=Paper=
{{Paper
|id=Vol-2608/paper72
|storemode=property
|title=Intelligent forecasting in multi-criteria decision-making
|pdfUrl=https://ceur-ws.org/Vol-2608/paper72.pdf
|volume=Vol-2608
|authors=Vladyslav Borysenko,Galyna Kondratenko,Ievgen Sidenko,Yuriy Kondratenko
|dblpUrl=https://dblp.org/rec/conf/cmis/BorysenkoKSK20
}}
==Intelligent forecasting in multi-criteria decision-making==
<pdf width="1500px">https://ceur-ws.org/Vol-2608/paper72.pdf</pdf>
<pre>
Intelligent Forecasting in Multi-Criteria Decision-Making

       Vladyslav Borysenko[0000-0002-9381-3080], Galyna Kondratenko[0000-0002-8446-5096],
          Ievgen Sidenko[0000-0001-6496-2469], Yuriy Kondratenko[0000-0001-7736-883X]

    Intelligent Information Systems Department, Petro Mohyla Black Sea National University,
                       68th Desantnykiv Str., 10, Mykolaiv, 54003, Ukraine,
            kalambyr11@gmail.com, halyna.kondratenko@chmnu.edu.ua,
        ievgen.sidenko@chmnu.edu.ua, yuriy.kondratenko@chmnu.edu.ua



         Abstract. In this paper, the subject of the research is the intelligent forecasting
         in multi-criteria decision-making, in particular, methods of prediction of time
         series: traditional models of autoregression, smoothing techniques and machine
         learning methods with the use of artificial neural networks and deep learning.
         Time series of Amazon share prices from the official sources over the past years
         serve as a base for exploration. The purpose of the work is to find out the pa-
         rameters that influence the efficiency and accuracy of models for analysis and
         forecast the share prices. Such assessment is complicated and vital because
         various types of criteria, in particular, sales and profitability, market indexes,
         exchange rates and general trends, etc., usually influence the decision-making
         process. The methods of research include the consideration and analysis of pre-
         diction methods using specific metrics. The relevance of the topic assumes that
         the accurate prediction at the financial market may contribute to the financial
         benefits for companies, government and other players on stock exchanges. A
         comparative analysis of the considered forecasting methods was conducted. It
         allows choosing the most appropriate intellectual method for increasing the ef-
         ficiency of a specific share price prediction. The further development of the
         subject of research includes ensemble-learning methods for neural networks,
         feature engineering, collecting a more extensive data set for forecasting.

         Keywords: time series, autoregression, smoothing, artificial neural networks,
         convolutional neural networks, recurrent neural networks, decision-making,
         multi-criteria approach, stock market, share price.


1        Introduction

The analysis of the behavior of stock prices is characterized by the ambiguous behav-
ior of the process, which is usually affected by many factors (trend, seasonality, the
geopolitical situation, etc.). Forecasting is a key point when making investment deci-
sions. The ability to predict the behavior of stock for making final decisions allows
you to make the best choice which otherwise might be unsuccessful [1].
   However, not all investors successfully profit from their investment. This is be-
cause the stock price is constantly fluctuating, and at any moment the price may fall

  Copyright © 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
below the price at which it was purchased. Therefore, to predict how the financial
market will behave is one of the most difficult tasks in the economy. In this predic-
tion, it is necessary to consider various factors, such as physical, psychological, ra-
tional and irrational behavior, and the like. All these aspects lead to the conclusion
that stock prices are very volatile and it’s difficult to predict them with a high degree
of accuracy. However, this task is urgent for the world and the whole international
economy, since the possibility of accurate prediction of the value of the stock is
closely linked to the financial gain of the companies, the government or personal
capital and the formation of the more rational financial behavior [2-3].
   Accurate predicting the value of assets on the exchange will also help to reduce in-
vestment risk and protect the investment income from the market volatility.


2      Related Works and Problem Statement

A stock market (hereinafter the SM) is an organized market where the securities own-
ers make the agreements of purchase and sale with the members of the SM as inter-
mediaries. The prices of these securities are determined by supply and demand, and
the process of sale is governed by the rules and regulations [4].
   There are three different approaches to research and predict the asset prices on the
stock market: technical, technological and fundamental analysis. In this paper, the
combination of the first two approaches was used.
   Technical analysis is the study of the dynamics of the main indicators of the market
with the help of graphical methods to predict the future direction of their movement.
A significant number of participants in stock and OTC markets use technical analysis
[5]. A successful trader O. Elder has spoken figuratively on this subject: "Technical
analysis is related to public opinion polls. It is a combination of science and art. The
scientific part consists in using statistical methods and computers; the creative part is
the interpretation of the data."
   Technological analysis originated in the era of the application of computer tech-
nology in business and other fields. The success of its application in solving complex
tasks is largely determined by the possibilities of modern information technologies.
The result of technological research is usually a selection of well-defined alternatives.
Therefore, the origins of this analysis, its methodological concepts lie in those disci-
plines that deal with decision-making: Theory of Operations and General Manage-
ment Theory.
   The technological analysis includes the ability to analyze, to predict, and to design
decision-making in complex systems of various nature, which are based on the time
series.
   A time series can be represented by four components: trend, seasonal variations,
cyclical variations and irregular factors [6]. The classical approach to the construc-
tion of the time series model is to schedule it into several components, each of which
examines specific methods. A trend is a general systematic linear or non-linear com-
ponent that can change over time. The seasonal and cyclical components are periodi-
cally repeating components. These components are not always present simultaneously
in the time series. In our case, there are no seasonal and cyclical components.


3      Basic Concepts and Intellectual Forecasting Methods for
       Stock Prices Prediction

Since a prediction based on statistical data collected with the same interval is re-
quired, we are dealing with the time series. Therefore, the time series models will be
considered, and those that suit the task of forecasting data will be selected. The most
popular forecasting time series models are autoregressive models, autoregressive
models with a moving average and models derived from them. Thus, we will focus on
these models [3].
    After selecting the specific methods of solving tasks, it is necessary to directly
conduct a prediction of selected characteristics. Constructing the desired models will
be implemented by their gradual complication, in other words, we will move from
simple to complex. In this paper, models and methods are evaluated using MAE and
MSE metrics [1, 7].
    MAE measures the average absolute value of errors in a set of predictions for con-
tinuous variables. Assuming that yˆ t is the values of the time series forecast in period
t, the metric is given by (1):

                                MAE     y  yˆ .
                                            t
                                                    t       t                        (1)


MAE is a linear score, which means that all individual differences are resolved to the
same average. MSE is the difference between the forecast and corresponding ob-
served values squared at each iteration. Because errors rise to the square before they
are averaged, MSE has a relatively high weight to large bias. This means that MSE is
most useful when large errors are particularly undesirable, which is consistent with
the objectives of this work:

                               MSE      y  yˆ  .
                                        t
                                                t       t
                                                                2
                                                                                     (2)


Both metrics, MAE and MSE, can take values from 0 to ∞ and do not take the direc-
tion of errors into account. The smaller is the value the index takes, the more accurate
is the forecast.
    Smoothing is an important and widespread method of financial market forecasting.
Smoothing methods are used to reduce the influence of random components (random
fluctuations) in time series. They provide an opportunity to obtain more “pure” values
that consist only of deterministic components. Some of the methods are aimed at re-
covering some of the components such as trend.
    The authors present 5 smoothing methods typically found when forecasting finan-
cial data: the simple moving average; the weighted moving average; the exponential
moving average; the double exponential moving average; the triple exponential mov-
ing average [8].
   The basic assumption of these methods is that the fluctuations in past values repre-
sent random deviations from a smooth curve, which can be extrapolated to create a
forecast.
   The autoregressive model is another effective tool for understanding and predicting
future values of the time series, which includes devolution of a variable on values of
series in the past. The importance of ARMA models lies in their flexibility and in
their ability to describe almost all features of the stationary time series. Autoregres-
sive of these models describe how consecutive observations in time affect each other,
while parts of the moving averages capture some possible unobserved upheavals, and
that allows simulating various phenomena that can be observed in a variety of fields
from biology to finance [1-3].
   The main idea of autoregression methods is that future values of the time series
cannot deviate to higher or lower than the previous values of the time series whatever
the reasons that caused those deviations are. The paper has presented such models of
autoregression as simple AR (Autoregression Model), ARMA (Autoregressive Mov-
ing Average Model) and ARIMA (Autoregressive Integrated Moving Average Model)
[1, 8].
   We’ve also proved that Artificial Neural Networks (ANN) have a significant ad-
vantage in time series forecasting because they are endowed with the capacity to solve
complex problems of forecasting [2, 9, 10].
   The output value of the neural network is defined mathematically as:
                                    q                        p                  
                      yt   0     g      y    ,
                                           j        0i                    ij i       t   (3)
                                   j 1                     i 1                

where p is the number of the input variables, q is the number of the hidden nodes,
 j and ij are the weights,  t is the random noise.
  As a function of g the following functions can be used [2, 11, 14].
  Sigmoidal function:

                                                         1
                                        f x                    .                      (4)
                                                   1  e x

In some literature, this is called a logistic function. This nonlinear function is one of
the most common activation functions for deep learning [12].
   Hyperbolic tangent:

                                                   e x  ex
                                    f x                            .                  (5)
                                                   e x  ex
This function gives the best result for multilayer neural networks compared with the
sigmoidal function. However, the function does not solve the problem of the vanish-
ing gradient. The main benefit offered by this function is that it is centered relative to
zero, which helps in the process of error backpropagation [11].
   Softmax function:

                                               e xi
                                   f x                  .                          (6)
                                              e
                                                      xj

                                               j


The Softmax function is another type of activation function used in neural computing.
It is used to calculate the probability distribution of a vector of real numbers and gets
the value range from 0 to 1.
    ReLU function:

                                  f x   max0, x  .                               (7)

This function is considered to be the most successful and widely used transfer func-
tion. ReLU shows the best productivity in deep learning compared to sigmoid and
hyperbolic tangent. ReLU is a nearly linear function and therefore preserves the prop-
erties of linear models, which make them easily optimized by methods of gradient
descent. The main advantage of using ReLU in calculus is that this function does not
require computing the exponent or dividing, therefore it guarantees a more rapid exe-
cution [11, 13].
    ANN has been and continues to be actively used in the financial markets; one of
the main advantages of ANN that make them so popular as harbingers of the market is
the natural nonlinearity that allows them to learn nonlinear mapping and correlation
of data. ANN also works on data, can be trained in real-time; they are highly adaptive,
easy to retrain in the event of market fluctuations and, finally, do well with data that
contain a certain number of errors [14-17]. ANN mechanism implies minimum par-
ticipation of the analyst in the model shaping, as far as the learning ability is charac-
teristic of all neural network models and learning algorithms adapt (adjust) the
weights according to the structure of the data presented for training.
    Consider the most common optimization techniques.
    Gradient descent [18]:

                                xt 1  xt    f xt  .                           (8)

According to this method, the steps proportional to the opposite gradient value are
taken for minimizing functions. The parameter to this method is a descent speed α.
When reaching a large value α will possibly make large steps for finding the mini-
mum, but there is a risk of skipping the lowest point. At a very low speed of learning,
the algorithm will move confidently in the direction of the negative gradient, but the
implementation, in this case, will take a long time.
   The method of stochastic gradient descent [19]. This is a type of gradient descent,
which handles 1 element for learning at each iteration. So, parameters are updated
even after one iteration, which processed only one value of the variable. This allows
optimizing the target function much faster than ordinary gradient descent. But if the
number of training data is very large, the number of iterations is therefore sufficiently
large.
   Mini-batch gradient descent [20]. This is the type of gradient descent, which is
faster than batch and stochastic gradient descent. Let there be the m number of input
data, then in one iteration of this method b<m elements will be processed. Conse-
quently, even if the number of training data is large, it is processed in smaller training
groups at once. Thus, the method works for a large amount of training data by opti-
mizing an objective function with less number of iterations.
   Gradient descent with momentum is the method that helps to accelerate a descent
in the corresponding direction and dampens the oscillations, approaching the local
minimum. This is implemented by adding a part of the update vector from the previ-
ous step to the vector of the current update.
   Adam (Adaptive Moment Estimation) [21]. Adam optimizer is one of the most
popular optimization algorithms of gradient descent since it is computationally effi-
cient and has very little memory requirements. This method calculates an individual
adaptive learning rate for each parameter from the evaluation of the first and second
moments of the gradients.
   In the simulation of isolated time series using ANN models a transformation of the
original data for increasing the number of input neurons and, consequently, increasing
predictive ability is allowed. Modeling of the time series using the ANN mechanism
consists in the formation of ANN as a certain structure, which describes the behavior
of the system under study in points in time, and forecasting is the prediction of the
future behavior of the system in the background.
   To sum it up, this work explains the mathematical foundations of neural networks,
namely multilayer perceptron of Rumelhart, convolutional neural networks and recur-
rent neural networks. The correctly chosen input is significant. In this paper, only the
value of the stock price in the past was taken as the initial data.


4      Practical Implementation of the Described Methods and
       Techniques

This section describes the architecture of the developed program and the final com-
parative results of all models. Compared to the classical mathematical methods, this
work has allowed confirming the viability and the feasibility of using artificial neural
networks for further study of their application in the financial markets [1].
   For this study, AMAZON stock statistics were taken from the official NASDAQ
(National Association of Securities Dealers Automated Quotation) website for the
past 5 years. Dataset (Fig. 1) is daily information on the price of shares at the begin-
ning and at the end of the day, the minimum and maximum value of a share during
the day, and the number of shares sold. The above parameters can be considered as
criteria since they have an optimization direction (maximum or minimum). Moreover,
the considered methods and approaches are also used to solve multi-criteria decision-
making problems with a more complex structure and the number of criteria. There-
fore, this task of forecasting the stock price with the considered methods and ap-
proaches can be considered in the context of multi-criteria decision-making.
   The moving average method was implemented for various values of the N parame-
ter of the previous time points number that had to be taken into account when building
a forecast (Fig. 2). It was checked that decreasing the N window length model shows
the more accurate result on the test sample, indicating that the latest data is the most
influential to the prediction, i.e. prediction considering the past 5 days is more accu-
rate than the forecast, which takes the last 20 days into account.




                               Fig. 1. Dataset as input data




                Fig. 2. The schedule of the simple moving average forecast

For the exponential moving average (Fig. 3) parameter is the level of the α smoothing
coefficient, which represents the degree of reduction of weighting from 0 to 1. The
lower the level of smoothing is, the more accurate is the forecast, as every previous
reading weighs more [2-3].
   This part of the paper also contains a comparison of the results of forecasting (by
terms of MAE and MSE) stock prices by various types of neural networks (MLP,
CNN, LSTM) with an additional comparison of their models, which have different
structures and parameters. Therefore, a detailed review of the architecture of neural
networks, justification for choosing the number of hidden layers, the number of neu-
rons in them, is not necessary.




            Fig. 3. The schedule of exponential moving average method forecast

For the data with a distinct trend, which corresponds to the input operation, the
method of double exponential smoothing that is the recursive application of exponen-
tial filter twice, gives the best result. This method has provided an additional parame-
ter β, which is responsible for the smoothing of the trend (Fig. 4). The combination of
the α and β pair has adjusted the accuracy and the quality of the forecast [7].




       Fig. 4. The schedule of the double exponential moving average method forecast

The results of implementation of all smoothing methods are summarized in Table 1.
   The autoregressive model [22] is an effective tool for understanding and predicting
future values of the time series, which includes the devolution of a variable on values
of series in the past. Autoregressive of these models describe how consecutive obser-
vations in time affect each other, while the parts of the moving averages capture some
possible unobserved upheavals. For the considered models, the autoregressive Akaike
coefficient allowed the software to choose the model that gives the best forecast.
Therefore, in the case of the dataset, it was the model ARIMA (2,1,1).

                  Table 1. The smoothing methods implementation results

       Method                  Specific parameters             Test MSE Test MAE
                                 Window N = 30                  0.0029   0.0455
    Simple moving
                                 Window N = 10                   0.0012  0.0281
       average
                                 Window N = 5                    0.0005  0.0180
                          The level of smoothing α = 0.1         0.0018  0.0340
  Exponential mov-
                          The level of smoothing α = 0.2         0.0011  0.0259
    ing average
                          The level of smoothing α = 0.6         0.00049 0.0165
   Double exponen-               α = 0.1, β = 0.3                0.00244 0.0388
   tial moving aver-             α = 0.2, β = 0.8                0.00163 0.0322
          age                    α = 0.6, β = 0.6                0.00054 0.0177

The execution results of all three models were summarized in a single comparative
Table 2.

                Table 2. The autoregressive models implementation results

               Model               Specific parameters         Test MSE Test MAE
    Simple autoregressive model            p=2                 0.000347 0.013322
   Autoregressive moving average
                                       p = 2, q = 1             0.000348    0.013346
               model
  Autoregressive integrated moving
                                    p = 2, q = 1, d = 1         0.000345    0.013335
          medium model

The simplest type of the neural network is a single-layer perceptron network (usually
a multilayer perceptron of Rumelhart), which consists of a single layer of output
nodes, and the inputs provided directly to the outputs via a series of weights. Univer-
sal approximation theorem for neural networks states that each continuous function,
which maps intervals of real numbers to a certain initial interval of real numbers, can
be arbitrarily closely approximated by a multilayer perceptron with only one hidden
layer [2, 12]. The main characteristic of the multilayer perceptron is its architecture,
namely the number of hidden layers, nodes and activation functions. Besides, since
the study outcome partially depends on the initialization of variables, each model was
trained separately 5 times, and all the characteristics for a comparative table are aver-
age values. For a more in-depth study of the model 5 architectures were implemented
in the paper, the accuracy of each of them is represented in Table 3.
   According to the comparative results (Table 3), the dependence of the quality of
the forecast on the number of nodes in the hidden layer, the optimization method and
the activation function is completely traced. The best result was the MLP 20-100-1
model with the Adam optimization algorithm and hyperbolic tangent as an activation
function. Here is a graph of the price of stocks sold predicted by this model (Fig. 5, a),
as well as a graph of the forecast error (Fig. 5, b).

                               Table 3. MLP learning results

     Model    Parame- Training Training Test Test Algo- Activa-
                ters   MAE       MSE    MAE MSE rithm     tion
              number                                    function
 MLP 20-10-1    221    0.0645   0.0074 0.090 0.014 adam   relu
 MLP 20-20-1    441    0.0462   0.0038 0.052 0.005 adam   relu
 MLP 20-100-1  2201    0.0438   0.0038 0.037 0.002 adam   relu
 MLP 20-100-1  2201    0.0035   0.0428 0.032 0.002 adam   tanh
 MLP 20-100-1  2201    0.0668   0.0078 0.093 0.014  sgd   relu




                  a)                                                     b)
               Fig. 5. MLP 20-100-1: a) price of stocks sold; b) forecast error

Convolutional neural network [23] is a type of artificial neural network, in which the
pattern of connectivity between neurons is inspired by the organization of the visual
cortex of animals, individual neurons that are arranged in such a way that they re-
spond to overlapping regions of the field. The main advantage of convolutional neural
networks is that we use the convolutional layers to identify the signs of a network that
allows training of a neural network without complex pre-processing since useful fea-
tures are learned during training [13]. Unlike multilayer perceptrons, convolutional
neural networks have a complex structure, since they need to gauge the number and
type of layers, the number of nodes in each layer, the optimization method, the activa-
tion function and the number of filters. So CNN-20-200-3 will designate 20 knots at
the entrance as the first layer, 200 filters sized 3x3, MaxPooling layer and 2 fully-
connected layers. In this work, 5 different CNN architectures were tested, the results
are shown in Table 4.
   The CNN-10-256-3 model with the Adam optimization algorithm and ReLU acti-
vation function showed the best result.
   A recurrent neural network (RNN) is an artificial neural network, the neurons of
which transmit feedback signals to each other [24]. The idea of RNN is to use consis-
tent information. In traditional neural networks, we assume that all the inputs (and the
outputs) are independent of each other. But for many tasks, this is not the best idea.
This type of neural network is well suited to predict the value of the stock, as further
steps may depend on the past.

                              Table 4. CNN learning results

     Model     Parame-        Train-     Train-    Test       Test    Algo-    Activa-
              ters num-         ing        ing     MAE        MSE     rithm     tion
                 ber          MAE         MSE                                 function
 CNN 10-256-3 198.657         0.0238     0.0011    0.096      0.013   adam      relu
 CNN 20-256-5 201.367         0.0182     0.0074    0.174      0.038   adam      relu
 CNN 20-256-3 199.937         0.0513     0.0050    0.290      0.110    sgd      relu
 CNN 20-500-3 756.501         0.0515     0.0050    0.259      0.089    sgd      relu
 CNN 20-300-3 273.901         0.0414     0.0030    0.255      0.079    sgd      tanh

For the implementation of the forecasting recurrent neural networks, the LSTM model
(Long Short-Term Memory Units) was chosen [25]. LSTMs help to save the bug,
which can be spread through time and layers. Maintaining a more constant error, they
allow the networks to continue exploring for many time steps [16]. To analyze the
effect of parameters on the prediction value of the shares 5 variants of the LSTM
architecture summarized in the following Table 5 were implemented. LSTM-20-30-
30-30-1 was used to identify neural LSTM networks with initial data of the previous
20 days, the number of nodes 30 on the first, the second and the third layers of LSTM
and one fully connected layer with the resulting output of one element.

                             Table 5. LSTM learning results

     Model        Parame- Train- Training Test       Test Algo- Activa-
                 ters num-   ing     MSE     MAE MSE rithm        tion
                    ber    MAE                                  function
     LSTM          18,631  0.0074 1.0564e-04 0.0198 0.0005 adam   relu
 20-30-30-30-1
     LSTM         22,681     0.0079 1.8716e-04 0.0188 0.0005 adam               relu
 20-30-30-40-1
     LSTM         18,631     0.0106 1.9467e-04 0.0159 0.0004 adam               relu
 30-30-30-30-1
     LSTM         18,631     0.0099 1.7086e-04 0.0150 0.0003 adam               relu
 10-30-30-30-1
     LSTM         18,631     0.0082 1.2895e-04 0.0152 0.0003           sgd      relu
 10-30-30-30-1

The best result was shown by the LSTM model 10-30-30-30-1 with the Adam optimi-
zation algorithm and ReLU activation function, and with the least lag window.
   The performance of classical algorithms and machine-learning algorithms in sec-
tion 4 can be summarized in a comparative Table 6.

                Table 6. The comparative results of the implemented methods

    Model/Method                Advantages                  Disadvantages
                    Ability to process trends of varyingVulnerability to extreme
     Smoothing
                    levels and seasonality components            values
                                                       Limitations in the assump-
    Autoregression          Easy to automation
                                                                  tions
                   Ability to handle complex nonlinear Require a large amount of
        ANN
                    patterns. High forecast accuracy              data


5       Conclusions

For investors, capital management is becoming increasingly important today. Profes-
sional investment managers and individual investors tend to have effective tools for
understanding the trends in the stock market, to minimize investment risk and in-
crease their profits. Some people believe that it is very difficult to predict stock prices.
However, in the real business world, successful traders carry out thousands of transac-
tions every year.
   Since the very beginning of the financial operations on the stock exchange, people
developed methods to predict the value of assets in the future. With advances in com-
puting technology and artificial intelligence, the accuracy of the methods is growing
every day.
   In this paper, the authors compared the indicators of forecasting between the neural
network and the classic time series forecast method, namely, the value of the com-
pany's shares on the stock exchange. The analysis showed that the neural network
models described in this study showed a very much better able to accurately predict,
and therefore confirmed the viability and the feasibility of using artificial neural net-
works for further study of their application in the financial markets. The predictive
power of the model was influenced by different factors, depending on the architecture
and input data.


References
 1. Kingdon, J.: Intelligent Systems and Financial Forecasting. Springer, London (1997).
 2. Palit, A., Popovic, D.: Computational Intelligence in Time Series Forecasting. Theory and
    Engineering Applications. Springer, London (2005). doi: 10.1007/1-84628-184-9
 3. Guerard, J.: Introduction to Financial Forecasting in Investment Analysis. Springer, New
    York (2013). doi: 10.1007/978-1-4614-5239-3
 4. Pavlov, V., Pilipenko, I., Krivov’yazyuk, I.: Securities in Ukraine: Students Book. Condor,
    Kyiv (2004). (in Ukrainian)
 5. Sokhatska, O.: Stock Exchange. Carte Blanche, Ternopil (2003). (in Ukrainian)
 6. Box, G., Jenkins, G., Reinsel, G.: Time Series Analysis: Forecasting & Control. John
    Wiley & Sons, New Jersey (2008)
 7. Wei, W.: Time Series Analysis: Univariate and Multivariate Methods. Pearson, New York
    (2005)
 8. Sitte, R., Sitte, J.: Neural Network Systems Technology in the Analysis if Financial Time
    Series. In: Leondes, C.T. (eds.) Intelligent Knowledge-Based Systems, pp. 1564–1615.
    Springer, Boston, MA (2005). doi: 10.1007/978-1-4020-7829-3_43
 9. Kondratenko, Y., Gordienko, E.: Neural Networks for Adaptive Control System of Cater-
    pillar Turn. In: Annals of DAAAM for 2011 & Proceeding of the 22th Int. DAAAM
    Symp. "Intelligent Manufacturing and Automation", pp. 0305–0306 (2011)
10. Gomolka, Z., Dudek-Dyduch, E., Kondratenko, Y.P.: From homogeneous network to neu-
    ral nets with fractional derivative mechanism. In: Rutkowski, L. et al. (eds.) International
    Conference on Artificial Intelligence and Soft Computing (ICAISC-2017), Part I, pp. 52–
    63, Zakopane, Poland (2017)
11. Nwankpa, C., Ijomah, W., Gachagan, A., Marshall, S.: Activation Functions: Comparison
    of Trends in Practice and Research for Deep Learning. arXiv:1811.03378 [cs.LG] (2018)
12. Robert, J.: The Application of Neural Networks in the Forecasting of Share Prices. Finance
    and Technology Publishing, New York (1996)
13. Di Persio, L.: Artificial Neural Networks architectures for stock price prediction: compari-
    sons and applications. International journal of circuits, systems and signal processing
    31(10), 404-405 (2016)
14. Kushneryk, P., Kondratenko, Y., Sidenko, I.: Intelligent dialogue system based on deep
    learning technology. In: 15th International Conference on ICT in Education, Research, and
    Industrial Applications: PhD Symposium (ICTERI 2019: PhD Symposium), vol. 2403, pp.
    53–62, Kherson, Ukraine (2019)
15. Sidenko, I., Filina, K., Kondratenko, G., Chabanovskyi, D., Kondratenko, Y.: Eye-tracking
    technology for the analysis of dynamic data. In: IEEE 9th International Conference on De-
    pendable Systems, Services and Technologies (DESSERT), pp. 479–484, Kiev, Ukraine
    (2018). doi: 10.1109/DESSERT.2018.8409181
16. Cryer, J., Chan K.-S.: Time Series Analysis. With Applications in R. Springer, New York
    (2008). doi: 10.1007/978-0-387-75959-3
17. Beran, J.: Mathematical Foundations of Time Series Analysis. A Concise Introduction.
    Springer, Cham (2017). doi: 10.1007/978-3-319-74380-6
18. Paper, D.: Gradient Descent. In: Data Science Fundamentals for Python and MongoDB,
    pp. 97–128. Apress, Berkeley, CA (2018). doi: 10.1007/978-1-4842-3597-3_4
19. Ketkar, N.: Stochastic Gradient Descent. In: Deep Learning with Python, pp. 113–132.
    Apress, Berkeley, CA (2017). doi: 10.1007/978-1-4842-2766-4_8
20. Danner, G., Jelasity, M.: Fully Distributed Privacy Preserving Mini-batch Gradient De-
    scent Learning. In: Bessani, A., Bouchenak, S. (eds.) Distributed Applications and Inter-
    operable Systems. DAIS 2015. Lecture Notes in Computer Science, vol. 9038, pp. 30–44.
    Springer, Cham (2015). doi: 10.1007/978-3-319-19129-4_3
21. Jiang, Y., Han, F.: A Hybrid Algorithm of Adaptive Particle Swarm Optimization Based
    on Adaptive Moment Estimation Method. In: Huang, D., Bevilacqua, V., Premaratne, P.,
    Gupta, P. (eds.) Intelligent Computing Theories and Application. ICIC 2017. Lecture
    Notes in Computer Science, vol. 10361, pp. 658–667. Springer, Cham (2017)
22. Olson, D., Wu, D.: Autoregressive Models. In: Predictive Data Mining Models. Computa-
    tional Risk Management, pp. 55–69. Springer, Singapore (2017)
23. Ayyadevara, V.: Convolutional Neural Network. In: Pro Machine Learning Algorithms,
    pp. 179–215. Apress, Berkeley, CA (2018). doi: 10.1007/978-1-4842-3564-5_9
24. Hammer, B.: Learning with recurrent neural networks. Springer, London (2000)
25. Manaswi, N.: RNN and LSTM. In: Deep Learning with Applications Using Python, pp.
    115–126. Apress, Berkeley, CA (2018). doi: 10.1007/978-1-4842-3516-4_9

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