Fig. 1. Heat tariff
Economic forecasting is of great importance when some economic mechanism is changed rapidly as in Russia nowadays. Any economic system is a deterministic-stochastic entity of great complexity. Because of this, informative models which offer the interplay of the most significant factors are inadequate for satisfactory long-term forecasting.
The paper describes a forecasting procedure based on the joint use of formalized method (numerical simulation) and adaptive method (simulation with a neural network) when the model structure is formed by incoming information.A combination of forecasts selected by experts allows one to make the most likely forecast from the “fan” of probable tracks.
An APL implementation of this procedure has been used for the forecasting of municipal expenditures and has brought significant economic benefits.
As is known, the penetration of the free market into a rigidly planned economy brings about stochastic upset and a continuously restored balance. Complexity, contradictoriness and uncertainty of the economic system’s behaviour considerably increase during the transitional period. Russia is an example of such transition. Classical economic theories fail to fit a basically bifurcational economic system in the Procrustean bed of meaningful analytical models [1]. Therefore, the development of non-formalised methods for forecasting economic processes is of current concern [2].
Obviously, long-term economic forecasting is only possible under the following conditions:
From this, the perspective application of adaptive methods (e.g., simulation with neural networks and group methods of data handling [3,4]) turns out to be successful in economics and finance.
Development of such methods doesn’t reject meaningful analysis, on the contrary, a combination of these approaches is rather useful. The more complete and better the input information for adaptive methods, the more accurate is the forecast output. On the other hand, currently accepted trajectories of development allow the singling out of the range of factors most significant for the period of forecasting, which makes it possible to construct an adequate model.
The economic mechanism is basically stochastic. This determines its current state as a “probability vector” in some phase space of events. So a single-value forecast of economic system behaviour is impossible. One should deal with a set of possible tracks. In this case, the forecast’s accuracy falls as its depth increases. In order to adopt the optimal economic strategy, it is important to choose the most probable trajectory. Expert analysis and forecast-weighting procedure appear to be the most acceptable methods here [5].
Thus, the method proposed for forecasting economic indicators comprises the following stages:
The report presents the technique in detail, describes its APL implementation and use of the developed software for forecasting economic indicators of actual municipal services.
At present Obninsk authorities exercise the right to freely use part of the tax revenue under the control of regional and federal government. However it’s rather problematical to finance the municipal economy from this source due to the imperfection of fiscal law in Russia. It results in the necessity of extra state financing, which is only possible on condition of strict justification of expenditures. Therefore the accuracy of forecasting the rise in prices is of paramount importance.
The heat tariff in Obninsk as a function of time has been chosen as the subject of inquiry. Available actual data covers the period from January 1993 to November 1995 and is presented in Fig. 1. The choice of this indicator may be attributed to its practical value, since heat expenditures are among the major components of the financial base of the city. On the other hand, these data characterize the transitional processes of market relations development in Russia in the conditions of resistance of the sluggish centralized planned economy.
Let us pick out demand pull and cost inflation as the factors influencing the dynamics of price formation. Money supply is the cause of demand pull. That is why centralized measures aimed at curbing inflation by means of reducing effective consumer demand and money supply are effective. In the case of cost inflation, money is less important as inflation of this type depends on production and market factors.A rise in prices caused by these factors requires expansion of the currency. Otherwise, the shortage of payment means and cost inflation result in cutting down on production [6].
Positive and negative effects of keeping back the money supply make tariffs fluctuate while the prices grow gradually (compare Fig. 1 and Fig. 2). In this case, the structure of the model is to represent a conservative tendency and take account of disturbing factors.
Let us take the following differential equation, describing the logistic function of growth of different economic indicators as a basic model [7]:
where 
is the limit response level.
The plot of this function is shown in Fig. 3.
Evidently, the heat tariff-time relation shows more complex behaviour. Input data and approximations (1) for the period up to August 1994 (a) and September 1995 (b) are shown in Fig. 4. One can see that the model does not fit the object in question adequately, and the rate of tariff variation is a more complex function of time. Assuming that disturbance of the system is of quasi-periodic nature, let us input in (1) a harmonic polynomial and write:
In search of the estimates vector 
, let us apply the procedure of nonlinear regression analysis. For an initial approximation 
, we use a random search method, which we’ll carry out in a predetermined domain 
We perform further revision of the estimates via the simplex-method of Nelder-Mead, which exploits only goal function calculation:
![Equation 3: Qindexi = SUM[sample size m, i=1] (y indexi - y-hat indexi)²](ob_eq3.gif)
where yi is the experimental value of response, 
is the approximation based on (2), m is sample size.
A combination of methods is the most effective technique at high uncertainty of the optimum point 
. Unlike typical regression problems, the model being estimated is given in differential form. So we use numerical integration of equation (2) applying the Runge-Kutta method of the fourth order [8] to calculate 
at every iteration.
To test the robustness of the estimates, verification of (2) has been made on samples of different size. The analysis has exhibited the decay of * (> 1020) and variation of parameter k* in a restricted interval of 0.87 to 0.95. The results obtained should be interpreted as a tariff increase according to exponential law with periodic disturbances.
Approximation based on equation (2) is shown in Fig. 5. Good agreement with real data is obvious. This trend was used as the first approximation for calculating the long-term tariff on heat with the aid of a neural network.
The neural network turns out to be very effective for solving determinate-stochastic problems since it is capable of transforming a set of input signals into a functional structure of internal connections by the adaptation of weight coefficients [9]. In our case, the four-layered network with different numbers of neurons in associative layers has been used. The input layer consisted of 2 neurons according to the number of variables applied to it: of current time and model approximation. Each network unit implemented the following function (e.g., [10]):
where z is the output signal of the neuron, y is the input signal of the neuron, A is an unknown parameter, A>0.
The adaptation of weight coefficients was based on the minimization of the residual function [11].
Starting values of the weights’ and parameters’ vectors were calculated by the random search method. The minimum number of neurons in each of the associative layers was 2, the maximum was 20. In the course of training, 17 available points of 24 ones were presented to each neural network. Seven points remained free for the combination of forecasts.
The combination of forecasts [5] is a means of achieving a compromise settlement under conditions of competition between particular structural models of the network, and is geared towards increasing the accuracy of the forecasts. Indeed, each particular model presents only one aspect of the global dynamic process. In order to produce a substantive forecast it’s necessary to combine particular ones. Let’s write the equation:
where 
is the combined forecast,
is the particular forecast,
is the vector of weights.
For procedure (5) to be effective, particular forecasts must be independent and unbiased; the sum of coefficients of weights must satisfy the condition of normalization [12]:
Taking into account these conditions and using a minimum forecast error variance criterion, we have this formula for calculating weights:
where 
is the covariance matrix of particular forecasts errors, In is a column of n 1’s, n is the number of forecasts.
To derive , we apply the linear method of least squares. Condition (6) is used as the selection criterion
The increase of long term forecast uncertainty, and the most probable plot of heat tariff against time is shown in Fig. 6. The method of constructing the graph is described above. Note the substantial range of forecast values at the end of the forecast interval, which is quite natural at considerable forecasting depth on a restricted data sample. The forecast can be adjusted as new data are received.
In order to develop a software package implementing the algorithms of the procedure described above, APL has been chosen. The reason is that it has a wide range of high-level possibilities of work with numerical data of large dimension and different structure. The authors’ experience has shown that APL is a very convenient and effective language for the realization of statistical algorithms. The graphical capabilities of APL Iverson Software Inc. offer ample scope for programming friendly user-interfaces, indispensable for work with experts [3,13].
Forecasting the budgeted expenses of one of Obninsk’s municipal ventures was performed in June 1995 for the period up to November 1995. The plots of recommended heat tariff forecast and experimental data to the end of 1995 are presented in Fig. 8. Note that in May of 1995 there was a dramatic rise in price and the forecast seemed hardly probable. Nevertheless, the suggested forecast has been accepted following fruitful discussions between experts. At the whole depth (6 months), the maximum relative deviation did not exceed 6.6%, which demonstrates the high accuracy of the algorithm.
The use of forecast tariff values on 10 types of public utilities for December resulted in a 30% rise in estimates for 1996 against that in June prices. Such an amount, though set too low, will considerably stabilize the city budget.
All these components are equally important. Combination of deterministic and adaptive methods allows the construction of a reliable long-term forecast of economic system behaviour as a complex deterministic-stochastic entity with limited experimental data. An APL implementation realizing the recommended technique has brought economic benefits to the municipal economy.