New approach to switching points optimization for segmented regression during mathematical model building

Abstract

Mathematical models building is widely used in diferent branches of human activity to describe statistical data obtained during observation of various phenomena. The main tool for this problem solution is approximation theory, especially ordinary least squares method. Basic goal during approximation is minimizing deviation between observed and estimated data. Analysis showed that providing given accuracy is possible based on usage of segmented regression models. Such models contain one or more switching points for segments connection. This paper deals with a problem of calculation of optimal values of switching point abscissa for segmented regression. Analytical expression for segmented regression was obtained using the Heaviside function. Switching point's determination is based on the usage of multidimensional optimization paraboloid. Paper presents the methodology for optimal segmented regression building. Simulation results and example of data processing proved increasing the accuracy of approximation in case of using the proposed methodology.

Keywords

• eol>mathematical model building
• approximation
• ordinary least squares method
• segmented regression
• optimization of switching point abscissa

1. Introduction

The mathematical models are used in many applications. Such models give the possibility to determine the mathematical relationship (formulas, logical dependency) for real world objects and phenomena. The one of the main motives to build mathematical models is: a) a greater understanding of researched phenomena, b) to analyze the object mathematically, c) to provide experimentation with model using simulation methods [ 1, 2 ].

The mathematical models building starts with experimental investigations and obtaining observations of some system, object or phenomenon. These operations form input data for model. According to these data, at the second stage mathematical formulations are carried out, and after those computational simulations are performed. Output data of simulation are used for model validation [ 3 ].

During mathematical models building, diferent models can be utilized. Researcher always tries to choose the best of them [ 4 ]. To do this the following criteria can be used: simplicity of mathematical equation with the given level of error, minimum number of coeficients in the mathematical equation, minimum sum of squared deviations between the predicted and empirical values and others [ 5 ].

The main algorithmic tool that is used to obtain information from mathematical models contains methods of linear algebra, data analysis, probability theory and mathematical statistics, functional analysis and others [ 6 ]. The mathematical models based on statistical data-driven approach can be built using the techniques of the approximation theory [ 7 ]. In case of approximation, spline functions or diferent polynomials are often used [ 8 ]. 2. Literature review and problem statement Nowadays, regression analysis becomes popular research tool for mathematical models building [ 9 ]. It allows to develop mathematical expressions to describe the behavior of some dependent random variable [ 10 ]. Regression analysis can be used to predict the value of dependent variable based on information of its previous realization trend.

The mathematical models building based on regression analysis can be used in diferent branches of human activity and scientific research: • in econometrics: to analyze economics behavior for certain country or city dependent on one or more factors [ 11, 12 ]; • in biology: to obtain regional models of biological processes [ 13 ]; • for electrical engineering: to describe realizations of electrical signals and parameters of electronic devices [ 14, 15 ]; • in reliability theory: to build the mathematical model for trends of reliability parameters and diagnostics variables [ 16, 17 ]; • in aviation system: to build the mathematical model for Unmanned Aerial Vehicle (UAV) and aircraft flight routes [ 18, 19 ], to analyze the possibilities of UAV cyber security hazards [ 20 ], to calculate the eficiency of functioning of aviation equipment [ 21, 22 ], and others; • for radar and navigation systems: to solve the problem of eficient target detection [ 23 ] and for approximation and prediction of data trends [ 24, 25, 26 ]; • during equipment operation: to calculate the optimal maintenance periodicity [ 27, 28 ] and to estimate the eficiency of diagnostics process [ 29, 30 ]; • for control systems: to find the correlation between statistical data for inertial stabilized platforms of ground vehicles [ 31 ] and to analyze possible control actions in case of aircraft departures and arrivals delays [ 32 ].

In practice, researchers apply simple linear regression [ 33 ] and more realistic nonlinear regression [ 34 ]. Considering nonlinear regression, it should be pointed that quadratic, cubic, exponential, segmented and even logistic regressions are widely used [ 35, 36 ]. Diferent software to implement such models was developed [ 37, 38 ].

As there are diferent types of regression curves, let (→,− ,) is set of one-dimensional functions, any of them depends on vecto→r− , of parameters and gives the estimate value for initial data in for two-dimensional array (, ) with sample size . According to existing ̂︀ results [ 9, 10, 33, 36 ], regression model with one independent variable can be presented as follows

= (→,− ,) + , where and are the dependent and independent variables, is an error of evaluation.

For simple linear regression model 1(→,− ,1) = 0,1 + 1,1, where 0,1 and 0,1 are parameters that must be determined [ 9 ].

To increase the accuracy of model, on the one hand, researchers use segmented regression techniques with several linear or parabolic sections for approximation empirical data [ 33 ]. On the other hand, additional analysis for heteroskedasticity in observed data trend is carried out [ 39, 40 ]. Literature analysis showed that unfortunately not enough attention is paid to another way of increasing the accuracy of model that is associated with calculation of optimal switching points (breakpoints or changepoints) between regression segments. To estimate the parameters of regression (including switching points), the maximum likelihood estimator (MLE) can be used [ 41, 42 ]. Moreover, paper [ 42 ] concentrates on replacing the traditional nonsmooth model with another that transitions smoothly at the switching point. Another approach can be based on Bayesian changepoint models [ 43, 44 ]. In some publications, there are attempts to solve this problem based on: 1) statistical simulation results using sequential search [ 45 ], 2) inverted F test confidence interval estimate for large sample sizes and bootstrapped confidence intervals estimate for small sample sizes [ 46 ]. Analysis of mentioned techniques for calculation of optimal switching points showed: a) MLEs require prior information on error distribution and approximate range of switching point, b) MLEs have bias of estimate, c) in some modifications MLE is the most computationally expensive, both in setup time and in run time, d) Bayesian estimators are more robust for dificult cases, but require additional prior limitations for model parameters. Moreover, the exact mathematical equations for optimal value of switching points in literature are not considered.

The aim of this paper is to develop a new approach to switching points optimization in case of segmented regression usage for mathematical models building. The calculation of the optimal values of abscissas of the switching points will give the possibility to increase the approximation accuracy and the possibility to improve the predictive properties.

From mathematical point of view, such problem can be considered as follows. At the first stage, it is necessary to choose the segmented approximation function (→,− ,) in such a way to minimize standard deviation between real values and estimates ̂︀ = (∀ : ((→,− ,)) ≤ ( (→,− , ))). (1)

At the second stage, it is necessary to carry out optimization of switching points abscissas and to find the corresponding values (1 , 2 , ..., ) = (1 , 2 , ..., ), (2) where is quantity of switching points in case of + 1 segments for regression usage.

3. Methodology

The best preferred statistical data processing algorithms can be used in the conditions of aprioristic uncertainty [ 47 ]. In this research some limitations about aprioristic information was made.

After observation of random phenomenon, the two-dimensional array (, ) with sample size is collected. Initial data are plotted in two-dimensional space in form of dependence. Based on visual analysis of data, researcher can identify geometrical structure of data trend and choose the appropriate approximation function. Assume that only segmented functions can be used. Such function contains two or more segment without discontinuities. The segments are connected in the switching points. The quantity of switching points or the quantity + 1 of segments is determined by researcher according to the analysis of geometrical structure of plotted data.

At the first step, type of segmented regression for data approximation is chosen. In authors opinion, it is enough to use one of three types of segmented regression: 1. Segmented linear regression

1() = 0,1 + 1,1 + ∑︁ +1,1( − )ℎ( − ), =1 (3) where ℎ( − ) is Heaviside step function.

In case of two segments usage, functional dependence (3) contains one switching point and three unknown coeficients. Equation (3) can be presented as follows

1() = 0,1 + 1,1 + 2,1( − 1 )ℎ( − 1 ).

Unknown coeficients 0,1, 1,1 and 2,1 are calculated according to ordinary least squares method in such a way

⎛ 0,1 ⎞ ⎛ = − 1, = ⎝ 1,1 ⎠ , = ⎝ 2,1 ∑︀=1 ∑︀=1 ∑︀=1( − 1 )ℎ1 ⎞ ⎠ , ℎ1 = ℎ( − 1 ), ⎡ = ⎣

∑︀1 2. Segmented parabolic regression ∑︀1( − 1 )ℎ1 ∑︀1( − 1 )ℎ1 ∑︀1( − 1 )ℎ1) ⎤ ∑︀1( − 1 )ℎ1⎦ .

∑︀1( − 1 )2ℎ1 2() = 0,2 + 1,2 + 2,22 + ∑︁ +2,1( − )2ℎ( − ).

=1 (4)

In the case of two segments usage, functional dependence (4) contains one switching point and four unknown coeficients. Equation (4) can be presented as follows 2() = 0,2 + 1,2 + 2,22 + 3,2( − 1 )2ℎ( − 1 ).

∑︀1 ∑︀1 2

Unknown coeficients squares method in such a way 0,2, 1,2, 2,2 and 3,2 are calculated according to ordinary least 3,2 3. Segmented linear-parabolic regression where () is sign function. This function is equal to zero, if the segment is linear, and is equal to one, if the segment is parabolic.

In the case of two segments usage with first parabolic and second linear segment, functional dependence (5) contains one switching point and three unknown coeficients. Equation (5) can be presented as follows

3() = 0,3 + 1,3 + 2,32 − 2,3( − 1 )2ℎ( − 1 ).

Unknown coeficients 0,3, 1,3 and 2,3 are calculated according to ordinary least squares method in such a way

⎛ 0,3 ⎞ ⎛ = − 1, = ⎝ 1,3 ⎠ , = ⎝ 2,3 ∑︀1 ∑︀1 ∑︀1 2 − ∑︀1 2 ℎ1 ⎞ ⎠ , ⎡ ∑︀1 ∑︀1 2 ∑︀1 3 − ∑︀1 2 ℎ1 = ⎣ ∑︀1 ⎦ .

∑︀1 2 − ∑︀1 2 ℎ1

At the second step, the quantity of switching points and the range of possible values of abscissas of switching points is selected subjectively based on visual analysis of observed data. For this approach, it is necessary to choose at least five possible values for each switching point. So matrix of vectors of possible abscissa values is generated in the following form ∑︀1 2 − ∑︀1 2 ℎ1 ∑︀1 3 − ∑︀1 2 ℎ1 ∑︀1 4 + ∑︀1(4 − 22 2 )ℎ1 ⎤ →(− 1 →,− 2 , ..→.,− ).

At the third step, regression coeficients and standard deviations between real values and estimates for all segmented regression types are calculated. Standard deviation is determined ̂︀ according to the equation

⎯⎸ 1 ∑︁( − )2, = ⎷⎸ − =1 ̂︀ where is a degree of freedom for selected model.

The standard deviation is calculated for all combinations of possible values of switching point abscissa. So at this step, the -dimensional dependence of (1 , 2 , ..., ) is obtained.

At the fourth step, the obtained dependence is approximated by -dimensional paraboloid based on ordinary least squares method. The general equation of -dimensional paraboloid where , , , are unknown coeficients need to be estimated, the sum is calculated only for < .

To simplify the calculation, it can be assumed that , = 0 and equation (7) will take a form (7) (8) In this case unknown coeficients can be found according to the following equation ⎛ 0 ⎞ ⎛ ∑︀1 ... ∑︀1 1,2,..., ⎞ ⎜ 1 ⎟ ⎜ ∑︀1 ... ∑︀1 211 1,2,..., ⎟⎟ = − 1, = ⎜⎝⎜⎜⎜⎜ ...1 ⎠⎟⎟⎟⎟⎟ , = ⎜⎝⎜⎜⎜ ∑︀1 ... ∑︀1 .2..1 1,2,..., ⎠⎟⎟

⎜ ∑︀1 ... ∑︀1 11 1,2,..., ⎟⎟ , = − 1 ∑︀1 ... ∑︀1 1 1,2,..., = ⎢⎢⎢⎡⎢⎣ ∑∑︀︀11...21111 ∑∑∑︀︀︀111...342111111 ∑∑∑︀︀︀111...23111111 ∑︀1 1 ∑︀1 211 1 ∑︀1 11 1 ... ∑︀1 1 ⎤ ... ∑︀ 211 1 ⎥

1 ... ∑︀1 11 1 ⎥⎥ . ... ... ⎥⎦ ... ∑︀1 21 where is quantity of chosen points in the range of possible values of abscissas of switching points.

At the fifth step, the minimum of -dimensional paraboloid is calculated to provide the criterion (2). For this purpose, the theory of optimization is used [48]. To find the minimum, it is necessary to solve the system of equations ⎧ (1 ,2 ,..., ) = 0, ⎪⎪ 1 ⎪⎪⎪ (1 ,2 ,..., ) = 0, ⎨ 2 (9) ⎪... ⎪ ⎪⎪ (1 ,2 ,..., ) = 0.

⎩⎪

In the case of -dimensional paraboloid (7) usage, the system of equations (9) turns to the system of linear equations that can be solved by one of known method. In case of simplified paraboloid (8) usage, the simple solution can be obtained in the following form (10)

At the sixth step, coeficients of segmented regression (3), (4) or (5) are recalculated, and resulting model is obtained. 4. Simulation results and numerical example Consider the problem of analysis of proposed methodology implementation based on the results of statistical simulation.

The statistical simulation starts with obtaining initial data set with two switching points. The data set contains deterministic and random components. The deterministic component can be presented as follows 1( ) = 0,1 + 1,1 + 2,1( − 1 )ℎ( − 1 ) + 3,1( − 2 )ℎ( − 2 ).

This dependence is converted into discrete form at the range [1; 100] with sampling interval = 1 and sample size = 100. The initial parameters of deterministic model can be diferent, but in this research, authors used the following initial numerical values: 0,1 = 500, 1,1 = 10, 2,1 = − 25, 3,1 = 20, 1 = 20 and 2 = 50.

Random component is generated at each sample point as additive Gaussian noise with zero expected value and standard deviation = 30. The number of procedures reiteration is 1000.

The example of one of data sets is given in table 1. The data in the table 1 present the values of dependent variable that was measured at points separated by sampling interval .

The graphical presentation of three examples of initial data set is shown in figure 1.

Visual analysis of data (figure 1) gives possibility to conclude that most convenient regression type for these data approximation is segmented linear regression with two switching points. Let = 5. The range of possible values of abscissas of switching points is 1 = (10, 15, 20, 25, 30), 2 = (40, 45, 50, 55, 60).

In this case it is necessary to calculate estimates of regression coeficients 0,1, 1,1, 2,1, 3,1 for all combinations of possible values of abscissas of switching points. After that, standard deviation (6) is determined for each option. The results of standard deviation calculation are given in table 2.

Data from table 2 are approximated by two-dimensional paraboloid based on ordinary least squares methods. For paraboloid types (7) and (8) following equations were obtained (1 , 2 ) = 364.893 − 6.6351 − 10.0122 + 0.11121 + 0.0922 + 0.0471 2 , (1 , 2 ) = 317.416 − 4.2611 − 9.0622 + 0.11121 + 0.0922 , The obtained paraboloids are shown in figure 2 and figure 3, respectively.

In the case of paraboloid (7) usage, it is necessary to solve system of equations (9) that takes a form

After derivatives calculation this system of equations turns to system of linear equations {︃ − 6.635 + 0.2221 + 0.0472 = 0, − 10.012 + 0.0471 + 0.182 = 0.

The solution of this system is ⎧ (1 ,2 ) = 0, ⎨ 1

(1 ,2 ) = 0. ⎩ 2

In the case of paraboloid (8) usage, the optimal values of abscissas of switching points are calculated according to equation (10). The results of calculation 1 = 18.941, 2 = 50.812.

/ 1 = 19.113,

/ 2 = 50.532.

Analysis showed that for this particular case simplified paraboloid gives greater accuracy of switching point’s abscissas estimates (relative error is 4.435 percent and 1.064 percent for the ifrst and second switching points, respectively).

Resulting segmented linear regressions for both optimization options (paraboloids (7) and (8)) are 1() = 484.143 + 11.397 − 25.025( − 18.941)ℎ( − 18.941)+

+18.021( − 50.812)ℎ( − 50.812), 1() = 484.987 + 11.26 − 25.073( − 19.113)ℎ( − 19.113)+

+18.155( − 50.532)ℎ( − 50.532).

The standard deviation for the first and second optimization options is 46.038 and 46.040, respectively. The results of approximation are shown in figure 4.

Resulting segmented linear regressions for both optimization options in figure 4 almost coincide and have approximately equal standard deviation.

Consider the statistical simulation results for 1000 reiteration procedures. Such simulation gives the possibility to build the probability density functions of estimates of switching point’s abscissas. Figure 5 shows the histograms for estimate of abscissa of the first (figure 5a) and second (figure 5c) switching point for paraboloid (7), the histograms for estimate of abscissa of the first (figure 5b) and second (figure 5d) switching point for paraboloid (8). Statistical characteristics (expected value, variance, minimum and maximum) of estimates for optimal values of abscissas of switching points using paraboloids (7) and (8) are given in table 3.

Analysis showed that general paraboloid (7) in average has greater accuracy for switching points abscissas estimation. In the case of the first switching points abscissas estimation, relative error is 3.63 and 4.32 percents for paraboloid (7) and (8), respectively. In the case of second switching points abscissas estimation, relative error is 0.968 and 1.376 percents for paraboloid (7) and (8), respectively. In addition, paraboloid (7) has greater scattering of estimate.

The simulation results give approximatly same eficiency of estimate and accuracy of mathematical model. So to simplify the calculation, optimizational paraboloid (8) can be used as more suitable during mathematical model building.

5. Conclusion

The paper considers new approach to switching point’s optimization for segmented regression during mathematical model building. The analytical equations for segmented linear, parabolic and linear-parabolic regressions are presented based on usage of Heaviside step function. To ifnd the optimal values of connection points between regression segments, multidimensional optimization paraboloid is used for describing the dependence of standard deviation on possible values of switching point’s abscissa. The proposed methodology, in contrast to the existing ones, allows to obtain the accurate mathematical formula for calculating the abscissa of switching points. Moreover, considered methodology has property of robustness for initial distribution of errors and dataset. The analysis of proposed methodology is carried out based on statistical simulation. The implementation of methodology is explained on numerical example for generated data set. Computations prove feasibility of proposed approach. The research results can be used to increase the accuracy of data approximation in mathematical model building.

Further research directions will be associated with a comparative analysis of the efeciency of the proposed methodology with other techniques for determining estimates of the abscissa of switching points (in particular, MLE and estimates based on the Bayesian approach) in the case of diferent limitations presence. conditions of aprioristic uncertainty, in: Proceedings of IEEE Ukrainian Microwave Week, 2020, pp. 418–423. doi:10.1109/UkrMW49653.2020.9252687. [48] G. V. Reklaitis, A. Ravindran, K. M. Ragsdell, Engineering Optimization. Methods and Applications, John Wiley and Sons, New York, 1983.

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