CN105321345B - A kind of road traffic flow prediction method filtered based on ARIMA models and kalman - Google Patents

Abstract

A kind of road traffic flow prediction method based on ARIMA models and kalman filtering algorithms, extracts road traffic historical data, the ARIMA models of highway traffic data in setup time sequence first；Then the ARIMA models of highway traffic data are combined with kalman filtering algorithms, obtain state equation, measurement equation and the renewal equation in kalman filterings；Road traffic real time data is finally extracted, based on the road traffic Flow prediction algorithm that ARIMA models and kalman are filtered, the real-time estimate of highway traffic data is realized.

Description

A kind of road traffic flow prediction method filtered based on ARIMA models and kalman

Technical field

The invention belongs to highway traffic data prediction field, it is related to the side of the treatment of highway traffic data and mathematical modeling Method, is a kind of Forecasting Methodology of road traffic flow.

Background technology

The prediction of road traffic flow be carry out traffic administration and control important prerequisite, be realize Traffic flow systems induction, Formulate the key of traffic safety strategy.Forecasting traffic flow is the important component of intelligent transportation, and the road of future time period can be predicted Road traffic behavior, to alleviating traffic congestion, effectively utilizing path resource important role.

In the Forecasting Methodology of existing road traffic flow, ARIMA models can well realize the pre- of short-term traffic flow Survey, but there is a problem of that lower-order model precision of prediction is low, high-order model parameter Estimation difficult.Kalman filter algorithm can be moved State changes forecast power, and accurate precision of prediction can be realized by recurrence equation, but based on kalman filter forecasting roads The state equation and measurement equation of traffic are difficult to obtain.

This patent proposes a kind of road traffic flow prediction method filtered based on ARIMA models and kalman.It is sharp first Setting up one with the highway traffic data in time series can reflect the low order ARIMA models that road traffic flow changes, Ran Houji Measurement equation, state equation and renewal equation in ARIMA model constructions kalman filtering, high order time is set up so as to solve Series model and the difficult problem of derivation kalman state equations, measurement equation, realize pre- to the high accuracy of road traffic flow Survey.

As intelligent transportation system is in the development of China, the realization of road traffic flow prediction can provide real for traveler When effective information, help them to select optimal path, realize road traffic paths chosen, reduce the travel time, alleviate traffic Congestion.

The content of the invention

In order to overcome the shortcomings of that existing road traffic flow prediction method cannot take into account high accuracy and real-time, the present invention is provided The road traffic flow prediction method filtered based on ARIMA models and kalman that a kind of precision of prediction is higher, real-time is good.

The technical solution adopted for the present invention to solve the technical problems is：

A kind of highway traffic data Forecasting Methodology based on ARIMA models and kalman filtering algorithms, comprises the following steps：

1) the highway traffic data ARIMA models in setup time sequence

Extract road traffic flow historical data, the highway traffic data ARIMA models in setup time sequence；

2) the road traffic Flow prediction algorithm filtered based on ARIMA models and kalman is built

Using the ARIMA models coupling kalman filtering algorithms of road traffic flow time series, build and be based on ARIMA models With the state equation during kalman filter forecasting road traffic flows, measurement equation and renewal equation；

3) highway traffic data real-time estimate is realized based on ARIMA models and kalman filtering

Road traffic real time data is extracted, it is real based on the road traffic Flow prediction algorithm that ARIMA models and kalman are filtered The real-time estimate of existing highway traffic data.

Further, the step 1) in, road traffic flow historical data is obtained, data prediction is carried out, based on pretreatment Highway traffic data afterwards, the general expression of structure ARIMA models are as follows：

Wherein,

Wherein, { x_tIt is the time series of highway traffic data, t=1,2 ...；It is autoregression；θ (b) is movement Average item；{e_tFor average is 0, variance is σ²Normal white noise process；It is autoregression term coefficient to be estimated, i=1, 2,…p；θ_jT () is moving average term coefficient to be estimated, j=1,2 ... q；B moves difference operator after being；K is difference operator；D is Difference order；P is Autoregressive；Q is moving average exponent number；

Then the road traffic state at t+1 moment can be predicted and be：

Wherein, x (t+1), x (t) ... x (t-p+1) represent t+1, t ... t-p+1 moment corresponding traffic data value respectively；Represent t autoregression term coefficient；θ₁(t),θ₂(t)...θ_qT () represents t moving average term system Number；E (t+1), e (t) ... e (t-q+1) are t+1, t ... t-p+1 moment corresponding noise figure, and Normal Distribution.

Further, the step 2) in, the observational equation and the measurement following formulae express of equation of Kalman filter：

X_k+1=AX_k+W_k (3)

Y_k=BX_k+V_k (4)

Wherein, X_k+1For the n of system ties up state vector, Y_kFor the m of system ties up observation vector, W_kBe system p dimensions it is random dry Disturb vector, V_kIt is the random m dimension observation noise vectors of system, A is n × n dimension state-transition matrixes of system, and B is the sight of system Survey matrix；

If x₁(t)=x (t), x₂(t)=x (t-1) ... x_p(t)=x (t-p+1), e₁(t)=e (t), e₂(t)=e (t- 1),…e_qT ()=e (t-q+1), the state of kalman filter forecasting algorithms is incorporated into by the ARIMA models of highway traffic data In equation and measurement equation, then ARIMA models can be expressed as：

Wherein, x₁(t),x₂(t),…x_pT () is illustrated respectively in t, highway traffic data sequence 1,2 ... p ranks it is right Should be worth；e₁(t),e₂(t)…e_qThe respective value of q ranks that t () is illustrated respectively in t, noise sequence 1,2 ....

x₂(t+1)=x₁(t),x₃(t+1)=x₂(t),…x_p+1(t+1)=x_p(t),e₂(t+1)=e₁(t),e₃(t+1)= e₂(t),…e_q(t+1)=e_q-1(t), then formula (4) be expressed as follows：

By (3), (4), (5), (6), equation can be measured：

Wherein, Y (t+1) represents t+1 moment corresponding highway traffic data value, x₁(t+1),x₂(t+1),…x_p(t+1) divide Not Biao Shi t+1 moment, highway traffic data sequence 1,2 ... the respective value of p ranks.

Further, the step 3) in, the highway traffic data filtered based on ARIMA models and kalman is calculated in advance Method, using the state equation shown in formula (6) and (7) and measurement equation, can obtain equation below：

P (t+1 | t)=A*P (t | t) * A'+R₁+R₂+…+R_q

Kg (t+1)=P (t+1 | t) * B'/(B*P (t+1 | t) * B'+Q)

X (t+1 | t+1)=X (t+1 | t)+Kg (t+1) * (Z (t+1)-B*X (t | t))

P (t+1 | t+1)=(I-Kg (t+1) * B) * P (t+1 | t)

Wherein, X (t+1 | t) is the highway traffic data value that t is predicted based on t, and P (t+1 | t) is X (t+1 | t) Corresponding covariance matrix；R₁,…,R_qIt is noise e₁,e₂,…,e_qCorresponding covariance matrix；Q is the association of observational equation noise Equation matrix；A is the state-transition matrix of system, and B is the observing matrix of system.

Can then obtain, the predicted value of the highway traffic data at t+1 moment is：

Wherein,It is the highway traffic data value at t+1 moment, and X (t+1 | t+1) it is t+1 moment highway traffic datas Optimal estimation matrix.

Beneficial effects of the present invention are mainly manifested in：Using the ARIMA models couplings of road traffic flow time series Kalman filter forecasting algorithms, build based on the state side during ARIMA models and kalman filter forecasting road traffic flows Journey, measurement equation and renewal equation；Road traffic real time data is extracted, based on the road that ARIMA models and kalman are filtered Forecasting traffic flow algorithm, realizes the real-time estimate of highway traffic data.

Brief description of the drawings

Fig. 1 is the flow chart of the road traffic flow prediction method based on ARIMA models and kalman filtering algorithms.

Specific embodiment

The invention will be further described below in conjunction with the accompanying drawings.

Reference picture 1, a kind of road traffic flow prediction method filtered based on ARIMA models and kalman, including following step Suddenly：

1) the step of setting up highway traffic data Time Series AR IMA models

Road traffic flow historical data is obtained, data prediction is carried out, based on pretreated highway traffic data, structure The general expression of ARIMA models is as follows：

Wherein,

Wherein, { x_tIt is the time series of highway traffic data, t=1,2 ...；It is autoregression；θ (b) is movement Average item；{e_tFor average is 0, variance is σ²Normal white noise process；It is autoregression term coefficient to be estimated, i=1, 2,…p；θ_jT () is moving average term coefficient to be estimated, j=1,2 ... q；B moves difference operator after being；K is difference operator；D is Difference order；P is Autoregressive；Q is moving average exponent number；

Then the road traffic state at t+1 moment can be predicted and be：

Wherein, x (t+1), x (t) ... x (t-p+1) represent t+1, t ... t-p+1 moment corresponding traffic data value respectively；Represent t autoregression term coefficient；θ₁(t),θ₂(t)...θ_qT () represents t moving average term system Number；E (t+1), e (t) ... e (t-q+1) are t+1, t ... t-p+1 moment corresponding noise figure, and Normal Distribution.

2) the observation side of the road traffic Flow prediction algorithm Kalman filter filtered based on ARIMA models and kalman is built Journey and the measurement following formulae express of equation：

X_t+1=AX_t+W_t(observational equation) (2)

Y_t=BX_t+V_t(measurement equation) (3)

Wherein, X_t+1For the n of system ties up state vector, Y_tFor the m of system ties up observation vector, W_tBe system p dimensions it is random dry Disturb vector, V_tIt is the random m dimension observation noise vectors of system, A is n × n dimension state-transition matrixes of system, and B is the sight of system Survey matrix.

If x₁(t)=x (t), x₂(t)=x (t-1) ... x_p(t)=x (t-p+1), e₁(t)=e (t), e₂(t)=e₁(t- 1),…e_qT ()=e (t-q+1), the ARIMA models of highway traffic data are incorporated into state equation and the survey of kalman filtering In amount equation, then ARIMA models can be expressed as：

Wherein, x₁(t),x₂(t),…x_pT () is illustrated respectively in t, highway traffic data sequence 1,2 ... p ranks it is right Should be worth；e₁(t),e₂(t)…e_qThe respective value of q ranks that t () is illustrated respectively in t, noise sequence 1,2 ....

x₂(t+1)=x₁(t),x₃(t+1)=x₂(t),…x_p+1(t+1)=x_p(t),e₂(t+1)=e₁(t),e₃(t+1)= e₂(t),…e_q(t+1)=e_q-1(t), then formula (4) be expressed as follows：

By (2), (3), (4), (5), equation can be measured：

Wherein, Y (t+1) represents t+1 moment corresponding highway traffic data value, x₁(t+1),x₂(t+1),…x_p(t+1) divide Not Biao Shi t+1 moment, highway traffic data sequence 1,2 ... the respective value of p ranks.

3) highway traffic data real-time estimate is realized based on ARIMA models and kalman filtering

Based on the highway traffic data prediction algorithm that ARIMA models and kalman are filtered, using shown in formula (5) and (6) State equation and measurement equation, can obtain equation below：

P (t+1 | t)=A*P (t | t) * A'+R₁+R₂+…+R_q

Kg (t+1)=P (t+1 | t) * B'/(B*P (t+1 | t) * B'+Q)

X (t+1 | t+1)=X (t+1 | t)+Kg (t+1) * (Z (t+1)-B*X (t | t))

P (t+1 | t+1)=(I-Kg (t+1) * B) * P (t+1 | t)

Wherein, X (t+1 | t) is the highway traffic data value that the t+1 moment is predicted based on t, P (t+1 | t) for X (t+1 | T) corresponding covariance matrix；R₁,…,R_qIt is noise e₁,e₂,…,e_qCorresponding covariance matrix；Q is observational equation noise Association's equation matrix；A is the state-transition matrix of system, and B is the observing matrix of system.

Can then obtain, the predicted value of the highway traffic data at t+1 moment is：

Y (t+1)=BX (t+1 | t+1) (7)

Example:A kind of road traffic flow prediction method based on ARIMA models and kalman filtering algorithms, including following step Suddenly：

1) the highway traffic data ARIMA models in setup time sequence

Because the road traffic flow in same section, correspondence time has a similitude, therefore selection Beijing classics Second Ring Road section (in Centre conservatory of music -- Western Informal Gate bridge), four day June in 2011 weekend (18,19,25,26) same test point actual measurement speed data (adopt Sample was at intervals of 2 minutes) as sample sequence { x_t}.Based on the road in the speed data setup time sequence of two days on the 18,19th Traffic data ARIMA models.

2) the road traffic Flow prediction algorithm filtered based on ARIMA models and kalman is built

To historical data time series modeling firstly the need of consider sequence stability, for jiggly sequence need into Row tranquilization is processed.Tranquilization treatment can carry out first-order difference treatment to it, sequence is changed into stationary sequence.According to minimum AIC criterion, sequence { x_tFinal mask structure determination be ARIMA (1,1,1).Because historical data modeling number is different, parameter Estimation also can be different, the general expression that model is drawn by arranging is：

Above formula can be expressed as

With reference to kalman filter forecasting algorithmic formulas, by the statement of formula (5), formula (9) can be deformed into

3) parameter determination based on ARIMA models and kalman filter forecastings

During based on ARIMA models and kalman filter forecastings, following parameter has been designed into：ARIMA moulds Shape parameter：Can be determined by model structure parameter (p, d, q) and historical data modeling number N. Kalman filtering parameters：State-transition matrix A, observing matrix B, state-noise vector W_k, observation noise vector V_k, Ke Yiyou ARIMA (p, d, q) determines.Related original state X (0), covariance matrix P (0 | 0) can be by empirically determined.For it is different when Carve history road traffic flow data, each not phase of the corresponding parameter (p, d, q, N) of optimal road traffic flow data model of acquisition Together.Here the parameter setting done is to the big of the road traffic flow prediction method of ARIMA models and kalman filtering algorithms General impact analysis.

Because these parameters respectively have an impact to the precision of algorithm, influence of each parameter to arithmetic accuracy is individually analyzed not The optimal of algorithm is can ensure that, therefore should be tied while considering that all parameters are predicted the road traffic flow when Algorithm Analysis is carried out The influence of fruit.

The absolute average relative error of prediction data, the influence to parameter to arithmetic accuracy is introduced to be analyzed：

Wherein,It is predicted value, Y (t) is measured value.NAME is the absolute average relative error of prediction data.

I.e. for different (p, d, q, N), there is corresponding NMAE.Therefore there is following equation：

NMAE=w (p, d, q, N)

I.e. there is certain distribution relation ω in (p, d, q, N) with NMAE, when finding NMAE minimums corresponding (p, d, q, N), i.e., For optimized parameter sets process.Therefore can obtain such as drag：

Min ω (p, d, q, N)

Finally the value of (p, d, q, N) can be determined by the training of road traffic historical data.

4) experimental result

Based on road traffic historical data, optimized parameter (p, d, q, N) is obtained.Car of this experimental result mainly for section Velocity amplitude is predicted.Road traffic real time data is extracted, the road traffic flow filtered based on ARIMA models and kalman is pre- Method of determining and calculating, realizes the real-time estimate of highway traffic data.It is comparative to have experimental result, by experimental result and single time Sequence sets up ARIMA models and carries out the statistics of road traffic flow prediction and contrasted.

Choose mean absolute relative error (marerr), maximum absolute relative error (mxarer) and relative error quadratic sum Average (rmrerr) as road traffic flow precision of prediction index, its computing formula difference it is as follows：

The experiment section statistical analysis that predicts the outcome of velocity amplitude of June 25,26 in 2011 is as shown in the table.

Table 1.

Claims (2)

1. a kind of road traffic flow prediction method filtered based on ARIMA models and kalman, it is characterised in that：Including following step Suddenly：

1) the highway traffic data ARIMA models in setup time sequence

Extract road traffic flow historical data, the highway traffic data ARIMA models in setup time sequence；

2) the road traffic Flow prediction algorithm filtered based on ARIMA models and kalman is built

Using the ARIMA models coupling kalman filter forecasting algorithms of road traffic flow time series, build and be based on ARIMA models With the state equation during kalman filter forecasting road traffic flows, measurement equation and renewal equation；

3) realize that highway traffic data real-time estimate extracts road traffic real time data based on ARIMA models and kalman filtering, Based on the road traffic Flow prediction algorithm that ARIMA models and kalman are filtered, the real-time estimate of highway traffic data is realized；

The step 1) in, road traffic flow historical data is obtained, data prediction is carried out, based on pretreated road traffic Data, build ARIMA models general expression it is as follows：

Wherein,

θ (b)=1- θ₁(t)b-θ₂(t)b²-…θ_q(t)b^q

K=1-b

Wherein, { x_tIt is the time series of highway traffic data, t=1,2 ...；It is autoregression；θ (b) is rolling average ；{e_tFor average is 0, variance is σ²Normal white noise process；It is autoregression term coefficient to be estimated, i=1,2 ... p；θ_jT () is moving average term coefficient to be estimated, j=1,2 ... q；B moves difference operator after being；K is difference operator；D is difference Exponent number；P is Autoregressive；Q is moving average exponent number；

Then the road traffic state at t+1 moment can be predicted and be：

Wherein, x (t+1), x (t) ... x (t-p+1) represent t+1, t ... t-p+1 moment corresponding traffic data value respectively；Represent t autoregression term coefficient；θ₁(t),θ₂(t)...θ_qT () represents t moving average term system Number；E (t+1), e (t) ... e (t-q+1) are t+1, t ... t-p+1 moment corresponding noise figure, and Normal Distribution；

The step 2) in Kalman filter observational equation and measurement the following formulae express of equation：

X_k+1=AX_k+W_k (3)

Y_k=BX_k+V_k (4)

Wherein, X_k+1For the n of system ties up state vector, Y_kFor the m of system ties up observation vector, W_kBe system p dimension random disturbances to Amount, V_kIt is the random m dimension observation noise vectors of system, A is n × n dimension state-transition matrixes of system, and B is the observation square of system Battle array；

If x₁(t)=x (t), x₂(t)=x (t-1) ... x_p(t)=x (t-p+1), e₁(t)=e (t), e₂(t)=e₁(t-1),… e_q(t)=e (t-q+1), by the ARIMA models of highway traffic data be incorporated into kalman filter forecasting algorithms state equation and In measurement equation, then ARIMA models can be expressed as：

Wherein, x₁(t),x₂(t),…x_pThe correspondence of p ranks that t () is illustrated respectively in t, highway traffic data sequence 1,2 ... Value；e₁(t),e₂(t)…e_qThe respective value of q ranks that t () is illustrated respectively in t, noise sequence 1,2 ...；

x₂(t+1)=x₁(t),x₃(t+1)=x₂(t),…x_p+1(t+1)=x_p(t),e₂(t+1)=e₁(t),e₃(t+1)=e₂ (t),…e_q(t+1)=e_q-1(t), then formula (5) be expressed as follows：

By (3), (4), (5), (6), equation can be measured：

$\[Y(t+1)\]=\left[\begin{array}{cccc}1& 0& ...& 0\end{array}\right]\left[\begin{array}{c}{x}_1(t+1)\\ x_2(t+1)\\ ...\\ x_p(t+1)\end{array}\right]---\left(7\right)$

Wherein, Y (t+1) represents t+1 moment corresponding highway traffic data value, x₁(t+1),x₂(t+1),…x_p(t+1) difference table The respective value of p ranks of showing t+1 moment, highway traffic data sequence 1,2 ....

2. as claimed in claim 1 to be based on the road traffic flow prediction method that ARIMA models and kalman are filtered, its feature exists In：The step 3) in, based on the highway traffic data prediction algorithm that ARIMA models and kalman are filtered, using formula (6) and (7) state equation and measurement equation shown in, can obtain equation below：

P (t+1 | t)=A*P (t | t) * A'+R₁+R₂+…+R_q

Kg (t+1)=P (t+1 | t) * B'/(B*P (t+1 | t) * B'+Q)

X (t+1 | t+1)=X (t+1 | t)+Kg (t+1) * (Z (t+1)-B*X (t | t))

P (t+1 | t+1)=(I-Kg (t+1) * B) * P (t+1 | t)

Wherein, X (t+1 | t) is the highway traffic data value that the t+1 moment is predicted based on t, and P (t+1 | t) is right for X (t+1 | t) The covariance matrix answered；R₁,…,R_qIt is noise e₁,e₂,…,e_qCorresponding covariance matrix；Q is the association side of observational equation noise Journey matrix；A is the state-transition matrix of system, and B is the observing matrix of system；

Can then obtain, the predicted value of the highway traffic data at t+1 moment is：

$\stackrel{\‾}{Y}(t+1=BX(t+1|t+1)---\left(8\right)$

Wherein,It is the highway traffic data value at t+1 moment, X (t+1 | t+1) is optimal for t+1 moment highway traffic datas Change estimated matrix.