The combination of linear and nonlinear methods is widely used in the prediction of time series data. This paper analyzes track irregularity time series data by using gray incidence degree models and methods of data transformation, trying to find the connotative relationship between the time series data. In this paper, GM
Track irregularity is a serious threat to the safety of train operation. Track irregularity data includes environmental variables (gauge, longitudinal level, cross level, alignment, and twist) and effective variables (vertical acceleration and horizontal acceleration). The developing and changing process of the track irregularity state is random, which cannot be defined by identified function. Generally, it can be researched with the combination of probability theory and analysis method within a certain range. Nowadays, most studies focus on the overall indicators which evaluate the changes of the track’s state, but a few studies focus on the changes of specific geometric parameters’ changes and the laws behind them. This is a basic difficulty.
Linear and nonlinear methods are two groups of models employed to estimate time series. DENG Julong [
In this paper, three aspects are studied on trends of track cross level state changes. First, it analyzes track irregularity time series data and tries to find the connotative relationships between time series data with the application of seven gray incidence degree theories; secondly, it predicts longterm track level changes at fixed measuring point; finally, it predicts changes of tracks over time at unit section in short term. This paper modifies and corrects the inadequacies in the GM
The idea of time series analysis has been applied in many areas of research, such as the relationship of following speed and spacing with driving time in driver’s safetyrelated approaching behavior [
Comparison of track cross level values at K550.00166 and K550.00191 mileage points.
It can be seen through Figure
In terms of the complicity of the relationships of time series curves, it is not easy to find a standard or a fixed formula to indicate the time series curve, but it can only give a complex evaluation on the changes and a developing tendency of the time series data. As a result, this paper analyzes and compares seven incidence degree algorithms. Certain relationships exist between track irregularity time series. Seven incidence degree [
Seven incidence degrees between cross level time series and reference cross level time series.
DID  AID  IAID  TID  SID  FODID  SODIG  

(1, 
0.9753  0.9774  0.5032  0.3758  0.9347  0.5811  0.1399 
(2, 
0.9779  0.9767  0.4867  0.3704  0.8977  0.5144  0.1874 
(3, 
0.6299  0.7450  0.5058  0.3897  0.8143  −0.3925  −0.1512 
(4, 
0.9707  0.9768  0.4604  0.3988  0.8718  0.5400  0.1737 
(5, 
0.9743  0.9768  0.4702  0.3790  0.8911  0.5150  0.1496 








The relationship of cross level data between adjacent hours is shown in Figure
It can be found from Figure
Scatter diagrams of detection data during the last tenday period of April and the first tenday period of May, 2008.
Track inspection data refers to the data obtained within a roughly fixed time interval (a half month), which is generated from geometry state detection along the mileage range of railway line. The time sequence of track geometry state changes with the following characteristics.
In the study of variation law of detection data, each detection data on a certain unit of section area is considered as a data unit. Data sequence consisted of data unit within a certain time frame is the object of study, forming a time series. Original time series data is described as follows:
In the formula,
Since each data unit is not a single data, but a data set of union section, rather than, therefore, it is necessary to transform processing in order to form data which can reflect the real characters of this section geometry state at
In the formula,
Changes of time series data at unit section.
In order to keep track status in good condition and to ensure operation safety, maintenance at regular intervals is needed as the track state changes. Only data from two maintenance operations can be seen as the objects of the study as well as time series data. It also means that this is a small data set within a short timespan. We need to find an effective forecasting method to realize our research goal even though historical data is limited.
As shown in Figure
Cyclical trend of track state condition change.
In this paper, track irregularity data by track inspection car in the experimentation is provided by State key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University. The cross level irregularity data is selected as the object of this research. The research selects the BeijingKowloon upline, the K550 + 000 to K550 + 075 mileage ranging from the second track inspection in late February 2008 to the second track inspection in late May 2009, a total of 31 inspection data as data object, each of which contains 300 cross level values and each data array contains 300 elements.
GM
Track cross level irregularity data
According to the analysis of the track cross level sequence of raw data, we find that
When
After transformation, let us solve differential equations
Then we can obtain the coefficients
Next, take the values of
With the application of the formula (
Comparison of actual value and prediction value at a fixed measuring point.
When gray model GM
Comparison of actual state and predictive state at unit section.
It can be found from Figures
Since the residuals are large, there will be a great inaccuracy in GM
Time series of the track geometry state changes has cyclical characteristics according to the analysis of the historical changing trend of cross level. We find that trigonometric function has obvious cyclical features. In this paper, trigonometric function is used to correct residuals of the prediction model. Here, the residual refers to the actual value minus the predicted value, that is,
In the formula,
With the principle of the minimum cumulative error of the fitted values and actual values, combined with the application of trigonometric wave mode matching method, we try to make sure that the posteriori error
Take
Combined with residual formula and the formula (
Let us predict track cross level state with the formula (
Comparison of actual value and revised predictive value at fixed measuring point.
When gray model GM
Comparison of actual state and prediction state at unit section after residual modification.
As can be seen from Figures
In gray forecasting, the prediction with good fitting and extrapolation leads to a smaller value
Comparison of model’s accuracy.
Test items  GM (1, 1)  GM (1, 1) after residual modification 


65%  43% 

86%  86% 
Through comparative analysis, the variance ratio of posteriori error of GM
Track cross level irregularity time series data is smooth and consistent with the characteristics of the stationary random sequence; so there is no need to eliminate the trend of the differential operator. Although there is no definite model in track state changes in the long run, the state change in a short period can still be considered as close to the linear model. In order to study the unit section of the overall level of state which changes over time, it is considered as onedimensional array data which contains 300 data at a select unit section. The track cross level irregularity time series data is
Then,
Autocovariance function refers to the random signal between the values of two different moments of the secondorder mixed central moments. Autocorrelation function depicts the incidence degree between adjacent variables of time sequence. The partial autocorrelation function was excluded from the impact of other intermediate variables; the two functions are closely related and can reflect the true incidence degree between two variables [
Values of autocorrelation function, autocovariance function, and partial incidence degree function.

0  1  2  3  4  5  6  7 


1  −0.5910  0.0835  0.0075  0.0040  −0.1106  0.3729  −0.4402 

20.7675  −12.2731  1.7341  0.1567  0.0821  −2.2967  7.7441  −9.1408 

1  −1.0356  −0.9201  −0.7259  −0.5022  −0.2565  0.1265  −0.0529 
It can be seen from Table
Through comprehensive analysis, the prediction model is defined as AR
We get
Thus, the AR
In the formula,
Taking estimated value on both sides of formula (
In formula, when
The predictive results of cross level irregularity data at late June 2009 and actual test data are shown in Figure
Comparison of predicted value and actual inspection data in late June 2009.
By contrasting the forecasted data with the actual inspection data, it can be found that the distribution characteristics of actual value and the predictive value can agree with each other well, and the data curves roughly coincide with each other.
Kalman filtering can be used to estimate the current state when the estimated state from the last time and the current state are known, needless to know historical information observations or estimates. In the absence of maintenance, changes of track geometry are closely related to the passing gross weight change; deviation of track geometry will be further from the standard value with the increase in gross weight; track geometry status is also affected by the impact of train speed. The higher the speed is, the greater force is exerted on the track and the greater influences on the track geometry status are. The track geometry (detection data), passing gross weight change, and train speed are used as the technical indicators for track state prediction, and the accumulation and analysis of historical data can be used for building track state prediction models.
With the application of Kalman filtering algorithm, in the track inspection data analysis and forecasting models,
In the formula,
Kalman filter model is applied to forecast the cross level status the next time when testing. The comparison of detection cross level value and the prediction value is shown in Figure
Comparison of detection value and the prediction value (ANN).
Artificial neural network (ANN) is widely used in function approximation, pattern recognition, and data compression [
In the formula,
Comparison of the detection value and the prediction value (Kalman filter model).
The specific error distribution of AR model, Kalman filtering model, is ANN model are shown in Table
Error distribution of forecasted data.
Models  Error range 





AR  Amount  44  104  82  70 
Percentage  15%  35%  27%  23%  
Kalman filtering  Amount  37  99  87  77 
Percentage  12%  33%  29%  26%  
ANN  Amount  36  65  99  100 
Percentage  12%  22%  33%  33% 
It can be seen from Table
After the comprehensive assessment of the incidence degrees of track irregularity between various indicators of factors, we find that when the associated values are higher, these correlated time sequences will normally have a higher degree of factors correlation or processes correlation. Meanwhile, the calculated results of incidence degree will be in a good agreement with the actual situation, which will provide a reliable basis for choosing modeling variables and analyzing factors. Improved GM
This research was supported by the National Natural Science Foundation of China (General Projects) (Grant no.: 61272029), National Key Technology R&D Program (Grant no.: 2009BAG12A10), China Railway Ministry Major Program (2008G017A), and State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (Contract no.: RCS2009ZT007).