Estimated-response error bounds

Figure 1. Plots showing the Calibration Process. A. Response transformation to constant variance Examples showing a. too little, b. appropriate, and c. too much transformation power. B. Amount Transformation in conforming to a (linear) model. C. Construction of p. confidence bands about the regressed line, q. response error bounds and intersection of these to determine r. the estimated amount interval.

Table XI. Estimated Values of the Response Error Bounds from Inverse Transformed Data. a 0.025 where 95% of Response Unknowns Will Lie within the Response Error...

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

Numerical experiments (Walsh, 1993) indicate that the peak-to-peak deviation of the exit concentration can be bounded, for fairly tightly tuned controllers, by calculating the response of the exit concentration to a reagent valve exhibiting a square wave oscillation with peak-to-peak amplitude equal to the deadband error. The exit concentration variation can therefore be estimated as... 

Another important role of the noise model in the iterative algorithm is to ensure whiteness of the residuals. This allows us to estimate the covariance of the FSF model parameter estimates and then to develop statistical confidence bounds for the corresponding step response estimates. In order to apply these results, it is important that the bias error in the model arising due to unmodelled dynamics be small relative to the variance error caused... 

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