Quantile Criterion Research Articles

On a numerical estimation method with a given accuracy of a quantile criterion in the case of a piece-linear loss function and a gaussian probability density

The solution of many practical problems leads to the calculation of the values of probabilistic criteria, the most common of which are the quantile and probability functionals. It is known that, under fairly general assumptions, methods suitable for solving problems of finding the values of a probabilistic criterion can be used to solve the problem of quantile analysis. The proposed method for solving the problem of quantile analysis is based on the use of the method of numerical multidimensional integration described in the previous works of the author. One of the important properties of this integration method is universality (when using it, we can set an arbitrary number of variables n and an arbitrary number of linear constraints r). The only limitation is the case of an unacceptably long solution time. Thus, the indicated universality is transferred to the solution of the considered problem of quantile analysis.

On the Hedging of Interest Rate Margins on Bank Demand Deposits.

In recent years, we have seen a very rapid increase in outstanding bank deposits. This increase has been particularly high since the outbreak of the COVID 19 pandemic, due to the lockdown that has, among other things, drastically reduced household consumption since the beginning of March 2020. This very sharp increase in outstandings increases the risk of banks which offer interest on deposits. In this paper, we deal with the mitigation of the risk contained in interest rate margins of demand deposits. We introduce and analyze hedging strategies of an asset and liability manager who focuses on the bank’s net operating income in a given quarter under standard accounting rules. Demand deposits are assumed to be correlated with market interest rate and to a commercial risk that cannot be fully hedged on the financial markets. We distinguish several types of dynamic hedging strategies based on both quadratic and quantile criteria. We provide explicit formula for all hedging strategies and we discuss their respective robustness. We show in particular that the quantile hedging criterion leads to somewhat riskier strategy since its gain may be nil, due to its knockout feature. We argue that our contribution establishes a stronger basis for the coverage of bank deposits, which is particularly important in the context of the COVID-19 pandemic and its economic consequences.

Transformation-Invariant Learning of Optimal Individualized Decision Rules with Time-to-Event Outcomes

In many important applications of precision medicine, the outcome of interest is time to an event (e.g., death, relapse of disease) and the primary goal is to identify the optimal individualized decision rule (IDR) to prolong survival time. Existing work in this area have been mostly focused on estimating the optimal IDR to maximize the restricted mean survival time in the population. We propose a new robust framework for estimating an optimal static or dynamic IDR with time-to-event outcomes based on an easy-to-interpret quantile criterion. The new method does not need to specify an outcome regression model and is robust for heavy-tailed distribution. The estimation problem corresponds to a nonregular M-estimation problem with both finite and infinite-dimensional nuisance parameters. Employing advanced empirical process techniques, we establish the statistical theory of the estimated parameter indexing the optimal IDR. Furthermore, we prove a novel result that the proposed approach can consistently estimate the optimal value function under mild conditions even when the optimal IDR is nonunique, which happens in the challenging setting of exceptional laws. We also propose a smoothed resampling procedure for inference. The proposed methods are implemented in the R-package QTOCen. We demonstrate the performance of the proposed new methods via extensive Monte Carlo studies and a real data application. Supplementary materials for this article are available online.

Two-Stage Stochastic Facility Location Model with Quantile Criterion and Choosing Reliability Level

Two-Stage Stochastic Facility Location Model with Quantile Criterion and Choosing Reliability Level

An Extension of the Quantile Optimization Problem with a Loss Function Linear in Random Parameters

This paper studies the stochastic programming problem with a quantile criterion in the classical single-stage statement under the assumption that the loss function is linear in random parameters. An extension of this problem is the minimax one in which the inner maximum of the loss function is taken with respect to the realizations of the vector of random parameters over the kernel of its probability distribution, and the outer minimum is taken with respect to the optimized strategy over a given set of admissible strategies. The extension principle of optimization problems is used to establish the following result: under a sufficient condition in the form of a certain probabilistic constraint, the optimal solution of this minimax problem is also optimal in the original problem with the quantile criterion.

Variable neighborhood search for stochastic linear programming problem with quantile criterion

We consider the stochastic linear programming problem with quantile criterion and continuous distribution of random parameters. Using the sample approximation, we obtain a stochastic programming problem with discrete distribution of random parameters. It is known that the solution to this problem provides an approximate solution to the problem with continuous random parameters if the size of the sample is large enough. Applying the confidence method, we reduce the problem to a mixed integer programming problem, which is linear with respect to continuous variables. Integer variables determine confidence sets, and we describe the structure of the optimal confidence set. This property allows us to take into account only confidence sets that may be optimal. To find an approximate solution to the problem, we suggest a modification of the variable neighborhood search and determine structures of neighborhoods used in the search. Also, we discuss a method to find a good initial solution and give results of numerical experiments. We apply the developed algorithm to solve a problem of optimization of a hospital budget.

Variable Neighborhood Search for a Two-Stage Stochastic Programming Problem with a Quantile Criterion

We consider a two-stage stochastic programming problem with a bilinear loss function and a quantile criterion. The problem is reduced to a single-stage stochastic programming problem with a quantile criterion. We use the method of sample approximations. The resulting approximating problem is considered as a stochastic programming problem with a discrete distribution of random parameters. We check convergence conditions for the sequence of solutions of approximating problems. Using the confidence method, the problem is reduced to a combinatorial optimization problem where the confidence set represents an optimization strategy. To search for the optimal confidence set, we adapt the variable neighborhood search method. To solve the problem, we develop a hybrid algorithm based on the method of sample approximations, the confidence method, variable neighborhood search.

Sample Average Approximation in a Two-Stage Stochastic Linear Program with Quantile Criterion

A two-stage stochastic linear program with quantile criterion is considered. In this problem, the first stage strategy is deterministic and the second stage strategy is chosen when a realization of the random parameters is known. The properties of the problem are studied, a theorem on the existence of its solution is proved, and a sample average approximation of the problem is constructed. The sample average approximation is reduced to a mixed integer linear program, and a theorem on their equivalence is proved. A procedure for finding an optimal solution of the approximating problem is suggested. A theorem on the convergence of discrete approximations with respect to the value of the objective function and to the optimization strategy is given. We also consider some cases not covered in the theorem.

A Bilevel Stochastic Programming Problem with Random Parameters in the Follower’s Objective Function

Under study is a bilevel stochastic linear programming problem with quantile criterion. Bilevel programming problems can be considered as formalization of the process of interaction between two parties. The first party is a Leader making a decision first; the second is a Follower making a decision knowing the Leader’s strategy and the realization of the random parameters. It is assumed that the Follower’s problem is linear if the realization of the random parameters and the Leader’s strategy are given. The aim of the Leader is the minimization of the quantile function of a loss function that depends on his own strategy and the optimal Follower’s strategy. It is shown that the Follower’s problem has a unique solution with probability 1 if the distribution of the random parameters is absolutely continuous. The lower-semicontinuity of the loss function is proved and some conditions are obtained of the solvability of the problem under consideration. Some example shows that the continuity of the quantile function cannot be provided. The sample average approximation of the problem is formulated. The conditions are given to provide that, as the sample size increases, the sample average approximation converges to the original problem with respect to the strategy and the objective value. It is shown that the convergence conditions hold for almost all values of the reliability level. A model example is given of determining the tax rate, and the numerical experiments are executed for this example.

On the task of allocating investment to facilities preventing unauthorized movement of road vehicles across level crossings for various statistical criteria

Aim. Railway transportation is affected by a whole range of transportation incidents, both related to rolling stock, i. e. vehicle-to-vehicle collisions, derailments, broken cast parts of bogies, etc. , and infrastructure, i. e. broken rail, fires at railway stations and terminals, broken catenary, etc. Among the above incidents, collisions at level crossings are the most likely to cause a public response, as collisions between trains and road vehicles often cause multiple deaths that are reported in national media, which entails significant reputational damage for JSC RZD. Additionally, collisions often cause derailment of vehicles, which may cause deaths and major environmental disasters, if dangerous chemical products are transported. Beside the reputational damage, collisions at level crossings cause significant expenditure related to the repair of damaged infrastructure and rolling stock, as well as damage caused by trains idling due to maintenance machines operation at the location of incident. That brings up the issue of optimal allocation of investment to facilities preventing unauthorized movement of road vehicles across level crossings (hereinafter referred to as protection systems). This problem is of relevance, as replacing level crossings with tunnels and viaducts is not going fast and does not imply the eventual elimination of all level crossing. Hence is the requirement for rational allocation of funds to the installation of protection systems over the extensive railway network. Given the above, the aim of this paper is to develop decision-making guidelines for the reduction of the number of transportation incidents in terms of statistical criteria, i. e. quantile and probabilistic. Methods . The paper uses methods of deterministic equivalent, of equivalent transformations, of the probability theory, of optimization . Results. The problem of maximizing the probability of no incidents is reduced to integer linear programming. For the problem of minimizing the maximum number of incidents guaranteed at the given level of dependability, a suboptimal solution of the initial problem of quantile optimization is suggested that is obtained by solving the integer linear programming problem through the replacement of binomially distributed random values with Poisson values. Conclusions. The examined models not only allow developing an optimal strategy with guaranteed characteristics, but also demonstrate the sufficiency or insufficiency of the investment funds allocated to the improvement of level crossing safety. Decision-making must be ruled by the quantile criterion, as the probability of not a single incident occurring may seem to be high, yet the probability of one, two, three or more incidents occurring may be unacceptable. The quantile criterion does not have this disadvantage and allows evaluating the number of transportation incidents guaranteed at the specified level of dependability.

Stochastic Model of the Electric Power Purchase System on a Railway Segment

Proposed was a mathematical optimization model of the power supply system of a railway segment where the electric power is bought from more than one supplier with allowance for the random demand. Consideration was given to several time periods with different tariffs for electric power. The system model represents a two-step problem of stochastic programming with a quantile criterion. At its first step, a primary purchase plan is generated. At the second step, an additional purchase is done to compensate for the lack of electric power arising because of the random demand. The model takes into consideration the losses arising at transmitting the electric power from the supplier to the segment, as well as the pent-up demand. The total operational costs of the power supply system under consideration are minimized. An algorithm was proposed to solve the problem by reducing the stated problem by discretizing the probabilistic measure and confidence method to the problem of mixed integer linear programming. A model example was discussed.

Just-in-Time Selection of Principal Components for Fault Detection: The Criteria Based on Principal Component Contributions to the Sample Mahalanobis Distance

Principal component analysis (PCA) has been widely used in the field of fault detection. A main difficulty in using PCA is the selection of principal components (PCs). Different PC selection criteria have been developed in the past, but most of them do not connect the selection of PCs with the fault detection. The selected PCs may be optimal for data modeling but not for fault detection. In this paper, the just-in-time cumulative percent contribution (JITCPC) criterion and the just-in-time contribution quantile (JITCQ) criterion are proposed to select PCs from the viewpoint of fault detection. In the JITCPC and JITCQ criteria, the contributions of PCs to the sample Mahalanobis distance are used to evaluate the importance of PCs to fault detection. The larger contribution the PC makes, the more important it is to detect a fault. The JITCPC criterion selects the leading PCs with the cumulative percent contribution (CPC) larger than a predefined threshold (e.g., 90%). The JITCQ criterion selects the PCs with...

The Decomposition Method for Two-Stage Stochastic Linear Programming Problems with Quantile Criterion

We consider the two-stage stochastic linear programming problem with quantile criterion in case when the vector of random parameters has a discrete distribution with a finite number of realizations. Based on the confidence method and duality theorems, we construct a decompositional algorithm for finding guaranteeing solutions.

On the Convergence of Sample Approximations for Stochastic Programming Problems with Probabilistic Criteria

We consider stochastic programming problems with probabilistic and quantile criteria. We describe a method for approximating these problems with a sample of realizations for random parameters. When we use this method, criterial functions of the problems are replaced with their sample estimates. We show the hypoconvergence of sample probability functions to its exact value that guarantees the convergence of approximations for the probability function maximization problem on a compact set with respect to both the value of the criterial function and the optimization strategy. We prove a theorem on the convergence of approximation for the quantile function minimization problem with respect to the value of the criterial function and the optimization strategy.

MODEL OF OPTIMIZATION OF PASSENGER TRANSPORTATION IN A LARGE TRANSPORT HUBS AT THE USE OF RAIL BUSES

In this paper, a mathematical model for optimizing the activity of a large transport hub is proposed. The proposed model takes into account only passenger transportation. The goal of optimization is to determine: the number of rail buses assigned to the node; Optimal tariffs for transportation; Distribution of transport units in directions depending on the random demand for transportation. The possibility of attracting additional transport units in case of inadequate own transport is taken into account. The model takes into account the impact of competitive motor transport.The proposed mathematical model is based on a two-stage two-level stochastic programming problem with a quantile criterion. This formulation of the problem takes into account several known approaches to modeling complex economic relations.The decision-making process in this model has a two-stage structure. To simulate the two-stage structure of decision-making, a two-stage problem of stochastic linear programming is used. The strategy of the first stage is the number of rail buses assigned to the node, and tariffs for transportation in each direction. At the second stage, the rail buses are distributed along the directions in the transport hub, depending on the realization of the random demand for transportation.In this case, there is the possibility of attracting for an additional fee rail buses, not assigned to the node. Rail buses serve both the directions of the transport hub and the directions from other transport nodes to the node that is being considered. So, the strategy of the second stage is the number of own rail buses allocated for transportation in each direction, and the number of additional provided rail buses in each direction.The strategy of the first stage is determined for several planned periods. In each of the planned periods, the strategy of the second stage is selected anew, depending on the implementation of random demand and the chosen strategy of the first stage.To take into account the influence of the competitor, a two-level task is used. It assumes the participation of two players in the market: the leader (transport hub) and the follower (competitor in the form of vehicles). The follower chooses his strategy, knowing the strategy of the leader. The follower strategy is the prices for transportation in each direction and the corresponding volumes of traffic. The leader in choosing his optimal strategy takes into account the optimal strategy of the follower.

Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed‐integer problem

The paper is devoted to the stochastic optimistic bilevel optimization problem with quantile criterion in the upper level problem. If the probability distribution is finite, the problem can be transformed into a mixed‐integer nonlinear optimization problem. We formulate assumptions guaranteeing that an optimal solution exists. A production planning problem is used to illustrate usefulness of the model. Copyright © 2017 John Wiley & Sons, Ltd.

Optimizing Quantiles in Preference-Based Markov Decision Processes

In the Markov decision process model, policies are usually evaluated by expected cumulative rewards. As this decision criterion is not always suitable, we propose in this paper an algorithm for computing a policy optimal for the quantile criterion. Both finite and infinite horizons are considered. Finally we experimentally evaluate our approach on random MDPs and on a data center control problem.

Stochastic problem of competitive location of facilities with quantile criterion

A stochastic problem of facility location formulated as a discrete bilevel problem of stochastic programming with quantile criterion was proposed. Consideration was given there to a pair of competitive players successively locating facilities with the aim of maximizing their profits. For the case of discrete distribution of the random consumer demands, it was proposed to reduce the original problem to the deterministic problem of bilevel programming. A method to calculate the value of the objective function under fixed leader strategy and procedures to construct the upper and lower bounds of the optimal value of the objective function were proposed.

Comparison of two algorithms for solving a two‐stage bilinear stochastic programming problem with quantile criterion

The paper is devoted to solving the two‐stage problem of stochastic programming with quantile criterion. It is assumed that the loss function is bilinear in random parameters and strategies, and the random vector has a normal distribution. Two algorithms are suggested to solve the problem, and they are compared. The first algorithm is based on the reduction of the original stochastic problem to a mixed integer linear programming problem. The second algorithm is based on the reduction of the problem to a sequence of convex programming problems. Performance characteristics of both the algorithms are illustrated by an example. A modification of both the algorithms is suggested to reduce the computing time. The new algorithm uses the solution obtained by the second algorithm as a starting point for the first algorithm. Copyright © 2015 John Wiley & Sons, Ltd.

Inter-ply stitching optimisation of highly drapeable multi-ply preforms

An efficient finite element model has been developed in Abaqus/Explicit to solve highly non-linear fabric forming problems, using a non-orthogonal constitutive relation and membrane elements to model bi-axial fabrics. 1D cable-spring elements have been defined to model localised inter-ply stitch-bonds, introduced to facilitate automated handling of multi-ply preforms. Forming simulation results indicate that stitch placement cannot be optimised intuitively to avoid forming defects. A genetic algorithm has been developed to optimise the stitch pattern, minimising shear deformation in multi-ply stitched preforms. The quality of the shear angle distribution has been assessed using a maximum value criterion (MAXVC) and a Weibull distribution quantile criterion (WBLQC). Both criteria are suitable for local stitch optimisation, producing acceptable solutions towards the global optimum. The convergence rate is higher for MAXVC, while WBLQC is more effective for finding a solution closer to the global optimum. The derived solutions show that optimised patterns of through-thickness stitches can improve the formability of multi-ply preforms compared with an unstitched reference case, as strain re-distribution homogenises the shear angles in each ply.