Model Browser User's Guide

Local Models

Modeling responses locally within a sweep as a function of the independent variable only. That is,

for

(6-1)

where the subscript i refers to individual tests and j to data within a test, is the j^th independent value, _i is a (rx1) parameter vector, is the j ^th response, and is a normally distributed random variable with zero mean and variance ². Note that Equation 6-1 can be either a linear or a nonlinear function of the curve fit parameters. The assumption of independently normally distributed errors implies that the least squares estimates of are also maximum likelihood parameters.

Local Covariance Modeling

The local model describes both the systematic and random variation associated with measurements taken during the i^th test. Systematic variation is characterized through the function f while variation is characterized via the distributional assumptions made on the vector of random errors e_i. Hence, specification of a model for the distribution of e_i completes the description of the intratest model. The Model-Based Calibration Toolbox allows a very general specification of the local covariance,

(6-2)

where C_i is an (n_i x n_i) covariance matrix, is the coefficient of variation, and _i is a (q-by-1) vector of dispersion parameters that account for heterogeneity of variance and the possibility of serially correlated data. The specification is very general and affords considerable flexibility in terms of specifying a covariance model to adequately describe the random component of the intratest variation.

The Model-Based Calibration Toolbox supports the following covariance models:

• Power Variance Model

(6-3)

• Exponential Variance Model

(6-4)

• Mixed Variance Model

(6-5)

where diag{x} is a diagonal matrix.

Correlation models are only available for equispaced data in the Model-Based Calibration Toolbox. It is possible to combine correlation models with models with the variance models such as power.

One of the simplest structures that can be used to account for serially correlated errors is the AR(m) model (autoregressive model with lag m). The general form of the AR(m) model is

(6-6)

where is the k^th lag coefficient and v_j is an exogenous stochastic input identically and independently distributed as . First- and second-order autoregressive models are implemented in the Model-Based Calibration Toolbox.

Another possibility is a moving average model (MA). The general structure is

(6-7)

where is the k^th lag coefficient and v_j is an exogenous stochastic input identically and independently distributed as . Only a first-order moving average model is implemented in the Model-Based Calibration Toolbox.

Response Features

From an engineering perspective, the curve fit parameters do not usually have any intuitive interpretation. Rather characteristic geometric features of the curve are of interest. The terminology "response features" of Crowder and Hand [7] is used to describe these geometric features of interest. In general, the response feature vector p_i for the i^th sweep is a nonlinear function (g) of the corresponding curve fit parameter vector _i, such that

(6-8)

Two-Stage Models for Engines

Global Models