A very interesting structure shown in Phase portraits is one related to the concept introduced into dynamical systems as an attractor. What is an attractor? An attractor for a dynamical system that manifests chaotic behavior is a point or set of points in the Phase Space, toward which such a system tends and from which it cannot escape as time passes. Indeed, in the case of strange attractors (such as the Lorenz Attractor), strange orbits appear in the space of the phases in which the system remains, without escape. One could therefore speak of a kind of steady state that can be found in the system, where a set of points of the Phase Space appears periodically. If the attractor is chaotic, then it is highly sensitive to initial conditions, however the state trajectories observed are confined within the limits of the attractor. Thus, a dynamic system characterized by a chaotic Attractor is locally unstable but completely stable since the system never moves away from the attractor Basin. It is obvious that in the case of a chaotic attractor, although the system is characterized by a degree of stability, it is nevertheless impossible to accurately predict the future state of the system when the initial conditions change even a little. In this case the displayed trajectories may show a large deviation, precluding predictability. Starting from a specific point (initial state of the system) in the phase space, it seems impossible to predict which trajectory within the boundaries of the attractor the evolution of the system will follow.

AIn this part of our analysis, we will focus on studying the time evolution of a time series and whether such a data series can manifest chaotic behavior and to what extent. We will therefore deal with the dynamic evolution of the process or system that produces the magnitude we observe and therefore we will consider that we observe the magnitude in time with a fixed time step, i.e. with a fixed sampling time. A set of such observations is called a time series. In some problems the sampling time may not be constant and then more specific processing of the time series is needed to perform the analysis. For example, the daily prices of the stock market constitute a time series with variable physical sampling time.

However, the sampling time of the measurements of the time series must remain constant if we want to draw safe conclusions about the various variables that affect it. If some measurements are further apart than the sampling time, the corresponding empty spaces may be filled by an interpolation method. It shall also be checked whether any data received has values that are much greater or less than the overall set of measurements, which may be due to errors in the measuring instruments or caused by the way in which they were obtained. Another issue concerns the amount of data that should be measured in combination with the time frame within which it was measured. Fewer points than are necessary is almost certain that will lead to erroneous conclusions.

The main characteristics to be studied for time series, before adapting a model that describes the evolution of it, are stagnation, trend, periodicity, determinism or stochasticity and of course linearity or non-linearity of it.

By stagnation, we mean to see whether or not the observed variations in time series values vary with time. In the case of a non-stationary time series, certain trends may occur, i.e. slow changes in its mean values over time, which should be checked. Also, in the case of a non-stationary time series, its values may display a periodicity, which may refer to specific time periods (per month, per quarter), they are also called seasonal. Such behaviors are important and should be included in our analysis, as they are important determinants of the evolution and predictability of the time series.

All the time series, which refer to real-world data, contain noise and in this sense can be characterized as stochastic. Any analysis of such data should therefore focus on investigating and identifying the deterministic part that directs the time series. When the cause of the evolution of its values is not clear due probably to noise and generally does not seem that this cause dominates its evolution, then the particular system is characterized as stochastic and only conclusions derived from a purely statistical study can be drawn. Again, if for some reason, we can assume that there is a deeper cause justifying the behavior of the system under study accompanied by some stochastic fluctuations, but not dominant in the evolution of the system, then we can apply some techniques involving deterministic dynamical systems, such as locating main periods if the system seems to exhibit a periodicity, or techniques for nonlinear potentials exhibiting a chaotic behavior.

Finally, the occurrence of linearity or non-linearity in such a system seems to be directly related to the deterministic or non-deterministic evolution of such a system. By the term linearity we define that all variables of the system can be expressed as linear combinations with respect to the parameters of the system. This means that for a linear system, the future evolution of the time series can be expressed as a linear combination of its previous observations, which cannot be true for a nonlinear system. In a nonlinear system, the nonlinear interaction between the parameters requires additional data based on the previous observations. Especially in the case of a linear deterministic system, with no noise, the analysis is relatively simple, giving solutions easily determinable (fixed point of convergence, periodic points or trajectories). In contrast, in the case of a nonlinear dynamical stochastic system, it is quite difficult to identify nonlinear dynamical relationships so that conclusions about the evolution of that system are quite precarious.

In order to be able to determine the chaotic nature of a time series and in addition to predict its evolution, the following conditions are necessary:

・The system under study should have a small number of variables (a system of few dimensions). As pointed out above, in chaotic systems where the number of variables is large, the usual mathematical techniques do not give reliable results so that the system is considered random and not chaotically deterministic.
・We should have a large number of observations (thousands) with a small amount of noise. Theoretically, we should have an infinite number of measurements, but this is obviously impossible. Small multitudes of measurements cannot give safe data of system behavior, not allowing an accurate description model to be derived.
・Also, an important factor is increased accuracy in observations, with as many decimal places as possible. The current computational power of computers allows processing such data accuracy by simultaneously extracting numerical results of the same accuracy.
・Finally, the time series should exhibit behavior of increased stagnation without long-term trends. What is desirable is for the series to keep its statistical properties (average, variance) relatively constant over whatever period of time we study it.

In classical statistical time series analysis linear features of the series such as mean values, dispersion and the so-called autocorrelation coefficient are studied. The study of the dynamic system is done using linear models such as ARMA (auto regulatory integrated moving average model), models which give satisfactory results in the case of linear processes but do not satisfactorily describe nonlinear processes. Applying linear analysis to the case of real time series with stochastic nonlinear features, much of the information is lost and received as noise. However, this piece of information can refer to a deterministic nonlinear behavior and can only be studied and retrieved using nonlinear methods of analysis. Of particular interest are nonlinear dynamic systems of small dimensions (with a small number of variables), such as pseudo-periodic systems and chaotic systems.

The dynamical systems that describe a chaotic time series are energy loss systems, i.e. systems where the initial "volume" of values of the system's variables in the Phase Space constantly shrinks. Each trajectory of evolution of these systems is attracted towards some invariant set of points, which is the attractor of the system. When studying such a chaotic time series, the nonlinear features of the system are sought, e.g. the dimension of the attractor of the system and its degree of complexity. These features are defined, based on the theory of nonlinear systems, as invariant measures of the system, which do not change over time as well as by the observation process.

For the analysis of a chaotic time series through the tools of nonlinear dynamics, the basic prerequisite is the reconstruction of the phase space of the system, without altering its characteristics, but its shape is not preserved during the reconstruction process. The theorem that provides the necessary conditions, under which we can have a smooth reconstruction of the observed Attractor, is the Takens theorem. What Takens demonstrated is that it is possible to derive the relationship between the dynamical behavior of the system and the observed variables, without necessarily knowing anything about the system under study. This can be done through the time delay method, which allows the construction of several time series of the system, using only the original time series. With this conversion, a variable of the system is converted into a series of variables of shorter time length, which however provide for each time a point in the new space that has been created. The method of time delays provides the possibility of extracting the "hidden" information found within the data of the time series. However, an incorrect choice of delay time and the dimension of the newly reconstructed phase space combined with the existence of extrinsic noise in the system can lead to incorrect conclusions about the evolution of the time series. Empirically, the choice of delay time should be such that it is the least possible integer where discernible changes are observed in the observed data. As for the new dimension of the phase space, it should be chosen so that the observed attractor retains its topological characteristics in the new space.

There are many classes of nonlinear models for describing a chaotic time series. These models are divided, according to how they are defined in the state space, into universal, semi-local, and local.

・Universal models have unique analytic expression for the entire domain of definition. Such models are polynomial and fractional.
・Semi-local models are analytically expressed like universal models but they consist of a set of basic functions, and for this their form changes in the different regions of the state space. Such models are neural networks and radial basis functions.
・Local models do not have unique analytic expression for the whole domain but are configured differently in each region of the state space. Such models are kernel models and local linear models.

These dominant types of models, although applied to the description of the evolution of a chaotic time series, sometimes give reliable results while in other cases they require improvement and adaptation by introducing parameters that substantially affect the system. The nonlinear behavior of data requires very careful application if we want a reliable though approximate forecasting. We would like to stress again that if the system is chaotic, then it is sensitive to initial conditions and therefore whatever is the suitability of the chosen model, the predictability is limited to a short time horizon, thus increasing uncertainty in forecasting its evolution.

Because the problem of predicting the evolution of a time series is the main purpose of analyzing them, the most prevalent prediction models are local models. In these, the analysis focuses on the part of the reconstructed space of the time series that we are interested in making the prediction. In this way, even if the entire series manifests a chaotic character, the trajectories that manifest in this limited space, that we are considering slowly and gradually evolve, allows us to make a short-term prediction. These models, strange as this may seem, have been widely accepted in the field of forecasting and have been improved in recent years by methods of optimizing the parameters that influence the future development of the system. It examined how to reconstruct the phase space that would not change the topological features of the original space and how the prediction time depth could be extended and what are the main factors determining this depth.

The issue of noise in the data is an important factor that should be taken into account. Noise affects the form of the reconstructed space, and therefore the proximity of the points we are considering. In local forecasting, in combination with a small number of adjacent points, it can be the main reason for failure in the forecast estimate, because in addition to the intrinsic indeterminacy of the reference point, the nearest neighbors will also have indeterminacy, which could not be adjacent points if the reconstruction was done without noise. For a better and more reliable forecast, a larger number of adjacent points should be taken. Another question that arises concerns the sampling time window and the process of reconstructing the Phase Space. If we increase the time window, will we achieve a better forecast? Is there an optimal reconstruction or a specific time window, with which we can achieve very good prediction regardless of the future time depth we want to predict? As shown by the study of such systems using local models, there is no such possibility of combination. It was observed that if we increase the time window no better predictions are achieved, rather the opposite.

Specifically, for the case of chaotic but deterministic time series, the models used to study and predict their evolution are based on the assumption that there is a stable relationship between the predictor variable (dependent variable) and certain parameters (independent variables) that affect it. This relationship can well be described as a cause-and-effect relationship. In these models, the existence of independent parameters, offer the researcher the possibility of finding many scenarios of evolution of the system with the ultimate goal of finding the optimal scenario. However, these models have serious drawbacks, such as needing a large amount of information about both the variables of the system and the parameters displayed in it. Finally, there are many cases where the production of complete forecasts from such a model requires previously estimating, on the basis of some other forecasting method, the future value of certain input variables.

Concluding this section on chaotic time series and the models that attempt to describe their behavior, we would like to comment on the criteria that determine the choice of optimal prediction. The main criterion for choosing a forecast method is the time horizon, i.e. the length of time to which the forecast will refer in the future. Qualitative methods are generally used for long-term forecasts, while quantitative methods are used for medium-and short-term forecasts. Also important is the number of periods for which provision is required. Some techniques are suitable for predictions corresponding to one or two periods after the most recent observation, while others to more. There are also techniques that combine prediction horizons with different lengths of time. Another important factor is the cost of a prediction method, which is determined by the amount of data required by the method and the complexity of its implementation. In many cases where choosing the appropriate forecasting method is not easy, it has proved preferable to combine the forecasts resulting from the application of different methods. This increases the accuracy of forecasts as the interval of variation of errors is significantly reduced.

In many cases where the choice of the appropriate forecasting method it is not easy, it has proved preferable to combine the predictions resulting from the application of different methods. This increases the accuracy of forecasts as the interval is significantly reduced variation of errors. Assuming stability of data behavior patterns, the optimal theoretical model would have to produce the most accurate predictions. But once these patterns change in practice, combining predictions from different models produces an average that is closer to reality than the theoretically more correct model. Using a combination of predictions is tantamount to assuming that it is impossible to find a model that optimally determines the behavior of historical data. But the study of the various techniques of combining predictions can ultimately lead to the identification of more appropriate patterns of behavior and therefore to the creation of better individual models. If the combination of different models produces more accurate predictions, it is theoretically possible to construct a model that makes optimal use of the different forms of information contained in the predictions of those models.