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 parameters 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 prediction 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 efficiency 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.
The analysis of the behavior of stock prices is characterized by the ambiguous
behavior of the process, which is usually affected by many factors (trend, seasonality, the
geopolitical situation, etc.). Forecasting is a key point when making investment
decisions. The ability to predict the behavior of stock for making final decisions allows
you to make the best choice which otherwise might be unsuccessful [
However, not all investors successfully profit from their investment. This is
because the stock price is constantly fluctuating, and at any moment the price may fall
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
prediction, it is necessary to consider various factors, such as physical, psychological,
rational 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 [
Accurate predicting the value of assets on the exchange will also help to reduce investment 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
owners make the agreements of purchase and sale with the members of the SM as
intermediaries. The prices of these securities are determined by supply and demand, and
the process of sale is governed by the rules and regulations [
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
[
Technological analysis originated in the era of the application of computer technology 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 disciplines that deal with decision-making: Theory of Operations and General Management 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 [
Basic Concepts and Intellectual Forecasting Methods for
Stock Prices Prediction
Since a prediction based on statistical data collected with the same interval is
required, 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 [
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 [
MAE measures the average absolute value of errors in a set of predictions for continuous variables. Assuming that yˆt is the values of the time series forecast in period t, the metric is given by (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 observed 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: (1) (2) Both metrics, MAE and MSE, can take values from 0 to ∞ and do not take the direction 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 recovering some of the components such as trend.
The authors present 5 smoothing methods typically found when forecasting financial data: the simple moving average; the weighted moving average; the exponential MAE yt yˆt .
t MSE yt yˆt 2 .
t
moving average; the double exponential moving average; the triple exponential
moving average [
The basic assumption of these methods is that the fluctuations in past values represent 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.
Autoregressive 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 [
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
Moving Average Model) and ARIMA (Autoregressive Integrated Moving Average Model)
[
We’ve also proved that Artificial Neural Networks (ANN) have a significant
advantage in time series forecasting because they are endowed with the capacity to solve
complex problems of forecasting [
The output value of the neural network is defined mathematically as:
q p yt 0 j g 0i ij yi t ,
j1 i1 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 [
Sigmoidal function:
e x ex .
(3)
(4)
(5)
In some literature, this is called a logistic function. This nonlinear function is one of
the most common activation functions for deep learning [
Hyperbolic tangent:
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
vanishing gradient. The main benefit offered by this function is that it is centered relative to
zero, which helps in the process of error backpropagation [
Softmax function:
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:
This function is considered to be the most successful and widely used transfer
function. ReLU shows the best productivity in deep learning compared to sigmoid and
hyperbolic tangent. ReLU is a nearly linear function and therefore preserves the
properties 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
execution [
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 [
Consider the most common optimization techniques.
Gradient descent [
The method of stochastic gradient descent [
Mini-batch gradient descent [
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 previous step to the vector of the current update.
Adam (Adaptive Moment Estimation) [
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 recurrent 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
comparative 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 [
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 beginning 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 decisionmaking problems with a more complex structure and the number of criteria. Therefore, this task of forecasting the stock price with the considered methods and approaches can be considered in the context of multi-criteria decision-making.
The moving average method was implemented for various values of the N
parameter 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
accurate than the forecast, which takes the last 20 days into account.
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 [
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
neurons in them, is not necessary.
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
exponential filter twice, gives the best result. This method has provided an additional
parameter 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 [
The autoregressive model [
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).
Convolutional neural network [
The CNN-10-256-3 model with the Adam optimization algorithm and ReLU activation 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
consistent 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.
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 [
Train- Training Test Test Algo
ing MSE MAE MSE rithm MAE 0.0074 1.0564e-04 0.0198 0.0005 adam The best result was shown by the LSTM model 10-30-30-30-1 with the Adam optimization algorithm and ReLU activation function, and with the least lag window.
The performance of classical algorithms and machine-learning algorithms in section 4 can be summarized in a comparative Table 6. For investors, capital management is becoming increasingly important today. Professional investment managers and individual investors tend to have effective tools for understanding the trends in the stock market, to minimize investment risk and increase 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 transactions 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 computing 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 company'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 networks 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. 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