Research

. The application and development of new technology make it possible to acquire real-time data of vehicles. Based on these real-time data, the behavior of vehicles can be analyzed. The prediction of vehicle behavior provides data support for the fine management of traffic. This paper proposes speed and acceleration have fractal features by R / S analysis of the time series data of speed and acceleration. Based on the characteristic analysis of microscopic parameters, the characteristic indexes of parameters are quantified, the fractal multistep prediction model of microparameters is established, and the BP (back propagation neural networks) model is established to estimate predictable step of fractal prediction model. The fractal multistep prediction model is used to predict speed acceleration in the predictable step. NGSIM trajectory data are used to test the multistep prediction model. The results show that the proposed fractal multistep prediction model can effectively realize the multistep prediction of vehicle speed.


Introduction
With an increase of tra c ow on highways all over the world, both the risk on tra c safety and the pressure of tra c management increase greatly.
e detection and warning of abnormal driving behavior is an indispensable part of intelligent tra c management and monitoring in that it detects abnormal driving behaviors, gives an early warning to possibly a ected vehicles, and e ectively avoids tra c accidents.With the rapid development of computer technology, image processing technology, and video detecting technology, the accuracy and content of video detection data can provide a solid foundation for abnormal driving behavior monitoring in motor vehicles ( [1][2][3][4]).
rough the application of Hidden Markov random eld ( [5]), on-board GPS data ( [6]), and mobile phone positioned data ( [7]), the detection and judgment of abnormal driving behavior becomes reality.Diaz Alvarez et al. present a system for estimating the remaining charge of an electric vehicle by considering the driving behavior measured using a smartphone, using speed and acceleration [8].Funfgeld et al. introduce a stochastic framework based on an explanatory model and stochastic processes to predict future vehicle dynamics with road network data [9].
Tra c microscopic parameters (instantaneous speed and acceleration) have strong nonlinear characteristics, and fractal theory is a good method to deal with time series data with self-similarity.Firstly, based on the chaos theory, the chaotic characteristics of the microvelocity and acceleration sequence data of the vehicle are analyzed.It is proved that the microparameters of the vehicle are predictable.en, based on R/S analysis, it is proved that the two parameters have fractal characteristics, and the prediction model of microscopic parameters is constructed based on the fractal theory.rough the quanti cation of the index of speed and acceleration time series data, neural network is used to estimate the predictable number of steps, and the prediction model of microparameter-predictable step is obtained.e multistep prediction of vehicle microtra c parameters is carried out based on fractal theory.e results show that the prediction model has a good performance.
is shows the model can be used in the prediction of driving behavior.You can develop a warning APP for vehicle abnormal driving behavior.Record the vehicle's real-time positioning data, fractal prediction, predict the driving behavior of the vehicle, identify abnormal driving behaviors such as speed of the vehicle, and provide warning.

Basic Data Sources
For the purpose of research, the data need to have the following characteristics: (1) freeway vehicle positioning time series data; (2) real-time data of vehicles, with short sampling period, high frequency and high accuracy.According to the predictable analysis of the characteristics of vehicle speed and acceleration, it is not necessary to contain all kinds of driving behavior when studying the prediction method of microscopic parameters.erefore, in order to reflect the general applicability of the forecasting method and avoid the particularity of the experimental vehicle, this paper chooses American NGSIM trajectory data to support the research.NGSIM stakeholder groups identified the collection of real-world vehicle trajectory data as important to understanding and researching driver behavior at microscopic levels.
e NGSIM datasets represent the most detailed and accurate field data collected to date for traffic microsimulation research and development.
e US Highway 101 (US 101) dataset was one of several datasets collected under the NGSIM program.e NGSIM trajectory data reflect the information on time series data of the speed or acceleration for the vehicle running on highway.e sampling interval is 0.1s (0.1 second), which can meet the research demand with its large amount of data, and the error of positioning and speed is small enough to be negligible.
e specific data attributes are shown in Table 1. is paper analyzes the speed and acceleration of the vehicle, based on two days' track data which come from the NGSIM trajectory data about I-80 section 05: 00∼17: 30 (June 15, 2005) and US-101 section 07: 50∼08:35 (April 13, 2005).As shown in Figure 1, the two sections consist of five main lanes, one collector-distributor lane, one entrance ramp, and one exit ramp.

Fractal Analysis of Microparameters
In this paper, the sampling interval is 0.1 s, 1 s, 2 s, and 4 s.As shown in Figure 2, with the increase of the sampling interval, the chaos of speed and acceleration show a decay trend, so the data of the small interval can be predicted based on the data prediction results of the large sampling interval.e self-similarity of fractal theory can achieve this process well, so the fractal theory can be used to predict the microscopic parameters.Because we have to judge whether the time series data have fractal characteristics, in this paper, we use R/S analysis to calculate Hurst index H [10].

Calculation of the Hurst Index to Determine Whether It
Has Fractal Characteristics.For the instantaneous speed or e standard deviation of the time series data is e accumulative dispersion of the series is e range of the accumulative dispersion of r time series is Hurst index H can be obtained using the following formula: Based on the value of H, whether or not the time series has fractal characteristics can be judged.When the value is between 0.5 and 1, this indicates that the time series has forward persistence; when the value is between 0 and 0.5, it indicates that it has reverse persistence; the time series is a random data sequence when it is equal to 0.5.erefore, when H is not equal to 0.5, the time series has fractal characteristics and makes fractal predictions.

Calculation of the Fractal Dimension of Microscopic
Parameters.
e fractal distribution curve is roughly similar to the exponential distribution, and the value of fractal dimension D is the value of the power exponent, which reflects the fractal strength of the data.e value of fractal dimension D can be obtained using the formula: D � 2 − H.

Prediction of Microparameters Based on
Fractal Theory Calculate the logarithmic scatter plot of S(i, j) and j, and obtain the formula: ln(S(i, j)) ln c + D ′ ln j.Use linear   e prediction is then made by the formula: ln(S(i, j)) � ln c + D ′ ln j. ereafter, decrease the predicted value S(i, j) step by step to get the forecast value to which the small interval corresponds.

Evaluation Indexes of the Predictive Model.
In this paper, the four indexes which are average absolute percentage error MAPE and RMSE are used to analyze the prediction effect of the fractal prediction model on vehicle speed and acceleration sequence.e calculation formulas of each index are as follows: where F i represents predictive value, R i represents actual value, and n represents the number of samples.

Quantification of the Microparameter Characteristic
Index.Affected by some factors, such as roads, traffic composition, traffic condition, and driver's behavior, the microscopic parameter of vehicles will show different dynamic characteristics at different times.Even if with the same prediction method and accuracy requirement, the predictable multisteps will still vary at different times.erefore, the prediction error will increase if we use a fixedstep prediction model.We can, instead, find the value within a predictable multistep model and decrease the prediction error caused by fixed-step prediction; we can also enhance the prediction accuracy if we analyze the status characteristics of parameters and determine the dynamic predictable steps in advance.
e estimation of step number for microparameter prediction must find the maximum predictable steps when the prediction accuracy can reach a certain requirement.
is reduces the prediction time and enhances the accuracy if we can determine the predictable steps of speed and acceleration in a certain state.Some researchers suggest that the predictable steps are approximately equal to the maximum value of the inverse of exponential of Lyapunov when predicting the time series data based on chaos theory.However, if we find the prediction parameters based on fractal theory, the predictable steps might be different, and it is hard to reflect the influence of time series data characteristics of speed and acceleration.erefore, this paper analyzes the time series characteristic of speed and acceleration and extracts the corresponding indicators and quantifies the indexes.By analyzing the correlation of those indexes and predictable steps, we establish the estimation model.
Microscopic traffic parameter data have obvious dynamic, periodic, trending, nonlinear, and uncertain characteristics, and these characteristics have a certain relationship with their predictable steps.Time series analysis is a statistical method of dynamic data processing.It can not only reveal the evolutionary law of a phenomenon itself but also describe the inherent quantitative relationship between a phenomenon and other phenomena from a dynamic perspective law.
erefore, this paper selects the average development level, average growth, average development speed, volatility, trend, uncertainty, and other indicators to describe the characteristics of short-term microscopic traffic parameter data series.In order to avoid redundancy, six parameters including average development level, average growth, average development speed, volatility, trend, and uncertainty are finally determined through correlation analysis.
Average level of development reflects the general development level of speed and acceleration in the period of time.e formula is as follows: Average growth reflects the average growth of each positioning over a period of time.e formula is as follows: Average development speed reflects the average degree of change in the period of time.e formula is as follows: e above time series characteristics can reflect the change trend of whole time series to a certain extent, but the dynamic fluctuation and uncertainty of microscopic parameters are not reflected.erefore, this paper proposes the volatility index, trend index, and uncertainty index of time series data of speed and acceleration.
For volatility characteristics of speed and acceleration sequence data, we use variance and coefficient of variation to reflect the degree of data discretization, where the variance reflects the absolute volatility, while the coefficient of variation is reflected in the relative volatility, as follows: In the formula, B(t) represents volatility index; n represents the total length of the time series data; x(t i ) represents the data when time window is equal to t i ; and x(t) represents the mean of all the data in the time window.
e movement of the vehicle has a change process, so the microscopic parameters of the vehicle also show different trend characteristics, such as rising trend, decreasing trend, and stable trend.For the different trends, the number of 4 Journal of Advanced Transportation predictable steps may be different.In this paper, the trend characteristics of the microscopic parameters are quantified, and the relationship between the predictable steps and microscopic parameters is analyzed: In the formula, Q(t) represents trend index and k represents the gradient of the data sequence in slope calculation.
Uncertainty index refers to the randomness of the data; generally, we use information entropy to reflect the uncertainty of the data sequence.is paper uses fuzzy entropy index to reflect the uncertainty of microscopic parameters.If the calculated length has n uncertainty indexes of the time data sequence, the ambiguity of each microparameter value is calculated first.e formula is as follows: en, the calculated uncertainty indexes based on the ambiguity of each value are as follows: In the formula, n represents the last n time series samples and p i represents the ambiguity the n data value.

Analysis of Characteristic Indexes.
It is possible to predict the data sequence based on the fractal prediction theory; however, due to the timeliness of the actual data, the prediction steps are related to the basic data, and the predictable steps of other data sequences are different.is paper quantifies the volatility, trend and uncertainty, and other indexes of the basic data series.American expressway data are used as an example to calculate the index.e results are shown in Figure 3.
It can be seen from Figure 3 that the curves of the other five indicators are different from the average level of development.
e average level of development reflects the developmental trend of the value during a certain period of time.
e figure tells us that vehicles have been in the accelerated trend in this period of time.
is paper uses the fractal prediction model to predict the sample speed data in Figure 3 and calculates the maximum number of predictable steps at each time interval.e error is less than 10%, and finally we get the actual predictable steps of speed at all times.e number of steps is shown in Figure 4.In conjunction with Figures 3 and 4, it can be seen that the other five parameters are inversely proportional to the number of predictable steps in addition to the average level of development.
e increase of the volatility index, the uncertainty index, trend index (slope), and speed would result in the decrease of predictable steps.
Similarly, by analyzing the acceleration of the data, the relationship between the acceleration variation characteristic and the acceleration predictable step number can be acquired.e actual predictable step number of the sample data-acceleration is shown in Figure 5. e conclusion is similar to the speed analysis, but the predictable step number of the acceleration is relatively smaller.erefore, in order to improve the prediction accuracy and predict the efficiency, the acceleration prediction value can be calculated according to the speed prediction value in practical application.

Predictable Step Estimation Model of Microscopic
Parameters.After determining the microparameter time series characteristics, this paper studies how to use these indexes to predict microparameters and make dynamic estimations.is is a one-to-many prediction problem to predict the predictable number of steps (an index) of the speed and acceleration based on the chosen six quantified indicators.In addition, the volatility, uncertainty, and trend of time series data are inversely proportional to the predictable steps of the data.
Because the artificial neural network algorithm can deal with many-to-one and nonlinear problems, we use the BP (back propagation) neural network model to predict the dynamic predictable steps of the microscopic parameters.Based on the time series value of speed and acceleration, this paper calculates the average development level, average growth rate, average growth rate, trend index, volatility index, and uncertainty index of the sequence and then standardizes these indexes and uses them as input parameters of the neural network model.e predicted number of steps are used as the output parameter.After, training and tests are conducted for the predictable step estimation model of speed and acceleration.Finally, the predictable step number estimation model of speed and acceleration is obtained.
e study shows that a three-layer BP neural network can deal with any nonlinear continuous function (see [11]).In this paper, we use six parameters which are average level of development, average development speed, average growth rate, trend index, volatility index, and uncertainty index as the input variables, and for the output variable, we use the actual predictable steps.
en, we construct a three-layer neural network and use the sigmoid function as the hidden layer function to select the optimal hidden layer number with the lowest error.
e evaluation index of the predictable step estimation model is similar to that of the microparameter prediction model, which uses the average absolute percentage error MAPE, and the formula is shown in equation (10).
e process of the predictable step number of the microscopic parameter is roughly divided into three steps.
ese include the online estimation of predictable steps of the microscopic parameters and are shown as follows: (  6 Journal of Advanced Transportation data within the number of steps are predicted in an iterative manner by the fractal prediction model.

Analysis of the Model and Case Verification
6.1.Fractal Analysis of Microparameters.According to equation ( 6), this paper, respectively, analyzes the characteristic of speed and acceleration and finally gets the analysis result of speed and acceleration of R/S. e result is shown in Figure 6.
According to the analysis results of R/S, the H value of the fitted speed time series is 0.99, and the H value of the fitted acceleration time series is 0.74.us, the data satisfy the range of 0.5 < H ≤ 1, indicating that it has fractal characteristics, and the time series is persistent and can be predicted accordingly.At the same time, we can see that the speed of H is closer to 1, indicating that the prediction accuracy of speed will be higher.

Fractal Prediction Model Analysis of Microparameters.
e fractal prediction model of microscopic parameters is used to predict the sample data.We need to locate the nearest r times, compute the fractal dimension D of the data, sum up the sequence, obtain S(1, j), S(2, j) and other data columns, ask for the dimensionality D′ of these data columns, and select the sequence D ′ corresponding to S(i, j), which is closest to D. is paper finds the cumulative sequence of the predicted values after prediction and reduces the cumulative sequence to obtain the predicted data sequence.erefore, the nearest time length r must be determined before parameter prediction.In the selection of r, the accuracy of data prediction and the calculation load should be considered.
e value r of this example is determined as 6 in this paper.In this paper, time series position data with steps of 2 to 10 are used for experimental comparison.e prediction accuracy and the number of predictable steps are comprehensively compared, and it is found that it is better to choose r � 6.
erefore, this paper selects position data with a time length r value of 6.According to the nearest 6 location data, the prediction of the positioning data at the next time is realized.Based on fractal theory, this paper forecasts the sample data of American expressway, and the fractal dimension of the last 6 location data are calculated, the result of which is shown in Figure 7, and the 4 cumulative sequence curves of speed is shown in Figure 8.
As shown in Figure 7, the D value of speed data is 1.06, and Figure 8 shows that the dimension D′ of S(2, j) is 1.0689, which is closest to that of D, so we use the ln S(2, j) � 1.0689 ln j + 3.6518 function to predict S(2, j + 1).
en, the speed prediction data of the next positioning time are calculated by the formula S(1, j + 1) � S(2, j + 1) − S (2, j). e same step is used to obtain the prediction data of other positioning times.e speed and acceleration prediction results of the three samples are shown in Figure 9.
It can be seen from Figure 9 that the speed prediction accuracy is high, but when the speed is abrupt, the speed prediction accuracy is weak, and the prediction results have a certain lag.e predictive evaluation index of the data sample is shown in Table 2.
From Table 2, it can be seen that the acceleration data are relatively large; this is mainly due to the acceleration value itself, which is relatively small and the denominator of the error index value is small.erefore, some subtle fluctuations can easily lead to large errors.It can be seen from the table that when the speed prediction accuracy of the sample data is high, the average error is low.Compared with the acceleration, it is obvious that the speed is more convenient to predict.In the simulation experiment, there is a large error in the speed prediction.e main reason for this error is that the emergency braking behavior of the vehicle cannot be better predicted.In general, the fractal theory can be used to predict the microscopic data of vehicles well, and the speed prediction is more accurate than the acceleration prediction.erefore, when predicting the value of acceleration, the acceleration prediction value can be calculated according to the predicted speed value.
e average acceleration value of the vehicles between the two locating moments can be calculated by the speed prediction value of two successive positions.
is value can then be used as a prediction of acceleration, so the prediction error of acceleration can be reduced, and the prediction performance can be improved.

Analysis of Predictable Step Estimation Model for Microparameters.
is paper predicts the data of NGSIM trajectory based on the fractal theory and determines the practicable step length in each time series, as well as the actual prediction step number.We then obtain 280 sets of basic speed time series values and their corresponding actual predictable step numbers.e 280 sets of data are used to calculate the speed characteristics of the 280 indicators, including volatility, trend, uncertainty, and six other indicators.We standardize the data set by maximum and minimum values, and 210 sets of data are trained in the predictable step estimation model to obtain the network model of 6 × 7 × 1 (6 input indicators in the input layer, 7 hidden neurons in the hidden layer, and 1 output parameter).e model can estimate the predictable steps of the time series data and obtain the predictable steps number.
e remaining 70 sets of data are tested on the model.e results are shown in Figure 10.
From Figure 10, we can see that the trained neural network has better predictable steps for time series data.e absolute error is less than 15%, and the average absolute percentage error is 9%.It also can be seen from the above figure that when the number of predictable steps of the dataset is small, the accuracy of the predictable step estimation model is relatively higher, but the prediction accuracy is decreased when the number of predictable steps increases.

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Conclusion
In this paper, the fractal characteristics of microscopic parameters are analyzed, and the fractal prediction model of microscopic parameters is proposed based on feature analysis.In order to improve the prediction speed, this paper analyzes the time series data characteristics of microscopic parameters, including quantitative volatility, trend, uncertainty, and other indicators; then, the fractal multistep prediction model of microscopic parameters is obtained.A related paper "Analysis and recognition of highway lane-changing behavior characteristics based on GPS location data" has been published in March [12].is published paper acquires the lane-changing parameters by exploiting on-board GPS data.
en, the parameters' statistical distribution characteristics of vehicles that change lanes and their surrounding vehicles are analyzed, and the characterization parameters of lane-changing behaviors are simultaneously extracted.On these grounds, the lanechanging identi cation model is established, haled on the hidden Markov model (HMM).e purpose of the published article and this paper both aim to realize the identi cation and monitoring of the microscopic state of highway tra c ow.However, the prediction methods used are di erent.e prediction model of the published paper is based on the hidden Markov model (HMM), while the model in this paper is based on fractal multistep prediction model.
1) e training of BP neural network: because there are some differences in the overall driving characteristics of vehicles in different regions and different sections, we first target the specific region and depend on the historical vehicle microparameter operation data analysis and then extract the characteristics of indexes and train offline BP neural network estimation Journal of Advanced Transportation model.In the training of BP neural network, the output variable comes from microscopic tra c parameter prediction model based on fractal theory.Determine predictable steps by judging whether the prediction accuracy (MAPE and RMSE) of the microscopic tra c parameter prediction model based on fractal theory meets the requirements.(2) e extraction of characteristic indexes: this paper extracts the dynamic characteristic index of the microscopic parameter based on time t and its previous time k series data.(3) Based on the predictable step number estimation model and the fractal prediction model, the fractal multistep prediction model of microscopic parameters can be carried out.e number of steps in which the sequence parameters of the current time can be e ectively predicted is obtained by the predictable step number estimation model, and then the

Figure 3 :Figure 4 :Figure 5 :
Figure 3: e analysis indexes of speed characteristic: (a) the average level of development of time sequentially, (b) the average development of time sequentially, (c) the average growth of time sequentially, (d) the tendency of time sequentially, (e) the volatility index of time sequentially, and (f ) the uncertainty of time sequentially.

Figure 7 : 2 Figure 8 :Figure 6 :
Figure 7: e last 6 data analyses results of R/S graph: (a) acceleration of R/S and (b) speed of R/S.
Finally, according to the time series characteristics of speed and acceleration, we extract the corresponding indicators, quantify all kinds of indicators, analyze the correlation between the indicators and the number of predictable steps, and establish the predictable step estimation model of microscopic parameters.US highway data are used to verify the model, and the results show that the proposed fractal multistep prediction model can e ectively realize the multistep prediction of vehicle speed.e model is based on the dynamic calculation of predictable steps, which can achieve as long a prediction as possible, while avoiding large errors caused by xed-step prediction.

Figure 10 :
Figure 10: e test results of predictable step model.

Table 2 :
Index table of data prediction and evaluation.