Model Browser User's Guide

Covariance Modeling

This frame is visible no matter what form of local model is selected in the list.

Covariance modeling is used when there is heteroscedasticity. This means that the variance around the regression line is not the same for all values of the predictor variable, for example, where lower values of engine speed have a smaller error, and higher values have larger errors, as shown in the following example. If this is the case, data points at low speed are statistically more trustworthy, and should be given greater weight when modeling. Covariance models are used to capture this structure in the errors.

You can fit a model by finding the smallest least squares error statistic when there is homoscedasticity (the variance has no relationship to the variables). Least squares fitting tries to minimize the least squares error statistic

, where is the error squared at point i.

When there is heteroscedasticity, covariance modeling weights the errors in favor of the more statistically useful points (in this example, at low engine speed N). The weights are determined using a function of the form

where is a function of the predictive variable (in this example, engine speed N).

There are three covariance model types.

Power

These determine the weights using a function of the form . Fitting the covariance model involves estimating the parameter .

Exponential

These determine the weights using .

Mixed

These determine the weights using . Note that in this case there are two parameters to estimate, therefore using up another degree of freedom. This might be influential when you choose a covariance model if you are using a small data set.

Correlation Models

These are only supported for equally spaced data in the Model-Based Calibration Toolbox. When data is correlated with previous data points, the error is also correlated.

There are three methods available.

MA(1) - The Moving Average method has the form .

AR(1) - The Auto Regressive method has the form .

AR(2) - The Auto Regressive method of the form

is a stochastic input, .

Local Model Class: Linear Models

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