ANOTHER PERSPECTIVE ON THE FORECASTING OF ECONOMIC PROCESSES PERSPECTIVE ON THE FORECASTING OF ECONOMIC PROCESSES

Pri analýze, modelovaní a prognózovaní ekonomických procesov sa v súčasnosti stretávame s rôznymi metódami, ktoré využívajú matematicko-štatistické metódy, ekonometrické postupy a systémový prístup. Na väčšinu sociálno-ekonomických procesov sa môžeme pozerať ako na deje, ktoré sa odohrávajú v systémoch najrôznejších vlastností, pričom sa vyznačujú istou zotrvačnosťou. Musíme pripomenúť, že sa nejedná len o zotrvačnosť hmoty, čo je zrejmé pri analýze fyzikálno-technických procesov, ale tiež o zotrvačnosť myslenia ľudí, čo sa prejavuje v správaní sociálnoekonomických štruktúr. Ak sme ochotní akceptovať zotrvačnosť pri analýze ekonomických štruktúr, potom sme nútení popisovať ich správanie pomocou zodpovedajúcich modelov. Prostredie, v ktorom sa odohrávajú ekonomické procesy je možné chápať ako dynamický systém, pričom vstupné veličiny majú vo väčšine prípadov náhodný charakter, a často je ich možné na vstupe systému kvantifikovať. Odozva systému na známe vstupné veličiny je závislá od statických a dynamických vlastností systému. Ak vlastnosti systému dokážeme s dostatočnou presnosťou popísať, potom pri známych vstupných veličinách dokážeme s danou presnosťou predpovedať správanie sa systému, prípadne predpovedať priebeh vybranej realizácie. S akceptovaním dynamiky sociálnoekonomických štruktúr sa stretávame v prípade analýzy časových radov, kedy na základe minulých hodnôt sa snažíme predpovedať hodnoty budúce. Odpoveď na otázku ako presné budú predpovedané hodnoty je závislá od dynamiky systému vzhľadom na dĺžku predpovedí a charaktere vstupných veličín. Na základe uvedeného sa pokúsme pozerať na ekonomické procesy ako na deje odohrávajúce sa v dynamickom systéme. This article describes methods used from dynamic systems theory in solving the tasks of forecasting economic processes. The possibility of using theoretical methods in forecasting is demonstrated with the example of the prediction regarding the development of the rate of exchange between the USD and the SK.


Introduction
In analysing and forecasting economic processes, there are various models which use mathematical-statistical methods, econometrical procedures and systematic approaches. The majority of social-economic processes are the actions located in the systems of various qualities with some inertia. We have to mention that it is not only the inertia of mass that is obvious in analysing economic structures, but it is also inertia of people's thinking that can be seen in the behaviour of socialeconomic structures. If we are able to accept the inertia in analysing the economic structures then we have to describe their behaviour with the help of corresponding models. The setting in which the economic processes are located is possible to understand as a dynamic system where the input quantities have, in the majority of cases, a random character. Usually, it is possible to quantify them in the input. The answer of the system to a known input depends on the static and dynamic qualities of the system. If we are able to describe the qualities of the system with accuracy, we will also be able to predict the future behaviour of the system with the same accuracy or to predict the course of a chosen realisation. In analysing time series, we have to accept the dynamism of social-economic structures when on the base of past values we are trying to predict future values. The answer to the question of how accurate will the predicted values be depends on the dynamism of the system connected with the length of predictions and with the character of input values. On the basis of these facts, we can try to look at economic processes as processes in a dynamic system.

Discrete dynamic system
On the basis of the definition of a dynamic system, the connection between input, state and output can be defined as : (1) where is the step response of state which shows how the state x(t 0 ) has changed by the operating of input u in the interval t 0 Ͻ Յ t to the state x(t) in time t. g is the function which determines the corresponding output to state, input and time.
In analysing economic processes, there is the information about the behaviour of the system in the form of time series. Consequently, it is suitable to describe the economic system with the help of the model in the form of a discrete dynamic system: where f and g have the same meaning as they have in the description of the continuous dynamic system. In the description of the linear time-variant discrete system's equations (3) and (4) will be changed to the form:

direct matrix between input and output
If we suppose the mathematical model of an economic system in the form of a linear discrete dynamic system (as is shown in Fig.1) it is necessary to determine the order of the system (the dimension of matrix of state, n), the relevant input information (matrix u(k)), the analysed output quantities (matrix y(k)) and the elements of the matrixes G(k), H(k), C(k) and D(k). In general, the order of the system depends on the dynamic qualities of the modelated real system. The matrix u(k) contains all accessible quantities which have influence on the behaviour of the system. In the analysing of economic processes there is a stage of choosing relevant information that is very difficult and depends on the experiences of the analysed economic phenomenon. The matrix y(k) contains the quantities which have no interesting course for us. If we have time-series with sufficient length which characterise the course of input quantities u(k) and corresponding hodnoty matíc tak, aby sa minimalizoval rozdiel medzi výstupom reálneho systému y r (k) a výstupom modelu y(k). Tento proces sa často nazýva nastavovaním modelu. Metódy priebežného nastavovania modelu sú uvedené napr. v [1] a [2]. Uvedený matematický model sa využíva najmä na hodnotenie vplyvu jednotlivých vstupných veličín u(k) na správanie sa systému. Je možné ho však využiť aj na riešenie úloh predikcie. V tomto prípade musíme poznať, prípadne predpovedať priebeh vstupných veličín u(k) v časovom rozsahu predikcie. Pri značnom zjednodušení môžeme hodnoty vstupných veličín u(k) považovať za nulové. Potom sa stretávame s dynamickým systémom, ktorého správanie je definované len počiatočnými podmienkami, hodnotami zložiek vektora x(k). Tento model potom odpovedá používaným autoregresným modelom časových radov. Vzťahy (5) a (6) prejdú od tvaru: Nech y(k) obsahuje len jednu zložku a stavový vektor x(k) bude tvorený predchádzajúcimi hodnotami výstupnej veličiny y(k) podľa vzťahu (9), potom nasledujúcu hodnotu y(kϩ1) určíme z n predchádzajúcich hodnôt na základe vzťahu (10), prípadne (11).
. . y(kϪnϪ1) time series of quantities y r (k) acquired from real process, it is possible to tune the relevant values of the matrix to minimise the difference between the output of the real system y r (k) and the output of model y(k). This process is named "continuous tuning" The methods of continuous tuning are described in [1] and [2]. The mathematical model mentioned is used to evaluate the influence of input qualities on the behaviour of the system. It is possible to use it also in the solutions of the tasks of prediction. In this case we have to know or to predict the course of input quantities u(k) in the time extend of prediction. After simplifying, we can consider the values of input quantities u(k) to 0. Then there is a dynamic system with its behaviour defined by the initial conditions -values of matrix x(k). This model corresponds to used autoregress models of time-series. The relationships (5) and (6) will be changed to: If y(k) contains only one element and state matrix x(k) previous values of output quantity y(k) according to (9) then next value y(kϩ1) can be determined from n -previous values on the basis of (10) or (11).
Priebežné nastavovanie parametrov modelu sa realizovalo v každom kroku na základe rekurentného predpisu: Predpovedaná hodnota o jeden krok sa vypočíta na základe vzťahu: pričom: xT(k) ϭ [ y a (k), y a (kϪ1), . . . y a (kϪnϪ1)], kde: Pričom m je okamžik, v ktorom bol na výstupe analyzovaného reálneho systému získaný posledný známy údaj. Od okamžiku m sa stretávame s prognózou. Na hodnotách matice C(k) je závislá rýchlosť konvergencie matice A(k), a tým aj rýchlosť prispôsobenia sa modelu časovo lokálnym zmenám v charaktere sle- After the mentioned simplifying the matrix A(k) will contain only one line with the elements a 1 (k) to a n (k). In the process of the continuous tuning of parameters of model A(k), we have to find the values of the elements of matrix A to minimise the initial criterion. Most often in the tasks of prediction we minimise the quadrate of difference between the predicted and the real value. The methods of tuning of matrix A are based on the theory of stochastic approximation or on the correlated determination between input and output values

Experimental verification
On the basis of the mentioned theoretical points, a simple mathematical model was created. With the help of this model, the chosen economic indicators were predicted [2]. To show the functioning of this model, we will use the model of prediction of the rate of exchange USD/SK from 1. 4. 96 to 23. 2. 98 . In this time period, 473 rate of exchange tickets were published. That is enough for the tuning of the model and also for the prediction. Continuous tuning of the parameters of the model was realised in each step on the basis of recursive relationship: Predicted value after the first step will be calculated on the basis of: where: kde k p je hodnota priemerného kurzu za celé sledované obdobie.

Záver
V súčasnosti sa stretávame s mimoriadne rýchlym rozvojom informačných technológií, ktoré nám sprístupňujú aktuálne informácie z najrôznejších oblastí ľudskej činnosti v dosiaľ nebývalom Fig.4 shows the predicted development of the rate of exchange USD/SK with a predicted interval of ten days. For the first tuning the first fifty elements of time series were used. The designed model predicts the value of the relative centred rate of exchange after ten days. As mentioned before, we see that the first predicted value of the rate of exchange is in point 60. The whole file contains 473 elements of sequence. Subsequently of it the last predicted value is in point 483.
For the evaluation of the success of the model, a relative error of forecast was created: where M is the number of elements of sequence (days) for the tuning of the parameters of the model, N is the number of all elements of the sequence, kp is the average rate of exchange y(i) forecasting for i-day and y r (i) is the real value. In Fig. 5

Conclusion
Today, information technologies are developing fast which makes possible the use of information from various areas of