Approximation and Extrapolation of Two-Dimensional Tables Текст научной статьи по специальности «Математика»

Approximation and Extrapolation of Two-Dimensional Tables

PhD V. V. Garbaruk, U. V. Terenteva Emperor Alexander I St. Petersburg State Transport University Saint Petersburg, Russia garbaruk@pgups.ru, trntv.u@gmail.com

Abstract. An approximation of a two-dimensional table by a function of two variables was carried out. As an example, experimental data characterizing the parameters of the storage battery are selected. First, various types of dependence on one parameter were considered with a constant value of the other. Power and exponential functions of one variable were chosen according to the greatest coefficient of determination R2. The desired function of two variables was presented as a product of these functions. The small relative error in the prediction of experimental data makes it possible to apply the obtained formula for further extrapolation.

Keywords: storage battery, two-dimensional table of experimental data, approximation, function of two variables, linear regression equation.

The task of establishing a functional relationship between data obtained experimentally often arises in scientific research. The approximation of numerical sequences by a function of one variable is well developed [1-3], but the fitting of a function of several variables is much more difficult. To process a three-dimensional data array, the «TableCurve 3D» program was created, which contains more than fifteen thousand [4] approximating functions. The use of this program makes it possible to quickly assess the quality of the choice of function, but the choice of the trait of the desired function of two variables remains with the researcher. In [5], based on a number of theorems by A. N. Kolmogorov and V. I. Arnold function of several variables is sought as a weighted sum of functions of one variable. To approximate the function of two variables, the product of two interpolation splines was also used [6]. The least squares method was applied [7] after representing the desired function as a sum of one-dimensional functions.

In this paper, we study a two-dimensional table of battery operation parameters. A function of two variables, approximating the table, is written as a product of two functions of one variable, obtained after consideration of various types of dependencies. The uninterrupted power supply of various microelectronic equipment is necessary to ensure the safety of train traffic. In the case of a loss of external power supply, the transition from the main power supply to the backup one is provided by the batteries of the railway station. One of the most important indicators of battery readiness for operation is its charge level, which is constantly checked during maintenance. It is known that the battery voltage varies depending on the degree of its charge. For various temperatures t, Table 1 shows the empirical values of the voltage H of the battery, which determine the percentage P of the degree of its charge after a special test.

When approximating the empirical points of the surface by a function of two variables H = f(P, t), we first considered various types of the dependence H = f(P, c) = g(P) for different

Table 1

Voltage H of the battery

P -15° 0° 25°

100 % 25,8 25,4 25,2

75 % 25,6 25,2 25,0

50 % 25,4 25,0 24,8

25 % 24,8 24,6 24,4

5 % 24,2 23,8 23,6

values of t = c (Fig. 1) and functions H = fc, t) = h(t) at constant values P = c (Fig. 2). According to the greatest coefficient of determination R2, the power g(P) = a x Pb and exponential h(t) = с x ekt functions were chosen.

Fig. 1. Power dependence of voltage on the degree of charge

* / \ H

''-1.2,8 .

' ^ 4 H= 12.757e

♦ ' R2 = 0.8947

" " ' - 12,6 . - ** •

!2,4 . - H= 12.557e --° 0006~

*'■ __ R2 = 0.8947

12.316e -»«»* -

Д2 = 0.9803 „

12.2 ~ ^ ■ ■ s

• 100%

♦ 50%

a 25%

-15 -5 5 15 25

Fig. 2. Exponential dependence of voltage on temperature

The desired function of two variables was presented as a product of power and exponential functions

H = d x Pb x ekt, (1)

because-when fixing one of the variables P or t, the character of the change in the value of H corresponds to the previously selected function of one variable.

The logarithm of this function ln d + b ln P + kt is linear with respect to the arguments ln P and t, what makes it possible to find out the factors ln d, b, k. Table 2 shows the logarithms of the voltage H of the battery, taken from Table 1.

Table 2

The values of the ln Н of the battery

t P -15° 0° 25°

100 % 3,25037 3,23475 3,22684

75 % 3,24259 3,22684 3,21888

50 % 3,23475 3,21888 3,21084

25 % 3,21084 3,20275 3,19458

5 % 3,18635 3,16969 3,16125

The coefficients ln d, b, k of the equation

(Fd = 0; f a11 x ln d + a12 x b + a13 x к = дг;

' Fb = 0; or j a2i x ln d + 022 x b + x к = д2; (2)

^Fk = 0 U31 xln d + 032 x b + 033 x к = дз,

where

5 3

а11 = 15, a12 = 3 x Z ln Pt, a13 = 5 x Z tj,

i=1 35

i=1

=!!*;

j=1 i=1

55

a-21 = 3 x Z lnPt, a22 = 3 x ^(lnPt)2,

i=1 1=1

3 5 3 5

= XZ/lnPi x lj),g2 = x lnpi);

j = 1 1=1 j = 1 1 = 1

3 3 5

a-31 = 5 x ,0.32 = ^^(lnPi x li),

j=1 j=1 1=1

3 3 5

P33 = 5 x Ztf, g3 = Z Z(zji x tj).

j=1 j=11=1

ln H = ln d + b ln P + kt

are found by the method of least squares, in which the sum of the squared deviations of the function ln H from the actual values Zji = \n(Hij) of Table 2 is minimized for the same values of the arguments.

Non-negative quadratic function

35

F(d,b,k) = ZZ(lnd + b x lnPt + к x tj — Zjt)2

j=1 i=1

has a single minimum point that satisfies the necessary condition

Algebraic linear system (2) has a unique solution b « 0,022; k и -0,00054; ln d и 2,445. After substituting the obtained coefficients into the formula ln H = ln d + b ln P + kt and potentiating this expression, we find the function of two variables

H = 11,53 x

p0,022

e0,00054t .

with a large coefficient of determination R2 = 0,98.

Table 3, in addition to the initial experimental data, contains the results of a forecast calculated by the formula obtained for the value of H, as well as the relative error of such the forecast.

Table 3

Relative error of the approximating formula

-15° 0° 25°

Exper. Forecast Error Exper. Forecast Error Exper. Forecast Error

100 % 25,80 25,68 0,46% 25,40 25,47 0,29% 25,20 25,13 0,26%

75 % 25,60 25,52 0,31% 25,20 25,32 0,46% 25,00 24,98 0,09%

50 % 25,40 25,30 0,40% 25,00 25,10 0,38% 24,80 24,46 0,16%

25 % 24,80 24,92 0,49% 24,60 24,72 0,50% 24,40 24,39 0,04%

5 % 24,20 24,07 0,54% 23,80 23,88 0,32% 23,60 23,56 0,18%

The maximum relative forecast error is only 0,54 %. Good approximation results make it possible to predict the battery voltage H in various ranges of P and t. For example, it becomes possible to extrapolate the experimental data of Table 1 to lower or higher temperatures. For example, for northern regions, Table 4 may be relevant.

Table 4

Extrapolation of the original table

\t P -35° -25° -15°

100 % 25,96 25,82 25,80

75 % 25,80 25,66 25,60

50 % 25,57 25,44 25,40

25 % 25,19 25,06 24,80

5 % 24,33 24,20 24,20

The formula (1) can be written in the form

p = (_K_)V‘.

\d x ekt)

If this formula is confirmed or refined by experimental studies, then it can be used to test the performance of batteries without resorting to Table 1.

References

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Аппроксимация и экстраполяция двумерных таблиц

к.т.н. В. В. Гарбарук, У. В. Терентьева

Петербургский государственный университет путей сообщения Императора Александра I

Санкт-Петербург, Россия garbaruk@pgups.ru, trntv.u@gmail.com

Аннотация. Проведена аппроксимация двумерной таблицы функцией двух переменных. В качестве примера выбраны экспериментальные данные, характеризующие параметры аккумуляторной батареи. Сначала были рассмотрены различные виды зависимости одного параметра при постоянном значении другого. По наибольшему коэффициенту детерминации R2 были выбраны степенная и показательная функции одной переменной. Искомая функция двух переменных была представлена в виде произведения этих функций. Малая относительная погрешность прогноза экспериментальных данных позволяет применять полученную формулу для дальнейшей экстраполяции.

Ключевые слова: аккумуляторная батарея, двумерная таблица экспериментальных данных, аппроксимация, функция двух переменных, линейное регрессионное уравнение.

Литература

1. Некрасов, О. Н. Аппроксимация экспериментальных данных на основе линеаризуемых функций // Научные и образовательные проблемы гражданской защиты. 2009. № 1. С. 78-85.

2. Крюкова, С. В. Оценка методов пространственной интерполяции метеорологических данных / С. В. Крюкова, Т. Е. Симакина // Общество. Среда. Развитие. 2018. № 1. С. 144-151.

3. Алюков, С. В. Аппроксимация ступенчатых функций в задачах математического моделирования // Математическое моделирование. 2011. Т. 23, № 3. С. 75-88.

4. Панфилов, Г. В. Аппроксимация трехмерных графических зависимостей / Г. В. Панфилов, А. В. Черняев // Известия Тульского государственного университета. Технические науки. 2017. Вып. 11-1. С. 194-206.

5. Бутырский, Е. Ю. Аппроксимация многомерных функций / Е. Ю. Бутырский, И. А. Кувалдин, В. П. Чалкин // Научное приборостроение. 2010 .Т. 20, № 2. С. 82-92.

6. Даугавет, В. А. Приближение функции двух переменных произведением функций одной переменной в заданной области / В. А. Даугавет, М. В. Киреева // Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия. 2010. № 3. С. 13-21.

7. Данилов, А. М. Интерполяция, аппроксимация, оптимизация: анализ и синтез сложных систем: Монография / А. М. Данилов, И. А. Гарькина. — Пенза: Пензенский гос. ун-т архитектуры и строительства, 2014. — 168 с.