Data grouping and modified initial condition in grey model improvement for short-term traffic flow forecasting

To improve the performance of the conventional grey model, emphasis should be based on the “new information prior using” principle. This paper presents detailed work on improving the precision of the conventional grey model by combining a data grouping technique with modification of initial condition to establish an optimized grey model. The data grouping technique and modification of initial condition methods have the advantage of adhering to the “new information prior using” principle. An empirical example of short-term traffic flow forecasting shows that the proposed optimized grey model, that is the modified initial condition grouped grey model, outperforms the existing models in both fitting and short-term forecasting. Moreover, the results demonstrate our claim that the distribution characteristic of the fitting error influences future short-term forecast accuracy. Now the proposed model can be of help in intelligent transportation systems for optimizing the use of existing infrastructure to enhance urban transportation systems in averting issues such as traffic congestion.


Introduction
Expansion of road capacity is a common but a "lengthyterm" approach to managing traffic congestion [1,2]. However, intelligent transportation systems (ITS) have solved and managed traffic problems by providing varied and advanced modes of transport [3][4][5]. For smart implementation of ITS, real-time traffic information is vital [4]. However, the provision of accurate real-time traffic information is an inherent problem in ITS [6]. Consequently, much effort to study on safety and efficiency of ITS has been intensified and devoted [3,7]. To improve the performance of ITS, agent-based evidential reasoning and "ZigBee" approaches have been proposed [6,7]. Furthermore, Amin et al. [8] introduced big data in shaping ITS. In this paper, we propose a combination of the data grouping technique (DGT) with modification of initial condition (MIC) to develop an optimized grey model (GM) which we refer to as the single-variable first-order modified initial condition grouped grey model and denote it as MICGGM (1,1). Where the first 1 stands for first-order and the second 1 stands for 1 variable under consideration in the grey differential equation. This optimized model can find application in advancing ITS. The Original Grey Model (OGM(1,1)), which we optimize in this paper, is well known and has been applied in modelling real-time events in other domains such as hepatitis B incidence and beef consumption forecasting [9,10].
Many intellectuals have investigated on enhancing the accuracy of the conventional GM(1,1) from various perspectives. For example, the function transformation technology, initial condition and background value modification have been considered to improve the accuracy of the OGM(1,1) [11][12][13][14][15][16]. Additionally, Thành [17] demonstrates that the application of the Fourier series in revising the grey model residual values provides more accuracy than the conventional model. From the grey system theory, to improve the precision of the grey model, emphasis should be based on the "new information prior using" principle [18]. Consequently, we note that most of the authors' efforts to improve the performance of the grey model have not concentrated on modification of the initial condition in adhering to the "new information prior using" principle. This has motivated us in proposing a combination of the DGT and MIC methods, which adheres to this principle, in improving the performance of the OGM(1,1). Therefore, this paper institutes and proves a new perspective of amalgamating the DGT and MIC in optimizing the accuracy of the OGM(1,1) [19,20].
The novelty of this paper is the proposed combining of the DGT with MIC in order to optimize the OGM(1,1). Getanda et al. [21] discussed the DGT and MIC in improving the OGM(1,1)'s accuracy but they never combined the two methods. This new strategy takes the advantage of the two methods in adhering to the "new information prior using" principle and in combination the accuracy of the proposed MICGGM(1,1) is generally good compared with existing models as shown later in this paper. And that is the crucial difference between the work of Getanda et al. [21] and the result of this paper. Thus, the proposed strategy is an extension of previous results [19,22]. Further, in this paper, we unveil the insight of the DGT as in how the grouping of raw data is done and how simulated data are superimposed to obtain overall fitted data. More importantly, we clearly show the adherence to the "new information prior using principle" by the DGT. Moreover, we analyze the relative fitting error and make judgment on its influence on future forecast results. Finally, we graphically demonstrate how the accuracy of the OGM(1,1) is improved from a lower accuracy range to an improved higher accuracy range by this new approach.

The grey system
The GM(1,1) is based on the grey system theory which was introduced in 1982 and has undergone several modifications being applied in numerous fields for analyzing, estimating, forecasting and modelling grey systems [9,10,[23][24][25][26]. The system extracts a governing relationship of a system and covers areas such as grey generating space, grey forecasting, grey control etc. [24,[26][27][28].

Grey generating operations and modelling algorithm
In grey modelling, the grey generating techniques develop a systematic series from a raw data series. In this paper, we present the raw data series as [9,26]: where m is the total number of data points. Accumulation of the series in (1) by: results in the following equation: which is the Accumulated Generating Operation (AGO). The AGO is a pre-processing technique which reduces the randomness of the actual data to improve data regularity and smoothness. From adjacent AGO neighbours, a Mean value Generating Operation (MGO) can be obtained as [24,26]: where z (1) (k) is the background value. The grey modelling algorithm has a unique characteristic of the first-order differential equation given as [26,29,30]: where X (1) (t) is a background grey value at time t, and a and b are the developing coefficient and grey input, respectively [9]. The parameters a and b are optimized and obtained as discussed in Section 2.2 [19,26]. The time response function of (5) is deduced as [29]: wherex (1) (r+1) is the prediction of the AGO and x (0) (1) of (6) is the initial condition which causes prediction error [18].

Estimating model parameters
The grey model parameter values a and b are such that the best-fit result minimizes the sum of squared errors or residuals which are the differences between the observed or experimental values and the corresponding fitted value given in the model. Thus, the ordinary leastsquare method best estimates these parameters which are calculated as [19,26,29]: where A is the data matrix and y is the measured vector and are given as: and (2) x (0) (3) . . .
And as long as the matrix A has a full rank and the inverse of A T A exists, then it is possible to compute "good" values of the parameters a and b that minimizes the sum of squared errors.

Proposed accuracy-improving methods
In this paper, we propose a combination of the DGT and MIC methods in boosting the accuracy of the OGM(1,1). Based on DGT, a Grouped Grey Model (GGM(1,1)) exists [22]. Furthermore, a modified initial condition grey model (MICGM(1,1)) based on MIC also exists [21]. Now in combination, the DGT and MIC methods result in an optimized Modified Initial Condition Grouped Grey Model referred to as MICGGM (1,1). Then the proposed model, MICGGM(1,1), is adopted for modelling short-term traffic flow forecasts and compared with the existing models for validation purposes. An additional existing model considered in this paper is the Modified Background Value Grey Model (MBVGM(1,1)) [23].

Data grouping technique
In this paper, we adopt the Strong Grouping (SG) technique in which the number of formed groups is given by [19,23]: where N is the number of groups, n is the total number of data used, and k is the number of data points in a group and it should be consistent throughout the grouping process. Figure 1 illustrates the SG in 4s (i.e. four data points per group) to form 19 groups as can be identified by 19 rectangles. Because of the dropping of old data and adding of new data in the grouping process, the SG technique has the advantage of adhering to the "new information prior using" principle [20,23]. The procedure of grouped grey modelling entails the repeated application of the conventional grey model on each group of data and averaging the predicted values at points of overlap [21]. Thus, grouped grey modelling inherently modifies the initial condition of (6) and this promotes accuracy improvement.

Modification of the initial condition (MIC)
As mentioned earlier, the initial condition of (6) is a cause of the precision error of the conventional grey model [18]. Thus, modification of this initial condition to enhance the model's accuracy is important and it is achieved as follows. As in Getanda et al. [20,21], MIC is adopted and by applying IAGO onx (1) (r) of (8), the restored (predicted) value of the raw data is obtained as: where Cis the optimized initial condition given as: Now by incorporation of this MIC into the OGM(1,1), an optimized model is established which has the advantage of improved accuracy compared with the OGM(1,1).
Thus, the process of time series grey prediction based on MIC can be outlined in the following four steps: (a) Calculation of the background value from the AGO sequence of (3) by (4), (b) Computation of the developing coefficient and grey input parameters by (9), (c) Computation of the optimized value of C by (14) and finally, (d) Computation of the restored (predicted) values by (13).

The proposed MICGGM(1,1)
Now to improve the OGM(1,1)'s accuracy in shortterm traffic flow forecast, we combine the DGT and the MIC methods in conventional grey modelling. The combination process involves the incorporation of MIC in each formed group data set. In other words, the raw time series data of Figure 2 is grouped before fitting and forecasting and the four-step process outlined above is adopted in the simulation processes. As illustrated by Figure 1, the DGT drops an old data point and adds a new data point in the formation of a  new group. This process keeps on modifying the initial condition of each formed group. Overall, the two methods have the advantage of modifying the initial condition as a requirement by the "new information prior using" principle, and in combination, we establish the proposed model denoted as MICGGM(1,1). Moreover, MICGGM(1,1) can achieve high accuracy compared with the existing models as demonstrated in Sections 5-7 of this paper. With this high accuracy, the MICGGM(1,1) can be used in advancing ITS by optimizing the use of existing infrastructure to enhance urban transportation systems in averting issues such as traffic congestion. The proposed model can be applied in both univariate and multivariate data prediction capturing the non-linear features of the traffic data, unlike other statistical methods such as the Autoregressive Integrated Moving Average (ARIMA) model which is univariate and does not support seasonal data. However, Seasonal Autoregressive Integrated Moving Average, SARIMA or Seasonal ARIMA, is an extension of ARIMA that explicitly supports univariate time series data with a seasonal component. Neural networks such as the recurrent neural networks have a short-term memory and this "forgetfulness" makes model training more difficult and time-consuming. With the proposed model, no data gets lost even though the modification of the initial condition may literally mean loss of information, but this is compensated by the adoption of the "new information prior using" principle. Generally, grey models require small amount of data to predict a system. Other models such as the ARIMA model and the long short-term memory neural network require large amount of historical traffic data to achieve shortterm traffic flow forecasting [31]. Further, the predictive effect of the ARIMA model depends on its parameters and orders. Thus, the proposed model is more advantageous in time-series data modelling and forecasting.

The origin of the data set
The two-hour vehicle traffic data set used in this paper was obtained from Ref. [19] as simulated in MITRAM [32,33]. The data are graphically presented in Figure 2. The number of vehicles is seen to have a trending pattern from six o'clock in the morning to the peak hour at around 7:20 AM. This is the morning peak when the traffic jam is high. Thereafter, the flow of traffic decreases to a lower value at around eight o'clock AM. These types of data are suitable for this research as our objective is to forecast by the proposed model for the purpose of its incorporation in ITS to avert traffic congestion. Thus, it is desirable that we subject our new model to such traffic congestion data.

Training and validation data sets
The original traffic flow data set of Figure 2 was subdivided into training and validation data sets as presented in Figure 3 for estimating and evaluating the proposed grey models, respectively [34]. The goal is to develop a trained (fitted) model for good generalization to new, unknown data. Then the fitted model is evaluated using "new" data from the held-out dataset (validation dataset) to estimate the model's accuracy in classifying the new data. The reason for sub-dividing the original data set is to reduce the risk of issues such as overfitting and, therefore, the data in the validation dataset are not used to train the models. Thus, in this paper, we employed the classical hold-out method.

Empirical application review
For analysis of the performance of the proposed grey model, we simulated a numerical example in MATLAB R2014a. The time series data of Figure 2 was subdivided into two parts as illustrated in Figure 3. Data from 6:00 AM to 7:45 AM are used for training the grey models, whilst data from 7:50 AM to 8:00 AM are used for testing the models.

Training the grey models
Based on the grey generating operations, we trained the OGM(1,1), MBVGM(1,1) and the MICGM(1,1) and tabulated their simulation values in Table 1. Figure 4 shows the plots for the real, simulated and residual data curves. The residual curve indicates the difference between the simulated and actual values.
In training for the proposed model, we adopt the DGT and MIC in the simulation as follows. First, we group the first 22 data points shown in Figure 2 into 19 groups of 4s based on (12) and Figure 1. These groups   are formed and tabulated as in Table 2 where DP stands for the data point. Second, we introduce MIC in the OGM procedure [9] and apply it to each group to obtain fitted data (FD) for each group as tabulated in Table 3. Note that each group will have unique values of the parameters a and b [19] which have not been provided in this paper. Consequently, the corresponding time functions are different. Also note that in Table 3, the FD for groups 6-14 is not shown because of wanting to reduce the size of Table 3.
Third, the overall simulation data sequence is obtained by superimposing the group simulation data at points of overlaps and this overall (final) sequence is as indicated in Table 3. For instance, groups 1 and 2 overlap once at data point 2 and, thus, the final FD is  Table 1.
Lastly, this proposed model, MICGGM(1,1), resul ted in simulation values tabulated in Table 1 and plotted in Figure 4. A close observation of the error curves in Figure 4 reveals that the proposed MICGGM(1,1) is the most accurate model in this fitting. In Sections 6 and 7 of this paper, we further evaluate the fitting errors and accuracies of the models as shown in Figures 7-9.

Validating the fitted grey models by short-term forecast
For forecasting in the near term (6) is extrapolated three points into the future. In testing the grey models, these short-term forecasting results are recorded in Table 1 and their accuracies are shown in Figure 9. Plotted in Figure 5 are the real, simulated and residual curves for the three grey models. The last three time sample points in Figure 5 (i.e. at 7:50, 7:55 and 8:00 AM) are the extrapolated points of focus. Hence to clearly show this short-term forecast, Figure 6 is an extracted part of Figure 5.

Error analysis
Our goal is to find the model that has the best performance on new data and the simplest approach is to evaluate the time-series fitting error involved in the models. The error of any particular model in forecasting future trends of a system is much influenced by its error in fitting the past data of that particular system. To validate this claim, we examine the distribution characteristic of the relative fitting errors over the full range of the data set under consideration. Table 4 is a record of OGM(1,1)'s, MBVGM(1,1)'s, MICGM(1,1)'s and MICGGM(1,1)'s relative fitting errors. MICGGM(1,1)'s relative errors are smaller. However, its large error part concentrates at the centre (middle) of the data set, whereas the errors of the OGM(1,1) are larger and distributed over the full range of the data set (Figure 7). At either ends of the data    set range, the accuracy of the proposed MICGGM(1,1) is high meaning that it can forecast the future trend of a system with good accuracy. For the OGM(1,1), MBVGM(1,1) and MICGM(1,1), the Lager error trend is likely to distribute into the future forecasting of a system and these models can forecast the future with lower accuracy. The role played by the DGT is to concentrate the fitting errors in the middle of the time series rather than spreading it along the entire data set.
The fitting accuracy at the middle is lower, whereas at the extreme ends, it is good and this is advantageous in forecasting the system behaviour at the ends. This is why the MICGGM(1,1) has a high fitting and good/reasonable forecasting accuracies compared with OGM(1,1), MBVGM(1,1) and MICGM (1,1). Similarly, we considered the relative fitting errors of the MICGM(1,1). Observe Figure 7 and note that MICGM(1,1), MBVGM(1,1) and OGM(1,1) have similar error distribution characteristics. Thus, they have almost the same fitting accuracies, see Figure  8. Although MICGM(1,1) and MBVGM(1,1) are improved grey models, their accuracies are much lower compared with that of the proposed model.

Accuracy improvement analysis
Measures of model performance, namely root mean square percentage error (RMSPE), mean absolute percentage deviation (MAPD), root mean square error (RMSE) and the mean absolute error (MAE) are adopted to evaluate the accuracy improvement of the proposed grey model [19].
In MATLAB R2014a, we computed the simulation errors, and in Excel, we plotted the fitting accuracies of each model as shown in Figure 8. Similarly, we plotted the short-term forecasting accuracies in Figure 9. It is evident from Figure 8 that the OGM(1,1) has lower fitting accuracy compared with MICGGM(1,1).  Note that the proposed MICGGM(1,1) has the highest fitting accuracy of 94.6987%. Thus, the OGM(1,1)'s accuracy of 77.802% has been improved to an accuracy of 94.6987% by MICGGM(1,1) as indicated by MAPD. See Figure 8. On the other hand, MICGM(1,1) and MBVGM(1,1) have fitting accuracies of 77.9812% and 77.7807 respectively.
Short-term forecast accuracy analysis is shown in Figure 9 in which the OGM(1,1) has lower shortterm forecast accuracy. On the other hand, the proposed MICGGM(1,1) has good forecast accuracy at 85.6143%, as computed by MAPD. Worthy is the improvement of the OGM(1,1)'s fitting and short-term forecast accuracies as shown in Figures 8 and 9, respectively. The criteria of MAPD and RMSPE are as tabulated in Table 5 [19,35,36] and it shows that the fitting accuracy of the proposed MICGGM(1,1) is high, whereas its short-term forecasting accuracy is good.

Computation time
We did the simulations in MATLAB R2014a environment and to measure the performance of our MATLAB code, we used the tic and toc performance functions to time how long the code takes to run. We ran the code multiple times and averaged and recorded the time in seconds as tabulated in Table 6. The fitting and forecasting time for the MICGGM(1,1) is longer compared with those of the OGM(1,1). This is because of the increased number of statements to be executed in the proposed model's MATLAB code. Nevertheless, the MICGGM(1,1) is more accurate and the increased computation time is insignificant. Moreover, timing is merely a performance metric, not a system correctness criterion. Thus, the proposed model can perform well in automated systems.

Conclusions and future focus
In this paper, we have analyzed and presented a new approach of amalgamating the DGT and MIC in optimizing the conventional GM(1,1) to establish a newly optimized model denoted as MICGGM(1,1). Further, with a numerical example, we did a performance analysis and the results have shown that the newly improved model, i.e., the MICGGM(1,1) has high fitting accuracy and good short-term forecasting accuracy as compared with the existing models. Moreover, we have shown that when large errors tend to concentrate at the middle of the data set, the near future can forecast with high accuracy meaning the new model has the best performance on new data. Thus, the distribution characteristic of the fitting error influences the future forecast accuracy of the model. We note that the proposed MICGGM(1,1) combines the advantage of DGT and MIC by adhering to the "new information prior using" principle in boosting the accuracy of the OGM(1,1). The proposed optimized GM can enhance the design of useful ITS's architecture through grey control and this is of help to transport management decisionmakers. Thus, it enriches and widens the application scope of the grey model. For instance, it can be useful in real-time proactive traffic control for large-scale adaptive traffic networks (such as those of city or mega-city scale) and this calls for multiple measurement points of data to be processed.
Our future work is to continue refining and developing the GM by combining the DGT with Fourier Series to ascertain their capability in enhancing the OGM(1,1)'s forecasting accuracy.