The use of the inverse transformation method for time series analysis Текст научной статьи по специальности «Математика»

UDC 338.27

Doi: 10.31772/2587-6066-2020-21-1-34-40

For citation: Shiryaeva T. A., Khlupichev V. A., Shlepkin A. K., Melnikova O. L. The use of the inverse transformation method for time series analysis. Siberian Journal of Science and Technology. 2020, Vol. 21, No. 1, P. 34-40. Doi: 10.31772/2587-6066-2020-21-1-34-40

Для цитирования: Ширяева Т. А., Хлупичев В. А., Шлепкин А. К., Мельникова О. Л. Метод обратного преобразования для анализа временных рядов // Сибирский журнал науки и технологий. 2020. Т. 21, № 1. С. 34-40. Doi: 10.31772/2587-6066-2020-21-1-34-40

THE USE OF THE INVERSE TRANSFORMATION METHOD FOR TIME SERIES ANALYSIS

T. A. Shiryaeva1, V. A. Khlupichev1, A. K. Shlepkin1*, O. L. Melnikova2

Krasnoyarsk State Agrarian University 90, Mira Av., Krasnoyarsk, 660001, Russian Federation 2Khakas State University 90, Lenin Av., Abakan, 655017, Russian Federation *E-mail:ak_kgau@mail.ru

In modern conditions of technology development, signs of systemacity are manifested to one degree or another in all areas, so the use of system analysis is an urgent task. In this case, the main factors in this situation are data processing and prediction of the state of a system. Mathematical modeling is used as a prediction method for a given subject area. A mathematical model is a universal tool for describing complex systems representing the approximate description of the class of phenomena of the external world expressed by mathematical concepts and language. The mathematical model can be represented as a set of systematic components and a random component. In this article, the object ofpre-diction is the irregular random component of a model, which reflects the impact of numerous random factors. The origin, nature and laws of variation of the random variable are known, therefore, to simulate its behavior or predict its future value, one needs high degree of certainty to establish the form of continuous distribution function of the random variable. The empirical distribution function is calculated using the sample of random variable values. This empirical function is close to the values of the desired unknown function of distribution. The resulting empirical function is discrete, therefore it is necessary to apply piecewise linear interpolation to obtain a continuous distribution function. The predicted random component of time series has been included in the initial regression model. In order to compare augmented and initial regression models, several values were excluded from the time series and new prediction was built. The value of the average approximation error for assessing the quality of the model is calculated. The augmented regression model proved to be more effective than the original one.

Keywords: forecasting, time series analysis, inverse transformation, system analysis.

МЕТОД ОБРАТНОГО ПРЕОБРАЗОВАНИЯ ДЛЯ АНАЛИЗА ВРЕМЕННЫХ РЯДОВ

Т. А. Ширяева1, В. А. Хлупичев1, А. К. Шлепкин1 , О. Л. Мельникова2

1 Красноярский государственный аграрный университет Российская Федерация, 660001, г. Красноярск, просп. Мира, 90

2Хакасский государственный университет имени М. Ф. Катанова Российская Федерация, 655017, г. Абакан, просп. Ленина, 90 *E-mail:ak_kgau@mail.ru

В современных условиях развития технологий признаки системности проявляются в той или иной степени во всех областях, поэтому использование системного анализа является актуальной задачей. При этом главными факторами в данной ситуации являются обработка данных и прогнозирование состояния системы. Для заданного объекта в качестве способа прогнозирования в данной работе применяется моделирование, а точнее математическое моделирование. Математическая модель - это универсальное средство исследования сложных систем, представляющее собой приближенное описание какого-либо класса явлений внешнего мира, выраженное с помощью математической символики.

Математическую модель можно представить как совокупность систематических компонентов и случайной составляющей. В данной статье регрессионная модель уже определена, а в качестве объекта прогнозирования рассмотрена остаточная нерегулярная компонента модели, которая отражает воздействие многочисленных факторов случайного характера.

Происхождение, природа и законы изменения данной случайной величины нам неизвестны, поэтому для моделирования ее поведения или предсказания ее будущих значений необходимо с высокой степенью достоверности установить вид непрерывной функции распределения данной случайной величины.

Для этого была рассчитана эмпирическая функция распределения с помощью выборки из значений случайной величины. Данная эмпирическая функция в определенной степени приближена к значениям искомой неизвестной функции распределения. Полученная эмпирическая функция носит дискретный характер, поэтому необходимо применить кусочно-линейную интерполяцию и таким образом получить непрерывную функцию распределения.

В исходную регрессионную модель была включена спрогнозированная случайная компонента временного ряда. Для того чтобы сравнить дополненную и исходную регрессионные модели, из динамического ряда были исключены несколько значений и построен новый прогноз. Рассчитано значение средней ошибки аппроксимации для оценки качества модели. Дополненная регрессионная модель показала себя эффективнее исходной.

Ключевые слова: прогнозирование, анализ временных рядов, обратное преобразование, системный анализ.

Introduction. For the specialists involved in data analysis, in most cases prediction may be said to be the main goal and task. Modern methods of statistical forecasting are often able to predict almost any possible indicators with high accuracy [1].

Forecasting is a system of scientifically based ideas about the possible conditions of an object in the future and alternative ways of its development [2]. There are no universal prediction methods for all occasions. Any practical forecasting problem can be satisfactorily solved only by a limited number of methods [3]. The choice of a forecasting method and its effectiveness depend on many conditions: the purpose of the forecast, the period of its lead prediction, the level of detailing and the availability of initial information [4]. The most commonly used forecasting method is mathematical modeling. A mathematical model is an approximate description of a specific process or phenomenon of the external world, expressed using a mathematical apparatus [5].

The components of the time series. Commonly, when studying a time (dynamic) series, it is depicted in the form of the following mathematical model:

Y = Y + Et

where Yt - time series value; Yt - systematic (deterministic) component of the time series; Et - random component of the time series [6].

The systematic component of the time series Yt is a result of the influence of constantly acting factors on the process being analyzed. Two main systematic components of the time series can be distinguished:

1. The trend of the time series.

2. The cyclic oscillations of the time series.

A trend is a general pattern of change in the indicators of the time series, stable and observed over a long period of time. A trend is described using some function, usually monotonic. This function is called 'trend function', or simply "trend" [7].

Among the factors that form the cyclical oscillations of the series, in turn, two components can be distinguished:

1) seasonality;

2) cyclicity.

Seasonality is a result of the influence of factors acting at a predetermined periodicity. These are regular fluc-

tuations that are periodic in nature ending within a year. The cyclic component is a nonrandom function that describes long (more than a year) periods of rise and fall [8; 9]

The random component of the time series Et is the component of the time series remaining after the allocation of systematic components. It reflects the effects of numerous random factors; it is a random, irregular component.

Random variables are diverse in nature and origin, although the distribution law can be written in a uniform universal form, namely, in the form of a distribution function that is equally suitable for discrete and continuous random variables [10].

Inverse Transformation Method. For forecasting purposes, as well as simulation, one may need a method for generating the random component of a time series. For this purpose, we use the inverse transformation method.

Let the random variable X have the distribution function F(x). We assume that F~:(x) is the inverse function of F(x). Then the algorithm for generating the random variable X with the distribution function F(x) will be the following:

1. Generate the value U having the uniform distribution over the interval (0;1);

2. Return X = F~l(U).

Fig. 2 depicts this algorithm graphically; the random variable corresponding to this distribution function can take either positive or negative values; it depends on the specific value of U. In Fig. 1, the random number U1 gives the positive value of the random variable X1 as a result, while the random number U2 gives the negative value of the random variable X2 as a result [11].

Estimating the distribution function of a random variable. Let us consider the time series as a sequence xu x2, ..., xn of independent and equally distributed, according to a certain law, random variables; this sequence is called the sample of volume n. Each xt(t = 1, 2, ..., n) is called variation. Having the sample, we do not have information about the form of the distribution function F(x). It is required to construct an estimate (approximation) for this unknown function.

The most preferred estimate of the function F(x) will be the empirical distribution function Fn(x).

/ 1 / 1 ^ 1 1 + u2 1 [ 1 ..... г

_____—

Х2 Xi

Fig. 1. Using the Inverse Transformation Method to generate a random variable

Рис. 1. Использование метода обратного преобразования для генерирования случайной величины

Fig. 2. Graph of the empirical distribution function Fn(e) Рис. 2. График эмпирическая функция распределения Fn(e)

The empirical distribution function (sampling distribution function) is the function Fn(x), which determines the relative frequency of the event X < x for each x value, i. e.

n

Fn (x) = ^, n

where nx - the number of xt values, less than x; n - sample size.

With a sufficiently large sample size, the functions Fn(x) and F(x) = P(X < x) differ insignificantly from each other.

The difference between the empirical distribution function and the theoretical one is that the theoretical distribution function determines the probability of the event

X<x, and the empirical function determines the relative frequency of the same event [12].

The empirical distribution function has all the properties of the integral distribution function:

1) the values of the empirical distribution function belong to the interval [0; 1];

2) Fn(x) is non-decreasing function;

3) Fn(x) = 0 at x < xmin, if xmin is the smallest variation;

Fn(x) = 1 at x > xmin , if xmin is the largest variation [6].

However, to use the inverse transformation method, it is convenient to have a continuous distribution function; therefore, it is necessary to interpolate the obtained empirical function.

Building a predicted model. We give an example of using the inverse transformation method when constructing a predicted model. As the initial data, we use the average monthly indicators of power consumption in the Krasnoyarsk Territory for 3 years from January 2009 to December 2011 [13].

Using some regression model, the forecast values of the time series were calculated. Actual Yt and predicted Yt values of the time series are presented in tab. 1

We build the empirical distribution function of the values of the deviations et of the predicted values Yt from the actual values Yt of the time series et = Yt - Yt. To do this, it is necessary to rank the sample {et}, thus obtaining the sample {e(t) = {e(1) < e(2) <... < e(n)} (tab. 2).

Since the frequency of each variation equals 1, the empirical function will have the following form:

Fn (е) =

0, e e[-»,вш);

t

— , e i n

[екгe(t+i)); t =0'1'-'n;

i,

e e[e(n), +ю) •

The graph of the function Fn(e) is shown in fig. 2. The obtained empirical function Fn(e) has a discrete form. We use piecewise linear interpolation to obtain the continuous distribution function of the random variable

F*(e). To do this, we use the equation of a straight line

passing through two points:

y = (x - xl) x| ^^^ |+ y1.

The continuous distribution function of the random variable Fn (e) will have the following form:

0 e eQ);

Fn (e) =

(ee(t))

(

V (n -1)

Л

V e(t+1) e(t) у

t -1

1 e e[e(t)'e(t+i)); t=

;[e( n), +4

Tablel

Actual Yt and predicted Yt values of the time series

t Yt Yt t Yt Yt t Yt Yt t Yt Yt

1 51-0123 53-09764 11 35-5809 43-42324 21 37-8690 44-58914 31 17-1468 15-15964

2 38-2345 38-37535 12 53-2584 54-26158 22 63-4957 57-61664 32 20-8548 21-45395

3 40-0023 35-24303 13 52-3887 52-69219 23 72-9843 72-28322 33 29-3791 31-55531

4 25-1288 32-13879 14 39-9125 41-30390 24 88-0214 83-09426 34 51-1710 44-09931

5 22-9338 27-08163 15 39-2113 32-14284 25 82-6095 79-01463 35 61-5869 59-79590

6 27-0146 20-64426 16 31-3420 24-38189 26 62-7282 63-15378 36 71-2594 73-02318

7 25-1154 17-77792 17 26-0102 19-52312 27 50-0250 48-20592

8 16-6987 19-20278 18 20-5578 17-87433 28 29-6211 34-02474

9 27-3114 23-86244 19 12-1214 22,62140 29 22-2954 22-75450

10 29-2400 31-48983 20 24-9374 32-44863 30 17-8092 15-25792

Table 2

Range of Values et and e,t)

t et e(t) t et e(t) t et e(t) t et e(t)

1 -2-085 -10-500 11 -7-842 -2,085 21 -6-720 1-791 31 1-987 6-370

2 -0-141 -7-842 12 -1-003 -1-764 22 5-879 1-819 32 -0-599 6-487

3 4-759 -7-511 13 -0-303 -1-391 23 0-701 1-987 33 -2-176 6-960

4 -7-010 -7-010 14 -1-391 -1-003 24 4-927 2-551 34 7-072 7-068

5 -4-148 -6-720 15 7-068 -0-599 25 3-595 2-683 35 1-791 7-072

6 6-370 -4-404 16 6-960 -0-459 26 -0-426 3-449 36 -1-764 7-337

7 7-337 -4-148 17 6-487 -0-426 27 1-819 3-595

8 -2-504 -2-504 18 2-683 -0-303 28 -4-404 4-759

9 3-449 -2-250 19 -10-500 -0-141 29 -0-459 4-927

10 -2-250 -2-176 20 -7-511 0-701 30 2-551 5-879

The graph of the function F36 (e) is shown in fig. 3.

Evaluating the predicted model. For further analysis of this method, let us consider several forecast models [14]:

1. We take the value Yt of the time series as a completely deterministic process, to carry out the forecast we use the values Yt calculated using the regression model;

2. The value Yt of the time series will be taken as a random variable for which we construct the distribution function F*(e) and calculate the predicted values ;

3. We will take the value Yt of the time series as a set of values Yt calculated using the regression model and the random component et, for which we construct the distribution function F*(e) and calculate the predicted values e't.

Let us make the operational prediction of energy consumption levels. For this purpose, we exclude from consideration the last 5 observations from the sample and we calculate new estimates of the parameters of the

regression model, as well as the new distribution functions F3* (x) and F3* (e).

We apply the inverse transformation algorithm to the obtained functions F3* (x) and F3* (e). To this end, we generate the sample {ut} of random numbers having a uniform distribution in the interval [0; 1], and return Y't = F3*-1 (ut) and e't = F311 (ut). The calculation results are presented in tab. 3.

When considering the obtained results, it is clear that the sum Yt + et is closer to the actual data than the predicted values calculated using the regression model. Thus, the predicted values smoothed out the predicted error to an extent (fig. 4).

As a criterion for assessing the quality of the model, we determine the value of the average approximation error, which is calculated using the formula [15]:

1 n

А = 11

iYact Ypr)

Yac

100%,

where Ypr - predicted value of time series; Yact - actual value of time series; n - size of time series [10].

Fig- 3- Graph of the continuous distribution function F36(e)

ж

Рис- 3- График непрерывной функции распределения F36 (e)

Results of applying the inverse transformation algorithm

Table 3

№ t Y Ut e; Yt + e; Yt'

1 32 25.1456 0.0608 -7.1360 18.0096 6.9693

2 33 37.1619 0.6514 1.9654 39.1274 14.9283

3 34 52.0351 0.6577 2.1085 54.1436 15.2489

4 35 70.5459 0.0515 -7.1507 63.3952 6.8725

5 36 86.8325 0.5448 0.3808 87.2133 13.4902

t=1

32 33 34 35 36

■■— Actual value О ■ Regression model —-A—- Predicted error

Fig. 4. Graph display of predicted results Рис. 4. Графическое отображение результатов прогноза

The values of the average approximation error for Yt,

Y{ and Yt + e't are the following:

1) A(Yt)«17,03%;

2) A(Yt')« 71,18%;

3) A(Yt + e't) « 15,59%.

The highest indicator of the average approximation error was obtained under the assumption that the time series Yt is a random variable. The average approximation error for the regression model is 54.15 % less, which tells us that the time series is a determinate value. As a result of including values et in regression, the average approximation error decreased by approximately 1.44 %.

Conclusion. The method presented above can be used to determine the continuous distribution function of a random variable and generate a random variable for predicting and simulation purposes.

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© Shiryaeva Т. А„ Khlupichev V. А., Shlepkin A. K., Melnikova O. L., 2020

Shiryaeva Tamara Alekseevna - Cand. Sc., Professor; Krasnoyarsk State Agrarian University. E-mail: info@kgau.ru.

Khlupichev Vladimir Aleksandrovich - Master's Student; Krasnoyarsk State Agrarian University. E-mail: vova.khlp@yandex.ru.

Shlepkin Anatoly Konstantinovich - Dr. Sc., Professor; Krasnoyarsk State Agrarian University. E-mail: ak_kgau@mail.ru.

Melnikova Olga Leonidovna - Cand. Sc., Professor; Khakas State University. E-mail: olga-l-melnikova@yandex.ru.

Ширяева Тамара Алексеевна - кандидат физико-математических наук, доцент, доцент кафедры информационных технологий и математического обеспечения информационных систем; Красноярский государственный аграрный университет. E-mail: info@kgau.ru.

Хлупичев Владимир Александрович - магистрант; Красноярский государственный аграрный университет. E-mail: vova.khlp@yandex.ru.

Шлепкин Анатолий Константинович - доктор физико-математических наук, профессор, профессор кафедры высшей математики и компьютерного моделирования; Красноярский государственный аграрный университет. E-mail: ak_kgau@mail.ru.

Мельникова Ольга Леонидовна - кандидат педагогических наук, доцент кафедры информационных технологий и систем; Хакасский государственный университет имени М. Ф. Катанова. E-mail: olga-l-melnikova@yandex. ru.