METHODS OF REGRESSION SIMULATION IN TECHNICAL DIAGNOSTICS

To ensure reliable operation of any technical object, certain parameters are regularly monitored, and their input and output values are determined from the measurement results in real time [1, 2]. To build a mathematical model, an adaptive regression modeling approach can be applied, implying research and selection of optimal methods for building a regression model. From the totality of the available data of each determining parameter, models are constructed and investigated. Models obtained by the least squares method can have insignificant parameters and the effect of multicollinearity [3]. In this case, you should investigate the possibility of applying a number of methods of adaptation to the identified violations. It is known that the presence in the file being processed of a column of identical values when computer processing data leads to an error and stopping the calculation.

Analyzing the models obtained for each response, taking into account the different sample sizes, one can speak of the instability of the models obtained [3, 4]. This can be indicated by a rather large scatter of parameter estimates for different samples. Internal criteria were used to assess the quality of the models obtained from samples.

Analyzing the models obtained by internal quality measures, it should be concluded about the size of the sample of observations, which is the most preferable. In the course of studies of the correlation matrix, the degree of interrelation between regressors on the subject of multicollinearity, which has very negative consequences for estimating regression coefficients, is determined [3, 5]. Remains for responses are examined for violations of the assumption of regression analysis about the normality of their distribution. To control the object under consideration in order to regulate the parameters, in some cases a model containing controlled parameters is necessary. To overcome multicollinearity and the abnormal distribution of residuals, the method of step-by-step regression is used. The independent variable most influencing this decrease is introduced into the regression. To cope with this problem, a robust regression technique was developed. The most common is the robust regression M-estimator method introduced by Huber. M-estimation is based on the idea of replacing the squares of the residuals by another residual function [5]. M-score is obtained by solving the system 'p' of nonlinear equations. The solution is not equivariant with respect to scaling. Thus, residues should be standardized using some estimate of the standard deviation σ, so that they should be evaluated simultaneously. Another one of the essential elements of this implementation is the addition to the algorithm of the operation of elimination of the regressor included in the model and worsening the value of the criterion by which the search for the optimal model is performed. Also, the analysis uses the method of random search with adaptation and random search with return [4, 6]. The task of finding the optimal set of regressors can be viewed as an optimization problem for functionals with Boolean variables equal to 0 or 1. Step-by-step regression and random search with adaptation allow obtaining models with significant parameters and good predictive properties, but not all of them contain the necessary controlled parameters. If you do not use the forced introduction of control parameters, then a suitable model using the method of step-by-step regression, as well as using the LSM, may fail. The ridge estimation allows obtaining models with small values of the coefficient of determination and the F-criterion, which does not meet the requirements of the optimal model [2, 4, 6].

When using the method of random search with adaptation, the models are obtained, characterized by a small standard error, comparable in value with the results of multiple and step-by-step regression. By the coefficient of determination and F-criterion, small improvements in the quality of the model are obtained. The main advantage of these models can be considered the fact that each of them contains controlled parameters that are necessary to regulate the state of a technical object. An analysis of the correlation relationships between the model parameters and the quality indicators of the technical object under consideration makes it possible to identify significant relationships that may require further research. Anomalous observations are required to be excluded from the initial data during further processing. Methods of step-by-step regression and random search with adaptation give models with good predictive abilities [2, 5, 6]. Choosing the best model for forecasting involves the use of various measures of model quality, both internal and external. To adapt to the detected violations, the methods of crest and robust estimation can also be used.

References:

1. Krestin E.A. Diagnostics of machines and equipment: Tutorial. SPb.: Lan, 2016. – 376 p.
2. Nosov V.V. Diagnostics of machines and equipment: Study Guide. SPb.: Lan, 2016. – 376 p.
3. Draper N. Applied Regression Analysis. M.: ID Williams, 2019. – 912 p.
4. Karlberg K. Regression analysis in Microsoft Excel. M.: Dialectics, 2019. – 400p.
5. Esaulov I.G. Regression analysis of data in the Mathcad package: Tutorial. SPb.: Lan P, 2016. – 224 p.
6. Voskoboinikov Yu. E. Regression analysis of data in the Mathcad package: Tutorial. SPb.: Lan, 2011. –224 p.