Group method of data handling information

Group method of data handling (GMDH) is a family of inductive algorithms for computer-based mathematical modeling of multi-parametric datasets that features fully automatic structural and parametric optimization of models.

GMDH is used in such fields as data mining, knowledge discovery, prediction, complex systems modeling, optimization and pattern recognition.^[1] GMDH algorithms are characterized by inductive procedure that performs sorting-out of gradually complicated polynomial models and selecting the best solution by means of the external criterion.

A GMDH model with multiple inputs and one output is a subset of components of the base function (1):

Y(x_{1},\dots ,x_{n})=a_{0}+\sum \limits _{i=1}^{m}a_{i}f_{i}

where f_i are elementary functions dependent on different sets of inputs, a_i are coefficients and m is the number of the base function components.

In order to find the best solution, GMDH algorithms consider various component subsets of the base function (1) called partial models. Coefficients of these models are estimated by the least squares method. GMDH algorithms gradually increase the number of partial model components and find a model structure with optimal complexity indicated by the minimum value of an external criterion. This process is called self-organization of models.

As the first base function used in GMDH, was the gradually complicated Kolmogorov–Gabor polynomial (2):

Y(x_{1},\dots ,x_{n})=a_{0}+\sum \limits _{i=1}^{n}{a_{i}}x_{i}+\sum \limits _{i=1}^{n}{\sum \limits _{j=i}^{n}{a_{ij}}}x_{i}x_{j}+\sum \limits _{i=1}^{n}{\sum \limits _{j=i}^{n}{\sum \limits _{k=j}^{n}{a_{ijk}}}}x_{i}x_{j}x_{k}+\cdots

Usually more simple partial models with up to second degree functions are used.^[1]

The inductive algorithms are also known as polynomial neural networks. Jürgen Schmidhuber cites GMDH as one of the first deep learning methods, remarking that it was used to train eight-layer neural nets as early as 1971.^[2]^[3]

^ ^a ^b Madala, H.R.; Ivakhnenko, O.G. (1994). Inductive Learning Algorithms for Complex Systems Modeling (PDF). Boca Raton: CRC Press. ISBN 978-0849344381.
^ Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. arXiv:1404.7828. doi:10.1016/j.neunet.2014.09.003. PMID 25462637. S2CID 11715509.
^ Ivakhnenko, Alexey (1971). "Polynomial theory of complex systems" (PDF). IEEE Transactions on Systems, Man, and Cybernetics. SMC-1 (4): 364–378. doi:10.1109/TSMC.1971.4308320.

[r1-1] Madala, H.R.; Ivakhnenko, O.G. (1994). Inductive Learning Algorithms for Complex Systems Modeling (PDF). Boca Raton: CRC Press. ISBN 978-0849344381.

[2] Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. arXiv:1404.7828. doi:10.1016/j.neunet.2014.09.003. PMID 25462637. S2CID 11715509.

[iva1971-3] Ivakhnenko, Alexey (1971). "Polynomial theory of complex systems" (PDF). IEEE Transactions on Systems, Man, and Cybernetics. SMC-1 (4): 364–378. doi:10.1109/TSMC.1971.4308320.

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