Data Interpretation Algorithm for Adaptive Methods of Modeling and
Forecasting Time Series
NATALIYA BOYKO
Artificial Intelligence Department,
Lviv Polytechnic National University,
12, Stepana Bandera Street, Lviv, 79013,
UKRAINE
Abstract: - The paper considers two forms of models: seasonal and non-seasonal analogues of oscillations. The
paper analyzes the basic adaptive models: Brown, Holt, and autoregression. The parameters of adaptation and
layout are considered by the method of numerical estimation of parameters. The mechanism of reflection of
oscillatory (seasonal or cyclic) development of the studied process through a reproduction of the scheme of
moving average and the scheme of autoregression is analyzed. The paper determines the optimal value of the
smoothing coefficient through adaptive polynomial models of the first and second order. Prediction using the
Winters model (exponential smoothing with multiplicative seasonality and linear growth) is proposed. The
paper proves that the additive model allows building a model with multiplicative seasonality and exponential
tendency. The paper proves statements that allow to choose the right method for better modeling and
forecasting of data.
Key-Words: - Average, Holt-Winters model, polynomial time series models, exponential smoothing
Received: September 25, 2022. Revised: April 13, 2023. Accepted: May 6, 2023. Published: May 18, 2023.
1 Introduction
Effective analysis, modeling, and forecasting of
financial and economic processes form the
foundation for making informed management
decisions across all levels of the economic
hierarchy. However, this task is inherently complex
and ambiguous, necessitating the use of advanced
models and methods to accurately capture the
nuances of modern financial and economic
processes, [1].
Most often, in the practical construction of
forecasts of economic indicators, their seasonality
and cyclicality are taken into account. Different
mathematical apparatus is used to predict non-
seasonal and seasonal processes. The dynamics of
many financial and economic indicators have a
stable fluctuating component. The study of monthly
and quarterly data is often observed within the
annual seasonal fluctuations, respectively, in the
period of 12 and 4 months. When using daily
observations, fluctuations with a weekly (five-day)
cycle are often observed. In this case, to obtain more
accurate forecast estimates, it is necessary to
correctly reflect not only the trend but also the
oscillating component. The solution to this problem
is possible only with the use of a special class of
methods and models, [1], [2], [3], [4], [5].
Seasonal models are based on their non-seasonal
counterparts, which are supplemented by means of
displaying seasonal fluctuations. Seasonal models
can reflect both a relatively constant seasonal wave
and a wave that changes dynamically depending on
the trend. The first form belongs to the class of
additive, and the second - to the class of
multiplicative models, [2]. Most models have both
of these shapes. The most widely used in practice
are Holt-Winters models, [6], and autoregressions,
[7].
In short-term forecasting, the dynamics of the
development of the studied indicator at the end of
the observation period is usually more important
than the trend of its development, which has
developed on average throughout the prehistory
period. The property of dynamic development of
financial and economic processes often prevails
over the property of inertia, so adaptive methods
that take into account information inequality of data
are more effective, [8].
Adaptive models and methods have a mechanism
of automatic adjustment to change the studied
indicator. The forecasting tool is a model, the initial
assessment of the parameters of which is carried out
on the first few observations. Based on it, a forecast
is made, which is compared with actual
observations. Next, the model is adjusted according
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
359
Volume 22, 2023
to the magnitude of the forecast error and is used
again to predict the next level, until all observations
are exhausted. Thus, it constantly "absorbs" new
information, adapts to it, and by the end of the
observation period reflects the current trend, [9],
[10]. The forecast is obtained as an extrapolation of
the latest trend. In different forecasting methods, the
process of setting up (adapting) the model is carried
out in different ways. Basic adaptive models are:
Brown model, [11];
Holt-Winters model, [6];
autoregression model, [7].
The first two models belong to the average mean
scheme, the latter to the autoregression scheme,
[12]. Numerous adaptive methods based on these
models differ in the way of numerical estimation of
parameters, determination of adaptation parameters,
and layout.
Under the moving average approach, the current
level estimation is a weighted average of prior
levels, with decreasing weights assigned to
observations as they become further removed from
the most recent level. In essence, observations
closer to the end of the observation period hold
greater informational value, [13].
According to the autoregression scheme, the
estimate of the current level is the weighted sum of
the orders of the models "p" of the previous levels.
The information value of observations is determined
not by their proximity to the simulated level, but by
the closeness of the relationship between them, [14],
[15], [16]. Both of these schemes have a mechanism
for reflecting the oscillating (seasonal or cyclical)
development of the studied process.
Autoregressive Integrated Moving Average
(ARIMA) is a popular method for forecasting time
series data using a single variable, [17]. The
problem with ARIMA is that it doesn't support
seasonal data. This is a time series with a repeating
cycle. ARIMA expects data that is not seasonal or
has a seasonal component removed, for example
seasonally adjusted using techniques such as
seasonal variance. This method supports direct
modeling of the seasonal component of the series
called Seasonal Autoregressive Integrated Moving
Average SARIMA, [14].
SARIMA, an extension of ARIMA, is
specifically designed to handle univariate time
series data that exhibit seasonal patterns. The model
incorporates seasonal terms that closely resemble
the non-seasonal components but account for
reversed shifts in the seasonal period.
Prophet is a technique for predicting time series
data through an additive model that captures non-
linear trends using yearly, weekly, and daily
seasonality, along with holiday effects. The method
is particularly effective for time series with
significant seasonal patterns and a considerable
historical data set. Prophet is highly resistant to data
gaps and trend shifts, and can typically handle
outlier values with ease, [18].
The purpose of the paper is to develop the
adaptive methods of modeling and forecasting the
time series based on a combination of the adaptive
methods of predictive modeling:
Holt-Winters model, [19],
moving average model, [20].
Time series analysis is predominantly concerned
with predicting real values, which can be
characterized as regression problems. As a result,
the evaluation metrics described in the paper will
concentrate on techniques for assessing the accuracy
of predictions for continuous variables.
The main contribution consists of the following:
the adaptive polynomial models used
sequentially allow to increase in the prediction
accuracy,
the data interpretation algorithm for
adaptive methods of modeling and forecasting time
series is developed,
the comparison between the Winters model
and the Tayle-Wage model shows the good quality
of the proposed predictive model.
This paper consists of several sections. In the
Methods and Means section, the data interpretation
algorithm for adaptive methods of modeling and
forecasting time series is given. The next section
presents the result of the calculation and data
interpretation. The last section concludes this paper
by containing the probable decision of appraisal
technique.
2 Methods and Means
The time series in adaptive models are presented in
the form (Formula 1):
(1)
where t – time indicator; –
coefficients of the adaptive model at the moment of
time t.
Depending on the shape of the trend and the
presence or absence of a periodic component, a
certain type of adaptive forecasting should be
chosen. To do this, you need to find the optimal
value of the smoothing parameters . They
should be used to calculate the coefficients
.
if the smoothing parameters change, the prediction
error increases. However, this approach will not
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
360
Volume 22, 2023
bring the quality of forecasting. The research
proposes an algorithm for determining the optimal
values of smoothing parameters.
Also, it is important to analyze the effectiveness of
the adaptive approach in other methods. Therefore,
it is proposed to develop an algorithm that allows
you to take into account the accuracy of the forecast,
the complexity of the model, and its adequacy and
compliance with the object under study.
There are two groups of adaptive models: linear
and seasonal.
According to Formula 2, the forecast of linear
growth models is shown, [31]:
(2)
where - the number of steps of the forecast;
- the coefficients of the adaptive model at a
moment of time t
Adaptive models of linear growth include the Holt
model, the Braun model, and the Box-Jenkins
model. The difference between linear growth
models lies in finding the parameters , [31].
The parameters of the Holt model are found in
Formula 3:
(3)
Formula 4 presents the calculation of parameters
according to the Tayle-Vage model, [31]:
,
(4)
where are the smoothing coefficients that
take values from 0 to 1, - the real value of the
series level at the t-th step, the predictive value
at the t-th step, - the error at the t-th step.
Characterizing the calculation of the parameters of
Formulas 3-4, it is possible to highlight a certain
feature of adaptive models. It is necessary to
calculate at each step. For the model to give
better results, it is necessary to find which
will most closely correspond to the time series.
The adaptive monoparameter Braun model is used
for stationary time series based on simple
exponential smoothing:
(5)
where is the prognostic value of time series
level in time (t+1), is exponential mean, is
adaptation coefficient, is the current time series
value.
In this context, the model's value is a weighted
average of both the current true value and past
model values. The weight, referred to as the
smoothing factor or alpha (α), dictates the rate at
which the model "forgets" the most recent actual
observation. A smaller α places more emphasis on
earlier model values, resulting in a smoother series.
Taking the adaptation coefficient α and the
warning period τ, it is necessary to approximate the
series using an adaptive polynomial model.
The Data Interpretation Algorithm for Adaptive
Methods of Modeling and Forecasting Time Series
(DIAAMMFTS) is developed in the paper.
DIAAMMFTS consists of the following steps:
Procedure 1:Zero order (р = 0);
Procedure 2:First order (р = 1);
Procedure 3:Second order (р = 2);
Procedure 4:Assess the accuracy and quality
of forecasts;
Procedure 5:Make a forecast.
All procedures of DIAAMMFTS are presented
below.
Procedure 1.
Procedure 1 developed as a sequence of the
following steps:
1. Let .
2. Append arrayusing the following formula:
where is an
actual value and is the previous number from
the prediction array.
3. Repeat step 2 for all values in a dataset.
Procedure 2.
So far, we have been able to get from our
methods at best a forecast only one point ahead (and
still nicely smooth the series), this is great, but not
enough, so we move to the expansion of exponential
smoothing, which will build the forecast two points
forward (and also nice to smooth out a number).
This will help us to divide the series into two
components - ℓ (level, intercept) and b(trend, slope).
The level, or expected value of the series, we
predicted using previous methods, and now the
same exponential smoothing can be applied to the
trend, naively or not very much believing that the
future direction of change of the series depends on
weighted previous changes.
(6)
The algorithm is the following:
1. Let , , and
, where is our initial dataset.
2. Define new level value using the formula:
.
3. Define new trend value using the
formula:.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
361
Volume 22, 2023
4. Defineour prediction .
5. Define and repeat steps 2-5 until
.
Procedure 3.
This technique involves introducing a third
component - seasonality - to the model. Thus, it can
only be applied when a specific seasonal pattern is
present, which is the case in our scenario. The
seasonal component accounts for cyclic fluctuations
around the trend and level, and is determined by the
length of the seasonal pattern, indicating the
duration after which the fluctuations repeat. For
each observation in the season, a corresponding
component is generated. For instance, if the
seasonal pattern is weekly (with a length of 7),
seven seasonal components are derived, each
representing a specific day of the week.
Therefore, a new system is defined:
(7)
The algorithm is the following:
1. Let , L = 24*7 , and
.
, where is
the initial dataset, and L is the length of the season
in our case we set it to count weeks and is the
number of seasons.
2. Defineavrg using this formula
3. Count n = n+1. Repeat step 2 until
n<s_num.
4. Defineusing formula
.
5. Define new level using the formula
6. Define new trend using the formula
7. Define new s using the formula
8. Define new result using the formula
9. Count x = x+1. Repeat steps 5-8 until
x<y.length.
10. Make a prediction using the formula
, where m is
the number that indicates how many steps forward
we want to predict.
The current level is now determined by
subtracting the corresponding seasonal component
from the current series value, while the trend
remains constant. Additionally, the seasonal
component is calculated based on the current series
value minus the level and the preceding component
value. With the inclusion of the seasonal
component, we can now make predictions for any
desired number of steps (m) into the future.
3 Results
The dataset consists of the dynamics of shares of a
company for 25 days, [21].
The time series xt of some economic indicators
consisting of n observations will be analyzed.
In Pandas, [22], there is a ready implementation
- DataFrame.rolling (window) .mean (). The more
we set the width of the interval - the smoother the
trend will be. If the data is very noisy, which is
especially common, for example, in financial terms,
such a procedure can help us see common patterns.
3.1 Adaptive Zero-Order Polynomial Model
The exponential mean has the form, [23]:
(8)
Taking the adaptation coefficient α = 0.5 and the
warning period τ = 1, it is necessary to approximate
the series using an adaptive polynomial model, [7],
[8], [9], [10].
The initial condition for (5) is given as follows:
, whereis an average value, for
example, the first five observations:
The forecast model value with the warning period τ
will be determined from the relation
The error is determined by the formula 9:
(9)
Using the Formula 8 first formula and the accepted
value of α = 0.5, calculate (Table 1).
For t = 1
t = 2
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
362
Volume 22, 2023
t = 3
Table 1. Predicting the time series xt one step
further (adaptive polynomial model of zero (p = 0)
order)
t
xt
P=0
St
Error
1
0
511
1
1
520
515.5
511
0.16
1
2
497
506.25
515.5
0.68
1
3
504
505.125
506.25
0.01
1
4
525
515.063
505.12
5
0.75
...
1
24
545
534.38
523.76
9
0.83
1
25
529
531.99
534.38
0.05
1
26
531.99
0.5
0.5
We have made a forecast for one step forward, but it
cannot be considered optimal. To obtain an adequate
forecast, it is necessary to choose such a value of α
that the sum of the squares of the deviations and the
error of the forecast was minimal. To determine the
optimal value of α, tabulate it from 0.1 to 0.9 in
steps of 0.1. Then each time we substitute it in the
calculation model to obtain the forecast and the
magnitude of the error. Thus, the value of α is
selected at which the error will also be minimal.
The distribution of the prediction error with
respect to the parameter α is shown in Figure 1.
Fig. 1: Dependence of forecasting error on α.
Figure 1 shows that the optimal value for the
zero-order model is α = 0.4, which is determined
based on the minimum total error E = 8.85. The
results of the forecast are shown in Figure 2.
Fig. 2: Forecasting results based on a zero-order
polynomial model (p = 0)
Numerical forecasting values are shown in
Table 2.
Table 2. The results of the forecast at α = 0.4
t
xt
P=0
St
Error
1
0
511.00
1
1
520
412.4
511.00
0.16
1
2
497
363.76
412.4
14.4
1
3
504
347.1
363.76
39.02
1
4
525
348.84
347.1
60.28
….
1
24
560
433.26
523.159
2.46
1
25
529
384.9
433.26
17.33
1
26
384.9
0.4
0.6
In Table 2 the results of the forecast are given.
They are not much different from our original
series.
3.2 Adaptive First-Order Polynomial Model
First, according to the time series xt, we find the
LSM (Least Squares Method), [24], estimate of the
linear trend:
Suppose, and.
To find the coefficients andon the graph of
the time series , the trend line is added (Figure 3).
In our case, the trend equation has the form:
where and.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
363
Volume 22, 2023
Fig. 3: Estimation of LSM regression line
Exponential averages of the 1st and 2nd order
are defined as
whereβ=1-α.
Hence the initial conditions are the following:
The estimation of the model (predicted) value
of the series with the warning period τ is equal to
Using this formula, the times series is given below:
The result of the calculation is given in Table 3. The
error value is lower than for the parameters
presented in Table 2.
Table 3. The results of calculations of the
predicted model at α = 0.5
t
xt
P=1
St
St[2]
Error
1
0
496.80
495.60
496.80
495.60
1
1
520
508.40
502.00
508.40
502.00
1
2
497
502.70
502.35
502.70
502.35
1
3
504
503.35
502.85
503.35
502.85
1
4
525
514.18
508.81
514.18
508.81
...
1
24
560
541.88
532.37
541.88
532.37
1
25
529
535.44
533.90
535.44
533.90
1
26
0.5
0.5
At t = 1 exponential mean levels are the following:
Based on this, the time series is given as:
The results of the calculations are shown in Table 3.
For the analyzed dataset, the predicted values are
first calculated at α = 0.5 and τ = 1.
Next, it is necessary to determine the optimal value
of α, based on the consideration of the minimum
total error. To do this, as in the first model, a value
of α with the minimum total error is selected.
Figure 4 shows the results of determining the
optimal smoothing parameter.
Fig. 4: Determination of the optimal value of α
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
364
Volume 22, 2023
Figure 4 shows that the minimum error of the
predicted model will be at α = 0.1.
The results of forecasting at the selected optimal
value of α are shown in Figure 5.
Fig. 5: Forecasting results based on a first-order
polynomial model (p = 1)
Thanks to this method, we obtained a smoother
series, based on which we were able to calculate
predictions for 1 step forward.
3.3 Adaptive Second-Order Polynomial
Model
According to the time series xt, we find the LSM
estimate of the parabolic trend, [25], [26]:
For the second-order model, the equation of the
parabolic trend has the form (see Figure 6):
.
Fig. 6: Finding the LSM estimate of the parabolic
trend according to the time series xt
Exponential averages of the 1st, 2nd and 3rd order
are the following:
Fig. 7: Determination of the optimal α
From the graph, it is seen that the optimal α is 0.25,
as at this value we get the smallest error.
The initial conditions are determined by the
following formulas:
The estimate of the model (prediction) with the
warning period τ is found in the expression
+
.
Next, we determine the optimal value of the
smoothing coefficient (see Figure 7). Taking into
account the optimally obtained value α = 0.25 (E =
9.06) the forecast is given (see Figure 8).
Fig. 8: Forecasting results based on a second-order
polynomial model (p = 2)
Next, the proposed model will be compared with
existing adaptive models. The Winters model and
Tayle-Vage are analyzed.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
365
Volume 22, 2023
3.4. Forecasting using the Winters Model
(Exponential Smoothing with Multiplicative
Seasonality and Linear Growth)
This model is convenient to use with a small amount
of initial data. The seasonal model of Winters with
linear growth has the form
where - original time series t = 1, 2, ..., n; - the
parameter characterizes the linear trend of the
process, ie the average values of the level of the
studied time series at time t; - seasonality
factor for phase of the -th cycle;
, where; l - the number of
phases in the full cycle (in monthly time series l =
12, in quarterly l = 4, etc.);εt - random error. It is
usually assumed that the vector
where; – unit
matrix with a size of (n × n).
The adaptive parameters of the model are
estimated using a recurrent exponential scheme
according to the time series xt, consisting of n
observations
Where - the increase of the average level of the
series from the moment t - 1 to the moment t;
- the calculated value of the time series, which
is determined for the time t with the warning period
τ, ie according to the moment (t -τ); , ,, -
parameters of adaptation of exponential smoothing,
and (0 <, , <1).
The increase in (j = 1,2,3) leads to an
increase in the weight of later observations, and a
decrease in αj leads to an improvement in the
smoothing of random deviations. These two
requirements are in conflict, and the search for a
compromise combination of values is the task of
optimizing the model.
Exponential alignment always requires a
preliminary estimate of the smoothed value. When
the adaptation process is just beginning, there
should be initial values prior to the first observation.
In our task it is necessary to define the initial
conditions:;;, where. Thus, the
calculated values ofare a function of all past
values of the original time series xt, parameters ,
and and initial conditions. The influence of
the initial conditions on the calculated value
depends on the value of the weights αj and the
length of the series preceding the moment t. Impacts
of;usually decrease faster than,
andare reviewed at each step, butonly
once per cycle.
First, by n = 8 observations of the time series
xt, we find the LSM estimate of the linear trend
. As a result of the calculation we have
Next, the initial conditions are defined:
Multiplicative zero-cycle seasonality
coefficients, [27]. define as the arithmetic mean
of seasonality indices
for-th phase in the
original time series (Figure 9):
Fig. 9: LSM assessment of a linear trend
We will perform calculations with adaptation
parameters ; ; and the
warning period τ = 1. Estimated values for the 1st
cycle (, ).
According to the formula for t = 1, we have
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
366
Volume 22, 2023
t = 2
t = 3
t = 4
Estimated values for the 2nd cycle ( = 2, =
t-4). Here we need the seasonality coefficients found
for the 1st cycle
t = 5
Since
refers to the 2nd cycle ( = 2) when
choosingbased on the fact that = 5-4 = 1
t = 6
t = 7
t = 8
t = 9 (forecast)
The calculated values and the forecast t, obtained
from the time series are presented in Figure 10.
Fig. 10: Forecasting results based on a third-order
polynomial model (p = 3)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
367
Volume 22, 2023
From the presented graph we can conclude that the
model of exponential smoothing with multiplicative
seasonality of Winters is better than the regression
model but worse than the proposed adaptive model.
The forecast results of Winters can be improved by
selecting the optimal values of α.
3.5 Production Forecast based on the Tayle-
Wage Model
Additive modeling is an approach of interest in
economic research, as it enables the construction of
a model featuring exponential trends and
multiplicative seasonality. This can be achieved by
converting the initial time series values into their
logarithmic equivalents, transforming the
exponential trend into a linear one, and the
multiplicative seasonality into an additive one.
Suppose the observation refer to the -th phase
of the -th cycle, where = t – l ( – l), l is the
number of phases in the cycle (for the quarterly time
series l = 4, and the monthly l = 12).
The model with additive seasonality and linear
growth can be represented as
Where - the average value of the level of the time
series at time t after excluding seasonal fluctuations;
- additive growth rate from time t-1 to time
t;– additive seasonality factor for the vt-th
phase of the kt-th cycle; – white noise.
Estimates of model parameters will be sought at
smoothing coefficients α1, α2, α3, where (0 <α1, α2,
α3<1) on the following adaptation procedures:
The initial conditions of exponential smoothing are
determined by the original time series xt (t = 1,2, …,
n).
First, on the time series xt, which contains n = 8
observations, we find the LSM - an estimate of the
linear regression equation
The calculated values of xt and deviations
are given below. Then the initial values of
additive seasonality coefficients are equal
We will perform calculations for adaptation
parameters α1 = 0.1; α2 = 0.4; α3 = 0.3 and the
warning period τ = 1.
First loop: vt = t; kt = 1; τ = 1,initial data for
calculation:
.
According to the formula for t = 1, we have
t = 2
t = 3
t = 4
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
368
Volume 22, 2023
Second loop: vt = t-4; kt = 2). Initial data for
calculation:
t = 5
t = 6
t = 7
t = 8
t = 9 (forecast)
As estimatestakes the average values of the
deviations, corresponding to the vt-th
phase of the original time series, where
.
Calculated according to the Tayle-Wage model, the
values of the time seriesare presented in Figure
11, where they are presented with the original time
series .
Fig. 11: Forecasting results using the Tayle-
Wage model
The graph shows that our forecast is not that far
from the original series and that it maintains the
trends.
The comparison of the proposed forecasting model
DIAAMMFTS with existing models is given in
Table 4. Mean Squared Error (MSE), [28], [29],
[30], is used for all the models.
Table 4. The comparison of forecasting
models.
Model
MSE
DIAAMMFTS
0.23
Tayle-Wage model
0.31
Winters model
0.39
The error values are expressed in squared units
of the predicted values. A mean squared error of
zero denotes flawless accuracy, or the absence of
errors.
4 Conclusion
In this work, the different adaptive methods are
analyzed. The Data Interpretation Algorithm for
Adaptive Methods of Modeling and Forecasting
Time Series (DIAAMMFTS) is developed in the
paper. This method is based on a 5-step procedure
and shows promising forecast skills.
Also, we implemented a program that builds
models using these methods. Based on the obtained
results and the characteristics of the models
calculated by the program, the results were analyzed
and a comparison of the methods used in the work
was carried out, based on which a conclusion was
made about the most efficient models for each
specific situation.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
369
Volume 22, 2023
The results of this work are the following:
time-series research and identification of
characteristics that affect the adequacy and
accuracy of models;
characteristics of time series dynamics that
influence the choice of forecasting model were
determined.
the new data interpretation algorithm for
adaptive methods of modeling and forecasting
time series is developed,
the comparison with Winters model and the
Tayle-Wage model shows the good quality of
the proposed predictive model;
there is implemented a program that builds
models and calculates forecasts by adaptive
methods;
the adaptive polynomial models used
sequentially allow to increase the prediction
accuracy.
The implemented program showed good results,
which allows us to conclude that these adaptive
models are effective in predicting economic or
conventional computational processes.
The model of exponential smoothing with
multiplicative seasonality of Winters is better than
the regression model but worse than the proposed
adaptive model. The forecast results of Winters can
be improved by selecting the optimal values of α.
References:
[1] D. A. Cranage, P. A. William, “A comparison
of time series and econometric models for
forecasting restaurant sales,” International
Journal of Hospitality Management, vol. 11,
issue 2, pp. 129-142, 1992.
[2] R. G. Fritz, Ch. Brandon, J. Xander
“Combining time-series and econometric
forecast of tourism activity,” Annals of
Tourism Research, vol. 11 issue 2, pp. 219-
229, 1984.
[3] S.S. Rangapuram, M.W. Seeger, J. Gasthaus,
L. Stella, B. Wang, T. Januschowski, “Deep
State Space Models for Time Series
Forecasting,” Neural Information Processing
Systems, 2018 [Online]. Available at:
https://d39w7f4ix9f5s9.cloudfront.net/0f/d8/8
8dcdaa144328a4fee9cb10275b7/8004-deep-
state-space-models-for-time-series-
forecasting.pdf
[4] L. Borgne, Y. Aël, S. Santini, G. Bontempi,
“Adaptive model selection for time series
prediction in wireless sensor networks,”
Signal Processing, vol. 87, issue 12, pp. 3010-
3020, 2007.
[5] K. Thiyagarajan, S. Kodagoda, Van L.
Nguyen, “Predictive analytics for detecting
sensor failure using autoregressive integrated
moving average model,” 12th IEEE
Conference on Industrial Electronics and
Applications (ICIEA), Siem Reap, June 2017.
2017, pp. 1926-1931 DOI: 10.1109/ICIEA
[6] Ch. Chatfield “The Holtwinters forecasting
procedure”, Journal of the Royal Statistical
Society: Series C (Applied Statistics), vol. 27,
issue 3, pp. 264-279, 1978
[7] Pedro A. Valdés-Sosa, Jose M. Sánchez-
Bornot, Agustín Lage-Castellanos, Mayrim
Vega-Hernández, Jorge Bosch-Bayard, Lester
Melie-García, Erick Canales-Rodríguez
“Estimating brain functional connectivity with
sparse multivariate autoregression,”
Philosophical Transactions of the Royal
Society B: Biological Sciences, vol. 360, issue
1457, pp. 969-981, 2005.
https://doi.org/10.1098/rstb.2005.1654
[8] H. Guney, A. Mehmet, H. A. Cagdas “A
novel stochastic seasonal fuzzy time series
forecasting model,” International Journal of
Fuzzy Systems, vol. 20, issue 3, pp. 729-740,
2018
[9] Hansun, S. E. N. G. “A new approach of
brown’s double exponential smoothing
method in time series analysis,” Balkan
Journal of Electrical & Computer
Engineering, vol. 4, issue 2, pp. 75-78, 2016
[10] Palmer, R. G., et al. “Artificial economic life:
a simple model of a stockmarket,” Physica D:
Nonlinear Phenomena, vol. 75, issue 1-3, pp.
264-274, 1994.
[11] B. G. Brown, W. Katz Richard, H. M. Allan
“Time series models to simulate and forecast
wind speed and wind power,” Journal of
Climate and Applied Meteorology, vol. 23,
issue 8, pp. 1184-1195, 1984
[12] Z. Cai, P. Jönsson, H. Jin, L. Eklundh,
“Performance of smoothing methods for
reconstructing NDVI time-series and
estimating vegetation phenology from
MODIS data,” Remote Sensing, vol. 9, issue
12, p. 1271, 2017
https://doi.org/10.3390/rs9121271
[13] G. A. N. Pongdatu, Y. H. Putra, “Seasonal
Time Series Forecasting using SARIMA and
Holt Winter’s Exponential Smoothing”. IOP
Conference Series, Materials Science and
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
370
Volume 22, 2023
Engineering Bandung, Indonesia, 9 May,
2018, 407, 012153.
DOI: 10.1088/1757-899X/407/1/012153
[14] K. Thiyagarajan, S. Kodagoda, R.
Ranasinghe, D. Vitanage, G. Iori, “Robust
Sensor Suite Combined With Predictive
Analytics Enabled Anomaly Detection Model
for Smart Monitoring of Concrete Sewer Pipe
Surface Moisture Conditions,” IEEE Sensors
Journal, vol. 20, issue 15, pp. 8232-8243,
2020. DOI: 10.1109/JSEN.2020.2982173.
[15] K. Thiyagarajan, S. Kodagoda, L. Van, R.
Ranasinghe, “Sensor Failure Detection and
Faulty Data Accommodation Approach for
Instrumented Wastewater Infrastructures,”
IEEE Access, vol. 6, pp. 56562-56574, 2018.
[16] K. Thiyagarajan, S. Kodagoda, N. Ulapane,
M. Prasad, “A Temporal Forecasting Driven
Approach Using Facebook’s Prophet Method
for Anomaly Detection in Sewer Air
Temperature Sensor System,” 2020.
TechRxiv. Preprint
Doi:10.13140/RG.2.2.31367.14245.
[17] E. Zunic, K. Korjenic, K. Hodzic, D. Donko,
“Application of Facebook's Prophet
Algorithm for Successful Sales Forecasting
Based on Real-world Data,” International
Journal of Computer Science and Information
Technology (IJCSIT), vol. 12, issue 2, pp. 23-
36, 2020. DOI: 10.5121/ijcsit.2020.12203
[18] E. Z. Martinez, E. A. S. Silva, “Predicting the
number of cases of dengue infection in
RibeirãoPreto, São Paulo State, Brazil, using a
SARIMA model,” Cadernos de
SaúdePública, vol. 27, pp. 1809–1818, 2011.
doi:10.1590/S0102-311X2011000900014.
[19] P. S. Kalekar, “Time series forecasting using
holt-winters exponential smoothing,” Kanwal
Rekhi School of Information Technology,
issue 4329008.13, pp. 1-13, 2004.
[20] Alysha M. De Livera, Rob J. Hyndman, Ralph
D. Snyder, “Forecasting time series with
complex seasonal patterns using exponential
smoothing,” Journal of the American
statistical association, vol. 106, issue 496, pp.
1513-1527, 2011
https://doi.org/10.1198/jasa.2011.tm09771
[21] Dataset Stock dynamics [Online]. Available
at: https://www.kaggle.com/econdata/stock-
dynamics
[22] Open Machine Learning Course: Time series
analysis in Python. [Online]. Available at:
URL: https://mlcourse.ai/articles/topic9-part1-
time-series/
[23] B. Seong, “Smoothing and forecasting mixed-
frequency time series with vector exponential
smoothing models,” Economic Modelling,
vol. 91, pp. 463-468, 2020 DOI:
10.1016/j.econmod.2020.06.020
[24] E. Ghaderpour, E. SinemInce, D. P. Spiros,
“Least-squares cross-wavelet analysis and its
applications in geophysical time series,”
Journal of Geodesy, vol. 92, issue 10, pp.
1223-1236, 2018.
[25] A. Corberán-Vallet, D. B. José, V. Enriqueta,
“Forecasting correlated time series with
exponential smoothing models,” International
Journal of Forecasting, vol. 27, issue 2, pp.
252-265, 2011.
[26] Ch. C. Holt, “Authorʹs retroperspective on
Forecasting seasonals and trends by
exponentially weighted moving averages,”
International Journal of Forecasting, vol. 20,
issue 1, pp. 11-13, 2004
[27] N. Boyko, “Application of mathematical
models for improvement of “cloud” data
processes organization,” Scientific journal
"Mathematical Modeling and Computing",
vol. 3, issue 2, pp. 111-119, 2016. doi:
https://doi.org/10.23939/mmc2016.02.111.
[28] A. Levin, W. Volker, C. W. John “The
performance of forecast-based monetary
policy rules under model uncertainty,”
American Economic Review, vol. 93, pp. 622-
645, 2003.
[29] S. Makridakis, S. Evangelos, A. Vassilios
“The M4 Competition: 100,000 time series
and 61 forecasting methods,” International
Journal of Forecasting, vol. 36, issue 1, pp.
54-74, 2020.
[30] N. Kunanets, O. Vasiuta, N. Boikо
“Advanced Technologies of Big Data
Research in Distributed Information
Systems,” Proceedings of the 14th
International Conference "Computer Sciences
and Information Technologies" (CSIT 2019),
Lviv, Ukraine, 17-20 September, 2019, pp.
71-76.doi: 10.1109/STC-CSIT.2019.8929756.
[31] A.O. Dolhikh, O.G. Baibuz “The software
development for time series forecasting with
using adaptive methods and analysis of their
efficiency,” Mathematical modeling, vol.
2(41), pp. 7-16, 2019.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
371
Volume 22, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed to the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research received no external funding.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.43
Nataliya Boyko
E-ISSN: 2224-2880
372
Volume 22, 2023
