Introduction

We consider the problem of inspecting and controlling a system. The system can be physiological, medical, technological, or an experiment such a clinical trial. The system state is described by a stochastic process Xt. The arguments are developed with few restrictions on the process. The arguments allow systems with monotone or non-monotone trajectories to be analyzed. The trajectories can be status measurements from clinical trials where the decision may be to stop the trial, choose between competing treatments [Bet98], or to change the treatment regimes. For an individual whose health status is being monitored the decision may be to make an intervention or to adjust the treatment regime. Typically if a status indicator crosses a critical level an intervention will be required. The extent of the intervention determines the future development of the system. We address the problem with two tools, firstly, the standard method of seeking the regeneration points of the stochastic process, secondly, by considering an associated stochastic real valued process. We thus begin with the underlying process Xt and work with a bivariate process (Xt,Yt).

The process Yt is generally constructed by applying a functional to the basic process, Yt = A(Xt), examples are the construction of a statistic or making a measurement. Within the models Yt can also describe a covariate process, an imperfectly observed version of Xt, or a associated process used as a surrogate. The advantages of the approach are that decisions can be based the entry of the pair (Xt, Yt) into a critical region, or on the associated process Yt alone. The model structure allows Xt to be unobservable. Depending on the context of the problem, either of Xt or Yt can be integrated out of the final results to give any desired marginal distribution.

The aim of the article is to show how optimal policies for a system described by the stochastic process Xt. Initially we make no strong assumptions about the process, only that it has the properties required to allow the necessary computations. The properties of the process are made explicit in particular examples. Examples arise in many ways, from the construction of statistics, for example to assist treatment comparisons [Bet98]; the consideration of marker processes in HIV infected patients [JK96]; in risk analysis for engineering projects [Noo96]; fatigue crack growth [Sob87, New91, New98]. The approach in this paper differs through the use of the bivariate process (Xt,Yt) motivated by an extension of the methodology in an earlier paper [New04]. The process can be monotone where the state is the level of a drug in the blood, whereas a measurement such as heart rate or blood pressure will be non-monotone. A simple example of the use of the maximum process is provided by temperature or blood pressure monitoring. It is a compromise between continuous monitoring and monitoring only at certain times. The associated process is frequently defined by a functional of the underlying process, namely Yt = A(Xt). Decision making is simplified if the process Yt has monotone sample paths; the map from Xt to Yt may be in two stages, the first an aggregation followed by taking the maximum of the aggregated process to guarantee monotonicity. Natural examples of an associated process are:

(c) an accumulation process Yt = J Xsds;

(e) errors in measurement, a distribution Fytixt describes the dependence of the observed process on the true process;

(f) covariate processes, a distribution FXt | yt describes the dependence of Xt on covariate Yt.

When the underlying process Xt is a Wiener process (a) above is well known, (b) is a Bessel process, and (c) is the Kolmogorov diffusion [McK63] which arises in a regression model [GJW99].

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