CN110020475B - Markov particle filtering method for traffic flow prediction - Google Patents
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Abstract
The application relates to a Markov particle filtering method for traffic flow prediction. The application combines the Markov chain and the particle filter algorithm, replaces the state space prediction model with Markov, determines the initial weight, and carries out repeated iterative updating through the particle filter algorithm to obtain the prediction result. The defect that Markov is not applicable to a nonlinear system and the prediction precision is insufficient is overcome. And carrying out error analysis on the prediction result to verify the applicability of the method. The application determines the traffic flow state division and can realize short-time traffic flow prediction. Can provide good theoretical support and decision basis for traffic control and induction.
Description
Technical Field
The application designs a prediction model, in particular to a traffic flow prediction model for Markov particle filtering.
Background
In an intelligent traffic system, short-time traffic flow prediction is one of key technologies for realizing advanced traffic control and traffic guidance. Aiming at the defects of the current Markov traffic flow prediction model in the aspect of precision and the characteristics of randomness and fluctuation of traffic flow, the Markov particle filter traffic flow prediction model is provided.
With the development of economy at a high speed, the number of motor vehicles is continuously increased, and a series of traffic problems such as traffic jams, traffic pollution, traffic accidents and the like affect the daily life of people. In recent years, traffic jam during peak morning and evening has become an unavoidable problem in life, especially during holidays, and traffic jam is a major factor affecting traffic capacity. In order to reasonably manage and control traffic, an effective control strategy is needed to be adopted to dredge traffic flow in the current time period so as to improve road traffic jam conditions and reduce environmental pollution. Short-term traffic flow prediction is an important research content.
The single traffic flow prediction method has special information variables and applicable conditions, and can only predict the flow from different angles, so that the single prediction method has certain limitation on traffic flow with strong random volatility and certain unilateral prediction result.
The uncertainty of traffic flow is strong, and the traffic flow has the characteristic of random nonlinearity and can be subjected to abnormal change of flow caused by external changes such as weather. The markov model is a powerful tool for measuring state space and analyzing time series data, but only can obtain rough prediction results, and is not applicable to a nonlinear system. Particle filtering techniques are highly adaptable to nonlinear systems and non-gaussian noise environments. Therefore, the Markov chain is combined with the particle filtering algorithm, the state space prediction model is replaced by Markov, the initial weight is determined, and then the particle filtering algorithm is used for carrying out repeated iterative updating to obtain the prediction result. The defect that Markov is not applicable to a nonlinear system and the prediction precision is insufficient is overcome. And carrying out error analysis on the prediction result to verify the applicability of the method.
Disclosure of Invention
In view of this, it is an object of the present application to provide a markov particle filtered traffic flow prediction model that determines traffic flow state partitioning and can implement short-term traffic flow prediction. Can provide good theoretical support and decision basis for traffic control and induction.
In order to achieve the purpose of the application, the technical scheme adopted is as follows:
before prediction, sample data is required to be preprocessed, null data caused by detector faults is repaired by adopting an adjacent period data averaging method.
The correction formula is as follows:
x k the k-time traffic flow is the flow of traffic at k times.
x k-1 The traffic flow at time k-1 is the traffic flow at time k-1.
x k+1 The traffic flow at time k+1.
Since the Markov model is a prediction of state transitions, it is necessary to attribute traffic flow to different states. The process is as follows:
representing traffic flow state by state set S, and the history sample data constitutes traffic flow state set s= { S 1 ,s 2 ,...,s n }。
And determining the traffic flow state by adopting a threshold method. Introduction of parameter mu 1 、μ 2 。
μ 1 ＝x k-1(min) :I:x k-1(max) (2)
μ 2 ＝{θ 1 ，θ 2 ，...，θ n } (3)
μ 1 The term "traffic flow" refers to a flow of traffic divided into a plurality of states at intervals of I. Typically take i=5.
μ 2 The threshold is saved.
x k-1(min) - - - - - - - - - - - - - - - - - - - - - - - - - - - - -, indicates the minimum value of the traffic flow at time k-1.
x k-1(max) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -, indicates the maximum value of the traffic flow at time k-1.
I represents the traffic flow dividing interval.
θ ι The threshold value represents a state boundary value, and a state has two boundary values, i=1, 2.
s i The expression interval is (θ) i-1 ,θ i ]I=1, 2,..n.
And (5) determining a state set. To the traffic x k-1 Sorting from big to small, calculating the number of statesIf h is not an integer, the state s is added h+1 As the last state. I.e. s h+1 ＝x k-1(max) The method comprises the steps of carrying out a first treatment on the surface of the The state set is s= { S 1 ,s 2 ,...,s h ,s h+1 }。
h represents the number of states.
s h+1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -.
In order to construct a Markov traffic flow prediction model, firstly, traffic states of sample traffic flows are determined, then a state transition matrix is obtained, and future traffic states are predicted according to the state transition matrix. The specific process is as follows:
and determining the state transition probability. The state transition matrix indicates no post-effect of markov, i.e. the state at time k is only related to the traffic state at time k-1. The traffic flow state is from the state s at the current k-1 moment i (k-1) transition to the State s at the next time, k j (k) Is uncertain, its likelihood is represented with probability as its state transition probability:
m i representing state s i Number of occurrences at different time periods.
m ij Representing the state s i Transition to state s j Is a number of times (1).
p(s i (k-1)→s j (k))、p(s j |s i )、p ij (k) Representing the state s i Transition to state s j Is a probability of (2).
And (5) determining a state transition matrix. Based on determining the state transition probability p ij (k) Then, a state transition matrix is constructed as follows:
p (k) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -.
Satisfy the following requirements
p j (k) The probability that k is in the j state at time is indicated.
And establishing a Markov particle filter prediction model. The method comprises the following steps:
and establishing a state equation.
And establishing an observation equation.
u 2 State boundary value of (k-1) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -, time) k-1.
Predicted value of time-k, i=1, 2.
Observations at time-k, i=1, 2.
Observation noise.
H, the observed value coefficient is set as the identity matrix E.
Principle of particle filtering algorithm. The particle filtering is a nonlinear filtering algorithm based on a sequential Monte Carlo method and a statistical method simulation method of recursive Bayesian estimation, the core idea of the particle filtering is to represent probability distribution of random state particles extracted from posterior probability, and the particle filtering is a sequential importance sampling method. For a real-time dynamic system, its dynamic spatial model is as follows:
determining a state equation and an observation equation
x k ＝f(x k-1 )+u k-1 (9)
y k ＝h(x k )+v k (10)
x k Predicted value of time-k.
y k Observations of time-k.
u k-1 Process noise.
v k-1 ----------observing noise.
f(x k-1 ) The system state equation at time k-1 is shown.
h(x k ) The system observation equation for time k is shown.
The prediction process comprises the following steps:
let z be k ＝{y 1:i I=1, 2,..k } is the set of all observations from the initial time to the time k.
p(x k |z k-1 )＝∫p(x k |x k-1 )p(x k-1 |z k-1 )dx k-1 (11)
p(x k |x k-1 ) The state transition probability density of the state equation is obtained from the state equation (10).
p(y k |x k ) Probability density of observation of the observation equation.
p(x k-1 |z k-1 ) The posterior probability distribution is obtained from sample data.
p(x k |z k-1 ) The prior probability is determined based on the state transition probability density p (x k |x k-1 ) The obtained product.
Status update process:
p(y k |z k-1 )＝∫p(y k |x k )p(x k |z k-1 )dx k (13)
equation (12) and equation (13) are only theoretical solutions and it is practically difficult to calculate the result, the basic principle of which is to generate a set of random sample particle sets, using the particle set vs. posterior probability distribution function p (x) k |z k ) Performing approximation processing to obtain predicted value of k moment and particles based on observed valueRepresenting the ith possible traffic flowQuantity (S)>Can be according to->Acquiring a state equation; />For the weight corresponding to the ith predicted traffic flow, i.e. importance weight,/>It is necessary to update and normalize each iteration. Can be expressed as:
the delta-function is a dirac delta function, which means that the function takes values equal to zero at points other than zero, and its integral over the whole definition domain is equal to 1.
x 0:k The state set from 0 to k is the state set.
The term "c" means a direct proportional function.
The normalized weight corresponding to the ith particle at time k.
The weight of the i-th particle at time k is satisfied>
Resampling:
the basic connotation of the particle filtering algorithm is iteration, so that the calculated center of gravity is placed on the particles with larger weight to improve the accuracy of the prediction result, therefore, the resampling algorithm is adopted, the idea is to copy the particles with larger weight and reject the particles with smaller weight, but the phenomenon of particle diversity shortage exists. The random reselection sampling method is provided, and concretely comprises the following steps:
generating n numbers of 0,1]Random number { d) uniformly distributed on l L=1, 2,..n }, the integer m satisfying the following equation (17) is found by the search algorithm.
Recording samplesAnd as new sample particles. Finally, interval [0,1 ]]Press->Divided into n cells, when the random number d l Falls within the nth interval (lambda n-1 ,λ n ]Copy the corresponding sample->
Drawings
FIG. 1 is a graph of two prediction methods versus actual traffic flow (throughout the day)
FIG. 2 is a graph of two prediction methods versus actual traffic flow (early peak)
FIG. 3 comparison of absolute error ER for two methods (all day)
FIG. 4 comparison of absolute error ER for two methods (early peak)
FIG. 5 comparative image of relative error RER for two methods (throughout the day)
FIG. 6 comparative graph of the relative error RER of the two methods (early peak)
Detailed Description
For further explanation of the technical scheme of the present application, the description is made herein with reference to the accompanying drawings and specific implementations. 1. Determining reference standard values of all main parameters:
step1: taking historical traffic flow as sample data at intervals of 5min, dividing traffic flow states according to the sample data, and determining a traffic flow state set S= { S 1 ,s 2 ,...,s n }。
Step2: and determining required parameters, wherein the number of particles is n, and h is the number of state sets.
Step3: traffic flow prediction is carried out according to a Markov prediction model, and n particle parameters are calculatedAnd->Setting the initial particle weight->
Predicted value of the ith particle at time-k.
Observed value of the ith particle at time-k.
Step4: and updating the particle weight. According to the formulaCalculating the weight corresponding to each particle>
Predicting the ith particle at time-k +.>When an observed value y is obtained k Is a probability of (2).
The weight normalization is carried out through a formula (16) to obtain
Step5: and judging a sample reselection process. And judging whether the particle sample is subjected to a sample re-selection process by adopting a similar efficiency method. Calculating an effective sampling scale N eff ，
N th The threshold is set to N and the threshold is set to N th =2n/3, n being the number of particles.
N eff The effective sampling scale is the same.
The effective sampling scale is smaller than the set threshold, namely N is satisfied eff ≤N th And performing reselection according to a random reselection method. Traffic flow is re-predicted using the new sample particles.
The effective sampling scale is larger than the set threshold, namely N is satisfied eff ＞N th At that time, the next step is performed.
Step6: and predicting an estimated value. The formula is as follows:
-----a predicted value of the ith particle at time-k.
The weight of the block is a weight after the treatment.
x k Predicted traffic flow at time-k.
2. Traffic flow sample determination:
(1) The experimental data is traffic flow data collected by a detector at a certain entrance direction of a certain intersection in the Changping area in Beijing city, and the collection interval is 5min.
(2) The data set comprises 24-hour 6048-group traffic data of working days (Monday to friday) of 21 days in 2017, 7 months, 5760-group traffic data of 20 days (3-7, 10-14, 17-21 and 24-28 days) are selected as training samples, the traffic state set is determined, and the flow of the whole day of 21 days (31 days) is predicted.
(3) Data from day 288 of the 21 st day was used as test samples, and data from 24 hours of the day and the early peak (7:00-8:55) period were processed for error analysis with the predicted results, respectively.
(4) In the experimental process, the prediction result of the interval i=5 is determined to be better.
3. Traffic flow prediction result analysis:
(1) And comparing the Markov particle filter prediction result and the traditional Markov prediction result with the acquired 21-day traffic flow test sample, and analyzing the traffic flow and the early peak traffic flow. As shown in fig. 1 and 2.
(2) As can be seen from fig. 1, the markov particle filter prediction method can be well fit to the actual situation, and has the same variation trend as the actual traffic flow. The traditional Markov prediction model better describes the fluctuation trend of the time period, but the prediction result is rough, and the error fluctuation is larger than that of the Markov particle filter traffic flow prediction model.
4. Traffic flow prediction error analysis:
(1) In order to further explain the accuracy and stability of the prediction result of the Markov particle filter model, the prediction result is compared with the prediction result of the traditional Markov prediction model, and absolute error ER, relative error RER, root mean square error RMSE and average error epsilon are adopted as evaluation indexes, wherein the formula is as follows:
x represents the original value of the traffic stream.
Traffic flow prediction.
Original traffic flow average.
n- -number of samples
The comparison and analysis are as follows fig. 3, 4, 5, 6.
As can be obtained from fig. 3 and fig. 4, the absolute error fluctuation ranges of the markov particle filtering prediction results are respectively within 0-60 and 2-10 with 1h and 5min as prediction intervals, and the absolute error fluctuation ranges of the traditional markov prediction results are respectively within 0-110 and 0-23. Therefore, the absolute error of the Markov particle filter prediction model at different prediction intervals is much smaller than that of the conventional Markov prediction model.
As can be seen from fig. 5 and fig. 6, with 1h and 5min as prediction intervals, the relative error of the markov particle filter prediction result is basically controlled to be below 0.28 and 0.15, while the absolute error of the conventional markov prediction result is below 0.65 and 0.4, and the markov particle filter traffic flow prediction model has small relative error and relatively gentle fluctuation.
The root mean square error and error analysis of the two algorithms are shown in tables 1 and 2.
Table 1 root mean square error of two algorithms
Table 2 error analysis of two algorithms
The root mean square error is very sensitive to extra-large or extra-small error values of the predicted data, and can well reflect the precision of the predicted result of the method. The results in table 1 show that the root mean square errors of the full-day and early-peak root mean square errors of the markov particle filter prediction model are respectively 32.94 and 5.24, which are smaller than those of the traditional markov prediction method.
The results in table 2 show that, with 1h and 5min as prediction intervals, the average errors of the markov particle filter prediction model are 6.04% and 6.41%, respectively, which are smaller than those of the conventional markov prediction method and the average errors of different time intervals have smaller differences.
And the prediction result is compared with the traditional Markov model for prediction precision and error analysis, and the result shows that the traffic flow prediction model based on the Markov particle filter has strong applicability and high prediction precision.
The application is not limited to the above-mentioned preferred embodiments, and any person who can obtain other various products under the teaching of the application can make any change in shape or structure, and all the technical solutions which are the same or similar to the application fall within the protection scope of the application.
Claims (1)
1. A Markov particle filtering method for traffic flow prediction is characterized in that:
before prediction, sample data are preprocessed, null data caused by detector faults are repaired by adopting an averaging method of adjacent period data;
the correction formula is as follows:
x k the k-time traffic flow is the flow of traffic at k times;
x k-1 -k-1 time traffic flow;
x k+1 -k+1 time traffic flow;
since the Markov model is a prediction of state transitions, it is necessary to attribute traffic flow to different states; the process is as follows:
representing traffic flow state by state set S, and the history sample data constitutes traffic flow state set s= { S 1 ，s 2 ，...，s n }；
Determining a traffic flow state by adopting a threshold method; introduction of parameter mu 1 、μ 2 ；
μ 1 ＝x k-1(min) ：I：x k-1(max) (2)
μ 2 ＝{θ 1 ，θ 2 ，...，θ n } (3)
μ 1 The term "dividing the traffic flow into a plurality of states at intervals of I"; taking i=5;
μ 2 -storing a threshold value;
x k-1(min) -means the minimum value of traffic flow at time k-1;
x k-1(max) -represents the maximum value of traffic flow at time k-1;
i represents traffic flow dividing interval;
θ ι -threshold value representing a state boundary value, one state having two boundary values, i=1, 2,..n;
s i the expression interval is (θ) i-1 ，θ i ]I=1, 2,., n;
determining a state set; to the traffic x k-1 Sorting from big to small, calculating the number of statesIf h is not an integer, the state s is added h+1 As the last state; i.e. s h+1 ＝x k-1(max) The method comprises the steps of carrying out a first treatment on the surface of the The state set is s= { S 1 ，s 2 ，...，s h ，s h+1 }；
h represents the number of states;
s h+1 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -;
in order to construct a Markov traffic flow prediction model, firstly determining the traffic state of a sample traffic flow, then solving a state transition matrix, and predicting the future traffic state according to the state transition matrix; the specific process is as follows:
determining state transition probability; the state transition matrix shows no post-effect of Markov, namely, the state at the moment k is only related to the traffic state at the moment k-1; the traffic flow state is from the state s at the current k-1 moment i (k-1) transition to the State s at the next time, k i (k) Is uncertain, its likelihood is represented with probability as its state transition probability:
m i representing state s i The number of occurrences at different time periods;
m ij representing the state s i Transition to state s j Is a number of times (1);
p(s i (k-1)→s j (k))、p(s j |s i )、p ij (k) Representing the state s i Transition to state s j Probability of (2);
determining a state transition matrix; based on determining the state transition probability p ij (k) Then, a state transition matrix is constructed as follows:
p (k) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -;
satisfy the following requirements
p j (k) -means the probability that k is in the j state at time;
establishing a Markov particle filter prediction model; the method comprises the following steps:
establishing a state equation;
establishing an observation equation;
u 2 a state boundary value of (k-1) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -, - - -1- -time;
predicted value of time of-k, i=1, 2,..n;
observations at time-k, i=1, 2,..n;
observation noise;
h, the observed value coefficient is set as the identity matrix E;
particle filtering, the dynamic spatial model of which is as follows: determining a status square
Equation and observation equation
x k ＝f(x k-1 )+u k-1 (9)
y k ＝h(x k )+v k (10)
x k Predicted value of time-k;
y k observations of time of k;
u k-1 process noise;
v k-1 observation noise;
f(x k-1 ) The system state equation for time k-1;
h(x k ) System observation equation for k time;
the prediction process comprises the following steps: let z be k ＝{y 1：i I=1, 2,..k } is all views from the initial time to the k time
A set of measured values;
p(x k |z k-1 )＝∫p(x k |x k-1 )p(x k-1 |z k-1 )dx k-1 (11)
p(x k |x k-1 ) The state transition probability density of the state equation is obtained from the state equation (10);
p(y k |x k ) The density of probability of observation of the observation equation;
p(x k-1 |z k-1 ) The posterior probability distribution is obtained from sample data;
p(x k |z k-1 ) The prior probability is determined based on the state transition probability density p (x k |x k-1 ) The obtained product;
status update process:
p(y k |z k-1 )＝∫p(y k |x k )p(x k |z k-1 )dx k (13)
equation (12) and equation (13) generate a set of random sample particle sets using the particle set versus posterior probability distribution function p (x) k |z k ) Performing approximation processing to obtain predicted value of k moment and particles based on observed valueRepresents the i-th possible traffic flow, < +.>According to->Acquiring a state equation; />For the weight corresponding to the ith predicted traffic flow, i.e. importance weight,/>Updating and normalizing in each iteration are needed; expressed as:
the delta-function is a dirac delta function, which means that the function takes values equal to zero at points other than zero, and its integral over the whole definition domain is equal to 1;
x 0：k state set from 0 to k times;
the terms "a" and "b" refer to a direct proportional function;
the normalized weight corresponding to the i-th particle at time k;
the weight of the i-th particle at time k is satisfied>
The process is as follows:
step1: taking historical traffic flow as sample data at intervals of 5min, dividing traffic flow states according to the sample data, and determining a traffic flow state set S= { S 1 ，s 2 ，...，s n }；
Step2: determining required parameters, wherein the number of particles is n, and h is the number of state sets;
step3: traffic flow prediction is carried out according to a Markov prediction model, and n particle parameters are calculatedAnd->Setting the initial particle weight->
Predicted value of the ith particle at time-k;
observations of particles at time i of k;
step4: updating the weight of the particles; according to the formulaCalculating the weight corresponding to each particle>
Predicting the ith particle at time-k +.>When an observed value y is obtained k Probability of (2); obtaining the +.A. through the normalization treatment of the weight value of the formula (16)>
Step5: judging a sample re-selection process; judging whether the particle sample is subjected to a sample re-selection process by adopting a similar efficiency method; calculating an effective sampling scale N eff ，
N th The threshold is set to N and the threshold is set to N th =2n/3, n being the number of particles;
N eff the effective sampling scale;
the effective sampling scale is smaller than the set threshold, namely N is satisfied eff ≤N th When in use, the reselection is carried out according to a random reselection method; predicting the traffic flow again by adopting new sample particles;
the effective sampling scale is larger than the set threshold, namely N is satisfied eff ＞N th When the method is used, the next step is carried out; the resampling process is specifically as follows:
generating n numbers of 0,1]Random number { d) uniformly distributed on l F=1, 2,..n }, finding an integer m satisfying the following equation (17) by a search algorithm;
recording samplesAnd as new sample particles; finally, interval [0,1 ]]Press->Divided into n cells, when the random number d l Falling within the nth interval lambda n-1 ，λ n Copy the corresponding sample->
Step6: predicting an estimated value; the formula is as follows:
predicted value of the ith particle at time-k;
weight after processing;
predicted traffic flow at time-k.
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