Discrete Conditional Phase-type models utilising classification trees: Application to modelling health service capacities

Discrete Conditional Phase-type models (DC-Ph) consist of a process component (survival distribution) preceded by a set of related conditional discrete variables. This paper introduces a DC-Ph model where the conditional component is a classification tree. The approach is utilised for modelling health service capacities by better predicting service times, as captured by Coxian phase-type distributions, interfaced with results from a classification tree algorithm. To illustrate the approach, a case-study within the healthcare delivery domain is given, namely that of maternity services. The classification analysis is shown to give good predictors for complications during childbirth. Based on the classification tree predictions, the duration of childbirth on the labour ward is then modelled as either a two or three-phase Coxian distribution. The resulting DC-Ph model is used to calculate the number of patients and associated bed occupancies, patient turnover, and to model the consequences of changes to risk status.     

Computational learning of the conditional phase-type (C-Ph) distribution

This paper presents a new algorithm for learning the structure of a special type of Bayesian network. The conditional phase-type (C-Ph) distribution is a Bayesian network that models the probabilistic causal relationships between a skewed continuous variable, modelled by the Coxian phase-type distribution, a special type of Markov model, and a set of interacting discrete variables. The algorithm takes a data set as input and produces the structure, parameters and graphical representations of the fit of the C-Ph distribution as output. The algorithm, which uses a greedy-search technique and has been implemented in MATLAB, is evaluated using a simulated data set consisting of 20,000 cases. The results show that the original C-Ph distribution is recaptured and the fit of the network to the data is discussed.     

On the approximation of empirical data for service system simulations

The difficulties associated with parameter estimation for phase-type approximations of empirical data distributions in queue modelling are well known. While significant progress has been achieved in improving such approximations, difficulties in parameter estimation still limit the extent to which queue modelling is applied in practice. This paper presents a simplified technique for approximating empirical data in service system simulations, based on the specialised Cox phase-type distribution. When utilised in simulation modelling of a service system, the specialised Cox distribution is shown to provide improved approximations to the combined waiting and service time distribution without the need for complex parameter estimation techniques. This approach should enable much greater flexibility in the application of queue modelling to service systems.     

Modelling nationwide hospital length of stay: opening the black box

Hospital length of stay is considered to be a reliable and valid proxy for measuring the consumption of hospital resources. Average length of stay, however, albeit easy to quantify and calculate, does not suitably reflect the nature of such underlying distributions and may therefore mask the effects that the different streams of patients have on the system. This paper uses routinely collected and readily available nationwide data on stroke-related patients, aged 65 years and over, who were discharged from English hospitals over a 1-year period. This will be the basis for a running example illustrating the alternative methods of analysis and models of patients' length of stay. The methods include statistical methods: survival analysis, mixed exponential and phase-type distributions; and decision modelling techniques: compartmental and simulation models. The paper concludes by summarizing these various modelling techniques and by highlighting the similarity of the estimated parameters of patient flow as calculated by the phase-type distribution and compartmental modelling techniques.     

Compartmental modelling of equipment subject to partial repair

A semi-Markov model for equipment that can be repaired and returned to service in a less than new state is developed, where several less than new states are possible before the equipment finally fails. A subclass of phase-type distributions is used to model the times spent in each of these states. Then, exploiting the phase-type structure, the composite distribution of the entire lifetime of the equipment can be constructed. Data from railway wagon wheel-sets are used to illustrate this modelling and data analysis.     

Histogram based bounds and approximations for production lines

We present a modelling method for the analysis of production lines with generally distributed processing times and finite buffers. We consider the complete modelling process, from the data collection to the performance evaluation. First, the data about the processing times is supposed to be collected in the form of histograms. Second, tractable discrete phase-type distributions are built. Third, the evolution of the production line is described by a Markov chain, using a state model.     

A case study of modelling preventive maintenance of a production plant using subjective data

In this paper we describe a subjective data based case study carried out at a company manufacturing copper products, which parallels a previously published objective data based study at the same plant. The purpose of this study is twofold. The first is to model the implementation of Planned Preventive Maintenance (PM) to an Extrusion Press using the delay time concept. The second is to test a method for estimation of model parameters from subjective data in the context of delay time modelling. The parameter values of the underlying fault arrival process and the delay time distribution were, unlike in a parallel objective study, initially estimated from subjective data obtained through a questionnaire survey. Since bias was present in the initial subjective estimate, a method of removing it was developed to improve the model fit. On the basis of the data analysis and delay time modelling, improved PM policy and procedures were proposed to increase the effectiveness and efficiency of PM.     

A Poisson process reparameterisation for Bayesian inference for extremes

A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more difficult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter m. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved by minimising the correlation between the asymptotic posterior distribution of the parameters. This choice of m ensures more rapid convergence and efficient sampling from the joint posterior distribution using Markov Chain Monte Carlo methods. Samples from the parameterisation of interest are then obtained by a simple transform. Results are presented in the cases of identically and non-identically distributed models for extreme rainfall in Cumbria, UK.     

拟合隐函数曲线的GNL法

１参数可线性化的曲线拟合问题假设我们已经获得ｎ组独立观测值（ｘｉ,ｙｉ）,其中是精确观测值,ｉ＝１,２,…,ｎ,下同．现欲在最小二乘意义下,拟合非线性模型并假设观测点都分布在模型函数曲线附近．其中为连续可导函数ｇ（ｙ）的反函数,且在我们所研究的ｙ变化范围内恒有ｇ’（ｙ）≠０,（ｊ＝１,２.....下同）为ｐ（＜ｎ）个待定参数．（ｘ）为区间Ｘ内ｐ个线性无关的已知连续函数．随机变量,即为也就是要求求解残差平方和Ｑ（）的无条件极值问题其中参数向量,残差向量,而残差都是微量．对这类常见问题,有以下几种算法．…     

Modelling activities at a neurological rehabilitation unit

A queuing model of a specialist neurological rehabilitation unit is studied. The application is to the Neurological Rehabilitation Centre at Rookwood Hospital (Cardiff, UK), the national rehabilitation unit for Wales. Due to high demand this 21-bed inpatient facility is nearly always at maximum occupancy, and with a significant bed-cost per day this makes it a prime candidate for mathematical modelling. Central to this study is the concept that treatment intensity has an effect on patient length of stay. The model is constructed in four stages. First, appropriate patient groups are determined based on a number of patient-related attributes. Second, a purpose-built scheduling program is used to deduce typical levels of treatment to patients of each group. These are then used to estimate the mean length of stay for each patient group. Finally, the queuing model is constructed. This consists of a number of disconnected homogeneous server queuing systems; one for each patient group. A Coxian phase-type distribution is fitted to the length of time from admission until discharge readiness and an exponential distribution models the remainder of time until discharge. Some hypothetical scenarios suggested by senior management are then considered and compared on the grounds of a number of performance measures and cost implications.     

Modelling the joint distribution of competing risks survival times using copula functions

The problem of modelling the joint distribution of survival times in a competing risks model, using copula functions, is considered. In order to evaluate this joint distribution and the related overall survival function, a system of non-linear differential equations is solved, which relates the crude and net survival functions of the modelled competing risks, through the copula. A similar approach to modelling dependent multiple decrements was applied by Carriere [Carriere, J., 1994. Dependent decrement theory. Transactions, Society of Actuaries XLVI, 45-65] who used a Gaussian copula applied to an incomplete double-decrement model which makes it difficult to calculate any actuarial functions and draw relevant conclusions. Here, we extend this methodology by studying the effect of complete and partial elimination of up to four competing risks on the overall survival function, the life expectancy and life annuity values. We further investigate how different choices of the copula function affect the resulting joint distribution of survival times and in particular the actuarial functions which are of importance in pricing life insurance and annuity products. For illustrative purposes, we have used a real data set and used extrapolation to prepare a complete multiple-decrement model up to age 120. Extensive numerical results illustrate the sensitivity of the model with respect to the choice of copula and its parameter(s).     

Modelling maintenance practice of production plant using the delay-time concept

In this paper we present a study carried out for a copper productsmanufacturing company, developing and applying the delay-timemodelling technique to model and thus optimize preventive maintenance(PM) of the plant. A key machine in the plant is used to illustratethe modelling process and management reaction. The parameter values of the process by which faults arise andof the delay-time distribution are estimated from maintenancerecord data of failures and faults found at PM, using the methodof maximum likelihood. A test of the model fit to data is carriedout. Based upon the estimated model parameters and the failuredelay time, an inspection model is proposed to describe therelationship between the total downtime and the PM interval.     

Asymptotics for the sojourn time distribution in the queue defined by a general QBD process with a countable phase space

We consider a FIFO queue defined by a QBD process. When the number of phases of the QBD process is finite, it has been proved that the stationary distribution of sojourn times in that queue can be represented as a phase-type distribution. In this paper, we extend this result to the case where the number of phases of the QBD process is countably many and obtain several kinds of asymptotic formula for the steady-state tail probability of sojourn times in the queue when the tail probability decays in exact exponential form.     

利用故障数据和估计的检查数据建立维修优化模型

利用时间延迟概念,根据故障记录数据和估计的检查数据建立了预防维修模型.通过对故障记录数据统计分析,提出了模型的假定条件.采用最大似然估计法,估计参数,包括缺陷发生率、不完全检查概率和时间延迟分布.建立了有关预防维修间隔期和总停机时间之间关系的检查模型,并根据估计参数和检查模型,计算最佳维修间隔期.     

二个四参数增长曲线模型参数初始值的确定方法与模型应用

对非线性回归模型进行非线性最小二乘估计一般需要确定参数初始值.在非线性回归模型中,General Logistic模型和Von Bertalanffy模型是二个含有四参数的增长曲线模型,对数据的拟合有较强的适应性,应用较为广泛.本文给出这两个模型参数初始值的确定方法,并应用于实际拟合,得到很好的效果.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.     

Least squares surface approximation to scattered data using multiquadratic functions

The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points (or base points for the multiquadratic basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadric parameter is explored, with some unexpected results in terms of “near repeated” knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.     

Minimizing transient times in a coupled solid‐state laser model

We consider a system of ordinary differential equations modelling the dynamics of two coupled solid‐state lasers. Under the dynamics, this system may execute transitions between in‐phase and out‐of‐phase states. For satellite communications and high‐speed data transfer the transition times should be reduced to their shortest possible duration. In this paper, we apply optimal control theory to find the values of various laser parameters (e.g. the amplitude of the injected field, detunings, and coupling constants) which minimize the transient times between out‐of‐phase and in‐phase states. The effect of each parameter is shown to be independent of the other two, and the transient time is shown to be a strictly increasing function of detuning and a strictly decreasing function of the coupling constant and amplitude of the injected field. The effect of initial conditions on transient times is also analysed. Published in 2005 by John Wiley & Sons, Ltd.     

Bayesian estimation of finite time ruin probabilities

In this paper, we consider Bayesian inference and estimation of finite time ruin probabilities for the Sparre Andersen risk model. The dense family of Coxian distributions is considered for the approximation of both the inter‐claim time and claim size distributions. We illustrate that the Coxian model can be well fitted to real, long‐tailed claims data and that this compares well with the generalized Pareto model. The main advantage of using the Coxian model for inter‐claim times and claim sizes is that it is possible to compute finite time ruin probabilities making use of recent results from queueing theory. In practice, finite time ruin probabilities are much more useful than infinite time ruin probabilities as insurance companies are usually interested in predictions for short periods of future time and not just in the limit. We show how to obtain predictive distributions of these finite time ruin probabilities, which are more informative than simple point estimations and take account of model and parameter uncertainty. We illustrate the procedure with simulated data and the well‐known Danish fire loss data set. Copyright © 2009 John Wiley & Sons, Ltd.     

带干扰phaes-type风险模型中的最大盈余及相关分布

本文考虑了索赔时间间距为phase-type分布时带干扰更新风险模型中的破产前最大盈余、破产后赤字的分布,建立了相应的积分-微分方程.最后,讨论了当索赔时间间距为Erlang(2)分布且索赔量满足指数分布时的特殊情形.