Piecewise Linear Approximation of Multivariate Functions

Linear approximation is a popular and economical way to gain appreciation of the behavior of functional relationships. Especially in higher dimensions, making sense out of a sample of data points in terms of the underlying multivariate relationship is greatly facilitated by some form of piecewise linear approximation. Such an approximation allows for estimation of the function at values not included in the sample, sheds light on the activity of the function at selected neighborhoods, and identifies regions where the relationship behaves interestingly. For the sake of concreteness let us consider a typical problem. A simulation model of a computer system has been given. The workload driving the system is described by two variables giving the respective 1463 percentages of two classes of users in the user population; there are three classes of users altogether. The two variables, then, form a bivariate parametrization of the workload. Six simulations were run, and the resulting mean response times are shown below. (This response time function will be denoted by h throughout this paper.) h (60, 25) = 5.5 /i (60, 7) = 2 . 1 h (40, 30) = 1.2 h (40, 7) = 1.0 h (20, 25) = 0.7 A (20, 15) = 0.7 It will be necessary to estimate the response times for many more workloads than for the six simulations already run. Yet, it would be inefficient to make a run for each possible parameter pair (jci, X2). It would also not be necessary, given the qualitatively simple relationship that generally exists between workloads and response times of computer systems.