Stochastic error temporal

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/models that are static or dynamic and linear or nonlinear. Only the stationary case win be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuals). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed models are in discretetime form (e.g., difference equations instead of differential equations, discrete summations instead of integrals). 

It is important to note that the approximations made in the derivation of Eqs. (2) and (4) are questionable for very short times. As a result, it is expected that short time dynamics from LD and BD simulations are likely to contain significant errors. For example, the equation of motion for BD simulations (Eq. (4)) neglects inertial terms. Because of this, all motions are damped by the same friction coefficient and must scale exactly with the viscosity. This is, of course, inconsistent with the experimental observation that high frequency motions such as bond vibrations are largely independent of the molecular environment. Although LD simulations retain the inertial term, spatial and temporal correlations in the stochastic forces at short times are neglected. Even GLE simulations, which contain some of these correlations, would not be expected to be quantitatively correct at short times. 

According to GUM, a measurement has imperfections which give rise to errors in the measurement result. A random error presumably arises from unpredictable or stochastic temporal and spatial variations of influence quantities. Although it is not possible to compensate for random error, it can usually be reduced by increasing the number of observations. Systematic error, like random error, cannot be eliminated but it too can often be reduced. Once the effect causing the systematic error has been recognized, the effect can be quantifled and a correction can be applied to compensate for the effect. The uncertainty of the result of a measurement reflects the lack of exact knowledge of the value of the measurand. 

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