CN109584552B - Bus arrival time prediction method based on network vector autoregressive model - Google Patents

Abstract

The invention discloses a method for predicting bus arrival time based on a network vector autoregressive model. Taking bus stops and intersections as nodes, an urban traffic network is constructed based on urban road traffic information and bus route planning, and extracted from an intelligent transportation system database. , Infer data such as the number of public facilities, travel speed between stations, traffic congestion degree, etc., the travel speed matrix between stations and the low-dimensional implicit factors of the urban transportation network, build a regression relationship between the implicit factors and predict the travel speed of the corresponding road section, and then based on The extended network vector space autoregressive model learns historical data, predicts the travel speed between stations in a certain period of time in the future, and then estimates the travel time between stations according to the distance between stations and the predicted travel speed. The bus arrival time, bus GPS positioning information and other data can effectively improve the prediction effect.

Description

A method for predicting bus arrival time based on network vector autoregressive model

Technical field:

The invention relates to the technical field of urban intelligent public transport information processing, in particular to a method for predicting bus arrival time based on a network vector autoregressive model.

Background technique:

In recent years, the rapid development of China's economy and the rapid progress of science and technology have promoted a substantial improvement in the level of urban public transportation. Among them, the bus is an important part of urban public transportation, and has become an indispensable means of transportation in people's modern life. With the continuous advancement of the urbanization process and the rapid expansion of the city scale, problems such as the rapid increase in the total number of passengers, the wide range of changes in the intensity of bus passenger flow, and the large differences in passenger transport effects at different time periods have become increasingly prominent. Accurate prediction of bus arrival time is an important means to relieve the pressure of urban public transport. On the one hand, bus arrival time prediction can provide decision support for bus passenger flow guidance, bus safety management and operation coordination, which is beneficial to improve the operation efficiency of urban bus network and reduce traffic congestion. On the other hand, it can provide passengers with the inquiry service of bus arrival time, help passengers make plans, and relieve the anxiety of waiting passengers.

Prediction of bus arrival time refers to using the data collected by the intelligent transportation system to model and predict the time when the bus arrives at the station. The corresponding modeling methods can be roughly divided into two types of strategies: time series analysis and machine learning. The time series analysis strategy extracts the travel time between historical bus lines and stops as a time series, and performs stationarity, randomness and other tests on it, and then selects an appropriate time series analysis model for prediction according to the test situation. The machine learning strategy treats the inter-site travel situation as an object, regards the inter-site travel time as a predictor variable, extracts the length of the inter-site travel section, congestion level, nearby weather conditions, POI conditions, upstream section travel time, etc. as features, and then randomly selects Forests, support vector machines, neural networks, etc. to build models. In general, existing methods cannot fully consider the impact of topological associations between urban road traffic networks on bus travel time. In addition, the collected bus arrival time often has a large number of missing data, and the existing work usually chooses to discard the missing data without proper processing.

Considering that the travel speed between stations can reflect the traffic situation and is directly affected by the travel speed of adjacent areas, the present invention converts the bus arrival time prediction into the travel speed prediction between stations. On this basis, a regression relationship is constructed by using the urban transportation network and the station travel speed matrix to fill in the missing historical data. Furthermore, a network vector autoregressive model is extended based on a partially linear single-index model to predict travel speed between sites. Finally, the travel time between stations is estimated based on the travel speed between stations, thereby predicting the time when the bus will arrive at the target station.

Invention content:

In order to overcome the defects of the prior art, the present invention considers the topological association of the urban traffic network, and makes full use of the bus arrival time, bus GPS positioning information and other data, and proposes a bus arrival time based on the network vector autoregressive model. The forecasting method effectively improves the forecasting effect.

The method for predicting the bus arrival time based on the network vector autoregressive model involved in the present invention comprises the following steps:

A. Data preprocessing for intelligent transportation system: take bus stations and intersections as nodes, build urban transportation network based on urban road traffic information and bus route planning, and extract and infer the number of public facilities and travel between stations from the intelligent transportation system database Speed, traffic congestion and other data;

B. Filling for missing travel speed between stations based on singular value matrix decomposition: For a certain period of time when travel speed is missing, extract the low-dimensional latent factor of the inter-station travel speed matrix and the urban transportation network in this period, and construct a regression between hidden factors relationship and predict the travel speed of the corresponding road segment;

C. Prediction of travel speed between sites based on the network vector partial linear autoregressive model: based on the extended network vector space autoregressive model to learn historical data, so as to predict the travel speed between sites in a certain period of time in the future;

D. Prediction and correction of bus arrival time: According to the distance between stations and the predicted travel speed, the travel time between stations is estimated, and then the travel time of each road section between the accumulated bus and the target station is estimated, and the correction is made with reference to historical data.

The step A involved in the present invention infers the travel relationship network between sites based on the urban road traffic network and bus route planning, and calculates the distance between sites according to the angle relationship between sites.

The step A involved in the present invention uses bus GPS data to infer the degree of congestion between stations.

The step B involved in the present invention constructs the topological association between the inter-site travel speed matrix and the inter-site travel relationship network, so as to fill in the missing travel speed between the sites.

The step C involved in the present invention expands the network vector space autoregression model based on the partial linear single-index model, so that it can handle the direct nonlinear relationship between the independent variable and the dependent variable.

Compared with the prior art, the present invention is reliable in principle, takes into account the topological correlation of the urban traffic network, makes full use of data such as bus arrival time, bus GPS positioning information, etc., effectively improves the prediction effect, and has a friendly application environment.

Description of drawings:

Fig. 1 is a flow chart of the bus arrival time prediction based on the network vector autoregressive model involved in the present invention.

Fig. 2 is a flow chart of filling missing values based on the singular value matrix decomposition method involved in the present invention.

Figure 3: Three cases of angular relationship between sites designed in Example 1

Detailed ways:

In order to express the objectives, technical solutions and advantages of the present invention more clearly, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

Example 1:

The scheme involved in this embodiment includes the following steps:

A. Data preprocessing for intelligent transportation systems

(1) Build a travel relationship network between sites

First, the intersection is placed as a node in the geodetic coordinate system according to the latitude and longitude, and the nodes are connected according to the urban road planning situation. Specifically, the network G=(V, L) can be used to describe; where V represents the intersection set, V={v ₁ , v ₂ ,...v _n }, n=|V| is the total number of intersections, L represents the set of road segments existing between intersections, L={<v _h , v _l >|v _h , v _l ∈V, 1<h, l <n}, the nodes in G are limited by the latitude and longitude, which can reflect the urban road conditions more realistically;

Then, add bus stops to the urban road traffic network according to the planning of bus routes; on this basis, redefine the node set as V=V ₁ ∪ V ₂ , where V ₁ represents the intersection set, and V ₂ represents the station set; The distance between intersections and the distance between stations and the nearest intersection are obtained from the database of the intelligent transportation system; further, to calculate the travel distance between stations, when both stations are close to a certain intersection, it is necessary to clarify the exact position relationship between the two stations and the intersection. Stations i and j, their azimuths Az _ij (0° < Az _ij < 360°) are calculated according to the following model

Among them, Wi represents the latitude of node _{i, J i represents} the longitude of node i, W _j represents the latitude of node j, and J _j represents the longitude of node j; the distance between sites can be calculated according to the included angle relationship, among which there are three cases as shown in the figure 3 shown

Figure 3-a shows that the azimuth angles of station A and station B relative to the intersection are not equal, and the distance D ^AB between station A and station B is the sum of the distances between the two stations and the intersection; Figure 3-b represents station A and station B The azimuth angles relative to the intersection are not equal, but the sum is 360°, and the distance D ^AB between station A and station B is the sum of the distances between the two stations and the intersection; Figure 3-c represents the distance between station A and station B relative to the intersection. The azimuths are equal, and the distance D ^AB between station A and station B is the absolute value of the distance difference between the two stations and the intersection. For the above three situations, calculate the distance between stations; then, generate the travel distance matrix D between stations according to the bus route planning;

Finally, the inter-station travel relationship network G _Bus = (V _Bus , L _Bus ) is extracted from G, where V _Bus represents the set of bus stations, V _Bus = {v ₁ , . . . , v _N }, N = |V _Bus | It is regarded as the number of bus stops in the travel relationship network between stations, L _Bus represents the set of adjacent road segments between stations, L _Bus = {<v _h , v _l >|v _h , v _l ∈ V _Bus , 1<h, l<N}, at the same time, the adjacency matrix of the inter-site travel relationship network G _Bus is generated according to the inter-site travel distance matrix as A=(a _ij )∈R ^N×N , where, when (v _h , v _l )∈L _Bus , a _ij =1, otherwise a _ij =0;

(2) Extract travel speed between sites

The traffic congestion situation of a certain road section will be affected by the adjacent road sections, and the travel time cannot directly reflect the traffic congestion situation; for this reason, in this embodiment, the travel time is extracted from the intelligent transportation system database, and then converted into the travel speed, and the travel speed is calculated. modeling forecast;

(2-1) The initial moment of the extracted data is used as the starting time, and the T time periods are divided at intervals by a fixed period of time;

代表t时段从站点i到站点j的平均旅行时间，因此，

构成了一个T维的高维度向量；(2-2) Y _t ∈R ^N×N is the travel time matrix of t period, and its elements are

represents the average travel time from site i to site j in period t, therefore,

A high-dimensional vector of T dimension is formed;

(2-3) Obtain travel speed data

可依据以下模型计算站点间旅行速度travel time between a given station

The speed of travel between sites can be calculated according to the following model

The inter-site travel time matrix is sequentially transformed into the inter-site travel speed matrix, and then a high-dimensional travel speed vector is generated

(3) Extract relevant covariates

The prediction of travel speed between stations should not only consider the topological correlation of the urban road traffic network, but also other factors that can affect the speed. Therefore, in this embodiment, the public infrastructure situation (POI, Point Of Interest) and the degree of traffic congestion are used as the coordination factors. variable;

(3-1) Public infrastructure

记录站点i到站点j间旅附近的公共设施数量；(采用

记录站点i到站点j间旅附近的公共设施数量)POI (Point Of Interest) represents the number of public infrastructure (such as schools, hospitals, shopping malls, cinemas) in the area where the bus stops are located; in this embodiment, the

Record the number of public facilities near the trip from site i to site j; (using

Record the number of public facilities near the trip from site i to site j)

(3-2) Traffic congestion level

并基于该路段公交车的历史数量序列

的最小值

第一四分位数

中位数

第三四分位数

最大值

将交通拥堵程度划分为四种级别：In this embodiment, the traffic GPS data is used to evaluate the congestion degree of the travel section; given two adjacent stations i and j, the number of buses between adjacent stations i and j in the t period is counted according to the GPS data.

And based on the historical number sequence of buses in this section

the minimum value of

first quartile

median

third quartile

maximum value

The degree of traffic congestion is divided into four levels:

Among them, 1 means unobstructed, 2 means relatively smooth, 3 means relatively congested, 4 means congested,

To sum up, (then) the covariate matrix model Z can be expressed as

Z=(Z ^POI , Z ^TPI ) ^T (4).

B. Filling the missing travel speed between stations based on singular value matrix decomposition

For the travel time speed matrix S _t ∈ R ^N×N in the t period, this embodiment extracts the low-dimensional latent factors of the travel speed matrix and the adjacency matrix of the travel relationship network between sites, and constructs a regression relationship between the latent factors to fill in the S _t The missing data includes the following three steps;

(1) Extract low-dimensional latent factors

The latent space network model involved in this embodiment extracts low-dimensional latent factors, and the latent space network model is

E_t是n×n白噪声矩阵，μ_t是整体均值，a_t、b_t代表节点的输出和接收效应，U_t、V_t代表交互效应，上述参数构成低维度隐含因子

它可以通过SVD模型估计in,

E _t is an n×n white noise matrix, μ _t is the overall mean, at and b _t represent the output and reception effects of nodes, and U _t and V _t represent the interaction effects _. The above parameters constitute low-dimensional implicit factors

It can be estimated by the SVD model

和

是是N×k非奇异矩阵，

是(k×k)对角元素为非零元素的对角矩阵，

n维向量

分别是

和

的列均值；进而，旅行时间速度矩阵S_t的低维隐含因子被

提取；类似的，可以提取站点间旅行关系网络邻接矩阵A的低维隐含因子，N_A＝[a_A，b_A,U_A,V_A]；in,

and

is an N×k nonsingular matrix,

is a (k×k) diagonal matrix with non-zero diagonal elements,

n-dimensional vector

respectively

and

The column _mean of

Extraction; similarly, the low-dimensional latent factor of the adjacency matrix A of the travel relationship network between sites can be extracted, N _A =[a _A, b _A , U _A , V _A ];

(2) Build a low-dimensional regression model between hidden factors

和

并构建如下回归模型First, obtain the row and column numbers of missing values in S _t , then delete the rows and columns corresponding to S _t and adjacency matrix A, and denote them as S _t ' and A'; then, extract their low-dimensional implicit factors

and

And build the following regression model

Wherein, the model f( ) can be a linear model, a nonlinear model or a non-parametric model, the random forest algorithm is adopted in this embodiment, and the number of decision trees is set to 200;

(3) Predict and fill in missing values

最后得出

的列均值

和

的列均值

代入

得出总体均值

后再代入以下模型First, obtain the row number and column number of missing values in S _t , then extract the row and column corresponding to S _t and adjacency matrix A, and denote them as S _t "and A", and then extract the low-dimensional implicit of A" Factor N _A″ = [a _A″ , b _A″ , U _A″ , V _A″ ]. Substitute NA _″ into the model (7) to obtain the corresponding low-dimensional implicit factor

Finally got

column mean of

and

column mean of

substitute

get the population mean

Then substitute the following model

根据行号和列号将

相应位置的数据代入S_t中，即得无缺站点间旅行速度矩阵

get

According to the row number and column number will

The data of the corresponding position is substituted into S _t , that is, the travel speed matrix between stations is obtained.

C. Inter-site travel speed prediction based on network vector partial linear autoregressive model

The network vector partial linear autoregressive model used in this embodiment is:

表示(与时间无关的公共设施数量、拥堵程度等特征变量)非线性变量对因变量的影响，

中的

代表站点i到站点j之间的相关协变量向量，g(z^ijγ)中的γ＝(γ₁，γ₂)^T是协变量系数即节点效应系数，

表示节点i连接到其他节点的总个数，模型中的

表示t-1时刻其他站点k对站点i的平均影响效应，模型中的

表示站点i到站点j路段前一刻旅行速度对当前旅行速度的影响，即t-1时刻的因变量对t时刻的因变量取值会有影响，

是误差项，它与协变量z^ij是相互独立的，且服从正态分布；它的期望和方差分别为

in,

Represents the influence of nonlinear variables on the dependent variable (characteristic variables such as the number of public facilities and the degree of congestion that are not related to time),

middle

Represents the correlated covariate vector between site i and site j, γ=(γ ₁ , γ ₂ ) in g(z ^ij γ) ^T is the covariate coefficient, that is, the node effect coefficient,

Indicates the total number of node i connected to other nodes, the model in the

represents the average influence effect of other site k on site i at time t-1, in the model

Represents the impact of the travel speed on the current travel speed at the moment before the section from station i to station j, that is, the dependent variable at time t-1 will have an impact on the value of the dependent variable at time t,

is the error term, which is independent of the covariate z ^ij and obeys the normal distribution; its expectation and variance are

Let β=(β ₁ , β ₂ ) ^T ,

Rewrite model (9) as:

可得:Let μ _i = z ^ij γ,

Available:

The steps for estimating the unknown parameter ξ=(γ ^T , β ^T ) ^T are as follows:

(1) Estimate g(·)

使用局部线性回归方法最小化如下的目标函数模型：for a given

Use the local linear regression method to minimize the following objective function model:

K(·)是核函数，h是带宽，K(·)是一有界、非负、有关于0对称的紧支撑且Lipschitz连续的密度函数in,

K( ) is the kernel function, h is the bandwidth, and K( ) is a bounded, nonnegative, compactly supported Lipschitz-continuous density function symmetric about zero

Get an estimate:

in,

(2) Estimate ζ

后，通过最小化以下的profile最小二乘函数得到

in getting (1)

Then, by minimizing the following profile least squares function to get

再重复(1)步骤，得到

然后再次重复(2)步骤，得到

不断重复，直至

get right

Repeat step (1) again to get

Then repeat step (2) again to get

repeat until

D. Prediction and correction of bus arrival time

In order to improve the prediction accuracy and correct the interference of the extension of the prediction period on the prediction result, in this embodiment, a correction coefficient α (0≤α≤1) is added to adjust the prediction result to improve the prediction accuracy.

进而，将

拆分为两个h维向量

和

其中，

然后根据模型(2)将

转换为站点间旅行速度向量后代入站点间旅行速度预测模型得到站点间旅行速度估计向量

并依次序根据下式计算站点间旅行时间There are l time periods from station i to station j, and travel time data will be extracted from the intelligent transportation system to form an l-dimensional vector

Further, will

split into two h-dimensional vectors

and

in,

Then according to model (2) the

Convert to the inter-site travel speed vector and enter the inter-site travel speed prediction model to obtain the inter-site travel speed estimation vector

and calculate the travel time between stations according to the following formula

get travel time prediction vector

Finally, let

And find the optimal correction coefficient α ₀ according to formula (15)

Then the travel time correction model output from site i to site j in t period is

After the data is obtained from the intelligent transportation system, the travel time of all road sections between station m and station n is calculated according to the above steps, then added and summed, and finally added with the departure time of station m and output, that is, the bus is completed. Arrival time prediction.

Claims (1)

1. a method for predicting bus arrival time based on network vector autoregressive model, is characterized in that, this method mainly comprises the following steps: A. Data preprocessing for intelligent transportation system: take bus stations and intersections as nodes, build urban transportation network based on urban road traffic information and bus route planning, and extract and infer the number of public facilities and travel between stations from the intelligent transportation system database Data on speed and traffic congestion levels; this includes the following steps: (1) Build a travel relationship network between sites First, the intersection is placed as a node in the geodetic coordinate system according to the latitude and longitude, and the nodes are connected according to the urban road planning situation, which is specifically described by the network G=(V, L); where V represents the intersection set, V={v ₁ , v ₂ ,...v _n }, n=|V| is the total number of intersections, L represents the set of road segments existing between intersections, L={<v _h , v _l >|v _h , v _l ∈V, 1<h, l< n}, the nodes in G are limited by latitude and longitude, reflecting the urban road conditions; then add bus stops to the urban road traffic network according to the bus route planning; redefine the node set as V=V ₁ ∪ V ₂ , V ₁ represents The intersection set, V ₂ represents the station set; the latitude and longitude of the station, the distance between the intersection, and the distance between the station and the nearest intersection are obtained from the database of the intelligent transportation system; then, the travel distance between the stations is calculated, and the two stations are both adjacent to a certain intersection. The exact location relationship of the intersection, given two stations i and j, their azimuths Az _ij (0°<Az _ij <360°) are obtained according to the following model: cos(c)=cos(90-W _i )×cos(90-W _j )+sin(90-W _i )×sin(90-W _j )×cos(J _i -J _j )

Among them, Wi represents the latitude of node _{i, J i represents} the longitude of node i, W _j represents the latitude of node j, and J _j represents the longitude of node j; the distance between sites can be calculated according to the included angle relationship, among which there are three cases: , the azimuths of station A and station B relative to the intersection are not equal, the distance D ^AB between station A and station B is the sum of the distances between the two stations and the intersection; the azimuth angles of station A and station B relative to the intersection are not equal, But the sum is 360°, the distance D ^AB between station A and station B is the sum of the distances between the two stations and the intersection; the azimuth angles of station A and station B relative to the intersection are equal, and the distance D between station A and station B ^AB is the absolute value of the distance difference between the two stations and the intersection; for the above three situations, the distance between the stations is calculated and obtained; then, the travel distance matrix D between the stations is generated according to the planning of the bus route; Finally, the inter-station travel relationship network G _Bus = (V _Bus , L _Bus ) is extracted from G, where V _Bus represents the set of bus stations, V _Bus = {v ₁ , . . . , v _N }, N = |V _Bus | It is regarded as the number of bus stops in the travel relationship network between stations, L _Bus represents the set of adjacent road segments between stations, L _Bus = {<v _h , v _l >|v _h , v _l ∈ V _Bus , 1<h, l<N}, the adjacency matrix of the inter-station travel relationship network G _Bus is generated according to the inter-station travel distance matrix as A=(a _ij )∈R ^N×N , where, when (v _h , v _l )∈L _Bus , a _ij =1, otherwise a _ij =0; (2) Extract travel speed between sites The travel time is extracted from the intelligent transportation system database, converted into travel speed, and the travel speed is modeled and predicted; the initial time of the extracted data is taken as the starting time, and T time periods are divided according to fixed time intervals; Y _t ∈ R ^{N ×N} is the travel time matrix of time period t, and its elements are

represents the average travel time from site i to site j in period t, therefore,

constitutes a T-dimensional high-dimensional vector; given the travel time between a certain station

The travel speed between sites can be obtained according to the following model

The inter-site travel time matrix is sequentially transformed into the inter-site travel speed matrix, and then a high-dimensional travel speed vector is generated

(3) Extract relevant covariates use

Record the number of public facilities near the trip from site i to site j; Use bus GPS data to evaluate the congestion degree of the travel section; given two adjacent stations i and j, count the number of buses between adjacent stations i and j in the t period according to the GPS data

And based on the historical number sequence of buses in this section

The minimum value (count ^ij _min ), the first quartile (count ^ij _0.25 ), the median (count ^ij _median ), the third quartile (count ^ij _0.75 ), the maximum value (count ^ij _max ) will be There are four levels of traffic congestion:

Among them, 1 represents unobstructed, 2 represents relatively unobstructed, 3 represents relatively congestion, and 4 represents congestion, then the covariate matrix model Z can be expressed as Z=(Z ^POI , Z ^TPI ) ^T (4); B. Filling for missing travel speed between stations based on singular value matrix decomposition: For a certain period of time when travel speed is missing, extract the low-dimensional latent factor of the inter-station travel speed matrix and the urban transportation network in this period, and construct a regression between hidden factors relationship and predict the travel speed of the corresponding road segment; it includes the following steps: For the travel time speed matrix S _t ∈ R ^N×N in the t period, extract the low-dimensional latent factors of the travel speed matrix and the adjacency matrix of the travel relationship network between sites, and construct the regression relationship between the latent factors to fill the missing data in S _t , which includes the following three steps; (1) Extract low-dimensional latent factors The latent space network model is used to extract low-dimensional latent factors, and the latent space network model is

in,

E _t is an n×n white noise matrix, μ _t is the overall mean, at and b _t represent the output and reception effects of nodes, and U _t and V _t represent the interaction effects _. The above parameters constitute low-dimensional implicit factors

It can be estimated by SVD model

in,

and

is an N×k nonsingular matrix,

is a (k×k) diagonal matrix with non-zero diagonal elements,

n-dimensional vector

respectively

and

The column mean of ; the low-dimensional implicit factor of the travel time velocity matrix S _t is

Extract, and then extract the low-dimensional latent factor of the adjacency matrix A of the travel relationship network between sites, N _A =[a _A , b _A , U _A , V _A ]; (2) Build a low-dimensional regression model between hidden factors First obtain the row and column numbers of missing values in S _t , then delete the rows and columns corresponding to S _t and adjacency matrix A, and denote them as S _t ' and A'; then extract their low-dimensional implicit factors

and

And build the following regression model

Among them, the model f( ) is one or more of a linear model, a nonlinear model or a non-parametric model, the random forest algorithm is used, and the number of decision trees is set to 200; (3) Predict and fill in missing values First obtain the row and column numbers of missing values in S _t , then extract the rows and columns corresponding to S _t and adjacency matrix A, and denote them as S _t "and A", and then extract the low-dimensional implicit factor N of A" _A″ = [a _A″ , b _A″ , U _A″ , V _A″ ], substitute N _A″ into the model (7) to obtain the corresponding low-dimensional implicit factor

Finally got

column mean of

and

column mean of

substitute

get the population mean

Then substitute the following model

get

According to the row number and column number will

The data of the corresponding position is substituted into S _t , that is, the travel speed matrix between stations is obtained.

C. Prediction of travel speed between sites based on the network vector partial linear autoregressive model: based on the extended network vector space autoregressive model to learn historical data, so as to predict the travel speed between sites in a certain period of time in the future; specifically including the following steps: The adopted network vector partial linear autoregressive model is

Among them, g(z ^ij γ) represents the influence of the number of time-independent public facilities and the non-linear variable of the congestion degree on the dependent variable, and g(z ^ij γ) in

Represents the correlated covariate vector between site i and site j, γ=(γ ₁ , γ ₂ ) in g(z ^ij γ) ^T is the covariate coefficient, that is, the node effect coefficient,

Indicates the total number of node i connected to other nodes, the model in the

represents the average influence effect of other site k on site i at time t-1, in the model

Represents the impact of the travel speed on the current travel speed at the moment before the section from station i to station j, that is, the dependent variable at time t-1 will have an impact on the value of the dependent variable at time t,

is that the error term and the covariate z ^ij are independent of each other and obey the normal distribution;

The expectation and variance of , respectively, are

make

Rewrite model (9) as:

Let μ _i = z ^ij γ,

Available:

The estimated unknown parameter ξ=(γ ^T , β ^T ) ^T includes: (1) Estimate g(·) for a given

A local linear regression method is used to minimize the following objective function model:

in,

K( ) is the kernel function, h is the bandwidth, and K( ) is a bounded, nonnegative, compactly supported Lipschitz-continuous density function symmetric about zero Get an estimate:

in:

(2) Estimate ξ in getting (1)

Then, by minimizing the following profile least squares function to get

get right

Repeat step (1) again to get

Then repeat step (2) again to get

repeat until

D. Prediction and correction of bus arrival time: According to the distance between stations and the predicted travel speed, the travel time between stations is estimated, and then the travel time of each road section between the bus and the target station is accumulated, and the correction is made with reference to historical data; the specific steps are: There are l time periods from station i to station j, and travel time data will be extracted from the intelligent transportation system to form an l-dimensional vector

again

split into two h-dimensional vectors

and

in,

Then according to model (2) the

Convert to the inter-site travel speed vector and enter the inter-site travel speed prediction model to obtain the inter-site travel speed estimation vector

and calculate the travel time between stations according to the following formula

get travel time prediction vector

Finally, let

And according to the model (15) to find the optimal correction coefficient α ₀

Then the travel time correction model output from site i to site j in t period is

The bus arrival time.

Images