# Ersatz estimation: gradient relationships in complex systems under uncertainty

*by Associate Professor Felisa Vazquez-Abad*

*Institution:*University of Melbourne

*Date: Wed 13th April 2005*

*Time: 11:00 AM*

*Location: Russell Love Theatre, Richard Berry Building*

*Abstract*: In this talk we introduce the problem of derivative (or more

generally, gradient) estimation, when the quantity of interest is the

expected cost incurred by a stochastic system, so that only noisy

measurements are observable. For the past three decades researchers have

tried to find ways to build statistics that will measure derivatives or

trends (sensitivities) of the expected cost, when one or several

parameters change values. Gradient estimation can be used for

optimization, estimation, or identification of parameters. To set the

ideas, we will first show simple examples where unbiased estimators of the

derivatives can be easily constructed.

In real systems, such as telecommunication networks, insurance portfolios,

and transportation systems, it is often either very difficult or very

costly to estimate gradients with respect to the control variables of

interest. In the 1990's, several researchers independently proposed ad-hoc

transformations of the derivatives to simplify the problem. Since that

time, I have tried to formulate a general methodology to justify such

transformations, which allow us to express the derivatives with respect to

the control variables of interest in terms of simpler derivatives,

obtained with respect to other parameters in the model. An \"ersatz\"

derivative is the estimator obtained via the inverse transformations. My

approach uses basic methods in Physics, such as time scale changes and

changes of variables, or of reference framework.

We will work out a complete example of a time scale change, discuss its

application to a queueing problem, an insurance problem and a

transportation problem, and then give a counter example where the

transformations do NOT work. For this latter example, we will apply our

general methodology to identify a correction term for the ersatz

derivative, to eliminate the bias. This example provides insight into

the effects of uncertainty and justifies the term \"ersatz\", meaning a

surrogate of lesser quality, because in many cases it is unrealistic to

compute the exact correction terms.

In this talk I will attempt to give a general idea of an on-going

research effort, rather than to present a finished Theory of gradient

estimation.

*For More Information:* Paul Norbury email: pnorbury@ms.unimelb.edu.au or Ole Warnaar email: warnaar@ms.unimelb.edu.au