Probabilistic vs. Deterministic Models - Modeling Uncertainty

Diagnosis Subtopics:

Diagnostic systems inherently make assumptions on uncertainty. The only question is whether this uncertainty is explicit, or is hidden inside of “black box” techniques, or is just part of engineering judgment during tuning.

Even in the simple case of manually setting a simple alarm/event threshold for event detection, probability is a factor. If the threshold is set too far from normal operations, you will fail to detect some problems. If the threshold is set too close to normal operations, you risk getting “false positives” during small transient disturbances. Statistically, this is balancing type I error vs. type II error. These error probabilities could be estimated using process data and knowledge of whether the problem actually existed. But usually this tuning setting is just based on engineering judgment and experience -- the probabilities in that case are only implicit in the designer’s head, not ever formally stated.

For models, we say they are deterministic if they include no representation of uncertainty. First principles, engineering design models generally are deterministic. But the uncertainty representations used for estimation and diagnosis are usually extensions the deterministic model. And in the case of empirically derived models such as regression models, the uncertainty is generally available as a byproduct of the regression or other procedures used.

Consider first the role of uncertainty when estimating variable values. Models used in estimating variable values include some form of uncertainty in the measurements, in the model, or both. Estimation is closely linked to diagnosis, and in many cases the techniques are extensions of estimation techniques. Diagnosis can be considered estimation of additional discrete states representing faults. 

In the case of estimation with a Kalman filter, the starting point is a deterministic model of the system state (differential equation or difference equations, plus an algebraic relation between states and measurements). But then the filter model adds in uncertainty in the measurements (“measurement noise”), and uncertainty in the model (“process noise”). The key tuning parameters (expressed as covariance matrices) are assumptions on the measurement noise and process noise. One derivation of the Kalman filter equations is as the solution to a least squares problem minimizing a weighted combination of measurement adjustments and model error calculations to achieve an optimal balance between the two. (The weighting is based on the inverses of the covariance matrices for the measurement noise and process noise. The static case of data reconciliation is a special case, using only algebraic equations, and asserting that the process noise is zero. (More general steady state estimation could also allow process noise, and really should in most cases, because of model uncertainty introduced in physical properties and other calculations).

The model for estimation depends on explicit assumptions on the noise. These can be estimated from historical data, or simply set as tuning parameters using engineering judgment and rules of thumb. For instance, for measurements, we normally assume independent measurements (diagonal covariance matrix) and “typical” variances (starting with typical standard deviations as a percent of the instrument range) for different types of sensors, possibly adjusting for the type of service they are in.

Diagnostic techniques based on these models typically look at model residuals or measurement adjustments. In either case, the tests are based on the assumed uncertainties.

Other forms of models are stated directly in probabilistic terms. An example of that is Bayesian modeling.

Copyright 2010 - 2013, Greg Stanley

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