CN113177259A - Extreme value theory-based rail irregularity peak value overrun management method - Google Patents

Extreme value theory-based rail irregularity peak value overrun management method Download PDF

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CN113177259A
CN113177259A CN202110402591.5A CN202110402591A CN113177259A CN 113177259 A CN113177259 A CN 113177259A CN 202110402591 A CN202110402591 A CN 202110402591A CN 113177259 A CN113177259 A CN 113177259A
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何庆
汪健辉
李晨钟
利璐
王平
柳恒
高文杰
李志强
黄波
王青元
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Abstract

The invention relates to the technical field of high-speed rail tracks, in particular to a track irregularity peak value overrun management method based on an extreme value theory, which comprises the following steps: firstly, establishing a peak value threshold value passing method model; secondly, parameter estimation; thirdly, drawing up management indexes of dynamic irregularity peak values and evaluating the rationality; the invention adopts a peak value threshold value passing method model in an extreme value theory, selects a threshold value according to a certain rule, models the data of which the historical detection value exceeds the threshold value, applies generalized pareto distribution research and determines the distribution function of a new sample sequence, and enables the existing peak values to have different standards and the likeThe allowable deviation management value of a stage is regarded as a loss value VaRPThe overrun probability is deduced reversely, so that the rationality of the existing peak value management standard is evaluated.

Description

Extreme value theory-based rail irregularity peak value overrun management method
Technical Field
The invention relates to the technical field of high-speed rails, in particular to a track irregularity peak overrun management method based on an extreme value theory.
Background
In recent years, rapid development of high-speed railways has put new demands on track smoothness. The smoothness of the track has important influences on driving safety, riding comfort, track service life, environmental noise and the like. The rail climbing and derailing of the train can be caused by the fact that the peak value of the track irregularity exceeds the limit, the running safety is greatly damaged, and the method is a common basic evaluation index for the existing line maintenance and repair of various countries.
At present, various scholars at home and abroad research the management value standard of the rail irregularity. For example, by analyzing simulation models under various working conditions, proposed management values of track irregularity of the Qin Shen special line test section are provided; establishing a whole vehicle model, solving a motion equation, and exploring and giving a height irregularity threshold value of a line with the speed per hour of 300 km/h; based on the idea of data classification, utilizing simulation data to obtain the boundary of the non-smooth peak security domain under different speed grades. The above research has performed detailed analysis on the control factors affecting the track smoothness, but it is not uncommon to make reasonable researches on the railway peak over-limit Risk Value (VAR) and peak management Value of the railway maintenance and repair basic work. VAR refers to the maximum possible loss at a certain confidence level and some period in the future, and is widely used in financial risk management and is gradually expanding towards other industries. The method is applied to railway and is used for evaluating the value of the risk of the track irregularity peak value overrun, and refers to the maximum peak value of a line which can appear in a future period under a given significant level.
Disclosure of Invention
The invention provides a method for managing the out-of-limit of the track irregularity peak value based on an extreme value theory, which can overcome some or some defects in the prior art.
The invention discloses an extreme value theory-based rail irregularity peak value overrun management method, which comprises the following steps of:
firstly, establishing a peak value threshold value passing method model;
the track inspection vehicle measured value sequence X, u is a threshold value, and the number of samples exceeding the threshold value u is nuF (x) is a distribution function; conditional distribution function Fu(y) is:
Fu(y)＝P(X-u≤y|X＞u)y≥0；
the conditional probability formula yields:
Figure BDA0003020987520000021
deformation can result in:
F(x)＝Fu(y)(1-F(u))+F(u)X≥u；
for a sufficiently large threshold u, the overrun yiApproximately obeying to a generalized pareto distribution Gξ,σ(y) then
Figure BDA0003020987520000022
In the formula: σ represents a scale parameter; xi represents the shape parameter, when xi is more than or equal to 0, y is more than 0 and infinity), when xi is less than 0,
Figure BDA0003020987520000023
probability density function g of the over-limit valueξ,σ(y) is：
Figure BDA0003020987520000024
Therefore, for the measured value sequence { x of the rail inspection vehicle1,x2,…,xn}, the log-likelihood function L (ξ, σ | y) may be expressed as:
Figure BDA0003020987520000025
secondly, parameter estimation;
determining a threshold u from the overrun expected map;
after u is determined, maximum likelihood estimation is carried out by utilizing the rail detection vehicle measured value sequence X according to a number likelihood function L (xi, sigma | y), and then estimated values of a scale parameter sigma and a shape parameter xi can be obtained;
thirdly, drawing up management indexes of dynamic irregularity peak values and evaluating the rationality;
after the threshold is determined, use (n-n)u) As an empirical estimate of F (u), the data distribution function for the threshold u can be found as:
Figure BDA0003020987520000031
after the estimated values of the parameters in the formula are obtained, the distribution function under the condition of passing a threshold u can be determined; and then according to the management values of different levels in the existing peak value management, the corresponding overrun occurrence probability can be obtained by utilizing the inverse function of the distribution function, so that the rationality evaluation of the existing dynamic management value is realized.
Preferably, the peak value threshold method model is analyzed at X>u, the overrun value yi＝X-u(i＝1,2,…,nu) Conditional distribution function Fu(y)。
Preferably, in the step two, the specific determination method of the threshold u is as follows: when the generalized pareto distribution has a shape parameter xi <1, the overrun expectation function e (m) is a linear function and can be expressed as:
Figure BDA0003020987520000032
an average remainder function of the form:
Figure BDA0003020987520000033
wherein n isu＝max{i|yi}；
If the sample has the thick tail distribution characteristic, the sample can be regarded as the generalized pareto distribution with xi <1, and the expected residual value is a linear function of u; therefore, the overrun expected graph obtained according to the measured rail inspection vehicle data can be selected by selecting the sufficiently large critical value u0So that when X is not less than u0Time e (x) is an approximate linear function used as a basis for determining the threshold value; meanwhile, the number n of samples exceeding the threshold value in the track inspection vehicle measured value sequence X can be obtainedu；
Preferably, in step three, let xmFor the management values of the rail inspection vehicle measured values under different levels when x is more than xmThe track dynamic irregularity influences comfort level, temporary repair or speed limit, and the probability is as follows:
Figure BDA0003020987520000041
determining the occurrence probability P of different grades according to the importance of indexes of different grades of dynamic irregularityα(ii) a Then for a given confidence level P, a value VaR may be lostPIs the P quantile of the loss distribution, i.e. xmIs estimated value of
Figure BDA0003020987520000042
Comprises the following steps:
Figure BDA0003020987520000043
aiming at the characteristics of low frequency, high loss and sparse data of the Over-limit of the rail irregularity peak value, the invention adopts a peak Over Threshold method (Peaks Over Threshold) model in the extremum theory, selects a Threshold value according to a certain rule, models the data of which the historical detection value exceeds the Threshold value, researches and determines the Distribution function of a new sample sequence by applying Generalized Pareto Distribution (Generalized Pareto Distribution), and takes the allowable deviation management values of the existing peak value standard in different grades as loss values VaRPThe overrun probability P is deduced reversely, so as to evaluate the rationality of the existing peak value management standard. The extreme value theory method based on the peak value threshold crossing method model has more advantages in the aspect of describing tail features of data distribution, focuses on paying attention to the distribution of the overrun data sequence, fully considers the possibility of occurrence of all large measured values, can reflect engineering practice more objectively, and is a relatively accurate quantile analysis method and a prediction tool.
Drawings
Fig. 1 is a flowchart of a method for managing an out-of-limit track irregularity peak based on an extremum theory in embodiment 1;
FIG. 2 is a histogram of probability distribution of high and low measured values in example 2;
FIG. 3 is a diagram of the expected overrun of the high and low measured value sequences in example 2;
FIG. 4 is a graph of cumulative distribution functions at different thresholds in example 2;
FIG. 5 is a graph of the risk value function at different thresholds in example 2;
fig. 6 is a time-series box diagram of measured acceleration values in example 2.
Detailed Description
For a further understanding of the invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples. It is to be understood that the examples are illustrative of the invention and not limiting.
Example 1
As shown in fig. 1, the present embodiment provides a method for managing an out-of-limit track irregularity peak based on an extremum theory, which includes the following steps:
firstly, establishing a peak value threshold value passing method model;
different from a general model, the peak value over-threshold model mainly studies data samples exceeding a set threshold value and observes the numerical value size and the overall distribution characteristics of the sequence. For the measured value sequence X of the rail inspection vehicle, assuming u as a threshold value, the number of samples exceeding the threshold value u is nuAnd F (x) is a distribution function. The peak value threshold method model analyzes that the value is in X>u, the overrun value yiConditional distribution function F of X-u (i ═ 1,2, …, nu)u(y) is:
Fu(y)＝P(X-u≤y|X＞u)y≥0 (1)
the conditional probability formula yields:
Figure BDA0003020987520000051
by transforming equation (2) we can obtain:
F(x)＝Fu(y)(1-F(u))+F(u)X≥u (3)
the Balkema-de Haan theorem indicates that for a sufficiently large threshold u, the overrun yiApproximately obeying to a generalized pareto distribution Gξ,σ(y) then
Figure BDA0003020987520000052
In the formula: σ represents a scale parameter; xi represents the shape parameter, when xi is more than or equal to 0, y belongs to [0, ∞), when xi<At the time of 0, the number of the first,
Figure BDA0003020987520000061
probability density function g of the over-limit valueξ,σ(y) is:
Figure BDA0003020987520000062
thus, for a given series of measured values { x } for a track inspection vehicle1,x2,…,xn}, the log-likelihood function L (ξ, σ | y) may be expressed as:
Figure BDA0003020987520000063
secondly, parameter estimation;
from the above formula, it can be known that the correct estimation of the threshold u, the scale parameter σ and the shape parameter ξ is the key to establishing the peak value over-thresholding model. According to repeated tests, the selection of the threshold u is directly related to whether the scale parameter and the shape parameter can be accurately estimated. An excessively large threshold value may result in too few overrun samples, affecting the estimation of the distribution function; if the threshold is too small, the convergence of the overrun data cannot be guaranteed, so that the deviation of parameter estimation is too large. Based on the characteristics of the measured value sequence of the rail inspection vehicle, the threshold value u is determined according to the overrun expected map.
When the generalized pareto distribution has a shape parameter ξ <1, its overrun expectation function e (m) is a linear function, which can be expressed as:
Figure BDA0003020987520000064
an average remainder function of the form:
Figure BDA0003020987520000065
wherein n isu＝max{i|yi}。
As can be seen from the equation (7), if the sample has the thick tail distribution characteristic, it can be regarded as being subjected to xi<1, the expected residual value is a linear function of u. Therefore, the overrun expected graph obtained according to the measured rail inspection vehicle data can be selected by selecting the sufficiently large critical value u0So that when X is not less than u0Time e (x) is an approximately linear function as a basis for determining the threshold. Meanwhile, the number n of samples exceeding the threshold value in the track inspection vehicle measured value sequence X can be obtainedu。
And after u is determined, performing maximum likelihood estimation according to the formula (6) by using the rail inspection vehicle measured value sequence X to obtain estimated values of the scale parameter sigma and the shape parameter xi.
Thirdly, drawing up management indexes of dynamic irregularity peak values and evaluating the rationality;
after the threshold is determined, use (n-n)u) As an empirical estimate of F (u), substituting equation (4) for equation (3) yields a data distribution function for threshold u as:
Figure BDA0003020987520000071
after the estimated values of the parameters in equation (9) are obtained according to the above method, the distribution function under the condition of passing the threshold u can be determined. And then according to the management values of different levels in the existing peak value management, the corresponding overrun occurrence probability can be obtained by utilizing the inverse function of the distribution function, so that the rationality evaluation of the existing dynamic management value is realized.
Let xmFor the management values of the measured values of the rail inspection vehicle under different levels, when x>xmThe track dynamic irregularity influences comfort level, temporary repair or speed limit, and the probability is as follows:
Figure BDA0003020987520000072
determining the occurrence probability P of different grades according to the importance of indexes of different grades of dynamic irregularityα(i.e., α), generally 1% to 5%. Then for a given confidence level P, a value VaR may be lostPIs the P quantile of the loss distribution, i.e. xmIs estimated value of
Figure BDA0003020987520000073
Comprises the following steps:
Figure BDA0003020987520000074
example 2
In the embodiment, a passenger dedicated line running at a speed of 300km/h is taken as an example, the measured data of the rail inspection vehicle of the line in the uplink section of 280km and 60 times in 2016-2019 are selected for analysis, and a mileage error correction model is utilized for preprocessing.
Thick tail detection
Taking the high-low half peak value (absolute value) in certain detection data as an example, a probability distribution histogram is drawn. As can be seen from fig. 2, the line peak-to-peak distribution has obvious non-normality. The distribution of the high and low peak values of the line can be verified to have relatively obvious thick tail through the inspection of an exponential quantile method, which shows that the probability of the high and low peak values occurring in the actual operation of the line is higher than that of the normal distribution. Therefore, a peak value threshold value method model is established through an extreme value theory to calculate the risk probability, and key analysis is carried out on thick tail distribution.
Threshold discussion and parameter estimation
Taking the high-low-half peak value (absolute value) as an example, the sample average remainder function is calculated according to equation (8), and an overrun expected graph is drawn (see fig. 3). In general, a suitable threshold u can divide about ten percent of the data into overrun samples, otherwise the characteristics of the sequence distribution cannot be grasped. After multiple tests, 7km is selected as an interval to be reasonably divided, and 4800 rail inspection vehicle height-actual measurement half-peak value sequences can be obtained. As can be seen from fig. 3, when the threshold u is around 3mm, the residual function e (u) has a smaller variation amplitude as an approximately linear function and a slope tends to increase in a forward direction, which indicates that the overrun data is thicker than the tail of a normal distribution, and is suitable for the pareto model to perform distribution fitting of a thick tail to describe the characteristics of the tail. At this time, only a small number of high/low measured value points are distributed at the tail of fig. 2, and the convergence rate is slow, which conforms to the characteristics of pareto distribution. According to McMeil (1997), the observed value can be initially determined to be the desired threshold u using a residual function. At this time, the number n of samples with the over-threshold u of 3mmu387, and a percentage of about 8%. It can be inferred that the threshold value is 3. It also allows the left high-low peak i level management value of the line to be properly relaxed, taking u-4, u-5 and comparing with u-3 (see fig. 4, fig. 5). Respectively calculating scale parameter and shape parameter estimation values under three different threshold values and corresponding cumulative distribution function and risk price by utilizing maximum likelihood estimationThe values are shown in Table 1.
TABLE 1 cumulative distribution function F (x) and VAR at different thresholdsP
Figure BDA0003020987520000081
Figure BDA0003020987520000091
As can be seen from fig. 4, the cumulative distribution function f (x) with the threshold u-3 is more than u-4 and u-5 contains more threshold probability of overrun, so that the estimated parameters are more accurate; as can be seen from FIG. 5, although the risk values of the same significance level are different at low significance levels (P < 90%), the risk values of the same significance level and different thresholds are very similar when the significance level is higher (P ≧ 90%). Therefore, from a strict point of view, it is scientifically reasonable to choose the threshold u as 3 (mm). For the index of dynamic irregularity level, the management value of the I level in the specification is 5, which is larger than other indexes of the same level. It was found that it is more reasonable to take the level threshold of 4 after the same analysis as above.
Peak value management value drawing up and evaluation
After the threshold value is determined, when the deviation grade of the high-low half peak value is grade I (frequently maintained) overrun probability is 5%, the grade I management value index obtained by using generalized pareto distribution on an overrun sample is 4.4mm, and is close to 4mm of the dynamic quality allowable deviation management value of a line track of 250 (including) -350 km/h in the specification. The combination formula (11) can reversely conclude that the probability of the line appearing in the high-low peak value deviation of grade I is 5.85%, the probability of appearing in the grade II (comfort level) is 3.14%, the probability of appearing in the grade III (temporary repair) is 1.72%, and the probability of appearing in the grade IV (speed limit (160km/h)) is 0.55%. The same treatment is carried out on the other indexes, and respective threshold values, scale parameters, shape parameters and overrun probabilities under different levels can be respectively obtained and are shown in tables 2 and 3.
TABLE 2 threshold values and parameter estimation values for different indexes
Figure BDA0003020987520000092
TABLE 3 over-limit probability of different index peak values of a certain operation speed per hour 300km/h line
Figure BDA0003020987520000101
The two tables can be used for obtaining the dynamic quality allowable deviation management value in the rail direction for the passenger dedicated line, the dynamic quality allowable deviation management value is relatively loose, the maintenance is needed frequently when the probability exceeds 9 percent, the speed limit is needed when the probability is about 5 percent, and the appropriate improvement standard according to the specification can be considered; the management values of the peak values of the grade II and above of the track gauge are relatively strict and have the probability of less than 1 percent, and the proper relaxation of the standard according to the specification can be considered; other index levels, distortion and peak value management values of height are relatively reasonable, and can be properly adjusted according to actual conditions on site.
Acceleration analysis
In "maintenance rules (trial) for ballastless track lines of high-speed railway", peak management also includes classification of vertical and lateral acceleration of the vehicle body, as shown in table 4. However, according to the actual on-site maintenance and repair investigation, the acceleration detection value is usually smaller than the I-level standard, and the purpose of maintenance cannot be achieved. Therefore, the present document performs box plot analysis on the existing vertical and horizontal field acceleration detection values according to the time series, and the result is shown in fig. 6. As can be seen from the figure, the range of the vertical acceleration is within-0.04/g, the range of the lateral acceleration is within-0.03/g, and no detection value exceeds the I-grade standard in the specification. This indicates that the acceleration index in peak management does not have a very good limiting effect during on-site maintenance and repair work. The peak value over-threshold method model based on the extreme value theory has a specific application range and is only suitable for describing and analyzing the generalized pareto distribution characteristics of over-threshold data, and the distribution type of the data sequence population is not required to be modeled.
TABLE 4250 (exclusive of) -350 km/h line track dynamic quality tolerance management values
Figure BDA0003020987520000102
Since the measured acceleration values of the line at different detection times are not in an overrun condition, the model is not suitable for the rationality evaluation of the existing acceleration management value of the line. Therefore, the acceleration index in the existing peak management needs to be properly normalized according to the actual situation of the site so as to reach the actual maintenance reference standard.
Conclusion
In the embodiment, the reasonability of the dynamic irregularity peak value management value of the track is researched by means of a generalized pareto distribution function based on a peak value threshold method model, and verification analysis is performed by combining specific measured data of the track inspection vehicle. The following conclusion is a summary of the advantages and disadvantages of the model and the rationality of the existing peak management indexes:
(1) the planning and evaluation of the rail dynamic irregularity peak management value are essentially considered extreme events. The peak value over-threshold method model only analyzes the generalized pareto distribution characteristics of over-threshold samples by adopting the division characteristics of the threshold values, and calculates the VaR of different grades according to the existing specificationsPThe over-limit probability of (2) realizes the rationality evaluation of the peak value management standard. The method fully considers the possible large measured value of the track dynamic irregularity, more comprehensively reflects the distribution characteristics of the measured data, and the obtained overrun probability can more objectively reflect the engineering practice.
(2) And the following data can be obtained by analyzing the measured data: the management value of the dynamic mass tolerance deviation of the vertical and transverse acceleration of the linear rail direction and the car body is looser and is about 50 percent higher than the actually measured peak value, and the appropriate standard improvement according to the specification and the actual situation on site can be considered; the rail direction management value is relatively loose, about 10 percent of actually measured peak value is overrun, wherein, the overrun probability of the IV-level speed limit standard is about 5 percent, which does not meet the actual line operation requirement, and the allowable deviation management value is properly improved according to the actual situation, thereby achieving the purpose of reducing the overrun probability; the management values of the peak values of the track gauge II and above are relatively strict, the overrun probabilities are less than 1%, and the proper relaxation standard can be considered; the peak management values of other index levels, heights and triangular pits are relatively reasonable and can be properly adjusted according to the actual situation on site.
(3) In the embodiment, for the purpose of simplifying analysis, the time dimension of the threshold is not considered, and only the numerical value is considered; in addition, it is empirical to assess the rationality of the management values according to the probability of occurrence of an overrun. These two problems are the problems that need to be intensively studied in the future in order to describe the reasonableness of the peak management value more objectively.
The present invention and its embodiments have been described above schematically, without limitation, and what is shown in the drawings is only one of the embodiments of the present invention, and the actual structure is not limited thereto. Therefore, if the person skilled in the art receives the teaching, without departing from the spirit of the invention, the person skilled in the art shall not inventively design the similar structural modes and embodiments to the technical solution, but shall fall within the scope of the invention.

Claims (4)

1. The extreme value theory-based rail irregularity peak value overrun management method is characterized by comprising the following steps of: the method comprises the following steps:
firstly, establishing a peak value threshold value passing method model;
the track inspection vehicle measured value sequence X, u is a threshold value, and the number of samples exceeding the threshold value u is nuF (x) is a distribution function; conditional distribution function Fu(y) is:
Fu(y)＝P(X-u≤y|X＞u)y≥0；
the conditional probability formula yields:
Figure FDA0003020987510000011
deformation can result in:
F(x)＝Fu(y)(1-F(u))+F(u)X≥u；
for a sufficiently large threshold u, the overrun yiApproximately obeying to a generalized pareto distribution Gξ，σ(y) then
Figure FDA0003020987510000012
In the formula: σ represents a scale parameter; xi represents the shape parameter, when xi is more than or equal to 0, y is more than 0 and infinity), when xi is less than 0,
Figure FDA0003020987510000013
probability density function g of the over-limit valueξ，σ(y) is:
Figure FDA0003020987510000014
therefore, for the measured value sequence { x of the rail inspection vehicle1，x2，...，xn}, the log-likelihood function L (ξ, σ | y) may be expressed as:
Figure FDA0003020987510000015
secondly, parameter estimation;
determining a threshold u from the overrun expected map;
after u is determined, maximum likelihood estimation is carried out by utilizing the rail detection vehicle measured value sequence X according to a number likelihood function L (xi, sigma | y), and then estimated values of a scale parameter sigma and a shape parameter xi can be obtained;
thirdly, drawing up management indexes of dynamic irregularity peak values and evaluating the rationality;
after the threshold is determined, use (n-n)u) As an empirical estimate of F (u), the data distribution function for the threshold u can be found as:
Figure FDA0003020987510000021
after the estimated values of the parameters in the formula are obtained, the distribution function under the condition of passing a threshold u can be determined; and then according to the management values of different levels in the existing peak value management, the corresponding overrun occurrence probability can be obtained by utilizing the inverse function of the distribution function, so that the rationality evaluation of the existing dynamic management value is realized.
2. The extreme value theory-based rail irregularity peak overrun management method according to claim 1, characterized in that: the peak value over-threshold method model analyzes the over-limit value y under the condition that X is more than ui＝X-u(i＝1，2，...，nu) Conditional distribution function Fu(y)。
3. The extreme value theory-based rail irregularity peak overrun management method according to claim 1, characterized in that: in the second step, the specific determination method of the threshold u is as follows: when the generalized pareto distribution has a shape parameter xi <1, the overrun expectation function e (m) is a linear function and can be expressed as:
Figure FDA0003020987510000022
an average remainder function of the form:
Figure FDA0003020987510000031
wherein n isu＝max{i|yi}；
If the sample has the thick tail distribution characteristic, the sample can be regarded as the generalized pareto distribution with xi <1, and the expected residual value is a linear function of u; therefore, the overrun expected graph obtained according to the measured rail inspection vehicle data can be selected by selecting the sufficiently large critical value u0So that when X is not less than u0Time e (x) is an approximate linear function used as a basis for determining the threshold value; meanwhile, the number n of samples exceeding the threshold value in the track inspection vehicle measured value sequence X can be obtainedu。
4. The extreme value theory-based rail irregularity peak overrun management method according to claim 1, characterized in that: in the third step, let xmFor the management values of the rail inspection vehicle measured values under different levels when x is more than xmThe track dynamic irregularity influences comfort level, temporary repair or speed limit, and the probability is as follows:
Figure FDA0003020987510000032
determining the occurrence probability P of different grades according to the importance of indexes of different grades of dynamic irregularityα(ii) a Then for a given confidence level P, a value VaR may be lostPIs the P quantile of the loss distribution, i.e. xmIs estimated value of
Figure FDA0003020987510000033
Comprises the following steps:
Figure FDA0003020987510000034
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