1
AE2
SDE3
RYSE4
A.1
16.4
59.1
61.4
15.1
60.3
62.2
A.2
12.4
49.3
50.8
12.3
49.9
51.4
B.l
23.3
106.6
109.2
18.1
102.3
103.9
B.2
39.3
123.1
129.2
37.1
172.5
176.5
c.1
48.6
121.7
131.1
45.5
117.6
126.1
c.2
50.3
139.8
148.5
48.9
132.2
140.9
1.
2. 3. 4.
AE
Balancing
SDE
RYSE
data sets, while A, 8, and C refer to the three different 1 and 2 refer to two alternative calibration methods (see text for details). AE = Average Error (equation 27) = Standard Deviation of the Error (equation 28) SDE RMSE = Root Mean Square Error (equation 25)
A note on the balancing factors of gravity models
production constrained gravity model is identical ?o, and therefore as efWeat ass, the Faess procedure, However, it does Rot necessady result in an improvement of the model predictions on a cell by cell basis. It is possible therefore, that the balancing of attractions may be zkn unnecessary step in the tise of the singly constrained ~a~~~~~e spatial ~~~~r~~~~~ m&et. Furtheremore, this research has shown that the baiancing procedure can be! interpreted as leading ta an arbitrary distartion of the calibrated friction factors, fn other: words, the simulated trip d~s~b~t~~ matrix after balancing Is not necessariiy consistent with the cat&rated friction factors. Therefore, the balancing of attractions in the commonly used production constrained formulation of the gravity model might be undesirable as well as
253
REFEREIKW 1. Federat Highway Ad~inis~r~tion~Computer programs for urban t~a~s~rtat~~~ planning. ~~er~~ ~~~o~of~u~ ~~u#~~~ US, Govt. Printing Office, Washington, DC. fB77].
2. A. W. Evans, Some propertiesof trip distribution methods.
Transm. Rex 4. 19-36 (1970). 3. F. J. ‘Cesario, Parameter Estimation in Spatial Ynteraction ~~e~~~~ Emmt Ph f. 503-51811973%. 4. F. J. C&ark, A new ~ot~r~re~at~o~ of &e ~~norrna~~~~~g~’ or “balancing” factors of gravity-type spatial mod&. SocioEcon. Plan. Sci. 11, 131-136(1977), S. M. A. Todes, A Regression Technisue for the Calibration of a
~R~CC~~~~~~ Ackfluwledgements-The research reported in this note was conducted while the authors were in the Department of Civil
6. F. S. Koppelman, &thodolog; for a;alysis of krrok in prediction with disaggregate choice models. Transprr,Rex Rec. 592 (1976).
Socio-Econ. Plan. Sci. Vol. IS, No. 5, pp. 249-253, 1981 Printed in Great Main
A NOTE ON THE BALANCING FACTORS OF GRAVITY MODELS ERIC I. PAS Duke University, Durham, NC 27706,U.S.A.
MARK A. TODES Technion-Israel Institute of Technology, Haifa, Israelt (Received
1 September
1980; in
revised form 19 March 1981)
Abstract-Application of the production constrained gravity-type spatial interaction model generally incorporates a “balancing of attractions” procedure to yield the fully constrained condition. Two techniques for balancing a singly constrained gravity model are compared in this note. These are the Federal Highways Administration adjusted attraction factor method and the Furness column and row balancing procedure. This comparison shows firstly that the two techniques are identical, and secondly that the balancing procedure results in an arbitrary distortion of the calibrated distribution function. Furthermore, some empirical results show that the balancing procedure does not necessarily improve the model’spredictionson a cell by cell basis. These results indicate that balancing may be an unnecessary as well as undesirable step in the application of the singly constrained gravity model. 1.
INTRODUCTION
constrained model is derived by arbitrarily setting Ci = 1 for all j,S which yields the following formulation for &:
In recent years the approach to analytical transport
planning has shifted largely towards the development of disaggregate behavioral models. Nevertheless, the conventional aggregate models are still being widely applied and elucidation of their appropriate use is therefore important. Perhaps the most widely applied of the conventional models is the well known gravity-type spatial interaction model for trip distribution. In an area with N origins and M destinations this model may be stated as follows: tii = B;C,PA,f(cij)
i = 172,. . . . ,N j=1,2 , . . . . . (M
Bi=[k$,Arf(cik)]-’
i=1,2 ,....,
N.
(4)
Substituting for Bi in eqn (1) yields the production con-
strained formulation of the gravity model: tij _
MP,Af(c,) AJ(Cik)
z,
(1)
(5)
Setting the factor Cj = 1 for all j is equivalent to relaxing the “attractions” constraint (3) shown above. Thus the gravity model shown in (5) will be production constrained-i.e. the model-predicted row totals will be equal to the input productions (Pi), but the predicted column totals will not necessarily be equal to the input attractions (Ai). Two alternative methods have been developed to “balance the attractions”. These are the Furness row and column balancing method and the Federal Highways Administration (FHWA) attraction factor method. This note presents a comparison of these two methods in which it is shown that they are, in fact, identical. The comparison also yields a formulation of the balanced (doubly constrained) model which enables further interpretation of the effect of the balancing factors.
where tii is the estimated number of trips made between origin i and destination j; Pi is the number of trips produced at origin i; Ai is the number of trips attracted to destination j; f(cij) is a function of the cost (generalized) of travel between zones i and j; and Bi and Ci are “balancing factors” which ensure that the origin and destination trip-totals constraints are satisfied. These constraints are:
ztij =
Pi
$+tij=Ap This note is concerned with the derivation, use and interpretation of these balancing factors. The most commonly applied form of the gravity model is the production constrained formulation. This singly
XMETHODS
OF BALANCING A SINGLY-CONSTRAINEDGRAVITYTYPEMODEL
The FHWA balancing method
Kurrent address: Murray, Biesenbach and Badenhorst Ing/Inc. Consulting Agricultural Engineers, P.O. Box 111571, Brooklyn 00111Pretoria. SOf course, if we set Bi = I for all i, we obtain the attraction constrained formulation, which is also singly constrained.
The computer programs for urban transportation planning distributed by the FHWA [ l] use the production constrained gravity model described above for the analysis of trip distribution. The distribution function [f(Cij)] is calibrated by an iterative procedure which aims to replicate the observed trip “length” frequency distribution. Once this is completed, the balancing of
249
E. I. PAS
250
and M. A. TODES
attractions is carried out by a second iterative process, using an adjusted attraction which can be expressed generally as follows: Aj
a,Ol’= afn-U x
I
2
I=1
(G[n-II) t1j
I,
PiaF’f
=
$,
The FHWA attraction factor, shown in eqn (6) can be reformulated as follows:
(6)
a,+l’
aj(“‘=Ajx F
where: a?) is the adjusted attraction in zone j at the nth balancing iteration; and tj~[“-” is the trips predicted by the gravity model at the (n - 1)th balancing iteration. Then the number of trips from i to j predicted at the nth balancing iteration are given by: pnl)
Comparison of the FHWA and Fumess balancing procedures
&ll,
but, a!“-‘) = Aj x I
and therefore,
(Cij)
(7) a (n-2)
a:“‘f(Gd
q@“=,J,x F
In this case, A, is the observed total trips attracted to zone j. When the model is used to predict future trip distribution, an initial predicted matrix is calculated using the calibrated distribution function and the attractions are balanced using the procedure described above. In this case, Aj is the predicted total trips attracted to zone j.
t;“-ll~
x
F
:,k21~~
Similarly, substituting for a/“-‘),. . . . , a?‘, and noting that aJo’ I = A.1,we obtain ay’=Ajx
The Fumess balancing method
The Furness method consists of alternate column and row balancing iterations which can be expressed in general terms as follows: and thus: (8)
a!“’ = AiRi I
(13)
where:
That is, Rj”) is an adjustment factor for the nth balancing iteration of the FHWA attraction factor method. Now, substituting eqn (13) into eqn (7) we obtain: t{x”l) I,
=
P.A. (cij)Ri(“) 1
(19
x AJ(Cik)R*‘“” k p+‘l’= t,
$x””
j
x T
(11)
;yl)
where: t!F[“” is the estimated trips between i and j after the nth “Furness” column balancing iteration; t::‘“” is the estimated trips between i and j after the nth “Furness” row balancing iteration; and t~~[ol)is the trips between i and j estimated originally by the gravity model. That is, ~~F[oI) _
_P,M(Cij)
3, AJ(cit)
.
(12)
Equation (15) is a general expression for the simulated trips from i to j after the nth balancing iteration of the FHWA attraction factor method. For n = 1, eqn (14) yields:
Rro) = 7 2&o,Y
Therefore; eqn (8) may be rewritten as: t
(16)
A note on the balancingfactors of gravity models and substituting from (16) and (5) into (9); we obtain:
251
Thus, in general: @‘“‘I = $rnl).
(24)
That is, the two alternative balancing procedures compared in this note lead to identical results. 3,DISCUSSIONAND INTERPRETATIONOF ANALYTIC RESULTS
Two significant points arise from the above analysis. First, the numerous properties of the Furness procedure that have been developed in the literature are equally Simplifying (17) yields: applicable to the FHWA adjusted attraction factor method. For example, Evans[Z] shows that if all rij > 0, pw, = PiAif( C,)R/” (18) then the Furness process will converge to a unique solution that has all rows and columns “balanced”. The I’ T AJ(Cik)R,‘“’ above analysis therefore implies that the FHWA technique will converge to the same unique, balanced soluEquation (18) represents the simulated trip distribution tion. It should be noted that the condition that all tii >O after 1 column and 1 row balancing iteration using the is a sufficient, but not a necessary condition for conFurness method. Now, substituting n = 1 in the general vergence. Cesario[3] shows that a matrix containing expression for the FHWA method feqn 15), we find some zero elements will still converge to the solution, as long as the sum of attractions in all columns other than the column containing a zero, is not less than the production in the row containing that same zero. Thus in large study areas, with large trip productions and attractions, the “zero elements” condition will not generally However, the r.h.s. of eqns (18) and (19) are identical. prevent convergence. In fact, the zero elements problem Thus, we have: does not arise with the FHWA method, since the gravity model can only produce zero trips between any zone pair r
m
J
=i PiAjf( Cij)Ri[2’ T AJ(cidRk”’
4. E~I~CALANALYS~ Gravity models were calibrated using three different sets of data and two different calibration techniques. The first technique was the FHWA iterative procedure based (23) on matching the observed and simulated trip “length”
E. I. PASand M. A. TODES
252
frequency distributions, and the second calibration method was a linear regression technique which is designed to derive zone-to-zone specific factors that minimize the sum of squares between the observed and simulated origin-destination matrix[5]. The data were obtained from origin-destination studies conducted in Cape Town, South Africa and Skokie, Illinois. The Skokie data consists of trips within a 22 zone area (data set A). The Cape Town data consists of trips to work by car in a 105 zone area, made by White and Coloured (mixed race) commuters (data sets B and C).t The origin-destination matrices predicted by the gravity models before and after balancing were compared with the respective observed matrices using the error measures described below. The error measures employed for comparing the results are those used by Koppelman[6]. The root mean square error (RMSE) is given by: RMSE = [i?
7 (REii)Z]I’*
(25)
where N is the total numper of cells in the distribution matrix (m X n); RE, is [(tii - fij)/tij]; tij is the observed trips between i and j; and tij is the model estimated trips between i and j. Koppelman shows that the RMSE can be separated into components representing the average error (AE) and the standard deviation of the error (SDE) as follows: RMSE* = AE* + SDE* where AE=$~RE, 1 I SDE = [ $F
F (REU- AE)2]“Y
(28)
tThe technical analyst finds it hard to avoid the reality of racially separated residential areas in South African cities.
This formulation enables broader interpretation of the errors. It can be seen from eqn (27) that AE is sensitive to the sign of RE. Thus if AE is positive, it indicates that the model has generally overpredicted the trjp interchange values since it implies that on average,(!,j - tij) is greater than zero. Similarly, if AE is negative, It Indicates that the model has generally underpredicted. The SDE provides a measure of the deviation of the error about AE. If for example, the AE is small but the SDE is large, this indicates that the model is generally providing good estimates, but that it is not well specified in certain cases. It can be seen in Table 1 that in all cases the average error (AE) was reduced by balancing the attractions. However, this does not necessarily imply an improved fit with the observed matrix, since it is possible that the balancing of attractions may also have resulted in a better balancing of overpredictions and underpredictions. In some cases the standard deviation of the error (SDE) was also reduced, thus indicating that there was in fact an overall improvement. However, in other cases the standard deviation of the error increased, indicating that the errors were generally larger after balancing. In most cases these changes were small and therefore not significant. In case B.2 however, the increase in the standard deviation of the error was significantly large. Thus the balancing procedure did not consistently produce an improved fit to the observed origin-destination matrix. Because of the disaggregation of the root mean square error into its two components, we can see that on the average the balancing procedure does improve the fit between the observed ans simulated trip distribution. However, we see also that the standard deviation of the error increases, indicating that the balancing procedure has the effect of producing large differences between observed and simulated trips for some origin-destination pairs. Furthermore, the results in Table 1 indicate that in those cases where the root mean square error is reduced by the balancing process, the change is relatively small. 5. CONCLUSIONS It has been shown that the balancing of attractions factor procedure commonly used in applications of the
Table I. Base year prediction errors of six gravity models Before
Balancing
After
MODEL1
AE2
SDE3
RYSE4
A.1
16.4
59.1
61.4
15.1
60.3
62.2
A.2
12.4
49.3
50.8
12.3
49.9
51.4
B.l
23.3
106.6
109.2
18.1
102.3
103.9
B.2
39.3
123.1
129.2
37.1
172.5
176.5
c.1
48.6
121.7
131.1
45.5
117.6
126.1
c.2
50.3
139.8
148.5
48.9
132.2
140.9
1.
2. 3. 4.
AE
Balancing
SDE
RYSE
data sets, while A, 8, and C refer to the three different 1 and 2 refer to two alternative calibration methods (see text for details). AE = Average Error (equation 27) = Standard Deviation of the Error (equation 28) SDE RMSE = Root Mean Square Error (equation 25)
A note on the balancing factors of gravity models
production constrained gravity model is identical ?o, and therefore as efWeat ass, the Faess procedure, However, it does Rot necessady result in an improvement of the model predictions on a cell by cell basis. It is possible therefore, that the balancing of attractions may be zkn unnecessary step in the tise of the singly constrained ~a~~~~~e spatial ~~~~r~~~~~ m&et. Furtheremore, this research has shown that the baiancing procedure can be! interpreted as leading ta an arbitrary distartion of the calibrated friction factors, fn other: words, the simulated trip d~s~b~t~~ matrix after balancing Is not necessariiy consistent with the cat&rated friction factors. Therefore, the balancing of attractions in the commonly used production constrained formulation of the gravity model might be undesirable as well as
253
REFEREIKW 1. Federat Highway Ad~inis~r~tion~Computer programs for urban t~a~s~rtat~~~ planning. ~~er~~ ~~~o~of~u~ ~~u#~~~ US, Govt. Printing Office, Washington, DC. fB77].
2. A. W. Evans, Some propertiesof trip distribution methods.
Transm. Rex 4. 19-36 (1970). 3. F. J. ‘Cesario, Parameter Estimation in Spatial Ynteraction ~~e~~~~ Emmt Ph f. 503-51811973%. 4. F. J. C&ark, A new ~ot~r~re~at~o~ of &e ~~norrna~~~~~g~’ or “balancing” factors of gravity-type spatial mod&. SocioEcon. Plan. Sci. 11, 131-136(1977), S. M. A. Todes, A Regression Technisue for the Calibration of a
~R~CC~~~~~~ Ackfluwledgements-The research reported in this note was conducted while the authors were in the Department of Civil
6. F. S. Koppelman, &thodolog; for a;alysis of krrok in prediction with disaggregate choice models. Transprr,Rex Rec. 592 (1976).