Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Grey box model
open-in-new
<h2 id="model-form">Model form</h2> <p>The general case is a <a href="/facts/Non-linear_system/4aYItALC">non-linear model</a> with a partial theoretical structure and some unknown parts derived from data. Models with unlike theoretical structures need to be evaluated individually,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> possibly using <a href="/facts/Simulated_annealing/JCmHqd3t">simulated annealing</a> or <a href="/facts/Genetic_algorithms/WP2AFWuW">genetic algorithms</a>. </p><p>Within a particular model structure, <a href="/facts/System_identification/v2xk1X0e">parameters</a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> or variable parameter relations<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> may need to be found. For a particular structure it is arbitrarily assumed that the data consists of sets of feed vectors f, product vectors p, and operating condition vectors c.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> Typically c will contain values extracted from f, as well as other values. In many cases a model can be converted to a function of the form:<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> m(f,p,q) <p>where the vector function m gives the errors between the data p, and the model predictions. The vector q gives some variable parameters that are the model's unknown parts. </p><p>The parameters q vary with the operating conditions c in a manner to be determined.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a><a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> This relation can be specified as q = Ac where A is a matrix of unknown coefficients, and c as in <a href="/facts/Linear_regression/5n998IhK">linear regression</a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> includes a constant term and possibly transformed values of the original operating conditions to obtain non-linear relations<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> between the original operating conditions and q. It is then a matter of selecting which terms in A are non-zero and assigning their values. The model completion becomes an <a href="/facts/Mathematical_optimization/oRn8Iv5I">optimization</a> problem to determine the non-zero values in A that minimizes the error terms m(f,p,Ac) over the data.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a><a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> </p> <h2 id="model-completion">Model completion</h2> <p>Once a selection of non-zero values is made, the remaining coefficients in A can be determined by minimizing <i>m</i>(<i>f</i>,<i>p</i>,<i>Ac</i>) over the data with respect to the nonzero values in A, typically by <a href="/facts/Non-linear_least_squares/3K5Nmz0S">non-linear least squares</a>. Selection of the nonzero terms can be done by optimization methods such as <a href="/facts/Simulated_annealing/JCmHqd3t">simulated annealing</a> and <a href="/facts/Evolutionary_algorithms/bo7eHlcA">evolutionary algorithms</a>. Also the <a href="/facts/Non-linear_least_squares/3K5Nmz0S">non-linear least squares</a> can provide accuracy estimates<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a><a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> for the elements of A that can be used to determine if they are significantly different from zero, thus providing a method of <a href="/facts/Model_selection/R11QK2GC">term selection</a>.<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a><a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> </p><p>It is sometimes possible to calculate values of q for each data set, directly or by <a href="/facts/Non-linear_least_squares/3K5Nmz0S">non-linear least squares</a>. Then the more efficient <a href="/facts/Linear_regression/5n998IhK">linear regression</a> can be used to predict q using c thus selecting the non-zero values in A and estimating their values. Once the non-zero values are located <a href="/facts/Non-linear_least_squares/3K5Nmz0S">non-linear least squares</a> can be used on the original model m(f,p,Ac) to refine these values .<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a> </p><p>A third method is <a href="/facts/Model_inversion/Fe01pQ9E">model inversion</a>,<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a><a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a><a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> which converts the non-linear m(f,p,Ac) into an approximate linear form in the elements of A, that can be examined using efficient term selection<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a><a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> and evaluation of the linear regression.<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a> For the simple case of a single q value (q = aTc) and an estimate q* of q. Putting dq = aTc − q* gives </p> m(f,p,aTc) = m(f,p,q* + dq) ≈ m(f,p.q*) + dq m’(f,p,q*) = m(f,p.q*) + (aTc − q*) m’(f,p,q*) <p>so that aT is now in a linear position with all other terms known, and thus can be analyzed by <a href="/facts/Linear_regression/5n998IhK">linear regression</a> techniques. For more than one parameter the method extends in a direct manner.<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a><a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a><a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a> 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 q to be able to be determined from an individual data set and the linear regression is on the original error terms<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a> </p> <h2 id="model-validation">Model validation</h2> <p>Where sufficient data is available, division of the data into a separate model construction set and one or two <a href="/facts/Cross-validation_(statistics)/pDWg8s0e">evaluation sets</a> is recommended. This can be repeated using multiple selections of the construction set and the <a href="/facts/Bootstrap_aggregating/EcIjuS4x">resulting models averaged</a> or used to evaluate prediction differences. </p><p>A statistical test such as <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared</a> on the residuals is not particularly useful.<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a> The chi squared test requires known standard deviations which are seldom available, and failed tests give no indication of how to improve the model.<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a> There are a range of methods to compare both nested and non nested models. These include comparison of model predictions with repeated data. </p><p>An attempt to predict the residuals m(, ) with the operating conditions c using linear regression will show if the residuals can be predicted.<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a><a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a> Residuals that cannot be predicted offer little prospect of improving the model using the current operating conditions.<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a> Terms that do predict the residuals are prospective terms to incorporate into the model to improve its performance.<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a> </p><p>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 <a href="/facts/Eigenvectors/8TjEoT8u">eigenvectors</a> of the <a href="/facts/Design_matrix/yRTdMhf4">regression matrix</a>. The values in A 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.<a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a> Extra parameters can be inserted into the model to make this test more comprehensive. </p> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Computer_experiment/7gConHHa">Computer experiment</a></li> <li><a href="/facts/Computer_simulation/Re6o8vzN">Computer simulation</a></li> <li><a href="/facts/Design_of_experiments/VMrNftbr">Design of experiments</a></li> <li><a href="/facts/Grey_box_testing/fkh2d1ay">Grey box testing</a></li> <li><a href="/facts/Mathematical_model/CGhyuxgz">Mathematical model</a></li> <li><a href="/facts/Nonlinear_system_identification/XnBWbGdF">Nonlinear system identification</a></li> <li><a href="/facts/Parameter_estimation/4nbgFDec">Parameter estimation</a></li> <li><a href="/facts/Research_design/hjmH1Yv5">Research design</a></li> <li><a href="/facts/Scientific_modelling/aNyvIyUQ">Scientific modelling</a></li> <li><a href="/facts/Simulation/Q4QJV1oI">Simulation</a></li> <li><a href="/facts/Statistical_model/jKkT5ftm">Statistical model</a></li> <li><a href="/facts/System_dynamics/EPI2wWsI">System dynamics</a></li> <li><a href="/facts/System_identification/v2xk1X0e">System identification</a></li> <li><a href="/facts/System_realization/RwYAYmQb">System realization</a></li> <li><a href="/facts/Systems_theory/C56Ck7DQ">Systems theory</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Bohlin, Torsten P. (7 September 2006). Practical Grey-box Process Identification: Theory and Applications. Springer Science & Business Media. ISBN 978-1-84628-403-8. <a href="978-1-84628-403-8" target="_blank">978-1-84628-403-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Grey-box model estimation". Mathworks 2. 2012. <a href="http://www.mathworks.com.au/help/ident/grey-box-model-estimation.html" target="_blank">http://www.mathworks.com.au/help/ident/grey-box-model-estimation.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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, Amsterdam. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Sohlberg, B., and Jacobsen, E.W., 2008. Grey box modelling - branches and experiences, Proc. 17th World Congress, Int. Federation of Automatic Control, Seoul. pp 11415-11420 <a href="https://www.sciencedirect.com/science/article/pii/S1474667016408025" target="_blank">https://www.sciencedirect.com/science/article/pii/S1474667016408025</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Draper, Norman R.; Smith, Harry (25 August 2014). Applied Regression Analysis. John Wiley & Sons. pp. 657–. ISBN 978-1-118-62568-2. <a href="978-1-118-62568-2" target="_blank">978-1-118-62568-2</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Weisberg, Sanford (25 November 2013). Applied Linear Regression. Wiley. ISBN 978-1-118-59485-8. <a href="978-1-118-59485-8" target="_blank">978-1-118-59485-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Heaton, J., 2012. Introduction to the math of neural networks, Heaton Research Inc. (Chesterfield, MO), ISBN 978-1475190878 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Stergiou, C.; Siganos, D. (2013). "Neural networks". Archived from the original on 2009-12-16. Retrieved 2013-07-03. <a href="https://web.archive.org/web/20091216110504/http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/cs11/report.html" target="_blank">https://web.archive.org/web/20091216110504/http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/cs11/report.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Lawson, Charles L.; J. Hanson, Richard (1 December 1995). Solving Least Squares Problems. SIAM. ISBN 978-0-89871-356-5. <a href="978-0-89871-356-5" target="_blank">978-0-89871-356-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). Numerical Recipes (3rd ed.). Cambridge University Press. ISBN 978-0-521-88068-8. <a href="978-0-521-88068-8" target="_blank">978-0-521-88068-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (1 November 2013). Bayesian Data Analysis, Third Edition. CRC Press. ISBN 978-1-4398-4095-5. <a href="978-1-4398-4095-5" target="_blank">978-1-4398-4095-5</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Bohlin, Torsten P. (7 September 2006). Practical Grey-box Process Identification: Theory and Applications. Springer Science & Business Media. ISBN 978-1-84628-403-8. <a href="978-1-84628-403-8" target="_blank">978-1-84628-403-8</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Mathworks, 2013. Supported grey box models <a href="http://www.mathworks.com.au/help/ident/ug/supported-grey-box-models.html" target="_blank">http://www.mathworks.com.au/help/ident/ug/supported-grey-box-models.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Hauth, J. (2008), Grey Box Modelling for Nonlinear Systems (PDF) (dissertation, Kaiserslautern University of Technology). <a href="https://kluedo.ub.uni-kl.de/files/2045/diss.pdf" target="_blank">https://kluedo.ub.uni-kl.de/files/2045/diss.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Hauth, J. (2008), Grey Box Modelling for Nonlinear Systems (PDF) (dissertation, Kaiserslautern University of Technology). <a href="https://kluedo.ub.uni-kl.de/files/2045/diss.pdf" target="_blank">https://kluedo.ub.uni-kl.de/files/2045/diss.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Nash, J.C. and Walker-Smith, M. 1987. Nonlinear parameter estimation, Marcel Dekker, Inc. (New York). <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>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), 129-148. <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Whiten, W.J., 1994. Determination of parameter relations within non-linear models, SIGNUM Newsletter, 29(3–4,) 2–5. 10.1145/192527.192535. <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Whiten, B., 2014. Determining the form of ordinary differential equations using model inversion, ANZIAM J. 55 (EMAC2013) pp.C329–C347. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7767" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7767</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Whiten, W.J., 1994. Determination of parameter relations within non-linear models, SIGNUM Newsletter, 29(3–4,) 2–5. 10.1145/192527.192535. <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Draper, Norman R.; Smith, Harry (25 August 2014). Applied Regression Analysis. John Wiley & Sons. pp. 657–. ISBN 978-1-118-62568-2. <a href="978-1-118-62568-2" target="_blank">978-1-118-62568-2</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Weisberg, Sanford (25 November 2013). Applied Linear Regression. Wiley. ISBN 978-1-118-59485-8. <a href="978-1-118-59485-8" target="_blank">978-1-118-59485-8</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Polynomial <a href="/wiki/Polynomial" target="_blank">/wiki/Polynomial</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Spline (mathematics) <a href="/wiki/Spline_(mathematics)" target="_blank">/wiki/Spline_(mathematics)</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Bohlin, Torsten P. (7 September 2006). Practical Grey-box Process Identification: Theory and Applications. Springer Science & Business Media. ISBN 978-1-84628-403-8. <a href="978-1-84628-403-8" target="_blank">978-1-84628-403-8</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>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), 129-148. <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Kojovic, T., and Whiten W. J., 1994. Evaluation of the quality of simulation models, Innovations in mineral processing, (Lauretian University, Sudbury) pp 437–446. ISBN 088667025X <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Kojovic, T., 1989. The development and application of Model - an automated model builder for mineral processing, PhD thesis, The University of Queensland. <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Xiao, J., 1998. Extensions of model building techniques and their applications in mineral processing, PhD thesis, The University of Queensland. <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). Numerical Recipes (3rd ed.). Cambridge University Press. ISBN 978-0-521-88068-8. <a href="978-0-521-88068-8" target="_blank">978-0-521-88068-8</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Nash, J.C. and Walker-Smith, M. 1987. Nonlinear parameter estimation, Marcel Dekker, Inc. (New York). <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Linhart, H.; Zucchini, W. (1986). Model selection. Wiley. ISBN 978-0-471-83722-0. <a href="978-0-471-83722-0" target="_blank">978-0-471-83722-0</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Miller, Alan (15 April 2002). Subset Selection in Regression. CRC Press. ISBN 978-1-4200-3593-3. <a href="978-1-4200-3593-3" target="_blank">978-1-4200-3593-3</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>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), 129-148. <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Kojovic, T., and Whiten W. J., 1994. Evaluation of the quality of simulation models, Innovations in mineral processing, (Lauretian University, Sudbury) pp 437–446. ISBN 088667025X <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>Kojovic, T., 1989. The development and application of Model - an automated model builder for mineral processing, PhD thesis, The University of Queensland. <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Whiten, W.J., 1994. Determination of parameter relations within non-linear models, SIGNUM Newsletter, 29(3–4,) 2–5. 10.1145/192527.192535. <a href="#fnref:43" class="footnote-back-ref">↩</a></p></li> <li id="fn:44"><p>Whiten, B., 2014. Determining the form of ordinary differential equations using model inversion, ANZIAM J. 55 (EMAC2013) pp.C329–C347. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7767" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7767</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></p></li> <li id="fn:45"><p>Linhart, H.; Zucchini, W. (1986). Model selection. Wiley. ISBN 978-0-471-83722-0. <a href="978-0-471-83722-0" target="_blank">978-0-471-83722-0</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></p></li> <li id="fn:46"><p>Miller, Alan (15 April 2002). Subset Selection in Regression. CRC Press. ISBN 978-1-4200-3593-3. <a href="978-1-4200-3593-3" target="_blank">978-1-4200-3593-3</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></p></li> <li id="fn:47"><p>Lawson, Charles L.; J. Hanson, Richard (1 December 1995). Solving Least Squares Problems. SIAM. ISBN 978-0-89871-356-5. <a href="978-0-89871-356-5" target="_blank">978-0-89871-356-5</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></p></li> <li id="fn:48"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></p></li> <li id="fn:49"><p>Whiten, B., 2014. Determining the form of ordinary differential equations using model inversion, ANZIAM J. 55 (EMAC2013) pp.C329–C347. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7767" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/7767</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></p></li> <li id="fn:50"><p>Whiten, W.J., 1994. Determination of parameter relations within non-linear models, SIGNUM Newsletter, 29(3–4,) 2–5. 10.1145/192527.192535. <a href="#fnref:50" class="footnote-back-ref">↩</a></p></li> <li id="fn:51"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></p></li> <li id="fn:52"><p>Deming, William Edwards (2000). Out of the Crisis p272. MIT Press. ISBN 978-0-262-54115-2. <a href="978-0-262-54115-2" target="_blank">978-0-262-54115-2</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></p></li> <li id="fn:53"><p>Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). Numerical Recipes (3rd ed.). Cambridge University Press. ISBN 978-0-521-88068-8. <a href="978-0-521-88068-8" target="_blank">978-0-521-88068-8</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></p></li> <li id="fn:54"><p>Kojovic, T., and Whiten W. J., 1994. Evaluation of the quality of simulation models, Innovations in mineral processing, (Lauretian University, Sudbury) pp 437–446. ISBN 088667025X <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></p></li> <li id="fn:55"><p>Kojovic, T., 1989. The development and application of Model - an automated model builder for mineral processing, PhD thesis, The University of Queensland. <a href="#fnref:55" class="footnote-back-ref">↩</a></p></li> <li id="fn:56"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></p></li> <li id="fn:57"><p>Kojovic, T., and Whiten W. J., 1994. Evaluation of the quality of simulation models, Innovations in mineral processing, (Lauretian University, Sudbury) pp 437–446. ISBN 088667025X <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></p></li> <li id="fn:58"><p>Whiten, B., 2013. Model completion and validation using inversion of grey box models, ANZIAM J.,54 (CTAC 2012) pp C187–C199. <a href="http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682" target="_blank">http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6125/1682</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></p></li> </ol>