456:
443:
determined in this manner need to be substituted into the nonlinear model to assess improvements in the model errors. The absence of a significant improvement indicates the available data is not able to improve the current model form using the defined parameters. Extra parameters can be inserted into
414:
on the residuals is not particularly useful. The chi squared test requires known standard deviations which are seldom available, and failed tests give no indication of how to improve the model. There are a range of methods to compare both nested and non nested models. These include comparison of
426:
using linear regression will show if the residuals can be predicted. Residuals that cannot be predicted offer little prospect of improving the model using the current operating conditions. Terms that do predict the residuals are prospective terms to incorporate into the model to improve its
46:
combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature. Thus, almost all models are grey box models as opposed to
386:
techniques. For more than one parameter the method extends in a direct manner. After checking that the model has been improved this process can be repeated until convergence. This approach has the advantages that it does not need the parameters
430:
The model inversion technique above can be used as a method of determining whether a model can be improved. In this case selection of nonzero terms is not so important and linear prediction can be done using the significant
605:
Kroll, Andreas (2000). Grey-box models: Concepts and application. In: New
Frontiers in Computational Intelligence and its Applications, vol.57 of Frontiers in artificial intelligence and applications, pp. 42-51. IOS Press,
908:
Whiten, W.J., 1971. Model building techniques applied to mineral treatment processes, Symp. on
Automatic Control Systems in Mineral Processing Plants, (Australas. Inst. Min. Metall., S. Queensland Branch, Brisbane),
175:
includes a constant term and possibly transformed values of the original operating conditions to obtain non-linear relations between the original operating conditions and
87:
with a partial theoretical structure and some unknown parts derived from data. Models with unlike theoretical structures need to be evaluated individually, possibly using
981:
Kojovic, T., and Whiten W. J., 1994. Evaluation of the quality of simulation models, Innovations in mineral processing, (Lauretian
University, Sudbury) pp 437–446.
17:
1002:
Kojovic, T., 1989. The development and application of Model - an automated model builder for mineral processing, PhD thesis, The
University of Queensland.
102:
or variable parameter relations may need to be found. For a particular structure it is arbitrarily assumed that the data consists of sets of feed vectors
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877:
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Xiao, J., 1998. Extensions of model building techniques and their applications in mineral processing, PhD thesis, The
University of Queensland.
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Whiten, W.J., 1994. Determination of parameter relations within non-linear models, SIGNUM Newsletter, 29(3–4,) 2–5. 10.1145/192527.192535.
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316:, that can be examined using efficient term selection and evaluation of the linear regression. For the simple case of a single
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Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (1 November 2013).
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to be able to be determined from an individual data set and the linear regression is on the original error terms
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Where sufficient data is available, division of the data into a separate model construction set and one or two
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243:
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Heaton, J., 2012. Introduction to the math of neural networks, Heaton
Research Inc. (Chesterfield, MO),
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that can be used to determine if they are significantly different from zero, thus providing a method of
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Nash, J.C. and Walker-Smith, M. 1987. Nonlinear parameter estimation, Marcel Dekker, Inc. (New York).
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is recommended. This can be repeated using multiple selections of the construction set and the
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122:, as well as other values. In many cases a model can be converted to a function of the form:
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18:
Knowledge talk:Articles for creation/Mathematical models: Grey box completion and validation
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619:, Proc. 17th World Congress, Int. Federation of Automatic Control, Seoul. pp 11415-11420
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techniques are much more efficient than most non-linear techniques. The model can be
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is now in a linear position with all other terms known, and thus can be analyzed by
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models that are purely theoretical. Some models assume a special form such as a
31:
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35:
941:
Determining the form of ordinary differential equations using model inversion
234:. Selection of the nonzero terms can be done by optimization methods such as
48:
203:
Once a selection of non-zero values is made, the remaining coefficients in
183:
are non-zero and assigning their values. The model completion becomes an
798:
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007).
144:
gives some variable parameters that are the model's unknown parts.
649:
Model completion and validation using inversion of grey box models
568:
Practical Grey-box
Process Identification: Theory and Applications
281:
and estimating their values. Once the non-zero values are located
75:(i.e. containing random components) depending on its planned use.
155:
in a manner to be determined. This relation can be specified as
766:
Lawson, Charles L.; J. Hanson, Richard (1 December 1995).
27:
Mathematical data production model with limited structure
312:) into an approximate linear form in the elements of
63:. These have special analysis methods. In particular
226:
over the data with respect to the nonzero values in
246:can provide accuracy estimates for the elements of
664:Draper, Norman R.; Smith, Harry (25 August 2014).
179:. It is then a matter of selecting which terms in
444:the model to make this test more comprehensive.
257:It is sometimes possible to calculate values of
617:Grey box modelling - branches and experiences
8:
1019:
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804:(3rd ed.). Cambridge University Press.
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407:or used to evaluate prediction differences.
187:problem to determine the non-zero values in
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651:, ANZIAM J.,54 (CTAC 2012) pp C187–C199.
571:. Springer Science & Business Media.
167:is a matrix of unknown coefficients, and
943:, ANZIAM J. 55 (EMAC2013) pp.C329–C347.
870:Grey Box Modelling for Nonlinear Systems
670:. John Wiley & Sons. pp. 657–.
615:Sohlberg, B., and Jacobsen, E.W., 2008.
140:, and the model predictions. The vector
878:Kaiserslautern University of Technology
565:Bohlin, Torsten P. (7 September 2006).
552:
694:Weisberg, Sanford (25 November 2013).
415:model predictions with repeated data.
277:thus selecting the non-zero values in
828:Bayesian Data Analysis, Third Edition
98:Within a particular model structure,
7:
418:An attempt to predict the residuals
151:vary with the operating conditions
118:will contain values extracted from
1024:Linhart, H.; Zucchini, W. (1986).
737:Stergiou, C.; Siganos, D. (2013).
285:can be used on the original model
261:for each data set, directly or by
136:gives the errors between the data
110:, and operating condition vectors
51:where no model form is assumed or
25:
1081:Deming, William Edwards (2000).
454:
296:, which converts the non-linear
207:can be determined by minimizing
496:Nonlinear system identification
191:that minimizes the error terms
1057:Subset Selection in Regression
1054:Miller, Alan (15 April 2002).
769:Solving Least Squares Problems
422:with the operating conditions
1:
370:q m’(f,p,q*) = m(f,p.q*) + (a
667:Applied Regression Analysis
593:"Grey-box model estimation"
410:A statistical test such as
1135:
853:Supported grey box models
265:. Then the more efficient
132:where the vector function
697:Applied Linear Regression
405:resulting models averaged
289:to refine these values .
283:non-linear least squares
263:non-linear least squares
244:non-linear least squares
232:non-linear least squares
269:can be used to predict
240:evolutionary algorithms
40:computational modelling
1084:Out of the Crisis p272
83:The general case is a
1119:Mathematical theorems
1114:Mathematical modeling
531:System identification
481:Design of experiments
964:Spline (mathematics)
595:. Mathworks 2. 2012.
511:Scientific modelling
501:Parameter estimation
476:Computer simulation
471:Computer experiment
236:simulated annealing
89:simulated annealing
939:Whiten, B., 2014.
867:Hauth, J. (2008),
647:Whiten, B., 2013.
536:System realization
491:Mathematical model
462:Mathematics portal
373:c − q*) m’(f,p,q*)
331:) and an estimate
292:A third method is
106:, product vectors
93:genetic algorithms
1094:978-0-262-54115-2
1067:978-1-4200-3593-3
1037:978-0-471-83722-0
851:Mathworks, 2013.
838:978-1-4398-4095-5
811:978-0-521-88068-8
801:Numerical Recipes
779:978-0-89871-356-5
739:"Neural networks"
707:978-1-118-59485-8
677:978-1-118-62568-2
578:978-1-84628-403-8
521:Statistical model
437:regression matrix
384:linear regression
366:q) ≈ m(f,p.q*) +
267:linear regression
173:linear regression
65:linear regression
57:linear regression
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486:Grey box testing
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439:. The values in
395:Model validation
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199:Model completion
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294:model inversion
230:, typically by
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195:over the data.
147:The parameters
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747:. Retrieved
743:the original
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749:2013-07-03
606:Amsterdam.
548:References
516:Simulation
100:parameters
79:Model form
73:stochastic
36:statistics
1030:. Wiley.
700:. Wiley.
287:m(f,p,Ac)
193:m(f,p,Ac)
53:white box
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