Прогнозування часового ряду за моделлю нормалізації
Empirical constructions of time series models based on the reduction of initial data to normally distributed values have been proposed. The goal of a normalization method is to construct an optimal forecast that is linear for the updated data, and the forecasted original data is recovered through th...
Saved in:
| Date: | 2025 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2025
|
| Subjects: | |
| Online Access: | https://journal.iasa.kpi.ua/article/view/336056 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | System research and information technologies |
| Download file: |  |
Institution
System research and information technologies| _version_ | 1866391928127356928 |
|---|---|
| author | Bondarenko, Viktor Bondarenko, Valeriia |
| author_facet | Bondarenko, Viktor Bondarenko, Valeriia |
| author_sort | Bondarenko, Viktor |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2025-07-25T15:56:08Z |
| description | Empirical constructions of time series models based on the reduction of initial data to normally distributed values have been proposed. The goal of a normalization method is to construct an optimal forecast that is linear for the updated data, and the forecasted original data is recovered through the inverse transformation. The different variants of such transformations have been considered, including the reduction of initial data to Gaussian fractional Brownian motion and a one-dimensional transformation using a strictly monotonic function. The computational experiment based on real data, which allows for a stationary model, confirms the higher quality of the forecast by the normalization method compared to traditional models. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2025.2.07 |
| first_indexed | 2025-07-27T04:04:09Z |
| format | Article |
| fulltext |
Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2025
106 ISSN 1681–6048 System Research & Information Technologies, 2025, № 2
UDC 519.24
DOI: 10.20535/SRIT.2308-8893.2025.2.07
TIME SERIES FORECASTING
USING THE NORMALIZATION MODEL
VIKTOR BONDARENKO, VALERIIA BONDARENKO
Abstract. Empirical constructions of time series models based on the reduction of initial
data to normally distributed values have been proposed. The goal of a normalization
method is to construct an optimal forecast that is linear for the updated data, and the
forecasted original data is recovered through the inverse transformation. The different
variants of such transformations have been considered, including the reduction of initial
data to Gaussian fractional Brownian motion and a one-dimensional transformation using
a strictly monotonic function. The computational experiment based on real data, which
allows for a stationary model, confirms the higher quality of the forecast by the
normalization method compared to traditional models.
Keywords: optimal forecast, stochastic model, parameter estimation, fractional
Brownian motion.
PRELIMINARY INFORMATION AND STATEMENT OF THE PROBLEM
Time forecasting is defined as estimation the future values of some function of a
time variable based on known observations up to the current moment. Other-
words, if we observe the trajectory )(tx , Tt 0 , then it is necessary to evaluate
the value ,)(sx . TsT The estimated values are called forecasts and are
denoted by ,)(ˆ sx . The stochasticity of the trajectory is an essential circumstance,
so )(x does not represent a deterministic function.
As a rule, the trajectory values are observed at discrete moments
,0 121 Ttttt nn
that is, the terms of the sequence are known },,,{ 21 nxxx , where )( kk txx , and
“future” values are to be assessed )(,,)( 1 rnn ttx ,
TtttT rnnn 21 .
If it is known a priori that ) (tx is an implementation of a random process )(t ,
which corresponds to a finite-dimensional distribution
})(,,)({ 11 mm xtxtP mmmm
xxx
dyyyyttfdydy
m
),..,,,..,( 21121
21
with a density ),,(),,,,,( 111 mmmmm xxfxxttf , then the optimal forecast
for r steps is the conditional average:
,),,,(),,,,,E(
ˆ
ˆ
211121
1
nnnrnnn
rn
n
xxxxx
g (1)
Time series forecasting using the normalization model
Системні дослідження та інформаційні технології, 2025, № 2 107
).( jj t
The coordinates of the conditional mean are calculated by the formula:
rnrjjn dydyxxyyypy
1121 ),,,,,(ˆ ,
where the conditional density
),,,,,( 121 nr xxyyyp
1
21121 )),,,((),,,,,,(
nnrnnnrn xxxfyyxxxf ,
rrnrnnn dzdzzzxxxfxxxf
112121 ),,,,,,(),,,( .
Optimality means that for random vectors
n
1
ξ ,
rn
n
1
η
and the fair ratio
22 )ξ(Eηmin)ξ(Eη hg
h
,
where the function g is defined by the equality (1), |||| is a Euclidian distance in nR .
As a rule, the density of the distribution mf is unknown, and to calculate the
forecast, it is necessary to create a stochastic time series model },,,,{ 21 nxxx
that is, to construct a description of the relationship between the values of the
series based on previous observations.
This model can be described
),,,( 11 kk ,
where },,,{ 21 n are the sequence of random variables for which the abserva-
tion kx is assigned by some value k , and is a random vector. Note that
standard models do not always take into account the specifics of observations and
the corresponding forecast turns out to be far from reality (as shown in the
example below).
The forecast constructions based on traditional time series models are
discussed in detail in the reference ]1[ .
The preferred option of forecasting is a situation where the time series
admits a Gaussian model, i.e. Q,0~
, where the matrix Q is depicted in
the form of
DC
BA
Q ,
and block elements DCBA , , , are determined by the ratios:
kjjk Ea , ,1 nkj ξ ;);0(~ A
knjnkjjk EEd , );0(~ ,1 Drkj
V. Bondarenko, V. Bondarenko
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 108
and the elements of matrix , , *CBCB are represent the mutual covariance of
,ξ and η so
;1, 1 . rknjEb knjjk
kjnjk Ec , .1, 1 nkrj
In this case, the optimal forecast is linear and is determined by the formula:
ξξ)(ηEˆ 1 CA│ . (2)
In particular, the sample },,{ 1 nxx can be considered the values of the frac-
tional Brownian motion ) (tBH , which is defined as a Gaussian random process
with characteristics
,0)(E tBH )(
2
1
)()(E
222 HHH
HH ststsBtB ,
where 10 H is called a Hurst exponent.
The properties of fractional Brownian motion are discussed in [2]. The
experience with temporal data shows that identifying a series as fBm values is a
rather rare phenomenon.
TIME SERIES NORMALIZATION METHOD
The idea of the method consists of transforming the original data },,, { 21 nxxx .
into Gaussian },,,{ 21 nuuu ..
In some cases, the original data can be converted to the fractional Brownian
motion values.
Let there be a continuous one-to-one mapping
nn RR : , ),,,(),,,( 2121 nn uuuxxx
such that the increments of the transformed data
,11 uz 1122 , , nnn uuzuuz
form a stationary sequence (i.e., they admit a stationary model).
The statistics
2
1
2 1
k
n
k
z
n
and 1
1
11
1
jj
n
j
zz
n
are consistent estimates of the variance of the increments and their one-step
covariance ([3, 4]).
Then consistent estimates of the correlation coefficient and Hurst parameter
are calculated by the formulas:
,
2
22
)1(
2
1
ln
ln
H
.
The necessary conditions for the hypothesis of “data },,,{ 21 nuuu form the
values of fractional Brownian motion” is the fulfillment of the limiting relations
for the statistics , nA , , nn DB nF , which are defined by the following relations:
Time series forecasting using the normalization model
Системні дослідження та інформаційні технології, 2025, № 2 109
2
31
1 3
1
1
4
kk
n
k
n zu
n
A ,
2
1
;0H ;
32
1
1
51
1
1
kk
n
k
Hn zu
n
B
22
1
;0~, 3
H
,
2
1
;0H ;
2
31
1 23
1
1
42
kk
n
k
Hn zu
n
D , ),1;0(~ ;1;
2
1
H
3
1
3
1
1
k
n
k
Hn z
n
F
,3
1;
2
1
H .
The proof of the limit relations is contained in 5 .
The standard algorithm for testing the specified hypothesis using known H
and is a following: let us assume that the hypothesis is fulfilled and we set the
significance level with comparing the value of the statistic with the tabular
value where .1)( F
In particular, for the marginal distribution function of statistics nD
(Н )5,0 :
1
3
2
Φ2)(
xxF , where Φ is a Laplace function.
Corresponding to the level of significance 05,0 the critical value 6 ,
and the hypothesis is accepted if .60 nD
If the hypothesis )( kHk tBu is true, then the optimal (linear) forecast for
},,,{ 221 nnn uuu in n steps is calculated by the formula (2):
nn
n
u
u
CA
u
u
1
1
2
1
ˆ
ˆ
,
DC
BA
Q ,
jkq )(
222 HHH kjkj , ,2,1 nkj
and the forecast of the primary variables
)ˆ,,ˆ()ˆ, ,ˆ( 21
1
21 nnnn uuxx
.
Note that the choice of transformation is a rather cumbersome procedure.
Let us consider another normalization method — the one-dimensional
transformation of the real data )( kk xu .
))((Φ)( 1 xFx
, (3)
where dz
z
x
x
2
exp
2
1
)Φ(
2
, F is a distribution function corresponding
to the sample },,,{ 21 nxxx .
V. Bondarenko, V. Bondarenko
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 110
Thereby, },,{ 1 nuu is a sample from the general population 1,0 and the
proposed procedure requires stationarity in the narrow sense of the real data and
requires an assessment of their distribution. The sample size should not be large
to prevent the detection of non-stationarity.
Actually, },,,{ 21 nxxx represents a sample of different random variables
)}(,),(),({ 21 nttt with the same distribution .F This disadvantage is partially
compensated by the dependence of the obtained data, which is determined by
their correlation matrix.
Let us note the following property of the transformation.
Proposition. Let is a strictly increasing function ξ t is a stationary
process in the narrow sense, i.e.
),,,,,(})(,,)({ 112111 nnnnn ttttxxFxtxtP .
Then the process ))(()( tt is also stationary in the narrow sense.
Proof. Under the condition of stationarity
),,,,,(})(,,)({ 112111 nnnnn ttttxxFxtxtP .
Distribution of the process )(t :
})(,,)({}))((, ,))(({ 1111 nnnn ytytPxtxtP
),,,,,( 1121 nnn ttttyyF , )(1
jj xy
also satisfies the definition of stationarity.
Let us formulate a modeling and forecasting algorithm using transfor-
mation (3).
1. Checking the data for stationarity, for example, using the Dickey–Fuller
criterion and determining the size of the training sample (in the case of checking the
adequacy of the model, determine the size of the training plus the predicted sample).
2. Estimation of the distribution function F of random value ξ by the
sample },,{ 1 nxx .
3. Construction of the sample },,{ 1 nuu . from a normal population )1,0(
by the formula
))((Φ 1
kk xFu
.
4. Calculation of the sample correlation coefficients
jkk
jn
k
j uu
jn
1
1
,
with consctruction of correlation matrix Q and defining the forecasting horizon .r
5. Construction of the forecast }ˆ ˆ{ ,,1 rnn uu of transformed data by the
formula (2).
6. Calculating predicted values }ˆ ˆ{ ,,1 rnn xx of primary data according to
the formula
))ˆ(Φ()ˆ(ˆ 11
kkk uFux
. rnn xx ˆ, ,ˆ 1 .
Time series forecasting using the normalization model
Системні дослідження та інформаційні технології, 2025, № 2 111
EXAMPLE OF FORECAST CALCULATION
The following example illustrates the application of the proposed model:
The meteorological data files Precipitation–Florida Climate Center.
The sample size is determined by stationarity, which is tested using the
Dickey–Fuller criterion.
In the given example, stationarity occurs in the interval 401 xx ; let us put
the data into the training sample .301 xx The data values and their graph are
shown in Table 1 and Fig. 1, respectively.
T a b l e 1 . The value of the series 401 xx
1x 2x 3x 4x 5x 6x 7x 8x 9x 10x
3.03 3.43 3.54 2.98 2.13 – 2.62 –0.61 –0.15 –2.15 –2.65
11x 12x 13x 14x 15x 16x 17x 18x 19x 20x
–3.58 –2.13 –2.32 –2.43 –2.8 –2.42 –3.15 –2.62 –2.81 –2.46
21x 22x 23x 24x 25x 26x 27x 28x 29x 30x
–1.2 –0.33 –1.81 –2.18 –0.2 0.6 3.07 5.48 6.34 8.81
31x 32x 33x 34x 35x 36x 37x 38x 39x 40x
6.2 4.04 2.86 1.53 0.702 1.7 3.72 5.33 6.27 3.32
If we consider the given data as values of one random variable, then from the
analysis of the sample },,,{ 21 nxxx the hypothesis follows that this random
variable is distributed by Gumbel’s law:
x
xF expexp)( ,
with the moments:
E , 577,0 is a Euler’s constant,
2E 22
2
2
0
)(
6
)(
dzelnz z ,
dzelnzE z3
0
3 )( ,)3(2)(
2
1
)( 3223
‐6
‐4
‐2
0
2
4
6
8
10
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Fig. 1. Meteorological data )( kk txx
V. Bondarenko, V. Bondarenko
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 112
)(s is a Reamann zeta function, .202,1)3(
The values of sample moments:
0037.0 x , 19.10
30
1 2
30
1
2
k
k
xx ;
3x 76,33
30
1 3
30
1
k
k
x ,
lead to estimates of the distribution parameters 49,2 ; 44,1 , so
49,2
44,1
expexp)(
x
xF .
Density distribution graph
49,2
1
)( xf )(
49,2
44,1
exp xF
x
shown in Fig. 2.
Let us construct the transformed data using the formula
)(Φ
49,2
44,1
expexpΦ 11
k
k
k y
x
u
,
where the values )(Φ 1
ky are calculated using the Laplace function table. The
result of the calculations is given in Table 2.
T a b l e 2 . Transformed data 401 uu
1u 2u 3u 4u 5u 6u 7u 8u 9u 10u
1.02 1.12 1.14 1.01 0.8 –0.84 –0.03 0.13 –0.63 –0.85
11u 12u 13u 14u 15u 16u 17u 18u 19u 20u
–1.32 –0.62 –0.65 –0.75 –0.92 –0.75 –1.09 –0.84 –0.92 –0.76
21u 22u 23u 24u 25u 26u 27u 28u 29u 30u
–0.25 0.07 –0.49 –0.64 0.11 0.37 1.03 1.56 1.72 2.13
31u 32u 33u 34u 35u 36u 37u 38u 39u 40u
1.685 –1.25 0.98 0.64 0.4 0.69 1.18 1.52 –1.71 1.09
Fig. 2. Density distribution )(xfξ
fζ
x
Time series forecasting using the normalization model
Системні дослідження та інформаційні технології, 2025, № 2 113
The evaluation of correlation coefficients
jkk
j
k
j uu
j
30
130
1
.
Leads to the results and for 7k the coefficients 0k .
1 2 3 4 5 6 7
0.77 0.57 0.43 0.27 0.144 0.036 0.02
The elements of the matrix Q are determined by the ratios:
,7,0
,7,
kj
kj
q kj
jk 37,1 kj , 10 , , jkjk qa 30,1 kj .
The forecast of transformed data in 7 steps is calculated by the formula (2)
30
1
1
37
31
ˆ
ˆ
u
u
CA
u
u
.
The forecasting results are shown in Table 3.
T a b l e 3 . The values of forecast in 7 steps
31u 32u 33u 34u 35u 36u 37u
1.685 1.25 0.98 0.64 0.4 0.69 1.18
Forecast
31û 32û 33û 34û 35û 36û 37û
1.682 1.155 0.838 0.443 0.205 0.6 0.32
31x̂ 32x̂ 33x̂ 34x̂ 35x̂ 36x̂ 37x̂
6.169 3.595 2.267 0.848 0.08 1.396 0.44
where the forecast of the initial data
3731 ˆˆ xx is calculated by the formula
))ˆ(Φ()ˆ(ˆ 11
kkk uFux
, 44,1)ln(ln49,2)(1
yyF .
Let us compare the quality of forecasting using a model, which have constructed
using the normalization method and four classical discrete time series models.
T a b l e 4 . Comparison of forecast quality
Time series values 3731 xx
31x 32x 33x 34x 35x 36x 37x
Actual data
6.2 4.04 2.86 1.53 0.702 1.7 3.72
The results of forecast Forecast
data 31x̂ 32x̂ 33x̂ 34x̂ 35x̂ 36x̂ 37x̂
Normalization
method 6.169 3.595 2.267 0.848 0.08 1.396 0.44
ARMA 7.16 9.53 9.902 7.286 9.672 10.06 9.9
Point Forecast 6.703 5.163 3.89 2,833 1.927 1.142 0.434
GARCH 8.379 8.908 7.437 7.565 7.9 8.02 9.21
ARIMA 8.652 8.063 7.44 6.85 6.328 5.808 4.35
V. Bondarenko, V. Bondarenko
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 114
CONCLUSIONS
Given the stationarity of the time series, modeling using the normalization
method, which is defined by the relation (3), provides higher forecast quality
compared to traditional forecasting methods.
REFERENCES
1. P.I. Bidyuk, O.S. Menyaylenko, O.V. Polovtsev, Forecasting methods. Luhansk:
Alma Mater, 2008, 604 p
2. Y. Mishura, “Stochastic Calculus for Fractional Brownian Motion and Related Proc-
esses,” Lecture Notes in Mathematics 1929, Springer 2008, 393 p. doi: 10.1007/978-
3-540-75873-0
3. K. Kubilius, Y. Mishura, K. Ralchenko, Parameter Estimation in Fractional Diffu-
sion Models, vol. 8, 2018, 390 p. doi: https://doi.org/10.1007/978-3-319-71030
4. V.G. Bondarenko, “On some statistics of fractional brownian motion,” System Research
& Information Technologies, no. 1, pp. 131–138, 2021. doi: https://doi.org/10.20535/
SRIT.2308-8893.2021.1.11
5. I. Nourdin, Selected Aspects of fractional Brownian motion. Milano: Springer, 2012,
124 p. doi: https://doi.org/10.1007/978-88-470-2823-4
Received 12.04.2024
INFORMATION ON THE ARTICLE
Viktor G. Bondarenko, ORCID: 0000-0003-1663-4799, Educational and Research Insti-
tute for Applied System Analysis of the National Technical University of Ukraine “Igor
Sikorsky Kyiv Polytechnic Institute”, Ukraine, e mail: bondarenvg@gmail.com
Valeriia V. Bondarenko, The University of the Littoral Opal Coast, France, e-mail: vale-
ria_bondarenko@yahoo.com
ПРОГНОЗУВАННЯ ЧАСОВОГО РЯДУ ЗА МОДЕЛЛЮ НОРМАЛІЗАЦІЇ /
В.Г. Бондаренко, В.В. Бондаренко
Анотація. Запропоновано емпіричні конструкції моделей часового ряду за
схемою зведення первинних даних до нормально розподілених. Метою такого
методу нормалізації є побудова оптимального прогнозу, який для оновлених
даних є лінійним, а прогнозовані первинні дані відновлюються через обер-
нене перетворення. Розглянуто варіанти таких перетворень — зведення пер-
винних даних до гаусівського фрактального броунівського руху та одно-
вимірне перетворення з використанням строго монотонної функції.
Обчислювальний експеримент на базі реальних даних, що допускають стаціо-
нарну модель, підтверджує вищу якість прогнозу методом нормалізації порів-
няно з традиційними моделями.
Ключові слова: оптимальний прогноз, стохастична модель, оцінювання пара-
метрів, фрактальний броунівський рух.
|
| id | journaliasakpiua-article-336056 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-17T09:26:03Z |
| publishDate | 2025 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/3a/5825d970a62ccc36717f4ec06c67953a.pdf |
| spelling | journaliasakpiua-article-3360562025-07-25T15:56:08Z Time series forecasting using the normalization model Прогнозування часового ряду за моделлю нормалізації Bondarenko, Viktor Bondarenko, Valeriia optimal forecast stochastic model parameter estimation fractional Brownian motion оптимальний прогноз стохастична модель оцінювання параметрів фрактальний броунівський рух Empirical constructions of time series models based on the reduction of initial data to normally distributed values have been proposed. The goal of a normalization method is to construct an optimal forecast that is linear for the updated data, and the forecasted original data is recovered through the inverse transformation. The different variants of such transformations have been considered, including the reduction of initial data to Gaussian fractional Brownian motion and a one-dimensional transformation using a strictly monotonic function. The computational experiment based on real data, which allows for a stationary model, confirms the higher quality of the forecast by the normalization method compared to traditional models. Запропоновано емпіричні конструкції моделей часового ряду за схемою зведення первинних даних до нормально розподілених. Метою такого методу нормалізації є побудова оптимального прогнозу, який для оновлених даних є лінійним, а прогнозовані первинні дані відновлюються через обернене перетворення. Розглянуто варіанти таких перетворень — зведення первинних даних до гаусівського фрактального броунівського руху та одновимірне перетворення з використанням строго монотонної функції. Обчислювальний експеримент на базі реальних даних, що допускають стаціонарну модель, підтверджує вищу якість прогнозу методом нормалізації порівняно з традиційними моделями. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2025-06-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/336056 10.20535/SRIT.2308-8893.2025.2.07 System research and information technologies; No. 2 (2025); 106-114 Системные исследования и информационные технологии; № 2 (2025); 106-114 Системні дослідження та інформаційні технології; № 2 (2025); 106-114 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/336056/324825 |
| spellingShingle | оптимальний прогноз стохастична модель оцінювання параметрів фрактальний броунівський рух Bondarenko, Viktor Bondarenko, Valeriia Прогнозування часового ряду за моделлю нормалізації |
| title | Прогнозування часового ряду за моделлю нормалізації |
| title_alt | Time series forecasting using the normalization model |
| title_full | Прогнозування часового ряду за моделлю нормалізації |
| title_fullStr | Прогнозування часового ряду за моделлю нормалізації |
| title_full_unstemmed | Прогнозування часового ряду за моделлю нормалізації |
| title_short | Прогнозування часового ряду за моделлю нормалізації |
| title_sort | прогнозування часового ряду за моделлю нормалізації |
| topic | оптимальний прогноз стохастична модель оцінювання параметрів фрактальний броунівський рух |
| topic_facet | optimal forecast stochastic model parameter estimation fractional Brownian motion оптимальний прогноз стохастична модель оцінювання параметрів фрактальний броунівський рух |
| url | https://journal.iasa.kpi.ua/article/view/336056 |
| work_keys_str_mv | AT bondarenkoviktor timeseriesforecastingusingthenormalizationmodel AT bondarenkovaleriia timeseriesforecastingusingthenormalizationmodel AT bondarenkoviktor prognozuvannâčasovogorâduzamodellûnormalízacíí AT bondarenkovaleriia prognozuvannâčasovogorâduzamodellûnormalízacíí |