A Combined Prediction Model for Subgrade Settlement Based...

Research ArticleA Combined Prediction Model for Subgrade SettlementBased on Improved Set Pair Analysis

Yafeng Li1 Wuming Leng12 Rusong Nie 12 HuihaoMei13 Siwei Zhou1 and Junli Dong1

1School of Civil Engineering Central South University Changsha Hunan 410075 China2National Engineering Laboratory for High Speed Railway Construction Central South University Changsha 410075 China3Department of Civil Construction and Environmental Engineering Iowa State University Ames IA 50010 USA

Correspondence should be addressed to Rusong Nie nierusong97csueducn

Received 4 January 2019 Accepted 1 April 2019 Published 18 April 2019

Academic Editor Giuseppe DAniello

Copyright copy 2019 Yafeng Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Prediction of subgrade settlement is a complex problem involving various uncertainty factors To overcome the defects andlimitations of the single prediction model a combined prediction model based on the improved set pair analysis was proposedto take into account the uncertainty and certainty of the single prediction model and make the combined prediction based on thecertainty degree and the criterion of set pair relationship was optimized In the model the set pair was first constructed to expressthe relationship between predicted andmeasured valuesThen the risk of the set pair relationship identificationwas expressed basedon the Bayesian decision theory and the optimal criterion of set pair relationship was obtained by the adaptive search algorithmNext the relationship between the predictionmodel and themeasured data was analyzed to get the certainty degree and the weightcoefficient was obtained according to the certainty degree Finally the combination of single prediction models was carried out Acase study and comparison with other methods were conducted to confirm the reliability and validity of the proposed model Theresult shows that thismodel fully considers the uncertainty and certainty of the single predictionmodel and also extends themethodof determining the criterion of set pair relationship which provides ideas for other combined prediction and evaluation problems

1 Introduction

Chinarsquos railway is entering a new stage whether it is a high-speed railway that aims to improve the speed of operation ora heavy-haul railway that aims to improve the axle load oflocomotives and vehicles it imposes stringent requirementson the stability of subgrade The postconstruction settlementof railway subgrade directly affects the safety stability andoperating life of the train therefore accurate settlement pre-diction is important for the safety evaluation reinforcementand maintenance of the subgrade Consequently accurateprediction of subgrade settlement has been a major concernin engineering research

Current prediction models for subgrade settlement aremainly divided into single prediction models and combinedprediction models The single prediction models are mainlydivided into three categories (1) curve fitting method suchas exponential curve method hyperbolic method Asaokamethod and Poissonmethod [1ndash5] (2) backanalysis method

such as the iterative backanalysis layer by layer methodthe modified layer-summation based on elastoplastic theoryand the viscous-elastic BIOT consolidation FEM based onmerchant model [6ndash9] (3) system theory method such astime series analysis gray theory and neural network method[10ndash13] However single prediction methods are often unsat-isfactory the parameters in the curve fitting method and thesystem theorymethod do not have definite physicalmeaningand the two methods analyze the settlement law of subgradefrom a specific aspect therefore it is inevitable to ignore orlose part of the information contained in the data [14] Forthe backanalysis method the solution of constitutive modelparameters requires a large amount of experimental andmeasured data and the validity of the data has a significantinfluence on the accuracy of the method which limits theapplication of this method In summary there are certaindefects and limitations in the single prediction models

To comprehensively analyze the subgrade settlement andcombine the advantages of the single prediction model the

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1064246 9 pageshttpsdoiorg10115520191064246

2 Mathematical Problems in Engineering

combined prediction model was introduced [15ndash17] Thecombinedpredictionmodel has the advantage of reducing thepredictive risk and improving the accuracy which have beenwidely applied and developed in the settlement predictionLeng [18] and Wu [19] proposed a combined predictionmodel based on the correlation coefficient reciprocal ofvariance and fitting error and introduced MSE MAE andMAP indicators to evaluate the prediction effect Zheng[20] and Zhao [21] proposed a combined prediction modelbased on the residuals and discussed the criteria for selectingbasic prediction models Different from the methods aboveLi [22 23] thought that the combined prediction basedon the squares of the errors or the absolute values of thedeviations was not necessarily the optimal combination andproposed the superior combination prediction model withthe maximum efficiency

The weight coefficient is the key of the combined pre-diction model [16] In the existing literature the weights aremostly based on the squares of residuals and dispersion thevalidity and the fitting error The methods of determiningthe weights are not uniform Moreover those combinedprediction models often neglect the relationship between theinformation contained in each single prediction models inthe process of portfolio analysis which restricts the in-depthmining of data information and fails to make full use ofexisting information for combination prediction

Subgrade settlement and its prediction is a complex anduncertain system problem affected by many factors Set pairanalysis is a novel analytical method for systematic prob-lems of uncertainty [24] It dialectically treats uncertaintyand certainty by identity-discrepancy-contrary analysis as awhole which can comprehensively describe the complexityand uncertainty problem This theory has been applied inthe rating risk assessment and correlation analysis in civilengineering [25ndash27] and also provides new methods andideas for studying the uncertain system problem of subgradesettlement and its prediction

The set pair relationship is the key of set pair analysis Inthe existingmethod of determining the identity-discrepancy-contrary relationship the subjective experiencemethod lacksscientificity and rationality and the association membershipmethod is difficult to apply The mean standard deviationmethod and the trisection method are commonly used todetermine the criterion of set pair relationship [28ndash30]but these methods only segment the sample numericallyand do not adequately reflect the information and internalrelationship contained in the data thus the informationcontained in single models has not been fully utilized andoptimally combined

The main objective of this paper is to introduce a com-bined prediction model based on set pair analysis and deci-sion theory for subgrade settlement In the proposed modelthe relationship of information from the each predictionmodel was fully considered based on set pair analysis and thecriterion of set pair relationship was reasonably determinedaccording to Bayesian decision theory and obtained usingself-adaption algorithm Based on the certainty degree theweight coefficient was obtained and the information fromeach prediction model was optimally combined Moreover

the feasibility and validity of the proposed method werefurther discussed by case study taking the subgrade set-tlement of Shuohuang heavy-haul railway as an examplefollowed by a comparison with single prediction modelsand a combined prediction model The proposed model alsoprovides new ideas for other combination prediction andevaluation problems

2 Theory

21 Set Pair Analysis Set pair analysis is a theory proposed byscholar Zhao for systematic problems of uncertainty [24 2731] The theory uses the certainty degree 120583 to quantitativelydescribe and analyze the relationship between certainty anduncertainty of systematic problemsThemathematical modelcorresponding to certainty degree 120583 is written as follows

120583 = 119886 + 119887119894 + 119888119895 (1)

where 120583 denotes the quantitative expression for set pairrelationship 119886 119887 and 119888 are identity degree discrepancydegree and contrary degree respectively and 119886 + 119887 + 119888 = 1Parameter 119894 is the discrepancy coefficient within [minus1 1] andparameter 119895 is the contrary coefficient and generally specifiedas -1

22 Bayesian Decision Theory Bayesian decision theory is apowerful tool that enables decision-making evenwith limiteddata succeeding where traditional statistical methods havefailed to capture inherent dynamics It is to make decisionswith the least risk or cost under probabilistic uncertaintyThismethod has been applied in many research fields such asimage processing and machine learning [32 33]

For a given object 119909 let 119882 = 1199081 1199082 119908119888 be afinite set of 119888 states and let 119860 = 1198861 1198862 119886119905 be a finiteset of 119905 possible actions Let 119901(119908119895 | 119909) be the conditionalprobability of an object 119909 being in state 119908119895 given that theobject is described by 119909 Let 120582(119886119894 | 119908119895) denote the loss orcost for taking action 119886119894 when the state is119908119895 For an object 119909suppose action 119886119894 is taken Since 119901(119908119895 | 119909) is the probabilitythat the true state is 119908119895 given 119909 the expected loss associatedwith taking action 119886119894 can be expressed as follows

119877 (119886119894 | 119909) =119888sum119895=1

120582 (119886119894 | 119908119895) 119901 (119908119895 | 119909) (2)

3 Combined Prediction Model Based onSet Pair Analysis

The combination prediction of subgrade settlement is essen-tially the analysis of relationship between each predictionmodel The proposed model uses the certainty degree 120583to express the certainty and uncertainty of the predictioninformation of each model and the certainty degree isobtained based on the set pairs and the criterion of set pairrelationshipNext set pairs are constructed using the absoluteerror Δ 119894 as shown below

Mathematical Problems in Engineering 3

119878 = 1199041 1199042 119904119863 is themeasured settlement ofmonitor-ing points and 119878119896 = 1199041198961 1199041198962 119904119896119863 is the predicted settle-ment of the 119896th prediction model Set pairs are constructedusing the absolute error Δ 119894 of the predicted and measuredvalues and the absolute errors are normalized for subsequentanalysis

Δ 119894 = 1003816100381610038161003816119904119896119894 minus 1199041198941003816100381610038161003816 (3)

119901119894 = Δ 119894 minus Δ119898119894119899Δ119898119886119909 minus Δ119898119894119899 (4)

The absolute errorΔ 119894 and the normalized value119901119894 expressthe accuracy and validity of the predicted value as well asthe set pair relationship between the predicted and measuredvalue If the error value is less than a limit value 1198791 it isconsidered that the prediction model has a good predictionat this point and the predicted value is in the identityrelationship with the measured value And if the error valueis greater than a limit value 1198792 then the prediction effect isbad and the predicted value is in the contrary relationshipwith the measured value Otherwise the prediction effectis in general and the predicted value and the measuredvalue are in a discrepancy relationship The total number ofsamples is119867 Based on the set pair relationship the numberof samples in which the relationship between the predictedand measured value is identity discrepancy and contrary is120576 120601 and 120593 respectively and the set pair relationship betweenthe measured values and the predicted values of the model 119896can be expressed by the certainty degree 120583119896 as follows

120583119896 = 120576119867 + 120601119867119894 +

120593119867119895 (5)

The certainty degree 120583119896 comprehensively reflects theaccuracy and effectiveness of prediction model 119896 The largerthe certainty degree 120583119896 the better the prediction effect andthe larger the weight coefficient of this prediction modelTherefore the certainty degree 120583119896 can be the basis fordetermining the weight Normalize the certainty degree 120583119896 toobtain the weights and the combination of single predictionmodels is given as

120596119896 = 120583119896sum119898119896=1 120583119896 (6)

119878119894 =119898sum119896=1

120596119896119878119896119894 (7)

4 Criterion of Set Pair Relationship Based onBayesian Decision Theory

From the set pair analysis above the set pair relationshipis the key But for the settlement problem in this paperthe criterion of set pair relationship is unknown The meanstandard deviation method and the trisection method arecommonly used for systematic problems of uncertaintywhere the criterion of set pair relationship is unknownbut the two methods determine the criterion numericallyand subjectively and do not consider the distribution and

intrinsic relationship of the sample dateTherefore this paperintroduces a new method to determine the optimal criterionof set pair relationship based on the minimum-risk Bayesiandecision theory

In order to apply Bayesian decision theory the meaningof normalized absolute error 119901119894 is explained here Thenormalized absolute error 119901119894 satisfies the form and meaningof probability in Bayesian theoryThe smaller the normalizedabsolute error the better the prediction effect and the greaterthe probability that the predictive value is in the identity rela-tionship with the measured value Conversely the greater theerror the worse the prediction and the greater the probabilitythat the predictive value is in the contrary relationship withthe measured value Otherwise the relationship between thepredicted and measured value is discrepancy Therefore thenormalized absolute error 119901119894 can be used as the subjectiveprobability in Bayesian decision theory for risk assessmentand parameter decision-making and finally to determine thecriterion of set pair relationship

For a given object 119909 let 119882 = 119908 minus119908 be the set of twostates Let119860 = 119886119868 119886119863 119886119862 be the set of three possible actionswhere 119886119868 119886119863 and 119886119862 respectively represent actions thatclassify the object 119909 into identity discrepancy and contraryrelations Let 119901 be the conditional probability of an object 119909being in state 119908 and the conditional probability of an object119909 being in state -119908 is 1-p The expected loss 119877 associated withtaking the individual actions and the expected loss 119877119905119900119897 oftaking actions on all samples can be described as follows

119877119868 = 120582119868119868 sdot 119901 + 120582119868119862 sdot (1 minus 119901)119877119863 = 120582119863119868 sdot 119901 + 120582119863119862 sdot (1 minus 119901)119877119862 = 120582119862119868 sdot 119901 + 120582119862119862 sdot (1 minus 119901)

(8)

119877119905119900119897 = sum119877119868 +sum119877119863 +sum119877119862 (9)

where 119877119868 119877119863 and 119877119862 are expected loss by determiningobject119909 into identity discrepancy and contrary relations120582119868119868120582119863119868 and 120582119862119868 denote the loss for taking actions of 119886119868 119886119863 and119886119862 respectively when the object 119909 in fact belongs to 119908 120582119868119862120582119863119862 and 120582119862119862 denote the loss for taking the same actionsrespectively when the object 119909 actually does not belong to119908

When the action taken on the object 119909matches the actualstate the expected loss isminimal and the criterion of set pairrelationship is themost reasonableTherefore the principle ofdetermining the set pair relationship can be expressed as thefollowing minimum-risk decision rules

119877119868 lt 119877119863119886119899119889 119877119868 lt 119877119862

119894119889119890119899119905119894119905119910119877119863 lt 119877119868

119886119899119889 119877119863 lt 119877119862119889119894119904119888119903119890119901119886119899119888119910


119877119862 lt 119877119868119886119899119889 119877119862 lt 119877119863

119888119900119899119905119903119886119903119910(10)

To simplify the decision rules we make some discussionsabout the relationship between loss functions Consider aspecial kind of loss functions with 120582119868119868 lt 120582119863119868 lt 120582119862119868 and120582119862119862 lt 120582119863119862 lt 120582119868119862 That is the loss of classifying an object119909 belonging to 119908 into identity relation is less than to the lossof classifying 119909 into the discrepancy relation and both of thetwo losses are strictly less than the loss of classifying119909 into thecontrary relation And the reverse order of losses is used forclassifying an object 119909 that does not belong to119908Then we canobtain the principle of determining the set pair relationshipexpressed in 119901 substituting formula (8) into formula (10)

119901 lt 120582119862119862 minus 120582119863119862(120582119863119868 + 120582119862119862) minus (120582119863119862 + 120582119862119868)119886119899119889 119901 lt 120582119862119862 minus 120582119868119862(120582119862119862 + 120582119868119868) minus (120582119868119862 + 120582119862119868)

119894119889119890119899119905119894119905119910119901 lt 120582119863119862 minus 120582119868119862(120582119863119862 + 120582119868119868) minus (120582119868119862 + 120582119863119868)

119886119899119889 119901 gt 120582119862119862 minus 120582119863119862(120582119863119868 + 120582119862119862) minus (120582119863119862 + 120582119862119868)119889119894119904119888119903119890119901119886119899119888119910

119901 gt 120582119863119862 minus 120582119868119862(120582119863119862 + 120582119868119868) minus (120582119868119862 + 120582119863119868)119886119899119889 119901 gt 120582119862119862 minus 120582119868119862(120582119862119862 + 120582119868119868) minus (120582119868119862 + 120582119862119868)

119888119900119899119905119903119886119903119910

(11)

This principle (11) can be written in simple form asfollows

119901 lt 120573119886119899119889 119901 lt 120574

119894119889119890119899119905119894119905119910119901 lt 120572

119886119899119889 119901 gt 120573119889119894119904119888119903119890119901119886119899119888119910

119901 gt 120572119886119899119889 119901 gt 120574

119888119900119899119905119903119886119903119910

(12)

where

120572 = 120582119863119862 minus 120582119868119862(120582119863119862 + 120582119868119868) minus (120582119868119862 + 120582119863119868)120573 = 120582119862119862 minus 120582119863119862(120582119863119868 + 120582119862119862) minus (120582119863119862 + 120582119862119868)120574 = 120582119862119862 minus 120582119868119862(120582119862119862 + 120582119868119868) minus (120582119868119862 + 120582119862119868)

(13)

Furthermore the cost of classifying an object 119909 into thediscrepancy relation is closer to the cost of a correct relationthan to the cost of an incorrect relation (120582119868119863minus120582119863119862)(120582119862119868minus120582119863119868)is the product of the differences between the cost of makingan incorrect action and cost of classifying an object into thediscrepancy relation and (120582119863119868minus120582119868119868)(120582119863119862minus120582119862119862) is the productof the differences between the cost of making a correct actionand cost of classifying an object into the discrepancy relationSo we can get the following inference

(120582119868119862 minus 120582119863119862) (120582119862119868 minus 120582119863119868) gt (120582119863119868 minus 120582119868119868) (120582119863119862 minus 120582119862119862) (14)

Based on the relationship between the loss functionsabove we can get the relationship between 120572 120573 and 1205740 le 120573 lt 120574 lt 120572 le 1 Furthermore we can simplify theprinciple (12) to the principle below Here we classify thesample at the critical point as identity or discrepancy relationnot discrepancy relation

119901 le 120573 119894119889119890119899119905119894119905119910120573 lt 119901 lt 120572 119889119894119904119888119903119890119901119886119899c119910119901 ge 120572 119888119900119899119905119903119886119903119910

(15)

The solution of the minimum value of expected loss119877119905119900119897 involves the six loss functions and parameters 120572 120573and 120574 and the six loss functions can be expressed in termsof the parameters 120572 120573 and 120574 So we make the followingassumptions and simplifications In six loss functions theloss 120582119868119868 generated by determining the object 119909 belongingto 119908 as the identity relation should be 0 and the loss 120582119862119862generated by determining the object 119909 not belonging to 119908as the discrepancy relation should also be 0 Let us set 120582119868119862equal to 1 so the six loss functions can be simplified to threeas follows

120582119862119868 = 1 minus 120574120574120582119863119862 = 120573 (120572 minus 120574)120574 (120572 minus 120573)120582119863119868 = (1 minus 120572) (120574 minus 120573)120574 (120572 minus 120573)

(16)

Assume that the number of samples which are deter-mined to be identity discrepancy and contrary relations is 120576120601 and 120593 respectively Substitute formula (16) into formulas


(8) and (9) and the expected loss 119877119905119900119897 caused by the actionsof all objects can be expressed as follows

119877119905119900119897 =120576sum119896=1

119877119868119896 +120601sum119896=1

119877119863119896 +120593sum119896=1

119877119862119896

= sum119901ge120572

(1 minus 119901) + sum119901le120573

(1 minus 120574120574 119901)

+ sum120573lt119901lt120572

[(1 minus 120572) (120574 minus 120573)120574 (120572 minus 120573) 119901 + 120573 (120572 minus 120574)120574 (120572 minus 120573) (1 minus 119901)]

(17)

When the expected loss 119877119905119900119897 is the minimum the actionson the set pair relationship determination for all samplesare the most appropriate and the parameter (120572 120573 120574) is theoptimal threshold for determining the criterion of set pairrelationship using formula (15)

5 Solution of Optimal Threshold Based onSelf-Adaption Algorithm

The minimum of the expected loss function 119877119905119900119897 involvesthree parameters 120572 120573 and 120574 The traditional direct solutionmethod has a large solution space and a heavy calculationThe self-adaption algorithm has the advantages of automat-ically adjusting the processing parameters and constraintsaccording to the characteristics of the data [34ndash36] It iswidely used for the solution of optimal values and parameterdetermination and provides a solution for the criterion of setpair relationship Since the parameter 120574 is not involved inthe criterion it only appears in the expected loss functionTo simplify the algorithm the parameter 120574 is discussed inadvance Without involving professional knowledge the lossgenerated by determining the object 119909 that actually belongsto 119908 as not belonging to 119908 is equal to the loss generated bydetermining the object 119909 that does not actually belong to 119908as belonging to 119908 that is 120582119868119862 = 120582119862119868 Then 120574 = 05 can beobtained from formula (16)

The algorithm is briefly described as follows

Input 119875 = 1199011 1199012 119901119898Output 120572 120573Step 1 Set the initial value 120572 = 09 120573 = 01 and 120574 = 05 119868 = 119863 = 119862 = and 119879 = 119868 cup 119863 cup 119862Step 2 For 1199011 isin 119875 let 119879 = 1199011 cup 119879 and calculate thecorresponding expected loss 119877119905119900119897 based on formula (17)

Step 3 On the premise of 120572 ge 120573 replace the parameters120572 120573 with p1 respectively and calculate the correspondingexpected loss 119877119905119900119897 Compare the expected loss values underdifferent (120572 120573) conditions and take the correspondingthreshold value (120572 120573) as the optimal parameter value whenthe expected loss 119877119905119900119897 is the minimum

Step 4 Continue to perform Step 2 and Step 3 on 1199012 untilall elements in set 119875 have completed the above steps and

select the final (120572 120573) to determine the criterion of set pairrelationship

6 Case Study

In order to verify the validity and feasibility of the proposedmodel the settlement data of Shuohuang heavy-hual railwaysubgrade were used to conduct the analysis The subgradeis in the transition section of the railway and bridge Thethickness of the ballast on the bridge and the transitionsection is about 145m and 05m respectively The filler ofthe subgrade is a silty clay with low liquid limit The profilediagram of the transition section is shown in Figure 1

The settlement data obtained is along the longitudinaldirection of the transition section In order to avoid themonitoring data affected by the vibration caused by trainpassing only the data of the skylight was selected for analysisAt the same time the information of the train passingthrough the monitoring point was recorded

In this paper the settlement at different monitoringpoints shows similar rules so one of the monitoring pointscan be selected as a demonstration for model calculation andverification (EL2 was selected in the paper)The settlement ofthe subgrade and the cumulative load of the passing train areshown in Table 1

For the existing heavy-haul railway with certain yearsof operation the instantaneous settlement and the primaryconsolidation settlement have been basically completedand the current settlement is composed of the secondaryconsolidation settlement and the permanent deformation ofsubgrade In this case the subgrade settlement is very smalltherefore the applicable prediction model for this situationis limited In this paper hyperbolic model exponential curvemodel GM (1 1)model Poisson model and neural networkmodel with better prediction effect were selected as basicsingle predictionmodels to conduct the combined predictionof subgrade settlement under cumulative train load Sincethe neural network method was carried out by numericalcalculation software the detailed formula is not given inTable 2 The basic prediction models are shown in Table 2

Based on the basic prediction models the predictedresults were obtained as shown in Table 4 Then the absoluteerrors of the predicted results were calculated and nor-malized Then we obtained the set 119875 which contains theconnotation of the set pair relationship and the probabilityThe expected loss function and the criterion of set pairrelationship were constructed using the method in section 4and the parameters (120572 120573) were solved using the algorithm inSection 5 The solution of parameters is shown in Figure 2and the criterion of set pair relationship is determined asformula (18)

119901 le 0385 1198941198891198901198991199051198941199051199100385 lt 119901 lt 0633 119889119894119904119888119903119890119901119886119899119888119910

119901 ge 0633 119888119900119899119905119903119886119903119910(18)

It can be seen from the criterion of set pair relationshipthat the prediction results of the single prediction models


EL5 EL4 EL3 EL2 EL1

Surface layer of subgrade bed

Bottom layer of subgrade bed

Embankment

Ballast

Frame-shaped culvert

Figure 1 Profile diagram of subgrade-bridge transition section

Table 1 Settlement and cumulative load of EL2

Cumulative load of passing train Qtimes107t 7 14 21 28 35 42 49Measured settlement Smm 0080 0120 0189 0238 0255 0271 0274Cumulative load of passing train Qtimes107t 56 63 70 77 84 91 98Measured settlement Smm 0297 0304 0307 0327 0331 0332 0335

Table 2 Single prediction model

Prediction model Regression formula

Hyperbolic model 119878119876 = 1198766432 + 227119876

Exponential curve model 119878119876 = 034 (1 minus 119890minus1198762695)GM (1 1)model 119878119896 = minus027119890minus024(119896minus1) + 035Poisson model 119878119896 = 0323

1 + 44119890minus0079119896Neural network method --

24

20

16

12

8

4

0

Num

ber o

f sam

ples

=0385 =0633

Normalized absolute error

[001

](0

102]

(020

3](0

304]

(0405

](0

506]

(0607

](0

708]

(080

9](0

910]

Figure 2 Sample information and threshold results

are mostly in the identity relationship with the measuredvalues and the prediction effect is good While a smallnumber of prediction results are in a discrepancy or contraryrelationship with the measured values and the predictioneffect is not satisfactory which lead to the occasional risk andvolatility of the single prediction model

Further the certainty degree is determined to quanti-tatively describe the prediction effect and the relationshipbetween the prediction model and the measured data Countthe number of samples in which the set relationship betweenthe predictive and measured values is the identity discrep-ancy and contrary and the certainty degrees120583119896 of three singlemodels are as follows

120583119878minus1198781 = 1114 +114 119894 +

214119895

120583119878minus1198782 = 1114 +114 119894 +

214119895

120583119878minus1198783 = 1014 +214 119894 +

214119895

120583119878minus1198784 = 314 +

914 119894 +

214119895

120583119878minus1198785 = 714 +

614 119894 +

114119895

(19)

Under the general conditions of existing research [24ndash30] the discrepancy coefficient 119894 and contrary coefficient119895 are usually specified as 05 and -1 respectively And thecertainty degree can be further calculated as 120583119878minus1198781 = 0679120583119878minus1198782 = 0679 120583119878minus1198783 = 0643 120583119878minus1198784 = 0393 and 120583119878minus1198785 =0643 Normalize the certainty degree and then obtain theweight coefficient and the combined prediction of subgradesettlement using formulas (6) and (7)Theweight coefficientsand combined prediction results are shown in Tables 3 and 4

In order to compare and analyze the prediction effects ofthe proposed model the prediction error was analyzed Thestandard error (SE) was selected as the accuracy evaluationindex of the prediction model and the relative standarderror (RSE) was selected as the stability evaluation indexThe


Table 3 Weight coefficient

Hyperbolic model Exponential curve model GM (1 1)model Poisson model Neural network model0224 0224 0212 0129 0212

Table 4 Prediction results of the proposed model and comparison with those of other models (mm)

Measuredsettlement 119878 Hyperbolic

modelExponentialcurve model GM (1 1)model Poisson

modelNeural network

model

Combinedpredictionmodel in

literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5

Hyperbolic modelExponential curve modelGM (11) modelPoisson modelNeural network modelCombined prediction model in literature [18] Proposed model

6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)

Figure 3 Errors of prediction models

proposed model and the comparison with other predictionmodels are shown in Figure 3 and Table 5

The single prediction models have good prediction effectfor most samples but there are some volatility overall Bycontrast the two combined prediction models show better

prediction effect they have significant advantages in the com-prehensive utilization of information and the improvementof prediction stability and accuracy Furthermore comparedwith the combined prediction model in literature [18] thestability and accuracy of the proposed model in this paperare higher and the prediction effect is better

7 Conclusions

The current combination prediction models for subgradesettlement neglect the relationship between the informationcontained in single prediction models and fail to makefull use of existing information for combined predictionSubgrade settlement and its prediction are a complex anduncertain system problem affected by many factors andset pair theory is a novel analytical method for systematicproblems of uncertaintyTherefore a combination predictionmodel based on set pair analysis was proposed in this paperFurthermore Bayesian decision theory is integrated into theset pair analysis to intelligently determine the criterion ofset pair relationship Finally the information contained insingle prediction models is optimally utilized and combinedWith the case study and the analysis of prediction effects thefollowing conclusions are obtained

Based on Bayesian decision theory the criterion of setpair relationship is determined which overcomes the subjec-tive shortcoming of current methods Furthermore the self-adaption algorithm is used to solve the optimal thresholdwhich improves the effective recognition of the boundarysamples and the accuracy of the set pair relationship


Table 5 Effects of prediction models

Prediction model 119878119864 119877119878119864Hyperbolic model 00102 00601Exponential curve model 00083 00447GM (1 1)model 00089 00467Poisson model 00122 00603Neural network model 00099 00497Combined prediction model in literature [18] 00087 00480Proposed model 00077 00420

The certainty degree expresses the certainty and uncer-tainty relationships between predicted and measured valuesand describes the accuracy and reliability of the predictivemodel Therefore the certainty degree was used as thebasis for determining the weight coefficient which providesnew standards and ideas for the determination of weightsThe case study and comparative analysis with other modelsshow that the combined prediction model based on set pairanalysis is effective and feasible for subgrade settlement Inaddition the method of determining the criterion of set pairrelationship based on decision-making method provides anew idea for the intelligent identification of set pair relationsand also provides theoretical support for the application of setpair analysis

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authorsrsquo research is supported by the National NaturalScience Foundation of China (nos 51878666 and 51678572)and the Innovation Project of Central South University(Grant no 2018zzts192)

References

[1] K Prakash N S Murthy and A Sridharan ldquoRectangularhyperbolamethod of consolidation analysisrdquoGeotechnique vol37 no 3 pp 355ndash368 1987

[2] A Asaoka ldquoObservational procedure of settlement predictionrdquoSoils and Foundations vol 18 no 4 pp 87ndash101 1978

[3] P F Verhulst ldquoNotice sur la loi que la population suit dans sonaccroissementrdquo Quetelet vol 10 no 10 pp 113ndash121 1837

[4] S-X Chen X-Y Wang X-C Xu F Yu and S-L Qin ldquoThree-point modified exponential curve method for predicting sub-grade settlementsrdquo Rock and Soil Mechanics vol 32 no 11 pp3355ndash3360 2011

[5] C F Huang Q Li S C Wu J Y Li and X L Xu ldquoAppli-cation of the Richards model for settlement prediction based

on a bidirectional difference-weighted least-squares methodrdquoArabian Journal for Science amp Engineering vol 43 no 2 pp 1ndash92017

[6] S N M Tafreshi T Shaghaghi G T Mehrjardi A R Dawsonand M Ghadrdan ldquoA simplified method for predicting the set-tlement of circular footings onmulti-layered geocell-reinforcednon-cohesive soilsrdquo Geotextiles and Geomembranes vol 43 no4 pp 332ndash344 2015

[7] M Ge N Li W Zhang J Zheng and C Zhu ldquoSettle-ment behavior and inverse prediction of post-constructionsettlement of high filled loess embankmentrdquo Yanshilixue YuGongcheng XuebaoChinese Journal of Rock Mechanics andEngineering vol 36 no 3 pp 745ndash753 2017

[8] C M Tan and S B Huang ldquoBack-analysis and prediction forsettlement of two-dimensional nonlinear viscoelastic soft claygroundrdquo Rock and Soil Mechanics vol 23 no 1 pp 67ndash71 2002

[9] S M He ldquoModified layer-summation based on elastopastictheoryrdquo Rock and Soil Mechanics vol 24 no 1 pp 88ndash92 2003

[10] H B Liu Y M Xiang and Y X Ruan ldquoA multivariable greymodel based on optimized background value and its applicationto subgrade settlement predictionrdquo Applied Mechanics andMaterials vol 34 no 1 pp 173ndash181 2013

[11] P-Y Chen and H-M Yu ldquoFoundation settlement predictionbased on a novel NGM modelrdquo Mathematical Problems inEngineering vol 2014 Article ID 242809 8 pages 2014

[12] X-W Ren Y-Q Tang J Li and Q Yang ldquoA prediction methodusing grey model for cumulative plastic deformation undercyclic loadsrdquo Natural Hazards vol 64 no 1 pp 441ndash457 2012

[13] H Gao Q-C Song and J Huang ldquoSubgrade settlement pre-diction based on least square support vector regession and real-coded quantum evolutionary algorithmrdquo International Journalof Grid and Distributed Computing vol 9 no 7 pp 83ndash90 2016

[14] L-Y Pan and X-Y Xie ldquoObservational settlement predictionby curve fittingmethodsrdquoYantu LixueRock and SoilMechanicsvol 25 no 7 pp 1053ndash1058 2004

[15] J N Bates andCW J Granger ldquoCombined forecastingrdquo Journalof Operational Research vol 20 no 4 pp 451ndash468 1969

[16] M P Clements and D I Harvey ldquoCombining probabilityforecastsrdquo International Journal of Forecasting vol 27 no 2 pp208ndash223 2011

[17] K F Lam H W Mui and H K Yuen ldquoA note on minimizingabsolute percentage error in combined forecastsrdquo Computers ampOperations Research vol 28 no 11 pp 1141ndash1147 2001

[18] W-M Leng Q Yang R-S Nie and J Yue ldquoStudy of post-construction settlement combination forecast method of high-speed railway bridge pile foundationrdquo Yantu LixueRock andSoil Mechanics vol 32 no 11 pp 3341ndash3348 2011

[19] Q H Wu and H F Li ldquoApplication of the changeable weightcombination model in building settlement predictingrdquo Journal


of Geomatics Science and Technology vol 26 no 2 pp 118ndash1202009

[20] J-G Zheng T Wang and J-W Zhang ldquoStudy of settlementprediction methods of loess subgraderdquo Yantu LixueRock andSoil Mechanics vol 31 no 1 pp 321ndash326 2010

[21] M-H Zhao Y Liu and W-G Cao ldquoStudy on variable-weightcombination forecasting method of S-type curves for soft clayembankment settlementrdquoYantu LixueRock and SoilMechanicsvol 26 no 9 pp 1443ndash1447 2005

[22] T Li Y-P Zhang and T-Q Zhang ldquoSuperior combinationforecasting settlement of soft clay embankmentsrdquo YanshilixueYu Gongcheng XuebaoChinese Journal of Rock Mechanics andEngineering vol 24 no 18 pp 3282ndash3286 2005

[23] H Y Chen and D P Hou ldquoResearch on superior combinationforecasting model based on forecasting effective measurerdquoJournal of University of Science and Technology of China vol 32no 2 pp 172ndash180 2002

[24] K Q Zhao Set Pair Analysis and Its Preliminary ApplicationZhejiang Science and Technology Press HangZhou China2000

[25] M Wang and G Chen ldquoA novel coupling model for riskanalysis of swell and shrinkage of expansive soilsrdquo Computersamp Mathematics with Applications vol 62 no 7 pp 2854ndash28612011

[26] M-W Wang P Xu J Li and K-Y Zhao ldquoA novel set pairanalysis method based on variable weights for liquefactionevaluationrdquoNatural Hazards vol 70 no 2 pp 1527ndash1534 2014

[27] M W Wang J L Jin and Y L Zhou Set Pair Analysis BasedCoupling Methods and Applications Science Press BeiJingChina 2014

[28] W S Wang H L Xiang Y Q Li and S J Wang ldquoA newapproach to annual runoff classification based on set pairanalysisrdquo Journal of Sichuan University vol 40 no 5 pp 1ndash62008

[29] T Yin X R Huang and W Wen ldquoCorrelation analysis ofencountering probability of runoff in the water-exporting andwater-importing area of western route of south-to-north watertransfer project by SPArdquo Advanced Engineering Sciences vol 41no 2 pp 48ndash52 2009

[30] M W Wang D F Wei X W Zhou and P C Wang ldquoCon-nectional matrix-based combination evaluation method forsurrounding rock stabilityrdquo Applied Mathematics amp Mechanicsvol 36 no 3 pp 294ndash302 2015

[31] H Garg and K Kumar ldquoAn advanced study on operationsof connection number based on set pair analysisrdquo NationalAcademy of Science Letters pp 1ndash4 2019

[32] J O Berger Statistical Decision Theory and Bayesian AnalysisSpringer Science amp Business Media New York NY USA 2013

[33] J V Stone Bayesrsquo Rule A Tutorial Introduction to BayesianAnalysis Sebtel Press 2013

[34] F Liang and Y H Yan ldquoResearch of dynamic self-adaptionintrusion detection model based of clusteringrdquo Computer Engi-neering amp Design vol 34 no 3 pp 814ndash820 2013

[35] J He ldquoAn experimental study on the self-adaption mechanismused by evolutionary programmingrdquo Progress in Natural Sci-ence Materials International vol 18 no 4 pp 481ndash486 2008

[36] G L Chen X F Wang Z Q Zhuang and D S Wang GeneticAlgorithm and Its Application Posts amp Telecom Press 1999

Hindawiwwwhindawicom Volume 2018

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Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


combined prediction model was introduced [15ndash17] Thecombinedpredictionmodel has the advantage of reducing thepredictive risk and improving the accuracy which have beenwidely applied and developed in the settlement predictionLeng [18] and Wu [19] proposed a combined predictionmodel based on the correlation coefficient reciprocal ofvariance and fitting error and introduced MSE MAE andMAP indicators to evaluate the prediction effect Zheng[20] and Zhao [21] proposed a combined prediction modelbased on the residuals and discussed the criteria for selectingbasic prediction models Different from the methods aboveLi [22 23] thought that the combined prediction basedon the squares of the errors or the absolute values of thedeviations was not necessarily the optimal combination andproposed the superior combination prediction model withthe maximum efficiency

The weight coefficient is the key of the combined pre-diction model [16] In the existing literature the weights aremostly based on the squares of residuals and dispersion thevalidity and the fitting error The methods of determiningthe weights are not uniform Moreover those combinedprediction models often neglect the relationship between theinformation contained in each single prediction models inthe process of portfolio analysis which restricts the in-depthmining of data information and fails to make full use ofexisting information for combination prediction

Subgrade settlement and its prediction is a complex anduncertain system problem affected by many factors Set pairanalysis is a novel analytical method for systematic prob-lems of uncertainty [24] It dialectically treats uncertaintyand certainty by identity-discrepancy-contrary analysis as awhole which can comprehensively describe the complexityand uncertainty problem This theory has been applied inthe rating risk assessment and correlation analysis in civilengineering [25ndash27] and also provides new methods andideas for studying the uncertain system problem of subgradesettlement and its prediction

The set pair relationship is the key of set pair analysis Inthe existingmethod of determining the identity-discrepancy-contrary relationship the subjective experiencemethod lacksscientificity and rationality and the association membershipmethod is difficult to apply The mean standard deviationmethod and the trisection method are commonly used todetermine the criterion of set pair relationship [28ndash30]but these methods only segment the sample numericallyand do not adequately reflect the information and internalrelationship contained in the data thus the informationcontained in single models has not been fully utilized andoptimally combined

The main objective of this paper is to introduce a com-bined prediction model based on set pair analysis and deci-sion theory for subgrade settlement In the proposed modelthe relationship of information from the each predictionmodel was fully considered based on set pair analysis and thecriterion of set pair relationship was reasonably determinedaccording to Bayesian decision theory and obtained usingself-adaption algorithm Based on the certainty degree theweight coefficient was obtained and the information fromeach prediction model was optimally combined Moreover

the feasibility and validity of the proposed method werefurther discussed by case study taking the subgrade set-tlement of Shuohuang heavy-haul railway as an examplefollowed by a comparison with single prediction modelsand a combined prediction model The proposed model alsoprovides new ideas for other combination prediction andevaluation problems

2 Theory

21 Set Pair Analysis Set pair analysis is a theory proposed byscholar Zhao for systematic problems of uncertainty [24 2731] The theory uses the certainty degree 120583 to quantitativelydescribe and analyze the relationship between certainty anduncertainty of systematic problemsThemathematical modelcorresponding to certainty degree 120583 is written as follows

120583 = 119886 + 119887119894 + 119888119895 (1)

where 120583 denotes the quantitative expression for set pairrelationship 119886 119887 and 119888 are identity degree discrepancydegree and contrary degree respectively and 119886 + 119887 + 119888 = 1Parameter 119894 is the discrepancy coefficient within [minus1 1] andparameter 119895 is the contrary coefficient and generally specifiedas -1

22 Bayesian Decision Theory Bayesian decision theory is apowerful tool that enables decision-making evenwith limiteddata succeeding where traditional statistical methods havefailed to capture inherent dynamics It is to make decisionswith the least risk or cost under probabilistic uncertaintyThismethod has been applied in many research fields such asimage processing and machine learning [32 33]

For a given object 119909 let 119882 = 1199081 1199082 119908119888 be afinite set of 119888 states and let 119860 = 1198861 1198862 119886119905 be a finiteset of 119905 possible actions Let 119901(119908119895 | 119909) be the conditionalprobability of an object 119909 being in state 119908119895 given that theobject is described by 119909 Let 120582(119886119894 | 119908119895) denote the loss orcost for taking action 119886119894 when the state is119908119895 For an object 119909suppose action 119886119894 is taken Since 119901(119908119895 | 119909) is the probabilitythat the true state is 119908119895 given 119909 the expected loss associatedwith taking action 119886119894 can be expressed as follows

119877 (119886119894 | 119909) =119888sum119895=1

120582 (119886119894 | 119908119895) 119901 (119908119895 | 119909) (2)

3 Combined Prediction Model Based onSet Pair Analysis

The combination prediction of subgrade settlement is essen-tially the analysis of relationship between each predictionmodel The proposed model uses the certainty degree 120583to express the certainty and uncertainty of the predictioninformation of each model and the certainty degree isobtained based on the set pairs and the criterion of set pairrelationshipNext set pairs are constructed using the absoluteerror Δ 119894 as shown below



Δ 119894 = 1003816100381610038161003816119904119896119894 minus 1199041198941003816100381610038161003816 (3)

119901119894 = Δ 119894 minus Δ119898119894119899Δ119898119886119909 minus Δ119898119894119899 (4)


120583119896 = 120576119867 + 120601119867119894 +

120593119867119895 (5)


120596119896 = 120583119896sum119898119896=1 120583119896 (6)

119878119894 =119898sum119896=1

120596119896119878119896119894 (7)







(8)

119877119905119900119897 = sum119877119868 +sum119877119863 +sum119877119862 (9)



119877119868 lt 119877119863119886119899119889 119877119868 lt 119877119862

119894119889119890119899119905119894119905119910119877119863 lt 119877119868

119886119899119889 119877119863 lt 119877119862119889119894119904119888119903119890119901119886119899119888119910


119877119862 lt 119877119868119886119899119889 119877119862 lt 119877119863

119888119900119899119905119903119886119903119910(10)



119894119889119890119899119905119894119905119910119901 lt 120582119863119862 minus 120582119868119862(120582119863119862 + 120582119868119868) minus (120582119868119862 + 120582119863119868)

119886119899119889 119901 gt 120582119862119862 minus 120582119863119862(120582119863119868 + 120582119862119862) minus (120582119863119862 + 120582119862119868)119889119894119904119888119903119890119901119886119899119888119910


119888119900119899119905119903119886119903119910

(11)


119901 lt 120573119886119899119889 119901 lt 120574

119894119889119890119899119905119894119905119910119901 lt 120572

119886119899119889 119901 gt 120573119889119894119904119888119903119890119901119886119899119888119910

119901 gt 120572119886119899119889 119901 gt 120574

119888119900119899119905119903119886119903119910

(12)

where


(13)




119901 le 120573 119894119889119890119899119905119894119905119910120573 lt 119901 lt 120572 119889119894119904119888119903119890119901119886119899c119910119901 ge 120572 119888119900119899119905119903119886119903119910

(15)



(16)




119877119905119900119897 =120576sum119896=1

119877119868119896 +120601sum119896=1

119877119863119896 +120593sum119896=1

119877119862119896

= sum119901ge120572

(1 minus 119901) + sum119901le120573

(1 minus 120574120574 119901)

+ sum120573lt119901lt120572


(17)









6 Case Study






119901 le 0385 1198941198891198901198991199051198941199051199100385 lt 119901 lt 0633 119889119894119904119888119903119890119901119886119899119888119910

119901 ge 0633 119888119900119899119905119903119886119903119910(18)



EL5 EL4 EL3 EL2 EL1



Embankment

Ballast










24

20

16

12

8

4

0

Num

ber o

f sam

ples

=0385 =0633


[001

](0

102]

(020

3](0

304]

(0405

](0

506]

(0607

](0

708]

(080

9](0

910]




120583119878minus1198781 = 1114 +114 119894 +

214119895

120583119878minus1198782 = 1114 +114 119894 +

214119895

120583119878minus1198783 = 1014 +214 119894 +

214119895

120583119878minus1198784 = 314 +

914 119894 +

214119895

120583119878minus1198785 = 714 +

614 119894 +

114119895

(19)









modelNeural network

model


literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5


6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)





7 Conclusions







Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018










Δ 119894 = 1003816100381610038161003816119904119896119894 minus 1199041198941003816100381610038161003816 (3)

119901119894 = Δ 119894 minus Δ119898119894119899Δ119898119886119909 minus Δ119898119894119899 (4)


120583119896 = 120576119867 + 120601119867119894 +

120593119867119895 (5)


120596119896 = 120583119896sum119898119896=1 120583119896 (6)

119878119894 =119898sum119896=1

120596119896119878119896119894 (7)







(8)

119877119905119900119897 = sum119877119868 +sum119877119863 +sum119877119862 (9)



119877119868 lt 119877119863119886119899119889 119877119868 lt 119877119862

119894119889119890119899119905119894119905119910119877119863 lt 119877119868

119886119899119889 119877119863 lt 119877119862119889119894119904119888119903119890119901119886119899119888119910


119877119862 lt 119877119868119886119899119889 119877119862 lt 119877119863

119888119900119899119905119903119886119903119910(10)



119894119889119890119899119905119894119905119910119901 lt 120582119863119862 minus 120582119868119862(120582119863119862 + 120582119868119868) minus (120582119868119862 + 120582119863119868)

119886119899119889 119901 gt 120582119862119862 minus 120582119863119862(120582119863119868 + 120582119862119862) minus (120582119863119862 + 120582119862119868)119889119894119904119888119903119890119901119886119899119888119910


119888119900119899119905119903119886119903119910

(11)


119901 lt 120573119886119899119889 119901 lt 120574

119894119889119890119899119905119894119905119910119901 lt 120572

119886119899119889 119901 gt 120573119889119894119904119888119903119890119901119886119899119888119910

119901 gt 120572119886119899119889 119901 gt 120574

119888119900119899119905119903119886119903119910

(12)

where


(13)




119901 le 120573 119894119889119890119899119905119894119905119910120573 lt 119901 lt 120572 119889119894119904119888119903119890119901119886119899c119910119901 ge 120572 119888119900119899119905119903119886119903119910

(15)



(16)




119877119905119900119897 =120576sum119896=1

119877119868119896 +120601sum119896=1

119877119863119896 +120593sum119896=1

119877119862119896

= sum119901ge120572

(1 minus 119901) + sum119901le120573

(1 minus 120574120574 119901)

+ sum120573lt119901lt120572


(17)









6 Case Study






119901 le 0385 1198941198891198901198991199051198941199051199100385 lt 119901 lt 0633 119889119894119904119888119903119890119901119886119899119888119910

119901 ge 0633 119888119900119899119905119903119886119903119910(18)



EL5 EL4 EL3 EL2 EL1



Embankment

Ballast










24

20

16

12

8

4

0

Num

ber o

f sam

ples

=0385 =0633


[001

](0

102]

(020

3](0

304]

(0405

](0

506]

(0607

](0

708]

(080

9](0

910]




120583119878minus1198781 = 1114 +114 119894 +

214119895

120583119878minus1198782 = 1114 +114 119894 +

214119895

120583119878minus1198783 = 1014 +214 119894 +

214119895

120583119878minus1198784 = 314 +

914 119894 +

214119895

120583119878minus1198785 = 714 +

614 119894 +

114119895

(19)









modelNeural network

model


literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5


6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)





7 Conclusions







Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018









119877119862 lt 119877119868119886119899119889 119877119862 lt 119877119863

119888119900119899119905119903119886119903119910(10)



119894119889119890119899119905119894119905119910119901 lt 120582119863119862 minus 120582119868119862(120582119863119862 + 120582119868119868) minus (120582119868119862 + 120582119863119868)

119886119899119889 119901 gt 120582119862119862 minus 120582119863119862(120582119863119868 + 120582119862119862) minus (120582119863119862 + 120582119862119868)119889119894119904119888119903119890119901119886119899119888119910


119888119900119899119905119903119886119903119910

(11)


119901 lt 120573119886119899119889 119901 lt 120574

119894119889119890119899119905119894119905119910119901 lt 120572

119886119899119889 119901 gt 120573119889119894119904119888119903119890119901119886119899119888119910

119901 gt 120572119886119899119889 119901 gt 120574

119888119900119899119905119903119886119903119910

(12)

where


(13)




119901 le 120573 119894119889119890119899119905119894119905119910120573 lt 119901 lt 120572 119889119894119904119888119903119890119901119886119899c119910119901 ge 120572 119888119900119899119905119903119886119903119910

(15)



(16)




119877119905119900119897 =120576sum119896=1

119877119868119896 +120601sum119896=1

119877119863119896 +120593sum119896=1

119877119862119896

= sum119901ge120572

(1 minus 119901) + sum119901le120573

(1 minus 120574120574 119901)

+ sum120573lt119901lt120572


(17)









6 Case Study






119901 le 0385 1198941198891198901198991199051198941199051199100385 lt 119901 lt 0633 119889119894119904119888119903119890119901119886119899119888119910

119901 ge 0633 119888119900119899119905119903119886119903119910(18)



EL5 EL4 EL3 EL2 EL1



Embankment

Ballast










24

20

16

12

8

4

0

Num

ber o

f sam

ples

=0385 =0633


[001

](0

102]

(020

3](0

304]

(0405

](0

506]

(0607

](0

708]

(080

9](0

910]




120583119878minus1198781 = 1114 +114 119894 +

214119895

120583119878minus1198782 = 1114 +114 119894 +

214119895

120583119878minus1198783 = 1014 +214 119894 +

214119895

120583119878minus1198784 = 314 +

914 119894 +

214119895

120583119878minus1198785 = 714 +

614 119894 +

114119895

(19)









modelNeural network

model


literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5


6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)





7 Conclusions







Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018










119877119905119900119897 =120576sum119896=1

119877119868119896 +120601sum119896=1

119877119863119896 +120593sum119896=1

119877119862119896

= sum119901ge120572

(1 minus 119901) + sum119901le120573

(1 minus 120574120574 119901)

+ sum120573lt119901lt120572


(17)









6 Case Study






119901 le 0385 1198941198891198901198991199051198941199051199100385 lt 119901 lt 0633 119889119894119904119888119903119890119901119886119899119888119910

119901 ge 0633 119888119900119899119905119903119886119903119910(18)



EL5 EL4 EL3 EL2 EL1



Embankment

Ballast










24

20

16

12

8

4

0

Num

ber o

f sam

ples

=0385 =0633


[001

](0

102]

(020

3](0

304]

(0405

](0

506]

(0607

](0

708]

(080

9](0

910]




120583119878minus1198781 = 1114 +114 119894 +

214119895

120583119878minus1198782 = 1114 +114 119894 +

214119895

120583119878minus1198783 = 1014 +214 119894 +

214119895

120583119878minus1198784 = 314 +

914 119894 +

214119895

120583119878minus1198785 = 714 +

614 119894 +

114119895

(19)









modelNeural network

model


literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5


6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)





7 Conclusions







Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018









EL5 EL4 EL3 EL2 EL1



Embankment

Ballast










24

20

16

12

8

4

0

Num

ber o

f sam

ples

=0385 =0633


[001

](0

102]

(020

3](0

304]

(0405

](0

506]

(0607

](0

708]

(080

9](0

910]




120583119878minus1198781 = 1114 +114 119894 +

214119895

120583119878minus1198782 = 1114 +114 119894 +

214119895

120583119878minus1198783 = 1014 +214 119894 +

214119895

120583119878minus1198784 = 314 +

914 119894 +

214119895

120583119878minus1198785 = 714 +

614 119894 +

114119895

(19)









modelNeural network

model


literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5


6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)





7 Conclusions







Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018














modelNeural network

model


literature [18]

Proposedmodel

0080 0087 0078 0080 0091 0090 0081 00820120 0146 0138 0137 0131 0116 0140 01340189 0188 0185 0182 0175 0173 0185 01870238 0219 0221 0217 0217 0234 0219 02260255 0244 0249 0245 0252 0247 0246 02490271 0263 0270 0267 0278 0258 0267 02680274 0279 0286 0284 0296 0262 0283 02790297 0293 0299 0297 0307 0308 0297 02960304 0303 0309 0308 0313 0314 0307 03050307 0314 0316 0316 0317 0316 0315 03120327 0322 0322 0323 0320 0321 0322 03240331 0330 0327 0328 0321 0323 0328 03290332 0336 0330 0332 0322 0323 0332 03330335 0342 0333 0335 0322 0324 0336 0337

40

30

20

0

10

1 2 3 4 5


6 7 8 9 10 11 12 13 14Samples

Abso

lute

erro

rs (1

0-3

mm

)





7 Conclusions







Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018












Data Availability




Acknowledgments


References















































Journal of












Journal of







Volume 2018






Volume 2018


































Journal of












Journal of







Volume 2018






Volume 2018








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