Algorithms precision

There is not always a clear answer or a strictly defined proper choice of a three-way model for a given application. Sometimes more than one three-way model has to be tried in order to find the proper one. In many cases, the models appear equally good and the selection may be based on practical criteria such as algorithm precision or speed. Knowing the palette of three-way models along with their properties and relationships serves as a basis for making objective choices. 

The same type of processing using digital means would require many digital channels and some fancy digital algorithms. Precision in the digital world translates into numbers of channels for... 

A straightforward derivation (not reproduced here) shows that the effect of the diree successive steps embodied in equation (b3.3.7), with the above choice of operators, is precisely the velocity Verlet algorithm. This approach is particularly usefiil for generating multiple time-step methods. 

The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

The Verlet algorithm has the numerical disadvantage that the new positions are obtained by adding a term proportional to Af to a difference in positions (2r — r, i). Since At is a small number and (2r, — r, i) is a difference between two large numbers, this may lead to truncation errors due to finite precision. The Verlet furthermore has the disadvantage that velocities do not appear explicitly, which is a problem in connection with generating ensembles with constant temperature, as discussed below. 

Comment The sequence of digits in each coefficient depends on the precision (e.g., three decimal places) and number of tabulated values (34, 50, or 64), the form of the optimization software used (Hewlett Packard HP71B Curve Fit Module), and the number of coefficients chosen (3. .. 8). Discrepancies between the approximated and the real table entries of up to j-LSD could be due either to insufficiencies of the algorithm or the rounding of table entries. The few LRR that are above 1% do not pose a risk for practical applications. 

Equations 13.14 to 13.16 constitute the well known recursive least squares (RLS) algorithm. It is the simplest and most widely used recursive estimation method. It should be noted that it is computationally very efficient as it does not require a matrix inversion at each sampling interval. Several researchers have introduced a variable forgetting factor to allow a more precise estimation of 0 when the process is not "sensed" to change. 

As it has been shown in this chapter knowing the concentrations of chemicals in the environment is a key aspect in order to carry out meaningful hazard and risk assessment studies. Predicting concentrations of chemicals can serve as a quick and robust way to produce an acceptable screening level assessment however if further precision is desired, the complexity of real environmental scenarios can make it a cumbersome and unaffordable task. Models improvement requires not only refining their computation algorithms but also and more important, implementing new inputs and processes in order to better describe real scenarios. 

This work covers only open loop response. Use of this algorithm alone on a real reactor would presuppose that the model is very precise — that a desired MW and S can be obtained merely by... 

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