CN105321345A - Road traffic flow prediction method based on ARIMA model and kalman filtering - Google Patents

Abstract

The invention provides a road traffic flow prediction method based on an ARIMA model and kalman filtering. Firstly, road traffic history data is extracted, and the ARIMA model of the road traffic history data in a time sequence is established; then the ARIMA model of the road traffic history data and a kalman filtering algorithm are combined, and state equations, measuring equations and update equations in a kalman filtering process are obtained; and finally, the road traffic real-time data is extracted. By adopting the road traffic flow prediction algorithm based on the ARIMA model and kalman filtering, real-time prediction of road traffic data is realized.

Description

A kind of road traffic flow prediction method based on ARIMA model and kalman filtering

Technical field

The invention belongs to highway traffic data prediction field, relating to the process of highway traffic data and the method for mathematical modeling, is a kind of Forecasting Methodology of road traffic flow.

Background technology

The prediction of road traffic flow is the important prerequisite of carrying out traffic administration and control, is the key realizing Traffic flow systems induction, formulate traffic safety strategy.Forecasting traffic flow is the road traffic state of the important component part of intelligent transportation, measurable future time period, to alleviation traffic congestion, effectively utilizes path resource important role.

In the Forecasting Methodology of existing road traffic flow, ARIMA model can realize the prediction of short-term traffic flow well, but has that lower-order model precision of prediction is low, the problem of high-order model parameter estimation difficulty.Kalman filter algorithm can on-the-fly modify forecast power, relies on recurrence equation can realize accurate precision of prediction, but is difficult to obtain with measurement equation based on the state equation of kalman filter forecasting road traffic.

This patent proposes a kind of road traffic flow prediction method based on ARIMA model and kalman filtering.First utilize the highway traffic data in time series to set up one and can reflect the low order ARIMA model that road traffic flow changes, then based on the measurement equation of ARIMA model construction kalman filtering, state equation and renewal equation, thus solve the problem set up high order time series model and derivation kalman state equation, measure equation difficulty, achieve the high-precision forecast to road traffic flow.

Along with intelligent transportation system is in the development of China, the realization of road traffic flow prediction can provide effective information in real time for traveler, helps them to select optimal path, realizes road traffic paths chosen, reduce the travel time, alleviate traffic congestion.

Summary of the invention

The deficiency of high precision and real-time cannot be taken into account to overcome existing road traffic flow prediction method, the invention provides that a kind of precision of prediction is higher, real-time is good, based on the road traffic flow prediction method of ARIMA model and kalman filtering.

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

Based on a highway traffic data Forecasting Methodology for ARIMA model and kalman filtering algorithm, comprise the steps:

1) the highway traffic data ARIMA model in sequence Time Created

Extract road traffic flow historical data, the highway traffic data ARIMA model in Time Created sequence;

2) the road traffic Flow prediction algorithm based on ARIMA model and kalman filtering is built

Utilize road traffic flow seasonal effect in time series ARIMA models coupling kalman filtering algorithm, build based on the state equation in ARIMA model and kalman filter forecasting road traffic flow process, measure equation and renewal equation;

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

Extract road traffic real time data, based on the road traffic Flow prediction algorithm of ARIMA model and kalman filtering, realize the real-time estimate of highway traffic data.

Further, described step 1) in, obtain road traffic flow historical data, carry out data prediction, the general expression based on pretreated highway traffic data, structure ARIMA model is as follows:

Wherein,

θ(B)＝1-θ ₁(t)B-θ ₂(t)B ²-…θ _q(t)B ^q

k＝1-B

Wherein, { x _t(t=1,2 ...) be the time series of highway traffic data; for autoregression item; θ (B) is moving average item; { e _tfor average is 0, variance is σ ²normal white noise process; for autoregression term coefficient to be estimated; θ _j(t) (j=1,2 ... q) be running mean term coefficient to be estimated; B moves difference operator after being; K is difference operator; D is difference order; P is Autoregressive; Q is running mean exponent number.

Then the road traffic state in t+1 moment is measurable is:

Wherein, x (t+1), x (t) ... x (t-p+1) represents t+1, t respectively ... the traffic data value that the t-p+1 moment is corresponding; θ ₁(t), θ ₂(t) ... θ _pt () represents t autoregression term coefficient; θ ₁(t), θ ₂(t) ... θ _pt () represents t running mean term coefficient; E (t+1), e (t) ... e (t-q+1) is t+1, t ... the noise figure that the t-p+1 moment is corresponding, and Normal Distribution.

Further again, described step 2) in, the observation equation of Kalman filter and the following formulae express of measurement 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 _kthe p dimension random disturbance vector of system, V _kbe the random m dimension observation noise vector of system, A is that the n × n of system ties up state-transition matrix, and B is the observing matrix of system;

If x ₁(t)=x (t), x ₂(t)=x (t-1) ... x _p(t)=x (t-p+1), e ₁(t)=e (t), e ₂(t)=e (t-1) ... e _q(t)=e (t-q+1), the ARIMA model of highway traffic data be incorporated into the state equation of kalman filter forecasting algorithm and measure in equation, then ARIMA model can be expressed as:

Wherein, x ₁(t), x ₂(t) ... x _pt () is illustrated respectively in t, highway traffic data sequence 1, and 2 ... the respective value on p rank; e ₁(t), e ₂(t) ... e _qt () is illustrated respectively in t, noise sequence 1, and 2 ... the respective value on q rank.

X ₂(t+1)=x ₁(t), x ₃(t+1)=x ₂(t) ... x _p+1(t+1)=x _p(t), e ₂(t+1)=e ₁(t), e ₃(t+1)=e ₂(t) ... e _q(t+1)=e _q-1(t), then formula (4) is expressed as follows:

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

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

Wherein, Y (t+1) represents the highway traffic data value that the t+1 moment is corresponding, x ₁(t+1), x ₂(t+1) ... x _p(t+1) represent that t+1 moment, highway traffic data sequence are 1 respectively, 2 ... the respective value on p rank.

Further, described step 3) in, based on the highway traffic data prediction algorithm of ARIMA model and kalman filtering, utilize formula (6) and the state equation shown in (7) and measure equation, following equation can be obtained:

P(t+1|t)＝A*P(t|t)*A'+R ₁+R ₂+…+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 based on t prediction t, and P (t+1|t) is the covariance matrix of X (t+1|t) correspondence; R ₁..., R _qfor noise e ₁, e ₂..., e _qcorresponding covariance matrix; Q is association's equation matrix of observation equation noise; A is the state-transition matrix of system, and B is the observing matrix of observation equation.

Then can obtain, the predicted value of the highway traffic data in t+1 moment is:

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

Wherein, for the highway traffic data value in t+1 moment, X (t+1|t+1) is t+1 moment highway traffic data optimal estimation matrix.

Beneficial effect of the present invention is mainly manifested in: utilize road traffic flow seasonal effect in time series ARIMA models coupling kalman filter forecasting algorithm, builds based on the state equation in ARIMA model and kalman filter forecasting road traffic flow process, measures equation and renewal equation; Extract road traffic real time data, based on the road traffic Flow prediction algorithm of ARIMA model and kalman filtering, realize the real-time estimate of highway traffic data.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the road traffic flow prediction method based on ARIMA model and kalman filtering algorithm.

Embodiment

Below in conjunction with accompanying drawing, the invention will be further described.

With reference to Fig. 1, a kind of road traffic flow prediction method based on ARIMA model and kalman filtering, comprises the following steps:

1) step of highway traffic data Time Series AR IMA model is set up

Obtain road traffic flow historical data, carry out data prediction, the general expression based on pretreated highway traffic data, structure ARIMA model is as follows:

Wherein,

θ(B)＝1-θ ₁(t)B-θ ₂(t)B ²-…θ _q(t)B ^q

k＝1-B

Wherein, { x _t(t=1,2 ...) be the time series of highway traffic data; for autoregression item; θ (B) is moving average item; { e _tfor average is 0, variance is σ ²normal white noise process; for autoregression term coefficient to be estimated; θ _j(t) (j=1,2 ... q) be running mean term coefficient to be estimated; B moves difference operator after being; K is difference operator; D is difference order; P is Autoregressive; Q is running mean exponent number.

Then the road traffic state in t+1 moment is measurable is:

Wherein, x (t+1), x (t) ... x (t-p+1) represents t+1, t respectively ... the traffic data value that the t-p+1 moment is corresponding; θ ₁(t), θ ₂(t) ... θ _pt () represents t autoregression term coefficient; θ ₁(t), θ ₂(t) ... θ _pt () represents t running mean term coefficient; E (t+1), e (t) ... e (t-q+1) is t+1, t ... the noise figure that the t-p+1 moment is corresponding, and Normal Distribution.

2) build based on the road traffic Flow prediction algorithm Kalman filter of ARIMA model and kalman filtering observation equation and measure the following formulae express of equation:

X _t+1=AX _t+ W _t(observation 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 _tthe p dimension random disturbance vector of system, V _tbe the random m dimension observation noise vector of system, A is that the n × n of system ties up state-transition matrix, and B is the observing matrix of system.

If x ₁(t)=x (t), x ₂(t)=x (t-1) ... x _p(t)=x (t-p+1), e ₁(t)=e (t), e ₂(t)=e ₁(t-1) ... e _q(t)=e (t-q+1), the ARIMA model of highway traffic data be incorporated into the state equation of kalman filtering and measure in equation, then ARIMA model can be expressed as:

Wherein, x ₁(t), x ₂(t) ... x _pt () is illustrated respectively in t, highway traffic data sequence 1, and 2 ... the respective value on p rank; e ₁(t), e ₂(t) ... e _qt () is illustrated respectively in t, noise sequence 1, and 2 ... the respective value on q rank.

X ₂(t+1)=x ₁(t), x ₃(t+1)=x ₂(t) ... x _p+1(t+1)=x _p(t), e ₂(t+1)=e ₁(t), e ₃(t+1)=e ₂(t) ... e _q(t+1)=e _q-1(t), then formula (4) is expressed as follows:

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

$\[Y(t+1)\]=\[\begin{array}{cccc}1& 0& \mathrm{...}& 0\end{array}\]\left[\begin{array}{c}{x}_1(t+1)\\ x_2(t+1)\\ \mathrm{...}\\ x_p(t+1)\end{array}\right]---\left(6\right)$

Wherein, Y (t+1) represents the highway traffic data value that the t+1 moment is corresponding, x ₁(t+1), x ₂(t+1) ... x _p(t+1) represent that t+1 moment, highway traffic data sequence are 1 respectively, 2 ... the respective value on p rank.

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

Based on the highway traffic data prediction algorithm of ARIMA model and kalman filtering, utilize formula (5) and the state equation shown in (6) and measure equation, following equation can be obtained:

P(t+1|t)＝A*P(t|t)*A'+R ₁+R ₂+…+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 based on the t prediction t+1 moment, and P (t+1|t) is the covariance matrix of X (t+1|t) correspondence; R ₁..., R _qfor noise e ₁, e ₂..., e _qcorresponding covariance matrix; Q is association's equation matrix of observation equation noise; A is the state-transition matrix of system, and B is the observing matrix of observation equation.

Then can obtain, the predicted value of the highway traffic data in 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 model and kalman filtering algorithm, comprises the following steps:

1) the highway traffic data ARIMA model in sequence Time Created

Because same section, the road traffic flow of corresponding time have similarity, therefore select classical Second Ring Road section (Central Conservatory of Music--Western Informal Gate bridge) in Beijing, weekend (18,19,25,26) same test point actual measurement speed data four day June in 2011 (sampling interval is 2 minutes) as sample sequence { x _t.Based on the highway traffic data ARIMA model in speed data sequence Time Created of two days on the 18th, 19.

2) the road traffic Flow prediction algorithm based on ARIMA model and kalman filtering is built

First the stability considering sequence needed to historical data time series modeling, jiggly sequence is needed to carry out tranquilization process.Tranquilization process can carry out first order difference process to it, makes sequence become stationary sequence.According to minimum AIC criterion, sequence { x _tfinal mask structure is defined as ARIMA (1,1,1).Because historical data modeling number is different, the estimation of parameter also can be different, show that the general expression of model is by arranging:

Above formula can be expressed as

In conjunction with kalman filter forecasting algorithmic formula, formula (9) can be deformed into by the statement of through type (5)

3) parameter based on ARIMA model and kalman filter forecasting is determined

In the process based on ARIMA model and kalman filter forecasting, be designed into following parameter: ARIMA model parameter: θ ₁t (), can be determined by model structure parameter (p, d, q) and historical data modeling number N.Kalman filtering parameter: state-transition matrix A, observing matrix B, state-noise vector W _k, observation noise vector V _k, can be determined by ARIMA (p, d, q).Relevant original state X (0), covariance matrix P (0|0) can be determined by experience.For not history road traffic flow data in the same time, the parameter (p, d, q, N) that the optimum road traffic flow data model of acquisition is corresponding is different.Here done setting parameter is just to the general impact analysis of the road traffic flow prediction method of ARIMA model and kalman filtering algorithm.

Because the precision of these parameters on algorithm respectively has impact, analyzing separately each parameter can not guarantee algorithm optimum on the impact of arithmetic accuracy, therefore should consider when carrying out Algorithm Analysis the impact that all parameters predict the outcome on this road traffic flow simultaneously.

Introduce the absolute average relative error of predicted data, the impact of parameter on arithmetic accuracy analyzed:

$NAME=\frac{\left|\stackrel{\‾}{Y}\left(t\right)-Y\left(t\right)\right|}{Y\left(t\right)}$

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

Namely for different (p, d, q, N), there is NMAE corresponding with it.Therefore there is following equation:

NMAE＝w(p，d，q，N)

Namely there is certain distribution relation ω in (p, d, q, N) and NMAE, and (p, d, q, N) corresponding when searching NMAE is minimum, is optimized parameter assignment procedure.Therefore can obtain as drag:

Minω(p，d，q，N)

$\begin{array}{cc}Where& NAME=\frac{\left|\stackrel{\‾}{Y}\left(t\right)-Y\left(t\right)\right|}{Y\left(t\right)}\end{array}$

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, obtain optimized parameter (p, d, q, N).This experimental result is predicted mainly for the toy vehicle velocity value in section.Extract road traffic real time data, based on the road traffic Flow prediction algorithm of ARIMA model and kalman filtering, realize the real-time estimate of highway traffic data.Comparative for making experimental result have, experimental result and single time series are set up the statistics that ARIMA model carries out road traffic flow prediction and contrast.

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

$mrerr=\frac{1}{N}{\Σ}_t\frac{|x_{pred}\left(t\right)-x_{real}\left(t\right)|}{x_{real}\left(t\right)}$

$mxarer=max\left|\frac{x_{pred}\left(t\right)-x_{real}\left(t\right)}{x_{real}\left(t\right)}\right|$

$rmrerr=\frac{1}{N}{\left\{\frac{x_{pred}\left(t\right)-x_{real}\left(t\right)}{x_{real}\left(t\right)}\right\}}^2$

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

Table 1.

Claims (4)

1., based on a road traffic flow prediction method for ARIMA model and kalman filtering algorithm, it is characterized in that:

1) the highway traffic data ARIMA model in sequence Time Created

Extract road traffic flow historical data, the highway traffic data ARIMA model in Time Created sequence;

2) the road traffic Flow prediction algorithm based on ARIMA model and kalman filtering is built

Utilize road traffic flow seasonal effect in time series ARIMA models coupling kalman filter forecasting algorithm, build based on the state equation in ARIMA model and kalman filter forecasting road traffic flow process, measure equation and renewal equation;

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

Extract road traffic real time data, based on the road traffic Flow prediction algorithm of ARIMA model and kalman filtering, realize the real-time estimate of highway traffic data.

2. as claimed in claim 1 based on the highway traffic data Forecasting Methodology of ARIMA model and kalman filtering, it is characterized in that: described step 1) in, obtain road traffic flow historical data, carry out data prediction, the general expression based on pretreated highway traffic data, structure ARIMA model is as follows:

Wherein,

θ(B)＝1-θ ₁(t)B-θ ₂(t)B ²-…θ _q(t)B ^q

k＝1-B

Wherein, { x _t(t=1,2 ...) be the time series of highway traffic data; for autoregression item; θ (B) is moving average item; { e _tfor average is 0, variance is σ ²normal white noise process; for autoregression term coefficient to be estimated; θ _j(t) (j=1,2 ... q) be running mean term coefficient to be estimated; B moves difference operator after being; K is difference operator; D is difference order; P is Autoregressive; Q is running mean exponent number.

Then the road traffic state in t+1 moment is measurable is:

Wherein, x (t+1), x (t) ... x (t-p+1) represents t+1, t respectively ... the traffic data value that the t-p+1 moment is corresponding; θ ₁(t), θ ₂(t) ... θ _pt () represents t autoregression term coefficient; θ ₁(t), θ ₂(t) ... θ _pt () represents t running mean term coefficient; E (t+1), e (t) ... e (t-q+1) is t+1, t ... the noise figure that the t-p+1 moment is corresponding, and Normal Distribution.

3., as claimed in claim 2 based on the highway traffic data Forecasting Methodology of ARIMA model and kalman filtering, it is characterized in that: described step 2) in Kalman filter observation equation and measure 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 _kthe p dimension random disturbance vector of system, V _kbe the random m dimension observation noise vector of system, A is that the n × n of system ties up state-transition matrix, and B is the observing matrix of system;

If

X ₁(t)=x (t), x ₂(t)=x (t-1) ... x _p(t)=x (t-p+1), e ₁(t)=e (t), e ₂(t)=e ₁(t-1) ... e _q(t)=e (t-q+1), the ARIMA model of highway traffic data be incorporated into the state equation of kalman filter forecasting algorithm and measure in equation, then ARIMA model can be expressed as:

Wherein, x ₁(t), x ₂(t) ... x _pt () is illustrated respectively in t, highway traffic data sequence 1, and 2 ... the respective value on p rank; e ₁(t), e ₂(t) ... e _qt () is illustrated respectively in t, noise sequence 1, and 2 ... the respective value on q rank.

X ₂(t+1)=x ₁(t), x ₃(t+1)=x ₂(t) ... x _p+1(t+1)=x _p(t), e ₂(t+1)=e ₁(t), e ₃(t+1)=e ₂(t) ... e _q(t+1)=e _q-1(t), then formula (5) is expressed as follows:

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

$\[Y(t+1)\]=\[\begin{array}{cccc}1& 0& ...& 0\end{array}\]\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 the highway traffic data value that the t+1 moment is corresponding, x ₁(t+1), x ₂(t+1) ... x _p(t+1) represent that t+1 moment, highway traffic data sequence are 1 respectively, 2 ... the respective value on p rank.

4. as claimed in claim 3 based on the highway traffic data Forecasting Methodology of ARIMA model and kalman filtering, it is characterized in that: described step 3) in, based on the highway traffic data prediction algorithm of ARIMA model and kalman filtering, utilize formula (6) and the state equation shown in (7) and measure equation, following equation can be obtained:

P(t+1|t)＝A*P(t|t)*A'+R ₁+R ₂+…+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 based on the t prediction t+1 moment, and P (t+1|t) is the covariance matrix of X (t+1|t) correspondence; R ₁..., R _qfor noise e ₁, e ₂..., e _qcorresponding covariance matrix; Q is association's equation matrix of observation equation noise; A is the state-transition matrix of system, and B is the observing matrix of observation equation.

Then can obtain, the predicted value of the highway traffic data in t+1 moment is:

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

Wherein, for the highway traffic data value in t+1 moment, X (t+1|t+1) is t+1 moment highway traffic data optimum estimation matrix.