The purpose of this paper is to present a general universal formula for <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;&...The purpose of this paper is to present a general universal formula for <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;">-variate survival functions for arbitrary <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> = 2, 3, <span style="font-family:;" "=""><span style="font-family:Verdana;">...<span style="font-family:Verdana;">, given all the univariate marginal survival functions. This universal form of <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;">-variate probability distributions was obtained by means of “dependence functions” named “joiners” in the text. These joiners determine all the involved stochastic dependencies between the underlying random variables. However, in order that the presented formula (the form) represents a legitimate survival function, some necessary and sufficient conditions for the joiners had to be found. Basically, finding those conditions is the main task of this paper. This task was successfully performed for the case <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> = 2 and the main results for the case <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> = 3 were formulated as Theorem 1 and Theorem 2 in Section 4. Nevertheless, the hypothetical conditions valid for the general <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> ≥ 4 case were also formulated in Section 3 as the (very convincing) Hypothesis. As for the sufficient conditions for both the <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:;" "=""><span style="font-family:Verdana;"> = 3 and <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> ≥ 4 cases, the full generality was not achieved since two restrictions were imposed. Firstly, we limited ourselves to the, defined in the text, “continuous cases” (when the corresponding joint density exists and is continuous), and secondly we consider positive stochastic dependencies only. Nevertheless, the class of the <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;">-variate distributions which can be constructed is very wide. The presented method of construction by means of joiners can be considered competitive to the <span style="font-family:Verdana;"><span style="font-family:Verdana;">copula<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> methodology. As it is suggested in the paper the possibility of building a common theory of both copulae and joiners is quite possible, and the joiners may play the role of tools within the theory of copulae, and vice versa copulae may, for example, be used for finding proper joiners. Another independent feature of the joiners methodology is the possibility of constructing many new stochastic processes including stationary and Markovian.展开更多
The probability analysis of ground deformation is becoming a trend to estimate and control the risk brought by shield tunnelling.The gap parameter is regarded as an effective tool to estimate the ground loss of tunnel...The probability analysis of ground deformation is becoming a trend to estimate and control the risk brought by shield tunnelling.The gap parameter is regarded as an effective tool to estimate the ground loss of tunnelling in soft soil.More specifically,x,which is a gap parameter component defined as the over(or insufficient)excavation due to the change in the posture of the shield machine,may contribute more to the uncertainty of the ground loss.However,the existing uncertainty characterization methods for x have several limitations and cannot explain the uncertain correlations between the relevant parameters.Along these lines,to better characterize the uncertainty of x,the multivariate probability distribution was developed in this work and a dynamic prediction was proposed for it.To attain this goal,1523 rings of the field data coming from the shield tunnel between Longqing Road and Baiyun Road in Kunming Metro Line 5 were utilized and 44 parameters including the construction,stratigraphic,and posture parameters were collected to form the database.According to the variance filter method,the mutual information method,and the value of the correlation coefficients,the original 44 parameters were reduced to 10 main parameters,which were unit weight,the stoke of the jacks(A,B,C,and D groups),the pressure of the pushing jacks(A,C groups),the chamber pressure,the rotation speed,and the total force.The multivariate probability distribution was constructed based on the Johnson system of distributions.Moreover,the distribution was satisfactorily verified in explaining the pairwise correlation between x and other parameters through 2 million simulation cases.At last,the distribution was used as a prior distribution to update the marginal distribution of x with any group of the relevant parameters known.The performance of the dynamic prediction was further validated by the field data of 3 shield tunnel cases.展开更多
文摘The purpose of this paper is to present a general universal formula for <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;">-variate survival functions for arbitrary <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> = 2, 3, <span style="font-family:;" "=""><span style="font-family:Verdana;">...<span style="font-family:Verdana;">, given all the univariate marginal survival functions. This universal form of <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;">-variate probability distributions was obtained by means of “dependence functions” named “joiners” in the text. These joiners determine all the involved stochastic dependencies between the underlying random variables. However, in order that the presented formula (the form) represents a legitimate survival function, some necessary and sufficient conditions for the joiners had to be found. Basically, finding those conditions is the main task of this paper. This task was successfully performed for the case <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> = 2 and the main results for the case <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> = 3 were formulated as Theorem 1 and Theorem 2 in Section 4. Nevertheless, the hypothetical conditions valid for the general <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> ≥ 4 case were also formulated in Section 3 as the (very convincing) Hypothesis. As for the sufficient conditions for both the <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:;" "=""><span style="font-family:Verdana;"> = 3 and <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;"> ≥ 4 cases, the full generality was not achieved since two restrictions were imposed. Firstly, we limited ourselves to the, defined in the text, “continuous cases” (when the corresponding joint density exists and is continuous), and secondly we consider positive stochastic dependencies only. Nevertheless, the class of the <span style="font-family:Verdana;"><span style="font-family:Verdana;">k<span style="font-family:Verdana;"><span style="font-family:Verdana;">-variate distributions which can be constructed is very wide. The presented method of construction by means of joiners can be considered competitive to the <span style="font-family:Verdana;"><span style="font-family:Verdana;">copula<span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;"> methodology. As it is suggested in the paper the possibility of building a common theory of both copulae and joiners is quite possible, and the joiners may play the role of tools within the theory of copulae, and vice versa copulae may, for example, be used for finding proper joiners. Another independent feature of the joiners methodology is the possibility of constructing many new stochastic processes including stationary and Markovian.
基金support provided by the National Natural Science Foundation of China(Grant Nos.52078236 and 52122806)Guangzhou Metro Group Co.,Ltd(JT204-100111-23001)Chongqing Urban Investment Infrastructure Construction Co(Grant No.CQCT-JS-SC-GC-2022-0081).
文摘The probability analysis of ground deformation is becoming a trend to estimate and control the risk brought by shield tunnelling.The gap parameter is regarded as an effective tool to estimate the ground loss of tunnelling in soft soil.More specifically,x,which is a gap parameter component defined as the over(or insufficient)excavation due to the change in the posture of the shield machine,may contribute more to the uncertainty of the ground loss.However,the existing uncertainty characterization methods for x have several limitations and cannot explain the uncertain correlations between the relevant parameters.Along these lines,to better characterize the uncertainty of x,the multivariate probability distribution was developed in this work and a dynamic prediction was proposed for it.To attain this goal,1523 rings of the field data coming from the shield tunnel between Longqing Road and Baiyun Road in Kunming Metro Line 5 were utilized and 44 parameters including the construction,stratigraphic,and posture parameters were collected to form the database.According to the variance filter method,the mutual information method,and the value of the correlation coefficients,the original 44 parameters were reduced to 10 main parameters,which were unit weight,the stoke of the jacks(A,B,C,and D groups),the pressure of the pushing jacks(A,C groups),the chamber pressure,the rotation speed,and the total force.The multivariate probability distribution was constructed based on the Johnson system of distributions.Moreover,the distribution was satisfactorily verified in explaining the pairwise correlation between x and other parameters through 2 million simulation cases.At last,the distribution was used as a prior distribution to update the marginal distribution of x with any group of the relevant parameters known.The performance of the dynamic prediction was further validated by the field data of 3 shield tunnel cases.
