519.2 Polynomial chaos and regression based on KolmogorovGabor polynomials: comparative modeling

Oblakova T. V. (Bauman Moscow State Technical University), Pham Q. (-)

POLYNOMIAL CHAOS AND REGRESSION BASED ON KOLMOGOROV-GABOR POLYNOMIALS: COMPARATIVE MODELING


doi: 10.18698/2309-3684-2023-4-93108


The application of the generalized expansion of polynomial chaos (PC) and models based on Kolmogorov-Gabor polynomials in regression problems is considered. When choosing PC expansion, the Wiener-Askey scheme was used, which sets the correspondence between the feature distribution law and the orthogonal polynomial basis. To calculate the expansion coefficients, non-intrusive methods were used: least squares, elastic network, as well as Ivakhnenko's inductive evolutionary algorithm. Kolmogorov-Gabor polynomials are used as a reference function of a polynomial neural network. Model errors and performance were calculated on a test set. Models were compared on a linear transport problem under uncertainty: the diffusion coefficient and drift were modeled by uniformly distributed random variables. It is shown that with a small interval of variation in the values of random variables, both models give good efficiency, but the PC model demonstrates a smaller spread of errors and is faster in time. For the de-cay equation with random coefficients distributed according to the Gaussian law, the influence of the correlation of these coefficients on the rate of convergence is studied. It is shown that with dependent coefficients, the best performance is observed in higher-order PC models. On the basis of comparative modeling, it has been established that the use of PC is unambiguously preferable in the following cases: a small dimension of the space of input features, a known law of distribution of input data, and correlated features. It is also shown that the use of PC with a large dimension of the space of input features is inefficient due to the rapid increase in the number of terms in the expansion, leading to a sharp increase in the time to process the task. In this case, the regression model based on the Kolmogorov-Gabor polynomials in combination with the GMDH turned out to be clearly preferable.


Xiu D., Karniadakis G. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM Journal on Scientific Computing, 2002, vol. 24, iss. 2, pp. 619–644. DOI: 10.1137/S1064827501387826
Wiener N. The Homogeneous Chaos. American Journal of Mathematics, 1938, vol. 60, pp. 897–936.
Cameron R.H., Martin W.T. The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals. Annals of Mathematics. Second Series, 1947, vol. 48, no. 2, pp. 385–392.
Ghanem R., Spanos P. Stochastic Finite Elements: A Spectral Approach. NewYork, Springer-Verlag, 1991, 214 p.
Ghanem R. Ingredients for a General Purpose Stochastic Finite Element Formulation. Computer Methods in Applied Mechanics and Engineering, 1999, vol. 168, no. 19, pp. 19–34.
Ghanem R. Stochastic Finite Elements with Multiple Random Non-Gaussian Properties. Journal of Engineering Mechanics, 1999, vol. 125, no. 1, pp. 26–40.
Xiu D., Karniadakis G. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 2003, vol. 187, pp. 137–167. DOI: 10.1016/S0021-9991(03)00092-5
Liu W., Dou Z., Wang W., Liu Y., Zou H., Zhang B., Hou S. Short-Term Load Forecasting Based on Elastic Net Improved GMDH and Difference Degree Weighting Optimization. Applied Sciences, 2018, vol. 8, iss. 9, art. no. 1603. DOI: 10.3390/app8091603
Ivakhnenko A.G. Polynomial Theory of Complex Systems. IEEE Transactions on Systems, Man, and Cybernetics, 1971, vol. SMC-1, no. 4, pp. 364–378.
Ivakhnenko A.G., Ivakhnenko G.A. The Review of Problems Solvable by Algorithms of the Group Method of Data Handling (GMDH). Pattern Recognition and Image Analysis, 1995, vol. 5, no. 4, pp. 527–535.
Onwubolu G.C. GMDH-Methodology and Implementation in MATLAB. London, Imperial College Press, 2016, 284 p.
Gu J., Chu L.L., Zhang Y.J, Shi W.G., Application of GMDH and variable cointegration theory in power load forecasting. Power System Protection and Control, 2010, vol. 38, no. 22, pp. 80–85.
Ahmadi M.H., Ahmadi M.A. Mehrpooya M., Rosen M.A. Using GMDH Neural Networks to Model the Power and Torque of a Stirling Engine. Sustainability, 2015, vol. 7, iss. 2, pp. 2243–2255. DOI: 10.3390/su7022243
Yang L.T., Yang H.G., Liu H.T. GMDH-Based Semi-Supervised Feature Selection for Electricity Load Classification Forecasting. Sustainability, 2018, vol. 10, iss. 1, pp. 217. DOI: 10.3390/su10010217
Najafzadeh M., Saberi-Movhed F., Sarkamaryan S. NF-GMDH-Based selforganized systems to predict bridge pier scour depth under debris flow effects. Marine Georesources & Geotechnology, 2018, vol. 36, iss. 5, pp. 589–602.
Zhang M.Z., He C.Z., Panos L.A. A D-GMDH model for time series forecasting. Expert Systems with Applications, 2012, vol. 39, iss. 5, pp. 5711–5716.
Petterson M.P., Iaccarino G., Nordstrom J. Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Springer International Publishing Switzerland, 2015, 379 p.
Strizhov V.V., Krymova E.A. Metody vybora regressionnyh modelej [Methods of selection of regression models]. Moscow, Computing Center of the Russian Academy of Sciences Pibl., 2010, 60 p.
Strizhov V.V. Porozhdenie i vybor modelej v zadachah regressii i klassifikacii [Generation and selection of models in regression and classification problems]. Diss. Dr. Sc. (Phys.-Math.), Moscow, 2014, 299 p.
Crestaux T., Maitre O.L., Martinez J.M. Polynomial chaos expansion for sensitivity analysis. Reliability Engineering & System Safety, 2009, vol. 94, iss. 7, pp.1161–1172.
Askey R., Wilson J. Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials. Memoirs of the American Mathematical Society, 1985, vol. 54, 55 p.
Popkova A.P. Linejnye regressionnye modeli na osnove polinomial'nogo haosa i ih primenenie [Linear regression models based on polynomial chaos and their application]. Bachelor's degree thesis, Moscow, 2022, 159 p.
Russikikh S.V., Shklyarchuk F.N. Application of the one-step Galerkin method for solving a system of ordinary differential equations with initial conditions.Mathematical Modeling and Computational Methods, 2022, no. 3, pp. 18–32.
Bazilevsky M.P. Analytical dependences between the determination coefficients and the ratio of error variances of the test items in Deming regression model. Mathematical Modeling and Computational Methods, 2016, no. 2, pp. 104–116.
Zou H., Hastie T. Regularization and Variable Selection via the Elastic Net. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 2005, vol. 67, no. 2, pp. 301–320.
Kadiev A.D., Chibisova A.V. Neural network methods for solving the problem of credit scoring. Mathematical Modeling and Computational Methods, 2022, no. 4, pp. 81–92.


Облакова Т.В., Фам Куок Вьет. Сравнительное моделирование на основе многочленов Колмогорова-Габора в задачах полиномиального хаоса и регрессии. Математическое моделирование и численные методы, 2023, № 4, с. 93–108.



Download article

Количество скачиваний: 192