CN109712389B - Path travel time estimation method based on Copula and Monte-Carlo simulation - Google Patents

Path travel time estimation method based on Copula and Monte-Carlo simulation Download PDF

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CN109712389B
CN109712389B CN201811534181.0A CN201811534181A CN109712389B CN 109712389 B CN109712389 B CN 109712389B CN 201811534181 A CN201811534181 A CN 201811534181A CN 109712389 B CN109712389 B CN 109712389B
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马晓磊
栾森
陈汐
鲁光泉
李萌
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Beihang University
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Abstract

The invention discloses a path travel time estimation method based on Copula and Monte-Carlo simulation, which comprises the following steps: s1, acquiring the travel time fluctuation rate of each path segment in one path; s2, performing edge distribution fitting on the path travel time fluctuation rate of each road section through maximum likelihood estimation; s3, fitting the path travel time fluctuation rate distribution function by using a Copula theory based on the path travel time fluctuation rate distribution function of each path section; s4, carrying out Monte-Carlo simulation on the basis of the Copula function to obtain a path travel time fluctuation rate sequence; s5, determining a confidence interval under a certain confidence level and calculating the expectation of the path travel time fluctuation rate; and S6, combining the expected value of the fluctuation rate of the path travel time with the path travel time of the previous time interval to estimate the path travel time of the current time interval. The invention considers the fluctuation of the travel time of each road section on the time sequence and also considers the space relation among the travel times of the road sections, deeply excavates the space-time characteristics of the travel time and has higher precision and reliability.

Description

Path travel time estimation method based on Copula and Monte-Carlo simulation
Technical Field
The invention relates to the technical field of intelligent traffic information processing, in particular to a path travel time estimation method based on Copula and Monte-Carlo simulation.
Background
At present, the Intelligent Transportation System (ITS) is proposed and applied to accelerate the development of urban economy and simultaneously make travelers put higher requirements on trip quality. Based on massive traffic data, the traffic state in the future is analyzed and predicted, so that the utilization of road traffic resources is maximized, and the travel time, traffic jam and traffic accidents are reduced, which is also one of the important purposes of the intelligent traffic system. The real-time and accurate road travel time prediction is a precondition for realizing traffic guidance and traffic control and is also a key factor for the transition of the intelligent traffic system from 'passive reaction' to 'active action'. Therefore, the analysis of the route travel time is estimated to be a hot spot problem in the traffic field. The route travel time is also referred to as travel cost from the starting point to the ending point. Due to the influence of random traffic demand changes and actual road conditions, the travel time has time-varying property, randomness and uncertainty, and shows complex fluctuation. The volatility of the travel time affects the reliability of the traffic conditions and also provides challenges for accurately predicting the travel time.
The travel time research in recent years has focused on analyzing the travel time distribution, and considers the sum of the link travel time distributions as the path travel time distribution. The premise of the algorithm idea is that the travel time of each road section is independently and uniformly distributed, but the premise obviously cannot meet the randomly fluctuating traffic state of each road section. Some scholars think that travel time on different road sections can have different distribution types, and the travel time distribution of each road section is skillfully combined by using Copula theory to obtain a combined distribution function capable of representing the travel time distribution of the whole path. Although the Copula function can represent the route travel time distribution and analyze the direct correlation structure of the road section travel time distribution, the Copula function also brings complicated multivariate integral calculation, and is difficult to implement and apply under a large-scale road network.
Therefore, how to perform calculation as simple as possible in a large-scale network and realize accurate travel time prediction is a problem that needs to be solved urgently by those skilled in the art.
Disclosure of Invention
In view of the above, the present invention provides a path travel time estimation method based on Copula and Monte-Carlo simulation, which obtains historical travel time data of a vehicle passing through each road section from a path composed of a plurality of road sections and calculates to obtain a road section travel time fluctuation rate sequence; secondly, performing edge distribution fitting on the path travel time fluctuation rate sequence, and determining the distribution type of the path travel time fluctuation rate sequence to obtain an edge distribution function; then constructing a multivariate joint distribution function of the path travel time fluctuation rate by using a Copula theory based on the path travel time fluctuation rate distribution function corresponding to each road section; then, converting the multivariate joint distribution into a value sequence with the joint distribution characteristics through a Monte-Carlo simulation method; and finally, path travel time estimation is realized through statistical analysis.
In order to achieve the purpose, the invention adopts the following technical scheme:
a path travel time estimation method based on Copula and Monte-Carlo simulation comprises the following steps:
step 1: obtaining driving travel time data in urban road network and using Ti d,tRepresenting the link travel time taken by the vehicle to travel the link i during the day d time period t; the rate of fluctuation r of the section travel time in said section ii tExpressed as:
Figure BDA0001906459980000021
wherein the road section travel time fluctuation rate r of a corresponding time period exists in different dates di tIs a sequence of values;
step 2: the path travel time fluctuation rate r corresponding to each road section ii tUsing a maximum likelihood estimation method to carry out edge distribution fitting to obtain an edge distribution function Fi(ri t)；
And step 3: using the edge distribution function F corresponding to each of the sections ii(ri t) Fitting Copula function with its corresponding parameters, using
Figure BDA0001906459980000022
Representing the road section travel time fluctuation rate distribution function obtained by fitting;
and 4, step 4: Monte-Carlo simulation is carried out on the basis of Copula function fitting, namely a road section travel time fluctuation rate distribution function F is subjected tocMonte-Carlo simulation is carried out to obtain the fluctuation rate of the path travel time
Figure BDA0001906459980000023
Sum path travel time fluctuation rate sequence
Figure BDA0001906459980000024
Step 5, determine confidence α, obtain
Figure BDA0001906459980000025
Is confidence interval of
Figure BDA0001906459980000026
Is calculated to obtain
Figure BDA0001906459980000027
The mathematical expectation within the confidence interval is calculated as follows:
Figure BDA0001906459980000031
Figure BDA0001906459980000032
wherein sup {. cndot } and inf {. cndot } respectively represent the supremum and infimum of the variable;
expected path travel time fluctuation rate ER at confidence level 1- αtThe following formula is used to obtain:
Figure BDA0001906459980000033
wherein, ERtIs that
Figure BDA0001906459980000034
A mathematical expectation within a confidence interval; 1 {. is an illustrative function;
step 6: the travel time of each section in the time period T-1 of the day is Ti d,t-1According to the desired path travel time fluctuation rate ERtFor the path travel time T in the T periodtThe estimation is carried out by the following calculation method:
Figure BDA0001906459980000035
wherein exp (. cndot.) represents an index with a base number eThe operation is carried out according to the operation parameters,
Figure BDA0001906459980000036
representing the path travel time estimate for the time of day t.
Preferably, Monte-Carlo simulation is performed on the basis of Copula function fitting, i.e. on the path travel time fluctuation rate distribution function FcMonte-Carlo simulation is carried out to obtain a path travel time fluctuation rate sequence
Figure BDA0001906459980000037
The specific calculation process is as follows:
step 41: constructing a correlation relation matrix sigma among the path travel time fluctuation rate sequences of the road sections, wherein the expression is as follows:
Figure BDA0001906459980000038
wherein, tauijThe correlation between the road section i and the road section j is represented, and the correlation is essentially Kendall's tau coefficient, and is calculated by means of a calculation formula of a Copula function as follows:
Figure BDA0001906459980000039
step 42: performing Cholesky decomposition on the correlation matrix Σ in step 41, and obtaining ∑ LLTDecomposing the formula to obtain a lower triangular matrix L;
step 43: if the fitting result of the Copula function is the road section travel time fluctuation rate distribution function FcFor a Normal distribution, an N-dimensional vector V with Copula characteristics is generated, where Z is an N-dimensional random vector Z ═ Z (Z, 1) following a standard Normal distribution N (0,1)1,Z2,..,Zn)；
Step 44: if the road section travel time fluctuation rate distribution function FcFor t distribution, an n-dimensional vector with Copula characteristics is generated
Figure BDA0001906459980000041
Wherein v is the degree of freedom under the t distribution obtained in the calculation process of the step 3, and Y is the chi-square-obedient distribution2(v) N-dimensional random vectors of (a);
step 45: generating path travel time fluctuation rates
Figure BDA0001906459980000042
The calculation formula is as follows:
Figure BDA0001906459980000043
wherein, Fi -1As a function of the edge distribution F in step 2i(ri t) The inverse function of (d); fcA path travel time fluctuation rate distribution function in the step 3;
step 46: repeating the steps 43-45, and obtaining the path travel time fluctuation rate sequence after executing k times
Figure BDA0001906459980000044
Preferably, in step 1, i is 1.
Preferably, the distribution function of the section travel time fluctuation rate in the step 2 includes three distribution types, namely normal distribution, t distribution and generalized hyperbolic distribution.
Preferably, the Copula function in step 3 has six types, i.e., Normal, T, Frank, Clayton, Gumbel and Joe.
Compared with the prior art, the path travel time estimation method based on Copula and Monte-Carlo simulation is improved from two aspects of data and model to overcome the defects of the prior art. The data aspect shows that the path travel time fluctuation rate is used for replacing the original travel time, the path travel time fluctuation rate is in a first-order logarithmic difference form of the travel time, the logarithmic time sequence reduces the interference of uncertain factors, and the method is more stable than the original travel time sequence and has better statistical characteristics. On the aspect of the model, a Monte-Carlo simulation method is further used for simplifying the multivariate probability integral calculation on the basis that the Copula describes the fluctuation rate of the path travel time. The method improves the applicability of the algorithm, and has important significance for the analysis and prediction and the practical application of the travel time under the large-scale road network.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the provided drawings without creative efforts.
FIG. 1 is a schematic diagram of a flow structure of a path travel time estimation method provided by the present invention;
FIG. 2 is a schematic structural diagram of an urban road composed of five road segments and a plot of historical travel time fluctuation rate thereof according to the present invention;
FIG. 3 is a schematic diagram of the correlation of the path travel time fluctuation rate based on Copula measurement provided by the present invention;
FIG. 4 is a diagram of frequency distribution histogram of analog value sequence of path travel time fluctuation rate.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The embodiment of the invention discloses a travel time estimation method based on Copula and Monte-Carlo simulation, which comprises the following steps:
s1: and calculating the fluctuation rate of the road section travel time. By Ti d,tTo representThe time taken by the vehicle to pass through the road section i in the d-th day time period t, the road section travel time fluctuation rate
Figure BDA0001906459980000053
Expressed as:
Figure BDA0001906459980000051
wherein, the corresponding road section travel time fluctuation rate r is formed by different datesi tIs a sequence of values.
S2: r for each road sectioni tFitting edge distribution by maximum likelihood estimation method, determining optimal distribution of travel time fluctuation rate of each path by using likelihood value (LL) and Chi chi cell information content (AIC) as fitting criterion, and using Fi(ri t) An edge distribution function obtained by fitting an edge distribution is shown.
S3: using F corresponding to each linki(ri t) And fitting the Copula function through the parameters of the distribution function obtained by the maximum likelihood estimation, wherein the fitting method still uses the maximum likelihood estimation and the evaluation standard still uses LL and AIC. Let ui＝Fi(ri t) I 1,2,3,4,5, the copula function is constructed as follows:
Figure BDA0001906459980000052
Figure BDA0001906459980000061
where θ represents a correlation parameter between variables. For convenience of presentation, use is made of
Figure BDA0001906459980000062
To express the road section travel time fluctuation rate distribution function, namely the Copula function obtained by fitting.
S4: Monte-Carlo simulations on a Copula basis toObtaining path travel time fluctuation rate
Figure BDA0001906459980000063
Sum path travel time fluctuation rate sequence
Figure BDA0001906459980000064
I.e. the distribution function F of the travel time fluctuation rate of the road sectioncMonte-Carlo simulations were performed.
S5 calculation of the path travel time fluctuation rate, firstly, the confidence coefficient α is determined and obtained
Figure BDA0001906459980000065
Is confidence interval of
Figure BDA0001906459980000066
And
Figure BDA0001906459980000067
the mathematical expectation within the confidence interval is calculated as follows:
Figure BDA0001906459980000068
Figure BDA0001906459980000069
wherein sup {. cndot. and inf {. cndot. represent the supremum and infimum, respectively, of the variable, the expected path travel time fluctuation rate ER at confidence levels 1- αtThis can be obtained by the following equation.
Figure BDA00019064599800000610
Wherein, ERtIs that
Figure BDA00019064599800000611
A mathematical expectation within a confidence interval; 1{·}Is an indicative function.
S6: path travel time estimation. In thatThe travel time of each section in the time period T-1 of the day is Ti d,t-1According to the desired path travel time fluctuation rate ERtFor the path travel time T in the T periodtThe estimation is carried out by the following calculation method:
Figure BDA0001906459980000071
wherein exp (·) represents an exponential operation with a base e,
Figure BDA0001906459980000072
representing the path travel time estimate for the time of day t.
S7: calculating the Absolute Percentage Error (APE) of the estimation result according to the following formula:
Figure BDA0001906459980000073
in order to further optimize the technical scheme, Monte-Carlo simulation is carried out on the basis of Copula to obtain a path travel time fluctuation rate sequence, and the specific calculation steps are as follows:
s41: constructing a correlation relation matrix sigma among path travel time fluctuation rate sequences, wherein the expression is as follows:
Figure BDA0001906459980000074
wherein, TijThe correlation between a link i and a link j is represented, which is essentially a Kendall's τ coefficient, and is calculated by means of a copula function as follows:
Figure BDA0001906459980000075
s42: performing Cholesky decomposition on the correlation matrix Sigma, wherein the correlation matrix Sigma is a positive definite matrix to obtain a lower triangular matrix L, and the Cholesky decomposition meets the requirement of LLT＝∑；
S43: if the fitting result of Copula function is a road section lineTravel time fluctuation rate distribution function FcFor a Normal distribution, an N-dimensional vector V with Copula characteristics is generated, where Z is an N-dimensional random vector Z ═ Z (Z, 1) following a standard Normal distribution N (0,1)1,Z2,..,Zn)；
S44: if the distribution function F of the road section travel time fluctuation ratecFor t distribution, an n-dimensional vector with Copula characteristics is generated
Figure BDA0001906459980000076
Wherein v is the degree of freedom under the t distribution obtained in the calculation process of S3, and Y is the chi-square distribution obedient2(v) N-dimensional random vectors of (a);
s45: generating path travel time fluctuation rates
Figure BDA0001906459980000077
The calculation formula is as follows:
Figure BDA0001906459980000081
s46: the above S41-S45 are a Monte-Carlo simulation process,
Figure BDA0001906459980000082
an estimated value representing the path travel time fluctuation rate is circularly executed for S43-S45, and a path travel time fluctuation rate sequence is obtained after k times of execution
Figure BDA0001906459980000083
In order to further optimize the technical scheme, the travel time of the vehicle passing through each road section in the step S1 is obtained by the map navigation software.
In order to further optimize the above technical solution, the fitting of the edge function obtained in S2 and the Copula function obtained in S3 both adopt a maximum likelihood estimation method, and can be obtained by professional analysis software, such as Matlab, R, and the like.
Examples
A travel time estimation method for Copula and Monte-Carlo simulation comprises the following steps:
1) as shown in fig. 2, for a map of a city road in China, fig. 2a shows a path composed of five road segments, and the number and the start and end points of the road segments are marked, and fig. 2b shows the road segment travel time fluctuation rate of each road segment from 7:01-7:30(t-1 period) to 7:31-8:00(t period) in 44 working days. The statistical information of the road section travel time fluctuation rate of the five road sections is shown in the following table:
road section Range of Mean value Median value Mean error Standard deviation of Deflection degree Kurtosis
L1 [-0.694,1.060] 0.325 0.352 0.053 0.352 -0.198 0.122
L2 [-1.259,0.895] 0.204 0.239 0.047 0.309 -2.034 9.325
L3 [-1.190,1.307] 0.457 0.480 0.060 0.397 -1.413 4.788
L4 [-0.258,0.666] 0.204 0.191 0.028 0.186 0.624 0.778
L5 [-0.276,0.926] 0.170 0.122 0.035 0.229 1.228 2.318
2) And (3) performing road section travel time fluctuation rate edge distribution fitting on the five road sections by utilizing R software, wherein the candidate edge distribution types are three: normal distribution, t distribution and generalized hyperbolic distribution, and the fitting result is shown in the following table:
Figure BDA0001906459980000084
Figure BDA0001906459980000091
the larger the LL value is and the smaller the AIC value is, the better the fitting effect is, and the last column in the table gives the optimal distribution type of the road section travel time fluctuation rate of each road section. For the determined distribution function of the link travel time fluctuation rate for each link, the following table shows the parameter μ for each corresponding distribution function:
Figure BDA0001906459980000092
3) and fitting the Copula function by using the road section travel time fluctuation rate distribution function of each road section and corresponding parameters thereof, wherein the results are as follows:
Figure BDA0001906459980000093
the estimated values of the parameters, the estimated standard deviations and the fitted LL and AIC for each Copula function are given in the table. Among them, two Copula functions, AMH and FGM, are not suitable for describing the joint distribution of multidimensional (n >2) variables and are therefore not used. By comparing LL and AIC, T-Copula can be determined as the optimal road section travel time fluctuation rate distribution function.
4) Calculating a correlation matrix of the path travel time fluctuation rates of the five road sections required by Monte-Carlo simulation according to the T-Copula function and corresponding parameters, wherein the corresponding results are as follows:
road section L1 L2 L3 L4 L5
L1 1.000 0.794 0.478 0.119 0.196
L2 0.794 1.000 0.681 0.273 0.335
L3 0.478 0.681 1.000 0.661 0.514
L4 0.119 0.273 0.661 1.000 0.724
L5 0.196 0.335 0.514 0.724 1.000
As can be seen from the correlation coefficient, the path travel time fluctuation rates of adjacent links have strong correlation, and the correlation tends to decrease as the distance increases. The correlation matrix is converted into a 3D diagram, as shown in fig. 3.
Repeatedly executing Monte-Carlo simulation on the basis of Copula function fitting to obtain a path travel time fluctuation rate sequence
Figure BDA0001906459980000105
S43-S45 in the specific calculation process, k times of obtaining the path travel time fluctuation rate sequence are executed. Each fluctuation rate represents a potential traffic state, so the larger the k value, the more likely all traffic states can be modeled. In this example, let k be 10000, and the obtained fluctuation ratio estimated value be [ -0.69, 1.58 [ ]]Within the range, the distribution histogram is shown in fig. 4.
5) The confidence level α is 0.05, and the result is obtained
Figure BDA0001906459980000101
With a confidence interval of (-0.053, 0.665), the expected path travel time fluctuation rate:
Figure BDA0001906459980000102
6) given that the sum of the travel times of the five road segments during the t-1 period on the 45 th working day is 295s, the estimated travel time of the path during the t period is:
Figure BDA0001906459980000103
7) finally, the absolute percentage error of the estimated path travel time is as follows:
Figure BDA0001906459980000104
the invention estimates the path travel time by analyzing the path travel time fluctuation rate, simplifies the calculation complexity in time estimation by applying a Monte-Carlo simulation method on the basis of the Copula theory, does not simply analyze the distribution of the travel time, analyzes the distribution of the travel time by excavating the space association rule and the time sequence fluctuation characteristic of the path travel time fluctuation rate, and has strong innovation significance.

Claims (4)

1. A path travel time estimation method based on Copula and Monte-Carlo simulation is characterized by comprising the following steps:
step 1: obtaining driving travel time data in urban road network and using Ti d,tRepresenting the link travel time taken by the vehicle to travel the link i during the day d time period t; the rate of fluctuation r of the section travel time in said section ii tExpressed as:
Figure FDA0002397014260000011
wherein the road section travel time fluctuation rate r of a corresponding time period exists in different dates di tIs a sequence of values;
step 2: r corresponding to each road section ii tUsing a maximum likelihood estimation method to carry out edge distribution fitting to obtain an edge distribution function Fi(ri t)；
And step 3: using the edge distribution function F corresponding to each road section ii(ri t) Fitting of Copula function was performed using
Figure FDA0002397014260000012
Representing the road section travel time fluctuation rate distribution function obtained by fitting;
and 4, step 4: Monte-Carlo simulation is carried out on the basis of Copula function fitting, and a road section travel time fluctuation rate distribution function F is subjected tocMonte-Carlo simulation is carried out to obtain the fluctuation rate of the path travel time
Figure FDA0002397014260000013
Sum path travel time fluctuation rate sequence
Figure FDA0002397014260000014
Obtaining a sequence of path travel time fluctuation rates
Figure FDA0002397014260000015
The specific calculation process of (2) is as follows:
step 41: constructing a correlation relation matrix sigma between the road section travel time fluctuation rate sequences, wherein the expression is as follows:
Figure FDA0002397014260000016
wherein, tauijThe correlation between the road section i and the road section j is represented, and is Kendall's tau coefficient, and the following calculation is carried out by using a calculation formula of a Copula function:
Figure FDA0002397014260000017
step 42: performing Cholesky decomposition on the correlation matrix Σ in step 41, and obtaining ∑ LLTDecomposing the formula to obtain a lower triangular matrix L;
step 43: if the fitting result of the Copula function is the road section travel time fluctuation rate distribution function FcFor Normal distribution, a Copula-specific distribution is generatedA characterized N-dimensional vector V, V ═ LZ, where Z is an N-dimensional random vector Z ═ Z (Z) following a standard normal distribution N (0,1)1,Z2,..,Zn)；
Step 44: if the road section travel time fluctuation rate distribution function FcFor t distribution, an n-dimensional vector with Copula characteristics is generated
Figure FDA0002397014260000021
V is the degree of freedom under t distribution obtained in the calculation process of the step 3, and Y is an n-dimensional random vector obeying chi-square distribution x 2 (v);
step 45: generating path travel time fluctuation rates
Figure FDA0002397014260000022
The calculation formula is as follows:
Figure FDA0002397014260000023
wherein, Fi -1As a function of the edge distribution F in step 2i(ri t) The inverse function of (d); fcA path travel time fluctuation rate distribution function in the step 3;
step 46: repeating the steps 43-45, and obtaining the path travel time fluctuation rate sequence after executing k times
Figure FDA0002397014260000024
Step 5, determine confidence α, obtain
Figure FDA0002397014260000025
Is confidence interval of
Figure FDA0002397014260000026
And is calculated to obtain
Figure FDA0002397014260000027
Calculating a mathematical expectation within a confidence intervalThe formula is as follows:
Figure FDA0002397014260000028
Figure FDA0002397014260000029
wherein sup {. cndot } and inf {. cndot } respectively represent the supremum and infimum of the variable;
expected path travel time fluctuation rate ER at confidence level 1- αtThe following formula is used to obtain:
Figure FDA00023970142600000210
wherein, ERtIs that
Figure FDA00023970142600000211
A mathematical expectation within a confidence interval; 1{·}Is an indicative function;
step 6: the travel time of each section in the time period T-1 of the day is Ti d,t-1According to the desired path travel time fluctuation rate ERtFor the path travel time T in the T periodtThe estimation is carried out by the following calculation method:
Figure FDA00023970142600000212
wherein exp (·) represents an exponential operation with a base e,
Figure FDA00023970142600000213
representing the path travel time estimate for the time of day t.
2. The method as claimed in claim 1, wherein in step 1, i-1.. n represents the number of road segments in a path, and n is greater than 2.
3. The path travel time estimation method based on Copula and Monte-Carlo simulation as claimed in claim 1, wherein the edge distribution types of the fluctuation rate of the segment travel time in step 2 include three distribution types of normal distribution, t distribution and generalized hyperbolic distribution.
4. The method according to claim 1, wherein the Copula function in step 3 has six types, i.e. Normal, T, Frank, Clayton, Gumbel and Joe.
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