Introduction

At the present stage, for effective management of enterprises it is necessary to be able to predict the major trends in social and economic systems, the main economic indicators characterizing financial position and efficiency of the use of companies' production resources.

Estimates and forecasts of the financial condition of the company make it possible to find additional resources, to increase its profitability and solvency.

Problems of the analysis and the forecast of financial condition of the company by means of corresponding indicators are an actual task, because on the one hand this is the result of the company, on the other it defines the preconditions for the development of the company. Qualitative forecast gives us an opportunity to develop reasonable strategic plans for economic activity of enterprises.

Under market conditions, the adequate forecasting and capacity planning of enterprises are impossible without working out economic and mathematical models that describe the use of available resources during the operation of enterprises.

To determine strategies for enterprise development, calculation of forecasts of economic indicators and factors of organizations plays an important role. If there is reliable information about the company in the past, mathematical methods can be applied to obtain necessary forecasts. These methods depend on the objectives and detailed forecast factors; they also depend on the environment.

Various aspects of the theory, practice, and forecast of financial condition of a company have been the subject of research of many domestic and foreign scientists, such as Blank I.A [1], Heyets V.M. [2], Zaychenko Y.P. [3], Ivakhnenko V.M. [4], Ivakhnenko O.G. [5], Yarkina N.M. [6], Tymashova L. [7], Stepanenko O.P. [8], Tkachenko R.O. [9], Matviichuk A.V. [10], Hanke J.E. [11], Lewis C.D. [12], Box G.E. [13].

When forecasting the indicators by which the financial position or efficiency of the company's production resources use are determined, it is impossible to point out a single "the best" method of prediction because the internal laws (trends) of various indicator systems are different and there arises the problem of choosing the method of forecasting the studied indicator system.

Therefore, the development of new forecasting models of corresponding systems of indicators is an actual and important problem.

The aim of the study is to develop an efficient scheme of time series prediction that automatically (in the course of its training) adjusts to the appropriate system of economic, social, environmental, and engineering parameters, and it can be successfully used in the development of high-quality strategic plans in the branch of economy, environment, and for forecast of different natural processes.

The research methodology includes the method of least squares, exponential smoothing method, iterative techniques of minimization of functionals, and methods of synthesis of neural-network schemes. t can be written as follows [14-16] ) , ,..., , ,...,

Synthesis of Forecasting Schemes of Time Series( ~1 1 t v v a a f v k t t r t    ,(1)

where 1 ,..., r aa are the model parameters, k is the depth of prehistory. To find the parameters a 1 ,...,a r , we constructed the functional

  , ) ,..., ( 2 1 1     n t t t r v v a a L (2)

which is usually to be minimized. Let

       n v v a a f v k t t r n (3)

where  is the step of the forecast. Depending on the type of the function f with the parameters ** 1 ,..., r aa , we have different optimal forecasting models of time series. To build a predictive scheme, at the beginning let us consider the autoregression method by means of which we define the optimal step of the prehistory * k  for the given time series t v with the fixed step of the forecast  . In the autoregression model, it is assumed that the indicator value

) ( 2 ) ( 1               k n k n n n v a v a v a v (4)

To determine the optimal values of the parameters ) ,..., 2 , 1 (

) *(   k t a t  for a fixed  t = t 0 ( ) , we minimize the functional   , ... ) ,...., (2 1 )( ) ( 1 ) ( ) ( 1            n k t k t k t t k v a v a v a a L           (5)

i.e. we solve the system of equations . ,..., 2 , 1 , 0

) (   k t a L i     (6) Let ) *( ) *( 1 ,...,    k a a

be a solution of the system (6). Then, according to (4) we have , ...

~1 ) *( 1 ) *( 2 ) *( 1                   k t k t t t v a v a v a v (7)

where .

    k t It is obvious that the variable t v ~ for a fixed value of    0    depends on the parameter ) 1 (       n k k

. To determine the optimal value of the prehistory of time series with the fixed step of the forecast  , i.e. models on the bases of which a new forecasting scheme are synthesized. Using the results of the forecasting models mentioned above on the time interval

parameter  k for 0    for the given time series t v , let us consider the variables       2 1 ) ( * ) ( * 1 2 2 1 ) ( * 2 ) ( * 1 2 2 1 ) ( * 1 1 ... , 1 1 , 1 v a v a v v a v a v n v a v n n n n n n t t t t n t t t                                                        Thus we obtain   * τ k τ n 2 1 δ δ ,..., δ , δn k n k n t , , 2 , 1 * *        

, we draw the following table Table 1

. The Prognostic Values of Time Series

Forecasting

Models

Elements of Time Seriest v 1 *    k n v 2 *    k n v  n v 1 M ) 1 ( 1 * ~   k n v ) 1 ( 2 * ~   k n v  ) 1 ( ~n v 2 M ) 2 ( 1 * ~   k n v ) 2 ( 2 * ~   k n v  ) 2 ( ~n v      q M ) ( 1 * ~qk n v    ) ( 2 * ~qk n v     ) ( ~q n v In each column n k n k n v v v ,..., , 2 1 * *      

of Table 1, we can find the least squared difference of the prognostic and the actual values of the corresponding time series terms. Mathematically this can be written as following:

let 1 * 1     k n j and  , ) ( ,..., ) ( , ) ( min 2 ) ( 2 ) 2 ( 2 ) 1 ( 1 1 1 1 1 1 1 q j j j j j j v v v v v v      2 * 2     k n j and   2 ) ( 2 ) 2 ( 2 ) 1 ( 2 ) ( ,..., ) ( , ) ( min 2 2 2 2 2 2 q j j j j j j v v v v v v      , ……………………………………………………… n j k  *  and  . ) ( ,..., ) ( , ) ( min 2 ) ( 2 ) 2 ( 2 ) 1 ( * q n n n n n n k v v v v v v       Define the sets * ,..., , 2 1  k I I I as follows             2 ) ( 2 ) ( 2 2 2 ) ( 1 1 ) ( ,..., 2 , 1 , ) ( ,..., 2 , 1 , ) ( ,..., 2 , 1 * * 2 2 1 1 i n n k k i j j i j j v v q i I v v q i I v v q i I                                

and draw the table Table 2. Parameters for Determining the Weighting Coefficients of the Model Forecasting Models

1 j 2 j  *  k j Resultant Column 1 M 11 a 12 a  * 1  k a 1 S 2 M 1 2 a 22 a  * 2  k a 2 S       q M 1 q a 2 q a  *  qk a q S where ). ,..., 2 , 1 , ,..., 2 , 1 ( , 1 0 , ,, 0 , , * 1 *... ~) ( ) ( ) ( ) 2 ( ) ( ) ( ) 1 ( ) ( ) ( 2 1 q n S S n S S n S S n v v v v q                   (8)

The coefficients of the forecasting models in the scheme (8) depend on the parameter  that determines the influence of the element t v upon the prognostic value

  n v ~. The more remote element t v is from the prognostic point   n

v ~, the less is its influence on the prognostic value

). 1 0 (   

In the case of 1   , all points of time series t v are equivalent, i.e. in the model ( 8) the distance of the element t v from the prognostic point

  n v ~

is not taken into account. Synthesis of the predictive scheme (8) will be completed in the course of training its concerning  . For this purpose, we construct the functional ), ( , )

... ( ) ( * 2 1 ) ( ) ( ) ( ) 1 ( ) ( ) ( * 1 i k n j v v v L i k i q j S S j S S j i r q i i                 

and minimize it by varying the value  . The interval (0,1] we divide into m equal subintervals and find the value

) ( i L  at the points ) ,..., 2 , 1 ( m i m i i   

. It is obvious that m gives the accuracy of the finding the minimum of the functional ) (

L . Let ) ( min * i m L   

. Then the forecast of time series we conduct according to the scheme (8), substituting * m  for  .

Implementation of Forecasting Schemes of Time Series in Artificial Neural Basis

The basis of all forecasting methods is an idea of extrapolation of patterns of the development of the process, which was formed by the time when the forecast came true for future period of time.

Let

n t v v v v

,...., ,..., , 2 1 is time series. For the synthesis of artificial neural-network forecasting scheme, there must exist a method (methods) of synthesis of neural elements that implement appropriate forecasting models, on whose basis a neural scheme should be constructed. For example, the following artificial neural element with linear activation function implements the autoregression model ... After the development of methods for the synthesis of neural elements that implement the optimal forecasting models in the corresponding classes of models, to predict the values ) ,..., 2 , 1

~1 ) ( 2 ) ( 1      n n n v w v w v    1 ) ( * * ...       k n k v w , with the( n i v i  at instants of time    n t

, let us design the following neural-network scheme m . In Block 2. m each neuron has two inputs and a weight vector (1,1), where the value

m k n v  

is given to the first input, and the prognostic value

) ( , ~ i m k n v  

is given to the 2 nd input, which is the output signal of the і th neuron of Block 1.m. Activation function of the і th neuron of Block 2.

m is set as follows

) ) ( exp( 2 ) ( ,  i m k n m k n v v      

. The neuron of the serial number i of Block 2.

m is related to i th neuron of the 3 rd layer in the following way: from the i th neuron of Block 2.

m to the m th input of the i th neuron of the 3 rd layer there is given the signal

) ( ,  i m f , where            otherwise. ), ) ( max(exp( arg if , , 0 1 2 ) ( , ) ( ,   i m k n m k n i m v v i f

Neurons of the 3 rd layer have the linear activation function, and each of the weighting coefficients of each neuron is equal to 1. At the output of the i th neuron of the 3 rd layer for the fixed  we obtain the number ) ( i w . The 3 rd layer, except for neurons with linear activation function, has one more BlokPROG containing exactly as many neurons as a Block of the 1 st layer contains. Neurons of this block implement corresponding forecasting model with the depth  and their serial numbers coincide with the numbers of neurons of Blocks of Layer 1.

The 4 th layer contains two linear neurons. The first neuron has s inputs, all its weighting coefficients are equal to 1, and it has activation function

) ( ) ( 2 ) ( 1 ...    s w w w    .

The second neuron of this layer has weighting coefficients

) ( ) ( 2 ) ( 1 ,..., ,    s w w w . If

the forecast result of the i th model of BlockPROG is denoted by , then at the output of the second neuron of Layer 4 we have

) ( ) ( ) 1 ( ) ( 1 ... ~s n s n v w v w        

. The 5 th layer contains one neuron that has two inputs, a weight vector (1.1), and the activation function .

... ... ~) ( ) (2) ( 1 ) ( ) ( ) 1 ( ) ( 1         s s n s n n w w w v w v w v          Blocks 2. m ( * ,...., 2 , 1  k m 

) determine the most effective basic forecasting models. At the output of the scheme we have a convex linear combination of the best forecasting models.

Effectiveness of the Constructed Forecasting Scheme

Following types of errors are often used in the implementation of forecasting time series forecasting

МАЕ -Mean Absolute Error     n t t t v v n MAE 1 1 (9)

where  t v is the values of the time series at time t;

 t v ~ predictable value t v .

The average absolute error of prediction ( 9) is an absolute measure of the quality of forecast, estimating it independently of the other predictions. It's enough to set a level of absolute error and compare the value of the specified error calculated by the formula (9).

To compare the quality of forecasting, it is often used the average relative error (MRE -Mean Relative Error) is often used

    n t t t t v v v n MRE 1 1 ,(10)

and the average square error (RMSE -Root Mean Square Error) is also used

  , 1 2 n v v RMRE n t t t     (11)

where t v are the terms of the time series, t v ~ are the prognostic values of t v . RMSE and MRE are relative errors, i.e. they can be used to compare two (or more) different time series prediction the best is the forecast whose value of MRE (10) or RMSE (11) is less.

According to the average relative error criterion, the quality of the forecast of the constructed predicting scheme is estimated by comparing its results with the results of main forecasting models on base of which it is synthesized. To perform this, we use data from the following Table 3 [17]. Having analyzed the data in Table 4, we see that the least average relative error occurs in the constructed forecasting scheme. In the two cases (for 1   ), the error of the scheme coincides with the error of autoregression method. Thus, in general, the scheme developed in this work is the most effective among the methods on which it is based. To obtain the average error (%) of the prediction methods for the given time series in percentage, one should multiply by 100% the corresponding values of quality from Table 4 Note. The constructed forecasting scheme is flexible. This means that a new model can be added to or excluded from basic models (on basis of which the predictive scheme is constructed) at any time. It should be noted that the method of synthesis of the very predictive scheme does not change.

Here are some results of the program implementation of developed forecasting scheme for determining the share of road passenger transport in Ukraine to all other types of transportation during time span since 1980 to 2013. Table 3 contains primary data of passenger traffic volume (period 1980-2013) and projections of passenger traffic (forecast period 2014-2018). On the base of this table it is evident that the average share of road passenger transport in Ukraine was 51.85% over the above mentioned period. Accordingly to the forecast this share will average 45.56% during the prediction period 2014-2018. Thus, the role of road passenger transport in Ukraine over the observable forecast period 2014-2018 is leading. Annual share of road passenger transport in Ukraine during the prediction period is shown on the following diagram:

Conclusions

A flexible scheme for forecasting of economic, social, environmental, engineering and technological indicators that can be successfully used in the development of reasonable strategic plans and decisions in the corresponding fields of human activity is worked out.

This forecasting scheme allows us to include new forecasting models of time series or to exclude a model or groups of models from it at any instant of time.

As for the models which remain in the scheme, the competition between them is made over a given period of time, and the final forecasting scheme represents a convex linear combination of models -winners with corresponding weighting coefficients.