=Paper=
{{Paper
|id=None
|storemode=property
|title=Synthesis of Time Series Forecasting Scheme Based on Forecasting
Models System
|pdfUrl=https://ceur-ws.org/Vol-1356/paper_41.pdf
|volume=Vol-1356
|dblpUrl=https://dblp.org/rec/conf/icteri/GecheKBGV15
}}
==Synthesis of Time Series Forecasting Scheme Based on Forecasting
Models System==
https://ceur-ws.org/Vol-1356/paper_41.pdf
Synthesis of Time Series Forecasting Scheme Based on
Forecasting Models System
Fedir Geche1, Vladyslav Kotsovsky2, Anatoliy Batyuk3, Sandra Geche4, and
Mykhaylo Vashkeba1
1 Uzhhorod National University, Department of Cybernetics and Applied Mathematics,
Uzhhorod, Ukraine
(fgeche@hotmail.com, vashkebam1991@gmail.com)
2 Uzhhorod National University, Department of Information Management Systems,
Uzhhorod, Ukraine
kotsavlad@gmail.com
3 Lviv Polytechnic National University, Department of Automatic Control Systems, Lviv,
Ukraine
abatyuk@gmail.com
4 Uzhhorod National University, Department of Economic Theory,
Uzhhorod, Ukraine
sandra.geche@gmail.com
Abstract. This article is dedicated to the development of time series forecasting
scheme. It is created based on the forecasting models system that determines
the trend of time series and its internal rules. The developed scheme is
synthesized with the help of basic forecasting models "competition" on a certain
time interval. As a result of this "competition", for each basic predictive model
there is determined the corresponding weighting coefficient, with which it is
included in the forecasting scheme. Created forecasting scheme allows simple
implementation in neural basis. The developed flexible scheme of forecasting
of economic, social, environmental, engineering and technological parameters
can be successfully used in the development of substantiated strategic plans and
decisions in the corresponding areas of human activity.
Keywords. Trend, forecasting model, time series, functional, step of forecast,
autoregression, neural element, neural network.
Key Terms. MachineIntelligence, DecisionSupport, MathematicalModel
1 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.
2 Synthesis of Forecasting Schemes of Time Series
Let v1 , v2 ,..., vt ,..., vn be a time series. Prognostic value v~t of the element vt at
the instant of time t can be written as follows [14-16]
v~t f (a1 ,..., ar , vt 1 ,..., vt k , t ) , (1)
where a1 ,..., ar are the model parameters, k is the depth of prehistory. To find the
parameters a1,...,ar , we constructed the functional
2
(2)
L(a1,..., ar ) vt v~t ,
n
t 1
which is usually to be minimized. Let a1* ,..., ar* are the values of parameters a1 ,..., ar
for which the functional L takes its minimum value. Then the prognostic value v~ n
of the model f with optimal parameters a1* ,..., ar* is determined as follows
v~n f (a1* ,..., ar* , vt 1 ,..., vt k , n ), (3)
where is the step of the forecast. Depending on the type of the function f with the
parameters a1* ,..., ar* , 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 vt with the fixed step of the forecast . In the autoregression
model, it is assumed that the indicator value vt at the instant of time t depends on
vt , vt 1 ,..., vt k 1 , where k is the parameter of the prehistory with fixed . The
prognostic value v~n by the autoregression method is found according to the
following model
v~n a1( )vn a2( )vn1 ... ak( )vnk 1. (4)
To determine the optimal values of the parameters at*( ) (t 1,2,...,k ) for a fixed
( t = t 0 ) , we minimize the functional
n 2
L(a1( ) ,...., ak( ) ) vt a1( )vt ... ak( )vt k 1 , (5)
t k
i.e. we solve the system of equations
L
0, t 1,2,..., k . (6)
ai( )
Let a1*( ) ,..., ak*( ) be a solution of the system (6). Then, according to (4) we have
v~t a1*( )vt a2*( )vt 1 ... ak*( )vt k 1, (7)
where t k .
It is obvious that the variable v~ t for a fixed value of 0 depends on the
parameter k (1 k n ) . To determine the optimal value of the prehistory
parameter k for 0 for the given time series v t , let us consider the variables
2
1 n
1 vt a1*( )vt ,
n t 1
vt a1*( )vt a2*( )vt 1 ,
2
1 n
2
n 1 t 2
n vn a1*( )vn ... an*()v1
2
Thus we obtain min δ1 , δ 2 ,..., δ n τ δ * . The variable k* determines the
kτ
optimal value of the prehistory parameter in the autoregression model for a fixed
0 .
After determining the k* for a fixed t 0 , consider the main base
forecasting models M1, M 2 ,...M q 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
t n k* 1, n k* 2,, n , we draw the following table
Table 1. The Prognostic Values of Time Series
Forecasting Elements of Time Series vt
Models
vn k * 1
vn k * 2
vn
M1 v~ (1) *
n k 1
v~ (1) *
n k 2
v~n(1)
M2 v~ ( 2) *
n k 1
v~ ( 2) *
n k 2
v~n( 2)
Mq v~ ( q ) *
n k 1
v~ ( q ) *
n k 2
v~n( q )
In each column vn k * 1, vn k * 2 ,...,vn 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 j1 n k* 1 and
1 min (v j1 v~j(11) ) 2 , (v j1 v~j(12) ) 2 ,...,(v j1 v~j(1q) ) 2 ,
j2 n k* 2 and
2 min (v j2 v~j(21) ) 2 , (v j2 v~j(22) ) 2 ,...,(v j2 v~j(2q) ) 2 ,
………………………………………………………
jk * n and
k * min (vn v~n(1) ) 2 , (vn v~n(2) ) 2 ,...,(vn v~n(q) ) 2 .
Define the sets I1 , I 2 ,...,I k * as follows
I1 i 1,2,...,q1 (v j1 v (ji ) ) 2 ,
1
I 2 i 1,2,...,q 2 (v j2 v (ji ) ) 2 ,
2
I k * i 1,2,...,q k * (vn vn(i ) ) 2
and draw the table
Table 2. Parameters for Determining the Weighting Coefficients of the Model
Forecasting j1 j2 jk *
Resultant
Models Column
M1 a11 a12 a1k *
S1
M2 a21 a22 a2 k *
S2
Mq aq1 aq 2 aqk*
Sq
where
k* s , if s I ,
a ps s
0, if s I s ,
k*
S p a pj ,0 1,( p 1,2,...,q, s 1,2,...,k* ).
j 1
q
With the help of S p S p ( ) and S ( ) S p ( ) we determine the weighting
p 1
coefficients of the forecasting models M p ( p q) , with which these models are
included in the following forecasting scheme
S ( ) S ( ) S ( )
v~n S1( ) v~n(1) S2( ) v~n(2) ... Sq( ) v~n(q) . (8)
The coefficients of the forecasting models in the scheme (8) depend on the
parameter that determines the influence of the element vt upon the prognostic
value v~ . The more remote element v is from the prognostic point v~ , the less is
n t n
its influence on the prognostic value (0 1). In the case of 1 , all points of
time series vt are equivalent, i.e. in the model (8) the distance of the element vt from
the prognostic point v~ is not taken into account.
n
Synthesis of the predictive scheme (8) will be completed in the course of training its
concerning . For this purpose, we construct the functional
k*
S ( ) S qr ( ) ~ ( q) 2
L( ) (v ji S1( ) v~j(1) ... S ( )
v j ) , ( ji n k* i),
i i
i 1
and minimize it by varying the value . The interval (0,1] we divide into m equal
i
subintervals and find the value L( i ) at the points i
(i 1,2,...,m) . It is obvious
m
that m gives the accuracy of the finding the minimum of the functional L( ) . Let
m* min L(i ) . Then the forecast of time series we conduct according to the scheme
(8), substituting m* for .
3 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 v1, v2 ,...,vt ,....,vn 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
v~n w1( )vn w2( )vn 1 ... ...wk(* )vn k * 1 , with the
Fig. 1. Neuron of the Optimal Autoregressive Model
optimal step k* of the prehistory and the step of the forecast if
w1( ) a1*( ) ,...,wk * ( ) ak** a1*( ) ,...,ak*(* ) are optimal values of parameters of the
autoregressive 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 vi (i 1,2,...,n) at instants of time t n , let us design the
following neural- network scheme
Fig. 2. Neuro-scheme for Time Series Prediction
All the blocks of the 1st layer contain the same number s of neurons, where each
neuron implements one of the forecasting models (autoregressive model, polynomial,
exponential, linear ones, Brown’s linear model, etc.). Neurons that implement the
same model in different blocks of this layer have the same serial number.
Each Block 2. m ( m 1,2,....,k ; k k* ) of the 2nd layer contains as much neurons
as in Block 1. m . In Block 2. m each neuron has two inputs and a weight vector (1,1),
where the value vn k m is given to the first input, and the prognostic value
v~n()k m,i is given to the 2nd input, which is the output signal of the іth neuron of
Block 1.m. Activation function of the іth neuron of Block 2. m is set as follows
~ ( )
exp( (vn k m v 2
. The neuron of the serial number i of Block 2. m is
n k m,i ) )
related to i th neuron of the 3rd 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 3rd layer there is given the signal
f m(,i) , where
~ ( )
1 , if i arg max(exp( (vn k m v 2
n k m , i ) ),
f m(, i)
0, otherwise.
Neurons of the 3rd layer have the linear activation function, and each of the
weighting coefficients of each neuron is equal to 1. At the output of the ith neuron of
( )
the 3rd layer for the fixed we obtain the number wi . The 3rd layer, except for
neurons with linear activation function, has one more BlokPROG containing exactly
as many neurons as a Block of the 1st 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 4th layer contains two linear neurons. The first neuron has s inputs, all its
weighting coefficients are equal to 1, and it has activation function
w1( ) w2( ) ... ws ( ) .
The second neuron of this layer has weighting coefficients w1( ) , w2( ) ,...,ws ( ) . If
the forecast result of the ith model of BlockPROG is denoted by , then at the
output of the second neuron of Layer 4 we have w1( ) v~n(1) ... ws ( ) v~n(s ) .
The 5th layer contains one neuron that has two inputs, a weight vector (1.1), and the
w ( ) v~ (1) ... ws ( ) v~n(s )
activation function v~n 1 ( ) n ( ) .
w1 w2 ... ws ( )
Blocks 2. m ( m 1,2,....,k* ) determine the most effective basic forecasting
models. At the output of the scheme we have a convex linear combination of the best
forecasting models.
4 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
1 n
MAE vt v~t (9)
n t 1
where vt is the values of the time series at time t;
v~t predictable value vt .
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
1 n vt v~t (10)
MRE ,
n t 1 vt
and the average square error (RMSE - Root Mean Square Error) is also used
vt v~t
n
2
RMRE t 1
, (11)
n
where vt are the terms of the time series, v~t are the prognostic values of vt . 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].
Table 3. The Original and Forecasted Volumes of Passenger Traffic
Under-
Automobile
Year Railway Sea River Aircraft ground
(coaches)1
railway
1980 648869 28478.4 24789 7801058 12492.4 430040
1981 653177 30705.6 27531.6 7794859 12720 473437
1982 656485 29362.2 26629.4 7874069 12728.7 515382
1983 668287 29690.2 26810.8 7876161 12711.6 520700
1984 687645 29228.8 24979.6 7998739 12777.8 551851
1985 695129 28660.6 23817.4 8076846 12616 602671
1986 734204 28681 21008.5 8230409 12797.5 598022
1987 717461 27567.3 18750.2 8383820 12670.4 590513
1988 711123 27961.5 20345.5 8552803 13065.3 634616
1989 704078 26524.3 20199.7 8382872 14299.6 648816
1990 668979 26256.7 19090.3 8330512 14833 678197
1991 537407 20786.5 18285.8 7450322 13959.6 595313
1992 555356 13139.5 11158 6464891 5669.3 610668
1993 501495 10497 8064.4 4795664 1947.4 644417
1994 630959 10358.2 6967.9 4039917 1673.3 684480
1995 577432 7817 3594.1 3483173 1914.9 561012
1996 538569 5044.6 2735.9 3304600 1724 536304
1997 500839 4311.3 2443.1 2512147 1484.5 507897
1998 501429 3838.3 2356.5 2403425 1163.9 668456
1999 486810 3084.3 2269.4 2501708 1087 724426
2000 498683 3760.5 2163.3 2557515 1164 753540
2001 467825 5270.8 2034.2 2722002 1289.9 793197
2002 464810 5417.9 2211.9 3069136 1767.5 831040
2003 476742 6929.4 2194.1 3297505 2374.7 872813
2004 452226 9678.4 2140.2 3720326 3228.5 848176
2005 445553 11341.2 2247.6 3836515 3813.1 886598
2006 448422 10901.3 2021.9 3987982 4350.9 917700
2007 447094 7690.8 1851.6 4173034 4928.6 931512
2008 445466 7361.4 1551.8 4369126 6181 958694
2009 425975 6222.5 1511.6 4014035 5131.2 751988
2010 427241 6645.6 985.2 3726289 6106.5 760551
2011 429785 7064.1 962.8 3611830 7504.8 778253
2012 429115 5921 722.7 3450173 8106.3 774058
2013 425217 6642 631.1 3343660 8107.2 774794
2014 424272.5 3490.2 453.8915 3059461.2 9308.8 816682.9
2015 414375.8 5373.2 406.4361 2645239.9 7243.7 876984.7
2016 425925.3 3847.1 369.0345 2641221.6 10609.4 972098.3
2017 420469.8 2975.1 233.0464 2395820.5 10870.7 1073108.1
2018 426849.1 3061.2 403.4616 2606148.8 12330.5 1205853.8
Table 4. Forecast Errors of Passenger Traffic according to MRE criterion
Forecasting methods Kinds of passenger traffic
Railway River Automobile
Step of the forecast
Autoregression method 0.0148 0.0115
0.0041
The method of least 0.7975 0.1680
squares with weights 0.015
Brown’s linear model 0.0358 0.0917 0.1478
Brown’s quadratic
model 0.0159 0.5516 0.086
Forecasting scheme 0.0039 0.0148 0.0115
Step of the forecast
Autoregression method
0.0045 0.0111 0.0233
The method of least
squares with weights 0.0048 0.0683 0.0595
Brown’s linear model 0.0585 0.0757 0.1482
Brown’s quadratic
model 0.0317 0.2295 0.0797
Forecasting scheme 0.0031 0.0108 0.0225
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. The quality of the prediction methods of passenger traffic for the
forecast period (2014-2018) with the steps of the forecast 1 and 5 is shown in
the following charts
Fig. 3. Forecasting errors of prediction methods with the step 1 (in %)
Fig. 4. Forecasting errors of prediction methods with the step 5 (in %)
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:
Fig.5. The share of road passenger transport in Ukraine over the period (2014-2018)
To compare the dynamics of changes of the volume of passenger traffic in Ukraine
for different types of vehicles (rail, river, road) we construct the following diagram.
Fig.6. Dynamics of passenger traffic in Ukraine (2014-2018)
5 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.
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