Model Browser User's Guide

Two-Stage Models for Engines

Lindstrom and Bates [6] define repeated measurements as data generated by observing a number of individuals repeatedly under various experimental conditions, where the individuals are assumed to constitute a random sample from a population of interest. An important class of repeated measurements is longitudinal data where the observations are ordered by time or position in space. More generally, longitudinal data is defined as repeated measurements where the observations on a single individual are not, or cannot be, randomly assigned to the levels of a treatment of interest.

Modeling data of this kind usually involves the characterization of the relationship between the measured response, y, and the repeated measurement factor, or covariate x. Frequently, the underlying systematic relationship between y and x is nonlinear. In some cases the relevant nonlinear model can be derived on physical or mechanistic grounds. However, in other contexts a nonlinear relationship might be imposed simply to provide a convenient empirical description for the data. The presence of repeated observations on an individual requires particular care in characterizing the variation in the experimental data. In particular, it is important to represent two sources of variation explicitly: random variation among measurements within a given individual (intraindividual) and random variation among individuals (interindividual). Inferential procedures accommodate these different variance components within the framework of an appropriate hierarchical statistical model. This is the fundamental idea behind the analysis of repeated measurement data.

Holliday [1,2] was perhaps the first to apply nonlinear repeated measurements analysis procedures to spark injection engine data. The focus of Holliday's work was the modeling of data taken from engine mapping experiments. In these experiments, engine speed, load, and air/fuel ratio were held constant while spark was varied. Various engine response characteristics, for example, torque or emission quantities, were measured at each spark setting. Holliday modeled the response characteristics for each sweep as a function of spark advance. Variations in the individual sweep parameters were then modeled as a function of the global engine operating variables speed, load, and air/fuel ratio. Conceptually, variations in the measurements taken within a sweep represent the intraindividual component of variance. Similarly, variation in the sweep-specific parameters between sweeps represents the interindividual component of variance. You can generalize these principles to other steady-state engine modeling exercises where the nature of data collection usually involves sweeping a single engine control variable while the remainder are held at fixed values. These points suggest that nonlinear repeated measurements analysis represents a general approach to the parameterization of mean value engines models for controls-oriented development.

Another application for models of this form is the flow equations for a throttle body. Assuming the flow equations are based upon the usual one-dimensional isentropic flow principle, then they must be modified by an effective area term, A_e, which accounts for the fact that the true flow is multidimensional and irreversible. You can map the throttle flow characteristics by sweeping the throttle position at fixed engine speed. This data collection methodology naturally imposes a hierarchy the analysis of which is consistent with the application of nonlinear repeated measures. Experience in modeling effective area suggests that free knot spline or biological growth models provide good local predictions. The global phase of the modeling procedure is concerned with predicting the systematic variation in the response features across engine speed. A free knot spline model has proven useful for this purpose.

Maximum Likelihood Estimation

Local Models