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A model for forecasting traffic accidents
A MODEL
FOR
FORECASTING
TRAFFIC
ACCIDENTS
N. 0. J@RGENSEN Danish Council of Road Safety Research, Copenhagen, Denmark
SCOPE
OF
THE
REPORT
IN THE planning
of the construction stages of a motorway scheme in Jutland, Denmark, it was considered necessary to evaluate the traffic safety implications of different choices of the construction stages. For that purpose a crude model for the occurrence of accidents on rural highways was set up and tested on existing data. This report describes mainly the statistical model developed for that purpose, the estimation of parameters in the model and tests on the geographical validity of the parameters. FORMULATION
OF
A STATISTICAL
MODEL
FOR
ACCIDENT
OCCURRENCE
The purpose of this work was to estimate the expected numbers of personal injury accidents on certain parts of the main road network in Jutland, in particular to estimate differences in these accident numbers, if different priorities were given to different parts of the motorway scheme. These estimates were used as on component in a comprehensive economic evaluation which was carried out by the Danish Ministry of Transport. No official report of this economic evaluation has been published so far. An academic work, which describes the basic systematic approach has been published by the Technical University of Denmark (Hamann, 1967). Figure 1 shows a map of Denmark with the motorway scheme and the trunk road AlO, which is the main existing north-south road in eastern Jutland. It appears from Fig. 1 that the opening of a certain part of the motorway in general will affect the traffic flow not only on A10 but on a number of connecting roads as well. Figure 2 shows a systematic diagram of the accident forecast. The basic data consists of fhe Police Accident Reports on all personal injury accidents on the total highway netyork in Denmark in the years 1962-1964. These reports were related to recordings of the physical data of the highway network such as recordings of the width of the roadway, the number of lanes, the existence of medians and the development along the road. Using these data 17 different road classes were defined. Within each class the network is subdivided into road sections of approximately 1 km and major highway intersections. For each section of a certain road class an annual average daily traffic (AADT) was estimated. Statistics of this kind are being worked out currently by the Ministry of Transport. A report on this statistical scheme by which accident data are related to data for the roads is given in an academic publication (Thorson, 1961). AS a crude model it was suggested that the accident density d (accidents per km per year) for each road class is a function of the traffic Aow: &a.N” (1) 261
= -=
-
o’i, o-,*. o
Motorways Trunk road Highways Secondary road Cities
Urban dlstrirts Urban distri&
lmXl ,nknk -._ +hnn ..-.. ,“__ ,,,,, ““.
with rmw
with 500 to 1000
inhab.
.
A model
for forecastin&
traffic
Existing
highway network physical data
Existing
Stotisticol
263
accidents
highway network
Future network olternatlves physical data
model of
FIG. 2. Systematic
diagram
of the accident
forecast
where N is the AADT and a and p are constants characterizing that road class. Preliminary graphical examinations based on a fairly large number of sections within each road class showed both a clear correlation and a considerable scatter of the observed densities around a regression line. In order to formulate the model more precisely it was suggested furthermore that the annual number of accidents on a certain road section is a Poisson distributed random variable. Several authors have used this assumption (Hondermarcq, 1966). Recently it has been examined in some detail in Denmark (Rasch, 1968). No special justification of this assumption is given in this paper. Under this assumption a more detailed statistical model for the occurrence of accidents appears. Let u,, be the number of accidents in a certain year on section numberj of road class i. It is assumed, that U,, is a Poisson variable such that
E(U,, )=a,.N,,l" 'L,, =a, .Y,,
(2)
where Y,, =z N,, “I . L ,, and N,, is the AADT
and
Pr(U,,
L,,is the length of road section (ii). Thus the distribution
=u)=(”
. Yfl)“exp(--n,. U!
is
Y,, ).
From the field data N,, and L,,are known for all road sections. This formulation covers road section. Intersections may be covered by omitting L ,,in equations (2) and (3). For certain reasons the major highway intersections were handled separately in the forecast. Note that all minor intersections are included in the data for the sections.
N 0
264
THE
BASIC‘
.IOII(;I:NSI:N
ASSUMPTION
IN
THE
FOREC‘AST
The basic assumptions of the accident forecast may now be specified in the following 2 points: 1. The general model, specified in (2) and (3) is accepted as a basis for estimating future traffic accidents. 2. The parameters u, and p# for road class i have numerical values which can be estimated from recent statistical information. These values are assumed to remain constant throughout the period of the forecast, i.o. - 10 years. It must be admitted at this point, that a more thorough justification of these basic assumptions has not been established. They were upheld mainly because graphical examinations of data through the years 190 1964 coincided well with the model for each year separately. It should be stressed that this is an utf hoc working model with obvious faults. As an example it could be noted, that when p is in the range 0~ /7< I then the curve (I) will have a vertical slope when N is very small. Consequently the accident rate, which is di N will take infinitely high values for very small values of N, which is obviously wrong. These difficulties are taken care of by appropriate groupings of observations along the N-axis. t:STIMATION
OF
TttF
PARAMETERS
A si~lult~~neoiis estimation of II, and pi in equation (2) for a certain observed values of U,, , and L ‘, is not possible by means of standard least Therefore a “trial and error” approach was used. For chosen values 0.6. . . .) u, was estimated. For given pI the corresponding N, is following equation c u,, (I, * -_. / u, . cy;,
(4)
-y,
Having estimated (I, the sum of squares of the deviations curve (2) and the observed values U,, was computed: SQ?,)~- Z(Li;,
road class from square methods. of pJ I$, -~ 0.5, estimated by the
u,.
S@, ) between
Y,, )“.
the estimated
(5)
That p, value which corresponds to the smallest SQ,) value is accepted as the optimal p, value. Table 1 shows examples of the results of the computations of S(p,) and u, values for different pi values and different road classes. GEOGRAPHtCAL
VALIDITY
OF
PARAMETER
VALIJES
The data given in Table 1 represent all accident records on all state and county highways in Denmark in 1963. The accident forecasts should only comprise a fairly limited part of the Danish highway network. Therefore it is quite likely that more realistic parameter values in (3) could be obtained if the parameter estimates were based only on data from the same part of the highway network. In order to examine this possibility the following test has been performed. The total number of accidents on road class i, Cf,, can be divided into two subgroups U,, = accidents on class i within a certain focal area U,, = accidents on class i in the remaining part of the country so that I/, - ci,, I U,., . Using the pi estimate for road class i, the two different estimates ii,, for the local area and CI,, for the remaining part of the country are
A model for forecasting
265
traffic accidents
TABLE 1. SAMPLE OF RESULTS FROM PARAMETER ESTIMATIONS
A. Example
1962 I963 1964
B. Parameter
of the variation of S(p) and a through different years Road class: 2-lane rural highway without cycle 1’ 0.5 0.6 0.7 0.8 .__~.. n 9.912 4.173 2.285 1.087 S(P) 2299 2226 2179 2162 a 9.036 4.312 2.045 0.965 Sip) 2036 1954 1894 1861 a 10,806 5.098 2.390 I.114 SiPI 2854 2773 2736 275 1
values used in the forecast
and estimated Rural 2.lane 2-lane 3-lane
tracks 0.9
I.0
0.514 2180 0.452 185X 0,516 2825
0.242 2240 0.21 I 1892 0.238 2968
All other p-values in the forecast are 1.00 as stated in the text. Note: If parameters are used for calculations according to equation accidents per km per year.
while according
0,113 2348 0,097 1969 0.109 3192
from the criterion min: Sip,) highways without cycle tracks Of8 with cycle tracks 0.7 with cycle tracks 1.1
Highways in developed areas 2-lane without cycle tracks 2-lane with cycle tracks 3-lane with cycle tracks
a,; ZL
I.1
(l), a factor
0.7 0.9 I.0
IO-3 must be applied
0.9u65 2.427 0.0728 6.947 1.238 0,437
to obtain
u
yr,
to equation
(4)
+u,,. a, &I=&, Y,, + y,, Y, The possibility of having different a-values in different parts of the country may now be tested. The test hypothesis is a ,, =a,, =a,. The alternative is ai, # a “, so that a double sided test should be applied. It may be shown (Hald, 1952; Thygesen) that the ratio R, where R=max(a,,,a,,) mm(a I,, a,, > is an F-distributed
variable
having
the degrees of freedom
.ri = 2(U,, + 1) f2 = 2u,, . A few tests along this line have been carried out. As the local area four police districts were chosen in southern Jutland along the A10 from the city of Kolding to the German border. These were chosen primarily because they were in the area along the A 10 which will be heavily influenced by the motorway, secondly because this area along the main road from Germany-with a fairly large proportion of international tourists-could be expected to have other accident patterns than the remaining part of the road network. The tests were carried out for 7 road classes with the results indicated in Table 2. From these results it was concluded, that no systematic deviations exist between this local area and the remaining part of the country. The fact that 2 out of 7 F-tests show differences
N. 0.
266
lOhi-data
Roadclass
.~OKC;ENSI-N
l?lr;tmctcr value\
I’
(‘I
Rural highways ?-lane without cycle tracks I-lane with cycle tracks 3.lane with cycle tracks
0.1) 0.x
0.553
I0
I ,4o‘l 0.0X?
Highways in developed arcas 2-lane without cycle tracks 7-lane with cvclc track5 i-lane withoit cycle track5 ?-lnnewith cvcle tracks
0.7 09 I .o
6.504 I I 00 0~4x0
14
O-296
Deer
C,
I.-
/,
01’ freedom
Signlficunl on ‘)i” ,
IL’CCI ‘I
i,
O.&&l I .040 0.17x
I.246 I 350 2 1%
778 02 x
2420 51x 476
>c\
7.00’) I .740
I -07x 1.i 2X ‘.7X3
3 IO hS
-3I 4 6
IlO
I 4KX
22
1~09~~ 0~440
2x
YOh
54 511
110
no
no
yes 110
that are significant on the 95 per cent level was not considered sufficient to prove that significant differences existed since the sign of the differences were not consistent in such a way that for some road classes we find N,, < u,, and for other road classes (I,, > (I,, . A possible explanation of the outcome of the tests is that the assumption (3) concerning the random variation of U,, underestimates the actual random variation. In this case, however. we consider fairly large parts of the total road network, and it has been shown by extensive studies (Rasch. 1968) that for such network parts the sum of all accidents does follow a Poisson distribution quite closely. C’ONCLUDING
REMARKS
In the planning situation described in this paper it was necessary to arrive at final data within a short time. For this reason it was considered a sufficiently good approximation to use the model specified in (2) and (3) with parameter values estimated from nationwide data. Some road classes~~particularly 4-lane roads of different categories-are so far not used extensively outside the big cities. The accident data for these classes were considered insufficient to give a meaningful estimate of the p-values through finding the minimal S(J,) value according to equation (5). For these road classes the simple assumption was used that the accident rate is constant independent of the AADT, which correspond to assuming p = 1.0. Having decided on parameter values for all road classes it was possible to carry out the accident forecasts according to Fig. 2. .,l~,X/~o,~,/c,cl~f,~~~~,~f/,~ Lecturer Inge Thygeaen of the Technical matical Statistics and Operational Research, has contributed formulation of the statistical model and suggesting the F-test
University of Denmark. lnstltute of Matheimportantly to this work by c!,:ril’yinp the
REFERENCES
HONDIXMARCQ, H. (1966). T&niqw C/C/(I c,rwukrrron. UniversitC de Liige. RASCH, G. (196X). En rcwnalysc afdanske og svenske forsdg over virkningen af hastighcdsgramxr pn tralikulykkrr. Not published. THOKSON. 01-r (I 967). TIX& cwcitlwlr.\ trnd rocrd /cI)~oN/. Thesis, Technical University of Denmark. Copenhagen. TH~G~SEN. 1%~: Private correspondence.
FOR
FORECASTING
TRAFFIC
ACCIDENTS
N. 0. J@RGENSEN Danish Council of Road Safety Research, Copenhagen, Denmark
SCOPE
OF
THE
REPORT
IN THE planning
of the construction stages of a motorway scheme in Jutland, Denmark, it was considered necessary to evaluate the traffic safety implications of different choices of the construction stages. For that purpose a crude model for the occurrence of accidents on rural highways was set up and tested on existing data. This report describes mainly the statistical model developed for that purpose, the estimation of parameters in the model and tests on the geographical validity of the parameters. FORMULATION
OF
A STATISTICAL
MODEL
FOR
ACCIDENT
OCCURRENCE
The purpose of this work was to estimate the expected numbers of personal injury accidents on certain parts of the main road network in Jutland, in particular to estimate differences in these accident numbers, if different priorities were given to different parts of the motorway scheme. These estimates were used as on component in a comprehensive economic evaluation which was carried out by the Danish Ministry of Transport. No official report of this economic evaluation has been published so far. An academic work, which describes the basic systematic approach has been published by the Technical University of Denmark (Hamann, 1967). Figure 1 shows a map of Denmark with the motorway scheme and the trunk road AlO, which is the main existing north-south road in eastern Jutland. It appears from Fig. 1 that the opening of a certain part of the motorway in general will affect the traffic flow not only on A10 but on a number of connecting roads as well. Figure 2 shows a systematic diagram of the accident forecast. The basic data consists of fhe Police Accident Reports on all personal injury accidents on the total highway netyork in Denmark in the years 1962-1964. These reports were related to recordings of the physical data of the highway network such as recordings of the width of the roadway, the number of lanes, the existence of medians and the development along the road. Using these data 17 different road classes were defined. Within each class the network is subdivided into road sections of approximately 1 km and major highway intersections. For each section of a certain road class an annual average daily traffic (AADT) was estimated. Statistics of this kind are being worked out currently by the Ministry of Transport. A report on this statistical scheme by which accident data are related to data for the roads is given in an academic publication (Thorson, 1961). AS a crude model it was suggested that the accident density d (accidents per km per year) for each road class is a function of the traffic Aow: &a.N” (1) 261
= -=
-
o’i, o-,*. o
Motorways Trunk road Highways Secondary road Cities
Urban dlstrirts Urban distri&
lmXl ,nknk -._ +hnn ..-.. ,“__ ,,,,, ““.
with rmw
with 500 to 1000
inhab.
.
A model
for forecastin&
traffic
Existing
highway network physical data
Existing
Stotisticol
263
accidents
highway network
Future network olternatlves physical data
model of
FIG. 2. Systematic
diagram
of the accident
forecast
where N is the AADT and a and p are constants characterizing that road class. Preliminary graphical examinations based on a fairly large number of sections within each road class showed both a clear correlation and a considerable scatter of the observed densities around a regression line. In order to formulate the model more precisely it was suggested furthermore that the annual number of accidents on a certain road section is a Poisson distributed random variable. Several authors have used this assumption (Hondermarcq, 1966). Recently it has been examined in some detail in Denmark (Rasch, 1968). No special justification of this assumption is given in this paper. Under this assumption a more detailed statistical model for the occurrence of accidents appears. Let u,, be the number of accidents in a certain year on section numberj of road class i. It is assumed, that U,, is a Poisson variable such that
E(U,, )=a,.N,,l" 'L,, =a, .Y,,
(2)
where Y,, =z N,, “I . L ,, and N,, is the AADT
and
Pr(U,,
L,,is the length of road section (ii). Thus the distribution
=u)=(”
. Yfl)“exp(--n,. U!
is
Y,, ).
From the field data N,, and L,,are known for all road sections. This formulation covers road section. Intersections may be covered by omitting L ,,in equations (2) and (3). For certain reasons the major highway intersections were handled separately in the forecast. Note that all minor intersections are included in the data for the sections.
N 0
264
THE
BASIC‘
.IOII(;I:NSI:N
ASSUMPTION
IN
THE
FOREC‘AST
The basic assumptions of the accident forecast may now be specified in the following 2 points: 1. The general model, specified in (2) and (3) is accepted as a basis for estimating future traffic accidents. 2. The parameters u, and p# for road class i have numerical values which can be estimated from recent statistical information. These values are assumed to remain constant throughout the period of the forecast, i.o. - 10 years. It must be admitted at this point, that a more thorough justification of these basic assumptions has not been established. They were upheld mainly because graphical examinations of data through the years 190 1964 coincided well with the model for each year separately. It should be stressed that this is an utf hoc working model with obvious faults. As an example it could be noted, that when p is in the range 0~ /7< I then the curve (I) will have a vertical slope when N is very small. Consequently the accident rate, which is di N will take infinitely high values for very small values of N, which is obviously wrong. These difficulties are taken care of by appropriate groupings of observations along the N-axis. t:STIMATION
OF
TttF
PARAMETERS
A si~lult~~neoiis estimation of II, and pi in equation (2) for a certain observed values of U,, , and L ‘, is not possible by means of standard least Therefore a “trial and error” approach was used. For chosen values 0.6. . . .) u, was estimated. For given pI the corresponding N, is following equation c u,, (I, * -_. / u, . cy;,
(4)
-y,
Having estimated (I, the sum of squares of the deviations curve (2) and the observed values U,, was computed: SQ?,)~- Z(Li;,
road class from square methods. of pJ I$, -~ 0.5, estimated by the
u,.
S@, ) between
Y,, )“.
the estimated
(5)
That p, value which corresponds to the smallest SQ,) value is accepted as the optimal p, value. Table 1 shows examples of the results of the computations of S(p,) and u, values for different pi values and different road classes. GEOGRAPHtCAL
VALIDITY
OF
PARAMETER
VALIJES
The data given in Table 1 represent all accident records on all state and county highways in Denmark in 1963. The accident forecasts should only comprise a fairly limited part of the Danish highway network. Therefore it is quite likely that more realistic parameter values in (3) could be obtained if the parameter estimates were based only on data from the same part of the highway network. In order to examine this possibility the following test has been performed. The total number of accidents on road class i, Cf,, can be divided into two subgroups U,, = accidents on class i within a certain focal area U,, = accidents on class i in the remaining part of the country so that I/, - ci,, I U,., . Using the pi estimate for road class i, the two different estimates ii,, for the local area and CI,, for the remaining part of the country are
A model for forecasting
265
traffic accidents
TABLE 1. SAMPLE OF RESULTS FROM PARAMETER ESTIMATIONS
A. Example
1962 I963 1964
B. Parameter
of the variation of S(p) and a through different years Road class: 2-lane rural highway without cycle 1’ 0.5 0.6 0.7 0.8 .__~.. n 9.912 4.173 2.285 1.087 S(P) 2299 2226 2179 2162 a 9.036 4.312 2.045 0.965 Sip) 2036 1954 1894 1861 a 10,806 5.098 2.390 I.114 SiPI 2854 2773 2736 275 1
values used in the forecast
and estimated Rural 2.lane 2-lane 3-lane
tracks 0.9
I.0
0.514 2180 0.452 185X 0,516 2825
0.242 2240 0.21 I 1892 0.238 2968
All other p-values in the forecast are 1.00 as stated in the text. Note: If parameters are used for calculations according to equation accidents per km per year.
while according
0,113 2348 0,097 1969 0.109 3192
from the criterion min: Sip,) highways without cycle tracks Of8 with cycle tracks 0.7 with cycle tracks 1.1
Highways in developed areas 2-lane without cycle tracks 2-lane with cycle tracks 3-lane with cycle tracks
a,; ZL
I.1
(l), a factor
0.7 0.9 I.0
IO-3 must be applied
0.9u65 2.427 0.0728 6.947 1.238 0,437
to obtain
u
yr,
to equation
(4)
+u,,. a, &I=&, Y,, + y,, Y, The possibility of having different a-values in different parts of the country may now be tested. The test hypothesis is a ,, =a,, =a,. The alternative is ai, # a “, so that a double sided test should be applied. It may be shown (Hald, 1952; Thygesen) that the ratio R, where R=max(a,,,a,,) mm(a I,, a,, > is an F-distributed
variable
having
the degrees of freedom
.ri = 2(U,, + 1) f2 = 2u,, . A few tests along this line have been carried out. As the local area four police districts were chosen in southern Jutland along the A10 from the city of Kolding to the German border. These were chosen primarily because they were in the area along the A 10 which will be heavily influenced by the motorway, secondly because this area along the main road from Germany-with a fairly large proportion of international tourists-could be expected to have other accident patterns than the remaining part of the road network. The tests were carried out for 7 road classes with the results indicated in Table 2. From these results it was concluded, that no systematic deviations exist between this local area and the remaining part of the country. The fact that 2 out of 7 F-tests show differences
N. 0.
266
lOhi-data
Roadclass
.~OKC;ENSI-N
l?lr;tmctcr value\
I’
(‘I
Rural highways ?-lane without cycle tracks I-lane with cycle tracks 3.lane with cycle tracks
0.1) 0.x
0.553
I0
I ,4o‘l 0.0X?
Highways in developed arcas 2-lane without cycle tracks 7-lane with cvclc track5 i-lane withoit cycle track5 ?-lnnewith cvcle tracks
0.7 09 I .o
6.504 I I 00 0~4x0
14
O-296
Deer
C,
I.-
/,
01’ freedom
Signlficunl on ‘)i” ,
IL’CCI ‘I
i,
O.&&l I .040 0.17x
I.246 I 350 2 1%
778 02 x
2420 51x 476
>c\
7.00’) I .740
I -07x 1.i 2X ‘.7X3
3 IO hS
-3I 4 6
IlO
I 4KX
22
1~09~~ 0~440
2x
YOh
54 511
110
no
no
yes 110
that are significant on the 95 per cent level was not considered sufficient to prove that significant differences existed since the sign of the differences were not consistent in such a way that for some road classes we find N,, < u,, and for other road classes (I,, > (I,, . A possible explanation of the outcome of the tests is that the assumption (3) concerning the random variation of U,, underestimates the actual random variation. In this case, however. we consider fairly large parts of the total road network, and it has been shown by extensive studies (Rasch. 1968) that for such network parts the sum of all accidents does follow a Poisson distribution quite closely. C’ONCLUDING
REMARKS
In the planning situation described in this paper it was necessary to arrive at final data within a short time. For this reason it was considered a sufficiently good approximation to use the model specified in (2) and (3) with parameter values estimated from nationwide data. Some road classes~~particularly 4-lane roads of different categories-are so far not used extensively outside the big cities. The accident data for these classes were considered insufficient to give a meaningful estimate of the p-values through finding the minimal S(J,) value according to equation (5). For these road classes the simple assumption was used that the accident rate is constant independent of the AADT, which correspond to assuming p = 1.0. Having decided on parameter values for all road classes it was possible to carry out the accident forecasts according to Fig. 2. .,l~,X/~o,~,/c,cl~f,~~~~,~f/,~ Lecturer Inge Thygeaen of the Technical matical Statistics and Operational Research, has contributed formulation of the statistical model and suggesting the F-test
University of Denmark. lnstltute of Matheimportantly to this work by c!,:ril’yinp the
REFERENCES
HONDIXMARCQ, H. (1966). T&niqw C/C/(I c,rwukrrron. UniversitC de Liige. RASCH, G. (196X). En rcwnalysc afdanske og svenske forsdg over virkningen af hastighcdsgramxr pn tralikulykkrr. Not published. THOKSON. 01-r (I 967). TIX& cwcitlwlr.\ trnd rocrd /cI)~oN/. Thesis, Technical University of Denmark. Copenhagen. TH~G~SEN. 1%~: Private correspondence.