2159:. The convergence of multilayered GMDH algorithms was investigated. It was shown that some multilayered algorithms have "multilayerness error" – analogous to static error of control systems. In 1977 a solution of objective systems analysis problems by multilayered GMDH algorithms was proposed. It turned out that sorting-out by criteria ensemble finds the only optimal system of equations and therefore to show complex object elements, their main input and output variables.
1887:
2149:. The problem of modeling of noised data and incomplete information basis was solved. Multicriteria selection and utilization of additional priory information for noiseimmunity increasing were proposed. Best experiments showed that with extended definition of the optimal model by additional criterion noise level can be ten times more than signal. Then it was improved using
2165:. Many important theoretical results were received. It became clear that full physical models cannot be used for long-term forecasting. It was proved, that non-physical models of GMDH are more accurate for approximation and forecast than physical models of regression analysis. Two-level algorithms which use two different time scales for modeling were developed.
1572:
1021:
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2142:
is characterized by application of only regularity criterion for solving of the problems of identification, pattern recognition and short-term forecasting. As reference functions polynomials, logical nets, fuzzy Zadeh sets and Bayes probability formulas were used. Authors were stimulated by very high
2206:
Another important approach to partial models consideration that becomes more and more popular is a combinatorial search that is either limited or full. This approach has some advantages against
Polynomial Neural Networks, but requires considerable computational power and thus is not effective for
2197:
with polynomial activation function of neurons. Therefore, the algorithm with such an approach usually referred as GMDH-type Neural
Network or Polynomial Neural Network. Li showed that GMDH-type neural network performed better than the classical forecasting algorithms such as Single Exponential
2207:
objects with a large number of inputs. An important achievement of
Combinatorial GMDH is that it fully outperforms linear regression approach if noise level in the input data is greater than zero. It guarantees that the most optimal model will be founded during exhaustive sorting.
2188:
There are many different ways to choose an order for partial models consideration. The very first consideration order used in GMDH and originally called multilayered inductive procedure is the most popular one. It is a sorting-out of gradually complicated models generated from
275:
First, we split the full dataset into two parts: a training set and a validation set. The training set would be used to fit more and more model parameters, and the validation set would be used to decide which parameters to include, and when to stop fitting completely.
707:
1882:{\displaystyle 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 }
501:
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The model is structured as a feedforward neural network, but without restrictions on the depth, they had a procedure for automatic models structure generation, which imitates the process of biological selection with pairwise genetic features.
2009:
2233:
In contrast to GMDH-type neural networks
Combinatorial algorithm usually does not stop at the certain level of complexity because a point of increase of criterion value can be simply a local minimum, see Fig.1.
2171:
the new algorithms (AC, OCC, PF) for non-parametric modeling of fuzzy objects and SLP for expert systems were developed and investigated. Present stage of GMDH development can be described as blossom out of
2062:
Criterion of
Minimum bias or Consistency – squared difference between the estimated outputs (or coefficients vectors) of two models fit on the A and B set, divided by squared predictions on the B set.
178:
308:
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More sophisticated methods for deciding when to terminate are possible. For example, one might keep running the algorithm for several more steps, in the hope of passing a temporary rise in
30:
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.
1492:
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2108:, they proposed "noise-immune modelling": the higher the noise, the less parameters must the optimal model have, since the noisy channel does not allow more bits to be sent through.
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702:
1016:{\displaystyle y\approx f_{(i,j);a,b,c,d,e,h}(x_{i},x_{j}):=a_{i,j}+b_{i,j}x_{i}+c_{i,j}x_{j}+d_{i,j}x_{i}^{2}+e_{i,j}x_{j}^{2}+f_{i,j}x_{i}x_{j}\quad \forall 1\leq i<j\leq k}
57:. 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
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Li, Rita Yi Man; Fong, Simon; Chong, Kyle Weng Sang (2017). "Forecasting the REITs and stock indices: Group Method of Data
Handling Neural Network approach".
2789:
Takao, S.; Kondo, S.; Ueno, J.; Kondo, T. (2017). "Deep feedback GMDH-type neural network and its application to medical image analysis of MRI brain images".
1071:
We do not want to accept all the polynomial models, since it would contain too many models. To only select the best subset of these models, we run each model
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on the validation dataset, and select the models whose mean-square-error is below a threshold. We also write down the smallest mean-square-error achieved as
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1902:
2304:— GMDH-based, predictive analytics and time series forecasting software. Free Academic Licensing and Free Trial version available. Windows-only.
2930:
2675:
Ivakhnenko, Aleksei G., and
Grigorii A. Ivakhnenko. "Problems of further development of the group method of data handling algorithms. Part I."
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Like linear regression, which fits a linear equation over data, GMDH fits arbitrarily high orders of polynomial equations over data.
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2193:. The best model is indicated by the minimum of the external criterion characteristic. Multilayered procedure is equivalent to the
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The GMDH starts by considering degree-2 polynomial in 2 variables. Suppose we want to predict the target using just the
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Other names include "polynomial feedforward neural network", or "self-organization of models". It was one of the first
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Instead of a degree-2 polynomial in 2 variables, each unit may use higher-degree polynomials in more variables:
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neuronets and parallel inductive algorithms for multiprocessor computers. Such procedure is currently used in
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is smaller than the previous one, the process continues, giving us increasingly deep models. As soon as some
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models. We now run the models on the training dataset, to obtain a sequence of transformed observations:
2626:(Modern Analytic and Computational Methods in Science and Mathematics, v.8 ed.). American Elsevier.
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2098:
496:{\displaystyle y\approx f_{a,b,c,d,e,h}(x_{i},x_{j}):=a+bx_{i}+cx_{j}+dx_{i}^{2}+ex_{j}^{2}+fx_{i}x_{j}}
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Inspired by an analogy between constructing a model out of noisy data, and sending messages through a
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And more generally, a GMDH model with multiple inputs and one output is a subset of components of the
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Fig.1. A typical distribution of minimal errors. The process terminates when a minimum is reached.
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parts of the observation, and using only degree-2 polynomials, then the most we can do is this:
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For the selected model of optimal complexity recalculate coefficients on a whole data sample.
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2461:"Learning polynomial feedforward neural networks by genetic programming and backpropagation"
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Generates subsamples from A according to partial models with steadily increasing complexity.
1526:
1386:
2755:"The Review of Problems Solvable by Algorithms of the Group Method of Data Handling (GMDH)"
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Pattern
Recognition and Image Analysis c/c of raspoznavaniye obrazov i analiz izobrazhenii
46:
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610:
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To choose between models, two or more subsets of a data sample are used, similar to the
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1520:) is discarded, as it has overfit the training set. The previous layers are outputted.
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accuracy of forecasting with the new approach. Noise immunity was not investigated.
61:. The last section of contains a summary of the applications of GMDH in the 1970s.
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Chooses the best model (set of models) indicated by minimal value of the criterion.
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This is the general problem of statistical modelling of data: Consider a dataset
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34:
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2319:
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Smooth, Double
Exponential Smooth, ARIMA and back-propagation neural network.
2691:
Perspectives of
Planing. Organization of Economic Cooperation and Development
2662:
2528:
Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview".
2484:
2445:
2341:
2004:{\displaystyle Y(x_{1},\dots ,x_{n})=a_{0}+\sum \limits _{i=1}^{m}a_{i}f_{i}}
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Algorithm on the base of Multilayered Theory of Statistical Decisions (MTSD)
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Estimates coefficients of partial models at each layer of models complexity.
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to predict. How to best predict the target based on the observations?
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on the validation set, as given above. The most common criteria are:
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External criteria are optimization objectives for the model, such as
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we should choose, so we choose all of them. That is, we perform all
17:
2637:
Ivakhenko, A.G.; Savchenko, E.A..; Ivakhenko, G.A. (October 2003).
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Heuristic Self-Organization in Problems of Engineering Cybernetics
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2119:
1292:
2310:— Commercial product. Mac OS X-only. Free Demo version available.
2722:
Inductive Method of Models Self-organisation for Complex Systems
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Pomekhoustojchivost' Modelirovanija (Noise Immunity of Modeling)
2354:
Python library of basic GMDH algorithms (COMBI, MULTI, MIA, RIA)
2353:
2133:
2021:
are elementary functions dependent on different sets of inputs,
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Calculates value of external criterion for models on sample B.
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Suppose that after this process, we have obtained a set of
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Inductive Learning Algorithms for Complex Systems Modeling
2891:
Self-Organizing Methods in Modelling: GMDH Type Algorithms
2382:
Inductive Learning Algorithms for Complex Systems Modeling
2124:
GMDH author – Soviet scientist Prof. Alexey G. Ivakhnenko.
68:
methods, used to train an eight-layer neural net in 1971.
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Basic Combinatorial algorithm makes the following steps:
1494:, the algorithm terminates. The last layer fitted (layer
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Divides data sample at least into two samples A and B.
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principle of "freedom of decisions choice", and the
2893:. New-York, Bazel: Marcel Decker Inc., 1984, 350 p.
2298:— Free upon request for academic use. Windows-only.
173:{\displaystyle \{(x_{1},...,x_{k};y)_{s}\}_{s=1:n}}
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2507:IEEE Transactions on Systems, Man, and Cybernetics
2003:
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2883:, Automatica, vol.6, 1970 — p. 207-219.
2639:"Problems of future GMDH algorithms development"
2032:is the number of the base function components.
2267:Pointing Finger (PF) clusterization algorithm;
8:
149:
91:
2753:Ivakhnenko, O.G.; Ivakhnenko, G.A. (1995).
2128:The method was originated in 1968 by Prof.
2584:Ivakhnenko, O.G.; Stepashko, V.S. (1985).
1487:{\displaystyle minMSE_{L+1}>minMSE_{L}}
1290:. The same algorithm can now be run again.
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2279:Group of Adaptive Models Evolution (GAME)
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1376:{\displaystyle minMSE_{1},minMSE_{2},...}
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1283:{\displaystyle z_{1},z_{2},...,z_{k_{1}}}
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2264:Objective Computer Clusterization (OCC);
2379:Madala, H.R.; Ivakhnenko, O.G. (1994).
2366:
2762:Pattern Recognition and Image Analysis
2622:Cybernetics and Forecasting Techniques
2500:"Polynomial theory of complex systems"
2459:Nikolaev, N.Y.; Iba, H. (March 2003).
2826:Pacific Rim Property Research Journal
2728:. Kyiv: Naukova Dumka. Archived from
2643:Systems Analysis Modelling Simulation
2618:Ivakhnenko, O.G.; Lapa, V.G. (1967).
2593:. Kyiv: Naukova Dumka. Archived from
1139:{\displaystyle f_{(i,j);a,b,c,d,e,h}}
7:
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2465:IEEE Transactions on Neural Networks
2420:Farlow, Stanley J. (November 1981).
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1061:{\displaystyle {\frac {1}{2}}k(k-1)}
697:{\displaystyle {\frac {1}{2}}k(k-1)}
28:Group method of data handling (GMDH)
2132:in the Institute of Cybernetics in
1961:
1803:
1781:
1759:
1699:
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1301:The algorithm continues, giving us
2863:Library of GMDH books and articles
2422:"The GMDH Algorithm of Ivakhnenko"
1068:polynomial models of the dataset.
989:
25:
2153:of General Communication theory.
2101:principle of external additions.
2348:Python library of MIA algorithm
2252:Objective System Analysis (OSA)
2048:Criterion of Regularity (CR) –
988:
245:{\displaystyle x_{1},...,x_{k}}
33:GMDH is used in such fields as
2902:. CRC Press, Boca Raton, 1994.
2896:H.R. Madala, A.G. Ivakhnenko.
2708:. London: English Univ. Press.
2438:10.1080/00031305.1981.10479358
2342:R Package for regression tasks
2292:— Open source. Cross-platform.
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1:
2931:Regression variable selection
2868:Group Method of Data Handling
2838:10.1080/14445921.2016.1225149
2261:Multiplicative–Additive (MAA)
2042:minimizing mean-squared error
252:observations, and one target
2791:Artificial Life and Robotics
2552:10.1016/j.neunet.2014.09.003
2246:Multilayered Iterative (MIA)
704:such polynomial regressions:
200:points. Each point contains
2655:10.1080/0232929032000115029
2498:Ivakhnenko, Alexey (1971).
2273:Harmonical Rediscretization
2076:train-validation-test split
600:{\displaystyle a,b,c,d,e,f}
546:{\displaystyle a,b,c,d,e,f}
2952:
2921:Artificial neural networks
2706:Cybernetics and Management
2081:GMDH combined ideas from:
1181:{\displaystyle minMSE_{1}}
81:This section is based on.
2926:Classification algorithms
2803:10.1007/s10015-017-0410-1
2515:10.1109/TSMC.1971.4308320
2426:The American Statistician
2385:. Boca Raton: CRC Press.
2270:Analogues Complexing (AC)
2195:Artificial Neural Network
2184:GMDH-type neural networks
2916:Computational statistics
2719:Ivahnenko, O.G. (1982).
2322:— Freeware, Open source.
2284:Software implementations
2477:10.1109/TNN.2003.809405
2509:. SMC-1 (4): 364–378.
2125:
2028:are coefficients and
2005:
1980:
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1551:{\displaystyle minMSE}
1514:
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1411:{\displaystyle minMSE}
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557:. Now, the parameters
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174:
2679:10.2 (2000): 187-194.
2316:— Commercial product.
2243:Combinatorial (COMBI)
2123:
2006:
1960:
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1802:
1780:
1758:
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1211:{\displaystyle k_{1}}
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503:where the parameters
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77:Polynomial regression
2332:plugin, Open source.
2314:PNN Discovery client
2130:Alexey G. Ivakhnenko
2052:on a validation set.
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72:Mathematical content
2693:. London: Imp.Coll.
2055:Least squares on a
1513:{\displaystyle L+1}
948:
914:
652:{\displaystyle i,j}
626:{\displaystyle i,j}
466:
445:
298:{\displaystyle i,j}
55:pattern recognition
39:knowledge discovery
2689:Gabor, D. (1971).
2258:Two-level (ARIMAD)
2202:Combinatorial GMDH
2126:
2083:black box modeling
2050:least mean squares
2001:
1879:
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1383:. As long as each
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59:external criterion
2936:Soviet inventions
2879:A.G. Ivakhnenko.
2704:Beer, S. (1959).
2649:(10): 1301–1309.
2290:FAKE GAME Project
2151:Shannon's Theorem
2087:genetic selection
2036:External criteria
1038:
674:
555:linear regression
265:{\displaystyle y}
193:{\displaystyle n}
16:(Redirected from
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2395:. Archived from
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2163:Period 1980–1988
2157:Period 1976–1979
2147:Period 1972–1975
2140:Period 1968–1971
2057:cross-validation
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47:complex systems
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2857:External links
2855:
2852:
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2832:(2): 123–160.
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2797:(2): 161–172.
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2775:10.1.1.19.2971
2768:(4): 527–535.
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2356:- Open source.
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2350:- Open source.
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2735:on 2017-12-31
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2191:base function
2183:
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2178:deep learning
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2174:deep learning
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2106:noisy channel
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1894:base function
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66:deep learning
62:
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56:
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40:
36:
31:
29:
19:
2898:
2890:
2829:
2825:
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2765:
2761:
2748:
2737:. Retrieved
2730:the original
2721:
2714:
2705:
2699:
2690:
2684:
2676:
2671:
2646:
2642:
2632:
2621:
2613:
2602:. Retrieved
2595:the original
2586:
2533:
2529:
2523:
2506:
2493:
2468:
2464:
2454:
2429:
2425:
2401:. Retrieved
2397:the original
2381:
2320:Sciengy RPF!
2232:
2209:
2205:
2190:
2187:
2168:
2167:
2162:
2161:
2156:
2155:
2146:
2145:
2139:
2138:
2127:
2110:
2103:
2089:of pairwise
2080:
2073:
2070:
2039:
2029:
2022:
2015:
2013:
1893:
1891:
1565:
1522:
1300:
1190:
1070:
278:
274:
83:
80:
63:
58:
51:optimization
32:
27:
26:
2887:S.J. Farlow
35:data mining
2910:Categories
2739:2019-11-18
2604:2019-11-18
2536:: 85–117.
2403:2019-11-17
2361:References
2302:GMDH Shell
2255:Harmonical
2238:Algorithms
2180:networks.
2169:Since 1989
1562:In general
1023:obtaining
49:modeling,
43:prediction
2846:157150897
2770:CiteSeerX
2663:0232-9298
2543:1404.7828
2485:1045-9227
2446:0003-1305
2336:R Package
1962:∑
1926:…
1877:⋯
1804:∑
1782:∑
1760:∑
1700:∑
1678:∑
1632:∑
1596:…
1050:−
1008:≤
996:≤
990:∀
715:≈
686:−
316:≈
2811:44190434
2568:11715509
2560:25462637
2091:features
2116:History
2095:Gabor's
180:, with
2844:
2809:
2772:
2661:
2566:
2558:
2483:
2444:
2389:
2099:Beer's
2093:, the
2014:where
2842:S2CID
2807:S2CID
2758:(PDF)
2733:(PDF)
2726:(PDF)
2598:(PDF)
2591:(PDF)
2564:S2CID
2538:arXiv
2503:(PDF)
2326:wGMDH
2296:GEvom
1896:(1):
2659:ISSN
2556:PMID
2481:ISSN
2442:ISSN
2387:ISBN
2330:Weka
2134:Kyiv
2067:Idea
2059:set.
1457:>
1002:<
53:and
18:GMDH
2834:doi
2799:doi
2651:doi
2548:doi
2511:doi
2473:doi
2434:doi
2912::
2889:.
2840:.
2830:23
2828:.
2805:.
2795:23
2793:.
2764:.
2760:.
2657:.
2647:43
2645:.
2641:.
2576:^
2562:.
2554:.
2546:.
2534:61
2532:.
2505:.
2479:.
2469:14
2467:.
2463:.
2440:.
2430:35
2428:.
2424:.
2412:^
2369:^
2328:—
2249:GN
2136:.
2078:.
1558:.
1188:.
805::=
388::=
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41:,
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2848:.
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2813:.
2801::
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2766:5
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2653::
2607:.
2570:.
2550::
2540::
2517:.
2513::
2487:.
2475::
2448:.
2436::
2406:.
2030:m
2025:i
2023:a
2018:i
2016:f
1997:i
1993:f
1987:i
1983:a
1977:m
1972:1
1969:=
1966:i
1958:+
1953:0
1949:a
1945:=
1942:)
1937:n
1933:x
1929:,
1923:,
1918:1
1914:x
1910:(
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