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Double-criterion Choice of the Optimal Model in GMDH Algorithms Ievgeniia Savchenko

Double-criterion Choice of the Optimal Model in GMDH Algorithms Ievgeniia Savchenko. International Research and Training Centre of Information Technologies and Systems of NAS of Ukraine savchenko@irtc.org.ua. Introduction.

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Double-criterion Choice of the Optimal Model in GMDH Algorithms Ievgeniia Savchenko

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  1. Double-criterion Choice of the Optimal Model in GMDH AlgorithmsIevgeniia Savchenko International Research and Training Centre of Information Technologies and Systems of NAS of Ukraine savchenko@irtc.org.ua

  2. Introduction The aim of this work is to develop the method of double-criterion selection of the optimal model in GMDH with sequence criteria applying. This method use in the real modelling tasks when optimal model of any system can’t be find by selection on Combinatorial GMDH algorithm.

  3. Outline • External GMDH criteria • Combined criteria • Double-criterion of the optimal model choice • Numerical example of double-criterion model choice • First variant of model search (AR – BS) • Second variant of model search (BS – AR)

  4. Outline • External GMDH criteria • Combined criteria • Double-criterion of the optimal model choice • Numerical example of double-criterion model choice • First variant of model search (AR – BS) • Second variant of model search (BS – AR)

  5. Choosing model by external criteria we can expect from model to be the most exact or the least bias. For this purpose the regularity or the bias criterion can be used. The regularity and bias criteria are orthogonal by nature. Combined criteria are used to simultaneously to take into account this features at model choice. • Calculation of the combined criteria requires assigning the weight coefficients of criteria according to what quality of the model it is needed to get. But such a procedure requires large computing time as values of criteria must be calculated for all generated models. The combined criteria are preferably because it simultaneously takes into account several section criteria. • Selection criteria are constructed so that they are reflected several obvious demands to model. For example, “the model must to give the well forecast” – regularity criterion, “coefficients of model must marginally depends on part of data sample” – bias criterion.

  6. A – learning data sample; B – testing data sample;C – examination data sample. • when recording means“the error on B model, coefficients of which are finding on A”

  7. The most widespread form of bias criterion is a minimum of bias decisions: • The minimum of bias errors is proposed: • when and is a total errors on the incorporated sample of model of the same structures with the parameters which estimated on samples A and Baccordingly:

  8. Outline • External GMDH criteria • Combined criteria • Double-criterion of the optimal model choice • Numerical example of double-criterion model choice • First variant of model search (AR – BS) • Second variant of model search (BS – AR)

  9. Combined criteria Either the combined criteria or a sequence of criteria usually are applied for the solution of a problem of double-criterion model choice. Often such problem is result of linear convolution of criteria particularly of the combined of criteria: when - weight coefficients; - value of criteria. However combined criterion calculation has some complexities: Needs the normalization (weightings) of criteria; Large computing time of value criteria calculation.

  10. Outline • External GMDH criteria • Combined criteria • Double-criterion of the optimal model choice • Numerical example of double-criterion model choice • First variant of model search (AR – BS) • Second variant of model search (BS – AR)

  11. Double-criterion of the optimal model choice • At the beginning some number of models is found by minimum of first criterion (regularity and bias). • Then the optimal model is selected by the second criterion form the set of best models by the first criterion.

  12. Outline • External GMDH criteria • Combined criteria • Double-criterion of the optimal model choice • Numerical example of double-criterion model choice • First variant of model search (AR – BS) • Second variant of model search (BS – AR)

  13. The output variable contains 30% noise in the data:

  14. First variant of model search (AR – BS)

  15. Optimal model is complexity . Value of criteria are ; . This model also have minimum of error on the examination data sample . The set of models chosen by GMDH

  16. Second variant of model search (BS – AR)

  17. Conclusion: • Examples have shown that the most reasonable sequence of the external GMDH criteria for the optimal model selection is the sequence when at beginning some set of models is selected by criterion regularity, then the one model is selected by bias criterion. Using of sequence BS – AR faces with loss of optimum model as models are selected more "rough". • It will lead to selection of simpler model in conditions of noise.

  18. Thanks you for attention Double-criterion Choice of the Optimal Model in GMDH Algorithms Ievgeniia Savchenko savchenko@irtc.org.ua

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