Τhis is the sixth, and last, part of our series. In this part we discuss concepts of time series analysis and the application of neural networks in their modeling and forecasting.

With the term time series, we mean a set of observable values of a variable x(t), each of which corresponds to a certain point in time t. We distinguish time series in discrete time series, where pairs of observations (t, x) are a discrete set and in continuous time series, where the pairs of values refer to a continuous interval of time values. The most common way to represent a time series is by using graphs, in which observations are reproduced diagrammatically over time. The main characteristics of time series are trend, cyclic, seasonal and outliers.

The term trend refers to the slow deviation observed in the average level of values of a time series, and which occurs over a long-term time horizon. This phenomenon is related to the structural characteristics of each time series, which is subject to examination and can have either an upward or downward trend or remain stable in relation to average values.

Cyclicality refers to a possible periodicity that the time series may present in relation to its evolution in time. The time intervals that this phenomenon is observed are quite long, for example every five or every ten years. Cyclicality is a characteristic that occurs with a fairly high frequency in economic time series and can be associated with economic phenomena, such as recessions and debt crises.

With the term seasonality, we refer to a factor of change, which arises due to seasonal or institutional changes and is observed at some point in the process of evolution of the time series. Its effect is determined by the magnitude of the significance of the change factor over the measured values (t, x).

Finally, the outliers are the observations that stand out to a large extent from the rest on the diagram of a time series and essentially indicate abrupt changes in the pattern of its behavior. Extreme values are usually impossible to predict, and their occurrence does not last particularly long in time. Special care is needed to treat outliers during modeling of a time series, since they almost always tend to significantly affect the results of this process. There is no specific approach to handling the extreme values of a time series, as each case is different, and their interpretation often lies in the experience and cognitive ability of the observer.

The main analysis objectives of a time series are as follows. Finding a statistical model, that simulates the existing data as best as possible and yields the fluctuations of them. The ability to predict the future values of the time series and finally the ability to check the validity and reliability of the results provided. Of course, the assurance, correctness and effectiveness of this model are judged by the prediction success based on the new future data as they will emerge based on the measurements.

Ιn everyday life, most processes are stochastic in their organization and evolution. We define as a stochastic process an event that develops in time following the laws of probability. A very important feature about the time series is stagnation, because due to this property, the researcher can construct a reliable statistical model of prediction in long term sense. The existence of stagnation is important because we can assume that certain structures or characteristics of our system remain constant in future predictions of our data values. For this reason, for the case of non-stationary time series (which constitute the majority), methods have been developed for the transformation of them into stationary ones. The use of artificial neural networks has contributed substantially to the process of predicting the evolution of time series. The reasons that contributed to their use are the following:

• Artificial neural networks are nonlinear. This means that nonlinear relationships between the input and output data can be processed.
• Artificial neural networks rely solely on data. There is no need to give some initial model that describes a certain relationship between inputs and outputs.
• Neural networks can be trained through machine learning methods, to deduce reliable results and then, based on their training, they can process any new data they are given.
• Finally, neural networks can adapt themselves using appropriate algorithms (as mentioned above) without the need for binding assumptions that classical statistical models need.

In particular, the evolution of time series concerning economic figures is of great interest. Neural networks can be applied to all kinds of financial problems, and in particular to predicting the course of stocks. Most common are forecasts for yield curves (that is, the relationship between the yield of bonds and their maturity dates at a given point in time), exchange rates, trading rates, etc. The main reasons that neural networks are used to predict the course of stock market are firstly that the characteristics of stocks are highly complex and difficult to shape, so a nonlinear model is imperative and secondly a large set of interacting data is often required to explain the course of a particular stock, which fits with neural network theory. As for the actual data, they are well documented although sometimes their quality is poor. Often one has to handle ellipses within the data or even discontinuities in the time series. The most appropriate way to address these difficulties is to let the network work with the incomplete data, by readjusting the weight ratios of the input data. Many times a significant part of the stock dynamics will be presented anyway.

In more detail, the steps of operating a neural network in order to predict a stock market product are the following:

• The collection of the data of the stock in question represents the entry of the data into the neural network. An input vector consisting of e.g. 1000 consecutive stock prices can be the training data and 1001st share price is the requested item.
• Feeding the neural network with the existing data and setting the parameters of the network so that the results we receive lead to the desired goal, reducing to a minimum any error that may occur during processing.
• Comparison of prediction results from the neural network with some measurable "future" data. If the network is properly "trained" then there will be apparent convergence and this means that the network is ready to provide future forecasts.

Concluding this brief presentation on artificial neural networks and their use in the field of forecasting and in particular in the field of stock products, I would like to point out that the study of the various theories and models of neural networks in stock forecasting in practice had particularly important and often positive points in the estimation of the course of prices of various stock products, whether they are stocks or derivatives. Certainly, some weak points were present in almost all models. However, it should not be overlooked that the course of the stocks is often influenced by unexpected data since it is based to a significant extent on the imponderable behavior of the human factor, where events either unrelated to the financial course of each company or regardless of the other economic environment affect unexpectedly on their course. An important success factor in the prediction provided by a neural network is the importance that will be given by each investor to the various parameters that will be entered into the system and the interpretation that will be given to the data that will be extracted from it. So while an artificial neural network can function flawlessly, there may be difficulty on the part of the researcher either to satisfactorily parameterize the system, or to correctly interpret the results of the network.