Mining Implicit Equations From Data Using Gene Expression Programming

Zhong, Jinghui; Yang, Jia-Quan; Chen, Yongliang; Liu, Wei-Li; Feng, Liang

doi:10.1109/tetc.2021.3068651

Cited by 3 publications

(2 citation statements)

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“…For solving this problem for vector α of n parameters, we can find the minimum of the quadratic form on the left-hand side of (39) subject to the normalizing condition on its right-hand side (needed for the identifiability of the parameters defined up to an arbitrary constant). It corresponds to presenting (39) as the Rayleigh quotient of two quadratic forms, α Cα/α Sα, reduced to the conditional least squares problem (10), and then to the generalized eigenproblem (11) for the nth order covariance matrices C and S of the variables and their errors, respectively. For unknown errors of observations, let us estimate the error's variance for each variable as the residual variance of this variable by all the rest of predictors, as it is considered in the case of two variables in the relation (22).…”

Section: Modeling For Many Variablesmentioning

confidence: 99%

“…Bifurcation is a function behavior where a number of solutions and their structure can change abruptly, and such modeling can be helpful in dealing with messy data characterized by a wide range of response variables at each point of the predictors' values, for instance, those known in advertising and marketing research. Various other applications of the implicit multi-value functions are also known [35][36][37][38][39]. Data fitting for implicit functions can be performed by various techniques of nonlinear statistical estimation available in modern software packages.…”

Section: Introductionmentioning

confidence: 99%

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Statistical Modeling of Implicit Functional Relations

Lipovetsky

2023

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This study considers the statistical estimation of relations presented by implicit functions. Such structures define mutual interconnections of variables rather than outcome variable dependence by predictor variables considered in regular regression analysis. For a simple case of two variables, pairwise regression modeling produces two different lines of each variable dependence using another variable, but building an implicit relation yields one invertible model composed of two simple regressions. Modeling an implicit linear relation for multiple variables can be expressed as a generalized eigenproblem of the covariance matrix of the variables in the metric of the covariance matrix of their errors. For unknown errors, this work describes their estimation by the residual errors of each variable in its regression by the other predictors. Then, the generalized eigenproblem can be reduced to the diagonalization of a special matrix built from the variables’ covariance matrix and its inversion. Numerical examples demonstrate the eigenvector solution’s good properties for building a unique equation of the relations between all variables. The proposed approach can be useful in practical regression modeling with all variables containing unobserved errors, which is a common situation for the applied problems.

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