Recent Developments of Self-Organising Modeling in Prediction and Analysis of Stock Market 

Ivakhnenko, A.G. 

Glushkov Institute of Cybernetics, Ukraina, Kyiv 34, PO Box 298-9, 
e-mail: Gai@gmdh.kiev.ua http://come.to/GMDH  
Muller, J.-A. 

Fachbereich Informatik/Mathematik, Hochschule fur Technik und Wirtschaft 

D 01069 Dresden, F.-List-Platz 1, Germany, e-mail: Muellerj@informatik.htw-dresden.de  



Review 

Abstract: At present, GMDH algorithms give us the only way to get accurate 
identification and forecasts of different complex processes in the case of 
noised and short input sampling. In distinction to neural networks, the results 
are explicit mathematical models, obtained in a relative short time. For ill-defined 
objects with very big noises better results should be obtained by analogues 
complexing methods. Neural nets with active neurones should be applied to rise 
up accuracy of complex objects modelling algorithms. 

1. Introduction 

Problems of complex objects modelling (functions approximation and extrapolation, 
identification, pattern recognition, forecasting of random processes and events) 
can be solved in general by deductive logical-mathematical or by inductive 
sorting-out methods. Deductive methods have advantages in the cases of rather 
simple modelling problems, when the theory of the object being modelled is known 
and therefore it is possible to develop a model from physically based principles 
employing the users knowledge of the process. 

Decision making in such areas as process analysis in macroeconomy, financial 
forecasting, company solvency analysis and another requires tools, which are 
able to get accurate models on basis of processes forecasting. However, arise 
problems that connected with large amount of variables, very small number of 
observations and unknown dynamical between these variables. Such financial 
objects are complex ill-defined systems that can be characterised by: 

· inadequate a priori information; 

· great number of immeasurable variables; 

· noisy and extremely short data samples; 

· ill-defined objects with fuzzy characteristics. 

Problems of complex objects modelling such as analysis and prediction of stock 
market and other, cannot be solved by deductive logical-mathematical methods 
with needed accuracy. In this case knowledge extraction from data, i.e. to 
derive a model from experimental measurements, has advantages in cases of rather 
complex objects, being only little a priori knowledge or no definite theory 
particularly for objects with fuzzy characteristics on hand. This is especially 
true for objects with fuzzy characteristics. 

The task of knowledge extraction from data is to select mathematical description 
from data. But the required knowledge for designing of mathematical models or 
architecture of neural networks is not at the command of the users. In 
mathematical statistics it is need to have a priori information about the 
structure of the mathematical model. In neural networks the user estimates this 
structure by choosing the number of layers and the number and transfer functions 
of nodes of a neural network. This requires not only knowledge about the theory 
of neural networks, but also knowledge of the object nature and time. Besides 
this the knowledge from systems theory about the systems modelled is not 
applicable without transformation in neural network world. But the rules of 
translation are usually unknown. 

GMDH type neural networks can overcome these problems - it can pick out 
knowledge about object directly from data sampling. The Group Method of Data 
Handling (GMDH) is the inductive sorting-out method, which has advantages in the 
cases of rather complex objects, having no definite theory, particularly for the 
objects with fuzzy characteristics. GMDH algorithms found the only optimal model 
using full sorting-out of model-candidates and operation of evaluation of them, 
by external criteria of accuracy or difference types [1,2]. 

2. Group Method of Data Handling (GMDH) 

2.1. Brief description 

The Group Method of Data Handling (GMDH) is self-organizing approach based on 
sorting-out of gradually complicated models and evaluation of them by external 
criterion on separate part of data sample. As input variables can be used any 
parameters, which can influence on the process. Computer is found structure of 
model and measures of selected parameters significance itself. That model is 
better that leads to the minimal value of external criterion. This inductive 
approach is different from commonly used deductive techniques or neural networks. 

The GMDH was developed for complex systems modelling, prediction, identification 
and approximation of multivariate processes, decision support after "what-if" 
scenario, diagnostics, pattern recognition and clusterization of data sample. It 
was proved, that for inaccurate, noisy or small data can be found best optimal 
simplified model, accuracy of which is higher and structure is simpler than 
structure of usual full physical model. 

There were defended more than 230 dissertations and published many papers and 
books devoted to GMDH theory and its applications. There are developed methods 
of mathematical induction for the solution of comparatively simple problems. 
GMDH can be considered as further propagation of inductive self-organising 
methods to the solution of more complex practical problems. It solves the 
problem of how to handle data samples of observations. The goal is to get 
mathematical model of the object (the problem of identification and pattern 
recognition) or to describe the processes, which will take place at object in 
the future (the problem of process forecasting). 

GMDH solves, by sorting-out procedure, the multidimensional problem of model 
optimization: 

, (1) 

where: G - set of considered models; CR is an external criterion of model g 
quality from this set; P - number of variables set; S - model complexity; x2 - 
noise dispersion; T - number of data sample transformation; V - type number of 
reference function. For definite reference function, each set of variables 
corresponds to definite model structure P = S. Problem transforms to much 
simpler one-dimensional 

, 

when x2= const, T = const, and V = const. 

Method is based on the sorting-out procedure, i.e. consequent testing of models, 
chosen from set of models-candidates in accordance with the given criterion. 
Most of GMDH algorithms use the polynomial reference functions. General 
connection between input and output variables can be expressed by Volterra 
functional series, discrete analogue of which is Kolmogorov-Gabor polynomial [1]: 

, 

where - input variables vector; 

- vector of coefficients or weights. 

Components of the input vector X can be independent variables, functional forms 
or finite difference terms. Other non-linear reference functions, such as 
difference, logistic, harmonic can also be used for model construction. The 
method allows to find simultaneously the structure of model and the dependence 
of modelled system output on the values of most significant inputs of the system. 

The GMDH theory solve the problems of: 

- long-term forecasting [3,18]; 

- short-term forecasting of processes and events [2]; 

- identification of physical regularities; 

- approximation of multivariate processes; 

- physical fields extrapolation [4]; 

- data samplings clusterization [5]; 

- pattern recognition in the case of continuous-valued or discrete variables; 

- diagnostics and recognition by probabilistic sorting-out algorithms [6]; 

- vector process normative forecasting [7]; 

- modeless processes forecasting using analogues complexing [8]; 

- self-organization of twice-multilayered neuronet with active neurones [9,10]. 

In [12] were obtained the theoretical grounds of GMDH effectiveness as adequate 
method of robust forecasting models construction. Essence of it consists of 
automatically generation of models in given class by sequential selection of the 
best of them by criteria, which implicitly by sample dividing take into account 
the level of indeterminacy. 

Since 1967 a big number of GMDH technique implementations for modelling of 
economic, ecological, environmental, medical, physical and military objects were 
done in several countries. Some outdated approaches are used in USA by Ward 
Systems Group, Inc. in "NeuroShell2", AbTech Corp. "ModelQuest", Barron 
Associates Co. "ASPN", and DeltaDesign Berlin Software "KnowledgeMiner" 
commercial software tools. 

Self-organising modelling is based on statistical learning networks, which are 
networks of mathematical functions that capture complex, non-linear 
relationships in a compact and rapidly executable form. Such networks subdivide 
a problem into manageable pieces or nodes and then automatically apply advanced 
regression techniques to solve each of these simpler problems. 

2.2. The "GMDH algorithms" and "algorithms of GMDH type" 

It's necessary to make difference between the original "GMDH algorithms" and the 
"algorithms of GMDH type" [11]. The first ones - work using the minimum of an 
external criterion (Fig.1) and therefore realise objective choice of optimal 
model. This original GMDH technique is based on inductive approach: optimal 
models are founded by sorting-out of possible variants and evaluated by external 
criterion. It is calculated on separate part of data sample, which is not used 
for model creation. That model is better which leads to minimal value of 
criterion. To make objective choice, selection is done without thresholds or 
coefficients in criterion. We recommend to calculate criteria two times: first 
to find the best models at each layer of selection for structure identification 
and second time to find the optimal model. Selection procedure is stopped when 
minimal criterion value is reached. 

Second is GMDH type algorithms - work on characteristic, expressed by words: "more 
complex is the model - more accurate it is". For it necessary to put definite 
threshold or to point out coefficients of weight for the members of the internal 
criterion formula, to find optimal model out in a some subjective way. But real 
problems usually are presented by short or noised data samples. Unfortunately, 
in almost all GMDH type software (ModelQuest, NeuroShell) and research works in 
USA and Japan this deductive approach is used, which is not effective for such 
kind of data. 

The inductive approach does not eliminate the experts or take them away from the 
computer, but rather assigns them a special position. Experts indicate the 
selection criterion of a very general form and interpret the chosen models. They 
can influence the result of modelling by formulating new criteria. Computer 
becomes an objective referee for scientific controversies, if criteria ensemble 
is coordinated between experts, which take part in discussion. 

Fig.1. External accuracy criterion minima values plotted against complexity of 
model structure S for different noise variance x2. 

LCM - locus of criterion minima line; 

--- - model choice by criterion minimum. 

The human element often involves errors and undesired decisions. Objective 
choice of optimal model by minimum of external criterion characteristic in 
actual GMDH algorithms often contradicts with the opinion of investigator. 
Objective algorithms give possibility to realise real artificial intelligence. 

2.3. Special GMDH peculiarities 

The main peculiarity of GMDH algorithms is that, when it uses continuous data 
with noise, it selects as optimal the simplified non-physical model. Only for 
accurate and discrete data the algorithms point out so-called physical model - 
the most simple optimal, from all unbiased models. 

It is proved the convergence of multilayered GMDH algorithms [25] and it is 
proved that shortened non-physical model is better than full physical model (for 
noisy and continuous data for prediction and approximation solving, more 
simplified Shannon’s non-physical models become more accurate [12]). It can be 
noted, that this conclusion has place in model selection on the basis of model 
entropy maximisation (Akaike approach), in average risk minimising (Vapnik 
approach) and in another modern approaches. The only way to get non-physical 
models is to use sorting-out GMDH algorithms. Regularity of optimal structure of 
forecasting models change in dependence on general indexes of data indeterminacy 
(noise level, data sample length, design of experiment, number of informational 
variables) was shown in [24,25,27]. 

The special peculiarities of GMDH are following: 

1) External supplement: Following S.Beer work [13], only the external criteria, 
calculated on new independent information, can produce the minimum of sorting-out 
characteristic. Because of this data sampling is divided into parts for model 
construction and evaluation. 

2) Freedom of choice: Following D.Gabor work [14], in multilayered GMDH 
algorithms are to be conveyed from one layer to the next layer not one but F 
best results of choice to provide "freedom of choice"; 

3) The rule of layers complication: Partial descriptions (forms of a 
mathematical description for iteration) should be simple, without quadratic 
members in them; 

4) Additional model definition: In cases, when the choice of optimal physical 
model is difficult, because of noise level or oscillations of criterion minima 
characteristic, auxiliary discriminating criterion is used [15]. The choice of 
the main criterion and constrains of sorting-out procedure is the main heuristic 
of GMDH; 

5) All algorithms have multilayered structure and parallel computing can be 
implemented for their realisation; 

6) All questions that arise about type of algorithm, criterion, variables set 
etc. should be solved by minimal criterion value. 

The main criteria used are: cross-validation PRR(s), regularity AR(s) and 
balance of variables criterion BL(s). Estimation of their effectiveness (investigation 
of noise immunity, optimality and adequateness) and their comparison with 
another criteria was done in detail in [24,25,26,15]. The conditions, under 
which GMDH algorithm produces the minimum of characteristics are following: 

a) criterion of model choice is to be external, based on additional fresh 
information, which was not used for model construction; 

b) the data sample is not to be too long. Such data sample produce the same form 
of characteristic as the exact data sample without noises; 

c) when difference type balance criterion BL(s) is used, small noise is 
necessary or the variables in the data sample should not be exactly measured [16]. 

Difference of the GMDH algorithms from another algorithms of structural 
identification, genetic and best regression selection algorithms consists of 
three main peculiarities: 

? usage of external criteria, which are based on data sample dividing and are 
adequate to problem of forecasting models construction, by decreasing of 
requirements to volume of initial information; 

? much more diversity of structure generators: usage like in regression 
algorithms of the ways of full or reduced sorting of structure variants and of 
original multilayered (iteration) procedures; 

? better level of automatization: there are needed to enter initial data sample 
and type of external criterion only; 

? automatic adaptation of optimal model complexity and external criteria to 
level of noises or statistical violations – effect of noiseimmunity cause 
robustness of the approach; 

? implementation of principle of inconclusive decisions in process of gradual 
models complication. 

2.4. Spectrum of GMDH algorithms 

Solution of practical problems and GMDH theory design lead to development of 
broad spectrum of software algorithms. Each of them corresponds to some definite 
conditions of it application [17]. Algorithms mainly differ one from another by 
the models-candidates set generator arrangement for given basic function, by the 
way of models structure complexing and, at last, by the external criteria 
accepted. Algorithm choice depends on specifics of the problem, noise dispersion 
level, sufficiency of data sample, and if data sample is continuous-valued only. 

Table 1. Spectrum of GMDH algorithms 

GMDH algorithms  
Variables  
Parametric  
Non-parametric  
- Combinatorial (COMBI)  
- Objective Computer  
- Multilayered Iterational (MIA)  
Clusterization (OCC);  
Continuous  
- Objective System Analysis (OSA)  
- "Pointing Finger" (PF)  
- Harmonical  
clusterization algorithm;  
- Two-level (ARIMAD)  
- Analogues Complexing (AC)  
- Multiplicative-Additive  
Discrete and binary  
- Harmonical Rediscretization  
- Algorithm on the base of Multilayered Theory of Statistical Decisions (MTSD)  

Most often criteria of accuracy, differential or informative type are used. The 
work of GMDH algorithms has a straightforward analogy with the work of gardener 
during selection of a new hybrid plant [11]. 

The basic parametric GMDH algorithms listed in table 1 have been developed for 
continuous variables. Among the parametric algorithms [1,9] the most known are: 

- the basic is Combinatorial (COMBI) algorithm. It is based on full or reduced 
sorting-out of gradually complicated models and evaluation of them by external 
criterion on separate part of data sample; 

- Multilayered Iteration (MIA) algorithm use at each layer of sorting procedure 
the same partial description (iteration rule). It should be used when it is 
needed to handle a big number of variables; 

- Objective System Analysis (OSA) algorithm. The key feature of it is that it 
examines not single equations, but systems of algebraic or difference equations, 
obtained by implicit templates (without goal function). An advantage of the 
algorithm is that the information embedded in the data sample is utilised better 
and we get relationships between variables; 

- Two-level (ARIMAD) algorithm for modelling of long-term cyclic processes (such 
as stock or weather). There are used systems of polynomial or difference 
equations for identification of models on two time scales and then choice of the 
best pair of models by external criterion value. For this can be used any 
parametric algorithm from described above [23]. 

Also less known parametric algorithms, which apply an exhaustive search to 
difference, harmonic or harmonic-exponential functions, and the Multiplicative-Additive 
algorithm, in which tested polynomial models are obtained by taking the 
logarithm of the product of input variables [18,19]. The parametric GMDH 
algorithms have proved to be highly efficient in cases where one is to model 
objects with non-fuzzy characteristics, such as engineering objects. In cases, 
where modelling involves objects with fuzzy characteristics, it is more 
efficient to use the non-parametric GMDH algorithms, in which polynomial models 
are replaced by a data sample divided into intervals or clusters. Such type 
algorithms completely solve the problem of coefficients estimates bias 
elimination. 

Non-parametric algorithms are exemplified by: 

- Objective Computer Clusterization (OCC) algorithm that operates with pairs of 
closely spaced sample points [5]. It finds physical clusterization that would as 
possible be the same on two subsamples; 

- "Pointing Finger" (PF) algorithm for the search of physical clusterization. It 
is implemented by construction of two hierarchical clustering trees and 
estimation by the balance criterion [20]; 

- Analogues Complexing (AC) algorithm, which use the set of analogues instead of 
models and clusterizations [8]. It is recommended for the most fuzzy objects; 

- algorithm, based on the Multilayered Theory of Statistical Decisions [6]. It 
is recommended for recognition of binary objects and for the variability of 
input data control to avoid the possible experts’ errors in it. 

Recent developments of the GMDH have led to neuronets with active neurons, which 
realise twice-multilayered structure: neurons are multilayered and they are 
connected into multilayered structure. This gives possibility to optimise the 
set of input variables at each layer, while the accuracy increases. The accuracy 
of forecasting, approximation or pattern recognition can be increased beyond the 
limits, which are reached by neuronet with single neurons, or by usual 
statistical methods [9,10,34]. In this approach, which corresponds to the 
actions of human nervous system, the connections between several neurons are not 
fixed but change depending on the neurons themselves. Such active neurons are 
able during the learning self-organising process to estimate which inputs are 
necessary to minimise the given objective function of neuron. This is possible 
on the condition that every neuron in its turn is multilayered unit, such as 
modelling GMDH algorithm. Neuronet with active neurons, which are described 
below, is considered as a tool to increase AI problems accuracy and lead-time 
with the help of regression area extension for inaccurate, noisy data or small 
data samples. 

The GMDH algorithms recently are applied in optimization to solve the problems 
of normative forecasting (after "what-if-then" scenario) and optimal control of 
multivariable ill-defined objects. Many ill-defined objects in macroeconomy, 
ecology, manufacturing etc. can be described accurately enough by static 
algebraic or by difference equations, which can be transformed into problems of 
linear programming by nomination of non-linear members by additional variables. 
GMDH algorithms are applied to evaluate deflections of output variables from 
their reference optimal values [7,21]. Examples of use of Simplified Linear 
Programming (SLP) algorithm should be used for expert computer advisor 
construction, normative forecasting and control optimization of averaged 
variables. An important example [10] gives the prediction of effects of 
experiments. The algorithm solves two problems: calculation of effects of a 
given experiment and calculation of parameters which are necessary to reach 
optimal results. It means, that the realisation of experiments can often be 
replaced by computer experiments. 

As already noted, considered GMDH algorithms have been developed for continuous 
variables. In practice, however the sample will often include variables 
discretized into a small number of levels or even binary values. To extend these 
GMDH algorithms to discretized or binary variables, the Harmonic 
Rediscretization algorithm has been developed [22]. 

The existence of a broad gamut of GMDH algorithms is traceable to the fact, that 
it is impossible to define the characteristics of the rest or controlled objects 
exactly in advance. Therefore, it can be good practice to try several GMDH 
algorithms one after another and to decide which one suits a given type of 
objects best. All the questions, which arise during modelling process, are to be 
solved by the comparison of the criterion values: that variant is better, which 
leads to more deeper minimum of basic external criteria. In this way, the type 
of algorithm is chosen objectively, according to the value of the discriminating 
criterion. 

Information about dispersion of noise level is very useful to decrease computer 
calculation time. For small dispersion level we can use the learning networks of 
GMDH type, based on the ordinary regression analysis using internal criteria. 
For considerable noise level the GMDH algorithms with external criteria are 
recommended. And for high level of noise dispersion non-parametric algorithms of 
clusterization or analogues complexing should be applied [8]. 

2.4.1. The Combinatorial GMDH algorithm (COMBI) 

The flowchart of the algorithm is shown in Fig. 2. The input data sample is a 
matrix containing N levels (points) of observations over a set of M variables. 
The sample is divided into two parts. Approximately two-thirds of points make up 
the learning subsample NA, and the remaining one-third of points (e.g. every 
third point) with same variance form the check subsample NB. Before dividing, 
points are ranged by variation value. The learning sample is used to derive 
estimates for the coefficients of the polynomial, and the check subsample is 
used to choose the structure of the optimal model, that is, one for which the 
external regularity criterion AR(s) takes on a minimal value: 

(2) 

or better to use the cross-validation criterion PRR(s) (it takes into account 
all information in data sample and it can be computed without recalculating of 
system for each checking point): 

To test a model for compliance with the differential balance criterion, the 
input data sample is divided into two equal parts. The criterion requires to 
choose a model that would, as far as possible, be the same on both subsamples. 
The balance criterion will yield the only optimal physical model solely if the 
input data are noisy. 

To obtain a smooth exhaustive-search curve (Fig. 1), which would permit one to 
formulate the exhaustive-search termination rule, the exhaustive search is 
performed on models classed into groups of an equal complexity. For example, the 
first layer can use the information contained in every column of the sample; 
that is full search is applied to all possible models of the form: 

, . (3) 

Non-linear members can be taken as new input variables in data sampling. The 
output variable is specified there in advance by the experimenter. At next layer 
are sorted all models of the form: 

, (4) 

The models are evaluated for compliance with the criterion, and so on until the 
criterion value decrease. For limitation of calculation time recently it was 
proposed during full sorting of models to range variables according to criterion 
value after some time of calculation or after some layers of iteration. Then 
full sorting procedure continues for selected set of best variables till the 
minimal value of criterion will be found. This gives possibility to set much 
more input variables at input and to save effective variables between layers to 
found optimal model. 

A salient feature of the GMDH algorithms is that, when they are presented 
continuous or noisy input data, they will yield as optimal some simplified non-physical 
model. If is only in the case of discrete or exact data that the exhaustive 
search for compliance with the precision criterion will yield what is called a 
physical model, the simplest of all unbiased models. With noisy and continuous 
input data, simplified (Shannon) models prove more precise [12,25] in 
approximation and for forecasting tasks. 

2 3 4 5 

Fig. 2. Combinatorial GMDH algorithm. 

1 - data sampling; 

2 - layers of partial descriptions complexing; 

3 - form of partial descriptions; 

4 - choice of optimal models; 

5 - additional model definition by discriminating criterion. 

. . . 

2 3 4 5 

Output model: Yk+1 = d0 + d1x1k + d2 x2k+ ... +dm xM k xM-1 k 

Fig.3 Multilayered Iterational algorithm: 

1 - data sampling; 

2 - layers of partial descriptions complexing; 

3 - form of partial descriptions; 

4 - choice of optimal models; 

5 - additional model definition by discriminating criterion; 

F1 and F2 - number of variables for data sampling extension. 

Calculations are faster when following techniques are used [24,25]: 

a) in all formulae informational array WTW is used instead of data sampling 
array W=(XY); 

b) model’s parameters are estimated by recursion method of "framing" which 
allows to use arrays calculated on previous steps; 

c) faster generation of variables ensemble is done using Garsaid binary counter, 
where current ensemble is differ from previous in one digit only. 

2.4.2. The Multilayered Iterative GMDH algorithm (MIA) 

As with the Combinatorial algorithm, the output variable must be specified in 
advance by the person in charge of modelling, which corresponds to the use of so-called 
explicit templates (Fig.4). In each layer, new output variables values, 
calculated by the F best models in each point are used to successively extend 
the data sample (Fig. 3). 

In Multilayered Iterative algorithm the iteration rule remains unchanged from 
one layer to next. As is shown in Fig. 3, the first layer tests models that can 
be derived from the information contained in any two columns of the sample. The 
second layer uses information from four columns; the third, from any eight 
columns, etc. The exhaustive-search termination rule is the same as for the 
Combinatorial algorithm: in each layer the optimal models are selected by the 
minimum of the criterion [16,25]. 

2.4.3. The Objective System Analysis algorithm (OSA) 

In discrete mathematics, the term template refers to a graph indicating which of 
the delayed arguments are used in setting up conditional and normal Gauss 
equations. A gradual increase in the structural complexity of candidate models 
corresponds to an increase in the complexity of templates whose explicit (a) and 
implicit (b) forms are shown in Fig. 4. 

When one uses implicit templates, one has, beginning from the second layer of 
the exhaustive search, to solve a system of equations and to evaluate the model, 
using a system criterion. 

The system criterion is a convolution of the criteria calculated by the 
equations that make up the system 

(5) 

where s is the number of equations in the system. The flowchart of the OSA 
algorithm is shown in Fig. 5. The key feature of the algorithm is that it uses 
implicit templates, and an optimal model is therefore found as a system of 
algebraic or difference equations. An advantage of this algorithm is that the 
number of regressors is increased and in consequence, the information embedded 
in the data sample) is utilised better. A disadvantage is that it calls for a 
large amount of calculations in order to solve the system of equations and a 
greater number of candidate models have to be searched. The amount of search can 
be reduced, using a constraint in the form of an auxiliary precision criterion. 

Fig.4. Derivation of conditional equations on a data sample 

Fig. 5. Objective System Analysis (OSA) algorithm 

In setting up the system of equations, one then discards the poorly forecasting 
equation (using equation only) for which the variation accuracy criterion for 
the forecast is less than unity (narrowing operation): 

(6) 

where: - is the variable values in the table; 
- is the value calculated according to the model and 
is the mean value. 

This criterion is recommended in the literature in order to evaluate the success 
of an approximation or of a forecast [15]. With d2 < 0.5, the result of 
modelling is taken to be good; with 0.5 < d2 < 0.8 it is taken to be 
satisfactory; with d2 > 1.0, modelling is considered to have failed, and the 
model yields misinformation.) 

2.5. Extended definition of the only optimal model by the theory of 
discriminating criteria 

It has been demonstrated theoretically and experimentally that the exhaustive-search 
curves shown in Fig. 1 are gradual and unimodal for the expected value of the 
criterion [25]. The number of candidate models tested in each exhaustive-search 
layer cannot be infinitely large. In other words, in constructing exhaustive 
search curves, the expected value of the criterion is in effect replaced by its 
mean (or least) value. Because of this, the curves take on a slightly wavy shape, 
and a small error may creep into the optimal model structure choice. 

The theory of discriminating criteria has been developed by Fedorov and 
Yurachkovsky [24] with special reference to experimental design. It has however 
proved its relevance to the self-organisation of models and active-neuron neural 
networks. The theory proceeds from the following premises: (I) there exists a "true" 
model represented in the data sample; (2) the assumed few object descriptions 
fit the model to a different degree; (3) the model that comes closest to the 
true model can be selected from its compliance with an auxiliary discriminating 
criterion. 

With such an approach, every GMDH algorithm consecutively uses two criteria. At 
first, an exhaustive search is applied to all candidate models for compliance 
with the main criterion, and a small number of models whose structure is close 
to optimal are selected. Then only one optimal model is selected that complies 
with a special discriminating criterion. The theory of optimal discriminating 
criteria is still in the developmental stage, but successful discriminating 
criteria are already known. 

In cases involving the selection of a structure for optimal polynomial models, 
the approximation or forecast variation criterion serves well. In the selection 
of optimal clusterization, good results are obtained with the symmetry criterion 
for the clusters distance matrix calculated relative to the secondary diagonal [21], 
etc. 

3. Data analysis: neural networks versus self-organising modelling 

The table 2 gives a comparison of both methodologies: neural networks and self-organising 
modelling in connection with their application to data analysis. 

Table 2. Neural networks versus self-organising modelling. 


Neural networks  
Statistical learning networks  
Data analysis  
universal approximator  
structure identificator  
Analytical model  
indirect by approximation  
direct  
Architecture  
unbounded network structure; experimental selection of adequate architecture 
demands time and experience  
bounded network structure [1]; adaptively synthesised structure  
A-priori-Information  
without transformation in the world of neural networks not usable  
can be used directly to select the reference functions and criteria  
Self-organisation  
deductive, given number of layers and number of nodes (subjective choice)  
inductive, number of layers and of nodes estimated by minimum of external 
criterion (objective choice)  
Parameter estimation  
in a recursive way; 

demands long samples  
estimation on training set by means of maximum likelihood techniques, selection 
on testing set (extremely short )  
Feature  
result depends from initial solution, time-consuming technique, necessary 
knowledge about the theory of neural networks  
existence of a model of optimal complexi-ty, not time-consuming technique, neces-sary 
knowledge about the task (criteria) and class of system (linear, non-linear)  

Results obtained by statistical learning networks and especially GMDH type 
algorithms are comparable with results obtained by neural networks [30]. In 
distinction to neural networks, the results of GMDH algorithms are explicit 
mathematical models obtained in a relative short time on the base of extremely 
short samples. The well-known problems of an optimal (subjective) choice of the 
neural network architecture are solved in the GMDH algorithms by means of an 
adaptive synthesis (objective choice) of the architecture. There are to estimate 
networks of the right size with a structure evolved during the estimation 
process to provide a parsimonious model for the particular desired function. 
Such algorithms combining the best features of neural nets and statistical 
techniques in a powerful way discover the entire model structure - in the form 
of a network of polynomial functions, difference equations and other. Models are 
selected automatically based on their ability to solve the task (approximation, 
identification, prediction, and classification). 

4. Nets of active neurons 

4.1. Self-organisation of twice-multilayered neural network 

A neural network is designed to handle a particular task. This may involve 
relation identification (approximation), pattern and situation recognition, or a 
forecast of random processes and repetitive events from information contained in 
a sample of observations over a test or control object. 

The present stage of computer technology allows a new approach in neural 
networks, which increases the accuracy of classical modelling algorithms. Such 
complex system can solve complex problems. We can use the GMDH algorithms as the 
complex neurons, where the self-organisation processes are well studied. 

Only by this inductive self-organising method for small, inaccurate or noisy 
data samples optimal non-physical model, accuracy of which is higher and 
structure is simpler than structure of usual full physical model can be found. 
GMDH algorithms are the examples of complex active neurons, because they choose 
the effective inputs and corresponding coefficients of them by themselves, in 
process of self-organisation. The problem of neuronet links structure self-organisation 
is solved in a rather simple way. 

Each neuron is an elementary system that handles the same task. The objective 
sought in combining many neurons into a network is to enhance the accuracy in 
achieving the assigned task through a better use of input data. As already noted, 
the function of active neurons can be performed by various recognition systems, 
notably by Rosenblatt's two-layer perceptrons - such neural network achieves the 
task of pattern recognition. In the self-organisation of a neural network, the 
exhaustive search is first applied to determine the number of neuron layers and 
the sets of input and output variables for each neuron. The minimum of the 
discriminating criterion suggests the variables for which it is advantageous to 
build a neural network and how many neuron layers should be used. Thus, the 
theory of neural network self-organisation is similar in many respects to that 
of each active neuron. 

Active neurons are able, during the self-organizing process, to estimate which 
inputs are necessary to minimise the given objective function of the neuron. In 
the neuronet with such neurons, we shall have twofold multilayered structure: 
neurons themselves are multilayered, and they will be united into a multilayered 
network. They can provide generation of new features of special type (the 
outputs of neurons from previous layer) and the choice of effective set of 
factors at each layer of neurons. The output variables of previous layers are 
very effective secondary inputs for the neurons of next layer. First layer of 
active neurons acts similar to Kalman filter: output set of variables repeated 
the input set but with filtration of noises. Number of active neurons in each 
layer is equal to number of variables given in initial data sampling. 

Neuronet structure is given in Fig. 6. Solely including the output calculated 
variables from each previous layer of neurons effects sample extension. The 
samples show the form of the discrete template used to teach the first neurons 
of a layer by the Combinatorial GMDH algorithm. In particular, when four input 
variables are used and two time delays are allowed for (t=2), the first template 
corresponds (to the following complete difference equation: 

The algorithm will suggest which of the proposed arguments should be taken into 
consideration and will help to estimate the connectivity coefficients. 

To begin with, we construct the first layer of neurons in the network. Then we 
will able to determine how accurate the forecast will be for all variables. For 
this purpose, we use a discrete template that allows a delay of one or two days 
for all variables. Then we add a second, a third, etc. layer to the neural 
network, as shown in Fig. 6, and go on doing so as long as this improves the 
forecast or decrease external criterion value. 

For each neuron, we have applied the extended definition procedure to one model 
(out of the five closest to the optimal one). For the optimal models, we have 
calculated the forecast variation criterion. It may be inferred, that there is 
no need to construct a neural network in order to form a forecast for those 
variables, for which variation criterion value takes on the least value in the 
first layer. It is advisable to use a neural network to form a forecast for the 
variables, for which the variation criterion takes on the least value in the 
last layers of neurons. 

The equations for the neurons of the network define the connections that must be 
implemented in the neural network; in this way they help achieve the task of 
structural self-organisation of the neural network. For brevity, the data sample 
in the above example is extended in only one way: tile output variables of the 
first layer are passed on as additional variables to the second, third, etc. 
layer of neurons. It is possible to compare different schemes of data sample 
extension by external criterion value. 

The task for self-organisation of such networks of active neurons by selection 
is to estimate the number of layers of active neurons and the set of possible 
potential inputs and outputs of every neuron. The sorting characteristic - ”number 
of neuronet layers - variables, given in data sample“ - defines the optimum 
number of layers for each variable separately. Neuronets with active neurons 
should be applied to raise the accuracy of short-term and long-term forecasts. 

Not only GMDH algorithms, but also many modelling or pattern recognition 
algorithms can be used as active neurons. Its accuracy can be increased in two 
ways: 

- each output of algorithm (active neuron) generate new variable which can be 
used as a new factor in next layers of neuronet; 

- the set of factors can be optimised at each layer. The factors (including new 
generated) can be ranked after their efficiency and several of the most 
efficient factors can be used as inputs for next layers of neurons. In usual 
once-multilayered ANN the set of input variables can be chosen once only. 

a) COMBI or b) OSA 

Fig 6. Schematic arrangement of the first two rows of a neural network. 

4.2. The search termination rule 

In self-organisation, the layers of neurons are extended as long as this 
improves the accuracy of the solution yielded by the neural network. This will 
be demonstrated later with reference to a relevant example. 

4.3. Group allowance for arguments 

We will call as the exhaustive-search characteristic of a neural network the 
graph that relates the main precision criterion for a specified variable to the 
layer number. This characteristic is similar to that of the GMDH algorithms. To 
obtain a smooth and unimodal curve, the exhaustive-search characteristic is 
calculated for many tools in the sample, and the results are averaged. 
Theoretically, the exhaustive-search characteristic has been investigated for 
the expected value of the criterion [24]. In practice, the exhaustive-search 
curve has to be constructed not for the expected value and even not for the mean 
value of the criterion. Rather, it is constructed for the best results of the 
exhaustive-search applied to a group for which the criterion takes on the least 
value. This exhaustive-search termination rule holds only when many 
approximation or forecast results are average. 

4.4. The selection of a discrete template 

What type of template to use depends on the task at hand (Fig. 4). In an 
approximation task, the template does not contain delayed arguments; in a 
forecast task, two or three delays have the be allowed for. In the former case, 
one obtains single-moment equations; in the latter, difference equations. 

4.5. Extended definition of one optimal model for each neuron in a network 

Self-organisation of each neuron taken separately uses the differential balance 
criterion or the regularity precision criterion. As already noted, the 
exhaustive-search curve approaches its minimum in a gradual manner, and the 
criteria of models close to the optimal one differ only slightly from one 
another in value. This explains why one has to use an extended definition 
algorithm. This algorithm, instead of one, selects several of the best models. 
From them chosen only one that complies with another variation discriminating 
criterion. 

4.6. Readout of modelling results 

Each layer in a neural network contains neurons, whose outputs correspond each 
to a particular specified variable: the output of the first neuron to the first 
variable, the output of the second neuron to the second variable, etc. Each 
column consists of neurons whose outputs correspond to one of the variables. 
From each column in turn, one neuron with a minimal variation criterion is 
selected. More specifically, one neuron having the best result is selected from 
the first column of neurons for which the output is the first variable; 
similarly, one neuron is selected from the second column of neurons for which 
the output is the second variable, etc. This selection procedure uniquely 
defines the number of layers for each variable and, thus, the structure of the 
neural network. 

4.7. The exhaustive search of methods for data-sample extension and narrowing 

The principal method of data-sample extension is by including the output 
variables from the previous layer that have complied with the criterion best of 
all. It will also be a good plan to test against the criterion the advisability 
of sample extension by simple non-linear transformations of input variables. In 
the example that follows, three variables are involved. They are x1, x2, and x3. 

(a) The extension using the covariance of the variables 

(b) The extension using the reciprocals of the variables 

The reciprocals should above all be proposed for the variables that take a minus 
sign in the equation; that is, they reduce the value of the output. 

4.8. Sample extension by consecutive elimination of the most efficient variables 

The diversity of the variables that come in for the exhaustive search (performed 
by each neuron) can further be increased by eliminating the most efficient 
variables, thus producing partial subsets. This can be best illustrated by an 
example. 

Let the input of a neural network accept a data sample containing just M=25 
variables. Suppose further that we have used the OSA algorithm and found in the 
first neuron an optimal system of forecasting difference equations in the 
variables x2 x12 x13 x18 x22 . These variables are least "fuzzy" and lend 
themselves to forecasting by this system of equation. We eliminate from the 
sample the variables thus found and apply the OSA algorithm to a second neuron. 
This yields a second optimal system of equations in the variables x3 x9 x14 x32. 
As a result, the minimum of the criterion increases (because the second set 
contains other than the best variables) and shifts to the left (Fig. 1). Now we 
eliminate from the sample the nine variables thus found, and apply the OSA 
algorithm to a third neuron. This yields an optimal system of equations in only 
three variables x5 x6 x11. The minimum of the criterion goes up still more and 
again shifts to the left etc. 

This shift in the minimum of the system criterion bears out the adequacy law, 
which states that for more fuzzy systems the optimal description (model) must be 
likewise more fuzzy and simple; that is, it must have a smaller number of 
equations [24]. Computer experiments confirm the above form of exhaustive-search 
curve. In the above example, the number of variables used for decision-making is 
increased from 5 (in the first neuron) to 5 + 4 + 3 + 2 + 1 =15 (in five neurons). 
Ten features are discarded as inefficient. So, we shall have 5 x 15 = 75 neurons 
in each layer. 

4.9. Simultaneous and successive algorithms for neural networks 

In a computer program, neurons can be implemented simultaneously or successively, 
using memory devices. 

4.10. Neuronets self-organisation and algorithms for optimization of control 
systems 

The principal roadblock in the use of linear and non-linear programming 
algorithms for complex system optimization is that it is often impossible to 
specify either the goal function or the applicable constraints with sufficient 
accuracy. Meanwhile, even minute inaccuracies in their specification may have a 
strong impact on the outcome of optimization. Active-neuron networks can be 
readily combined with linear and non-linear programming algorithms. 

One of the output functions is taken as the objective function, the equations of 
the other output variables can serve as equality-type constraints. This removes 
the subjective factor from the specification of the goal function and 
constraints. The human operator defines criteria for their choice, and not the 
objective function and constraints themselves [21]. 

6. Examples of applications. 

Besides the applications of commercial GMDH software there were a lot of 
implementations made in very different fields. Many of them are described in the 
ukrainian journal "Avtomatica" (translated in "Soviet Automatic Control", "Soviet 
Journal of Automation and Information Sciences" and then in "Journal of 
Automation and Information Sciences" in full size). The basic GMDH technique 
applications include the studies on: economical systems (analysis and 
forecasting of macroeconomy parameters, decision support and optimization), 
ecological systems analysis and prediction (forecasting of oil fields and river 
flow, harvest analysis and ionosphere state definition), environment systems 
analysis, medical diagnostics, demographic forecasting, weather modelling, 
econometric modelling and marketing, manufacturing, planning of physical 
experiments, materials estimation, multisensor signal processing, microprocessor-based 
hardware, eddy currents, x-ray, acoustic and seismic analysis and widely in 
military systems (radar, infrared, ultrasonic and acoustics emission, missile 
guidance). 

6.1. Prediction of characteristics of stock market 

Currency, international stock trading and derivatives contracts play an 
increasing role for many investors. Commonly used a portfolio consisting of a 
number of contracts. Assets returns must be predicted and controlled by a 
prediction/control module. Control of risk via prediction/control module of 
individual investments returns inside the portfolio provides the most likely 
process. 

It is known that in most economic applications i.e. financial risk control, 
neural networks give success only of 70-80%. By means of the new approach of 
GMDH twice-multilayered neural networks it will be improved by 5-10%. Prediction 
accuracy for short and very noised data also increases in short and long-time 
predictions by 10-50% in comparison to statistical methods and neural networks, 
especially for stochastic processes [30,31]. On the base of predictive control 
it increases the results of a repetitive control. 

As an example prediction of the activity on the stock exchange in New York was 
considered in [10]. In the following on the base of observations in the period 
of February 22 up to June 14, 1995 in seven periods 7 variables of the stock 
market (DAX, Dow Jones, F.A.Z., Dollar and other) are predicted. In the 
information base delays of all variables up to 35 are included. Also there were 
used not only linear reference functions to describe the variables, but also non-linear. 
It was to model and to predict 7 time series not independently as time series 
models but rather as highly interactions network (input - output - model). Table 
3 shows the accuracy of predictions for all variables (mean MAD [%]). 

Using the results of model generation (at first level of neuronet) it is 
possible to improve the accuracy of models in a second model generation, where 
are used the model outputs from previous for models generation of optimal 
complexity. This procedure can be continued up to decreasing accuracy of models. 

Table 3. Observation and prediction periods. 

Observation period  
Long-term prediction period  
Model  
Prediction  
Period up to  
Days  
Begin  
End  
Days  
Max delay  
Mean MAD [%]  
a  
March, 17  
18  
March,20  
March, 31  
10  
5  
0.985 %  
b  
March, 31  
28  
April,3  
April, 18  
10  
10  
2.055 %  
c  
April, 18  
38  
April, 19  
May, 3  
10  
15  
0.809 %  
d  
April, 28  
46  
May, 2  
May, 15  
10  
20  
1.642 %  
e  
May, 15  
56  
May, 16  
May, 30  
10  
26  
1.217 %  
f  
May, 30  
66  
May, 31  
June ,14  
10  
30  
1.206 %  
g  
June, 14  
76  
June, 16  
June, 29  
10  
35  
0.760 %  

Table 4 shows the resulting model error (MAPE [%]) and prediction error (MAD [%]) 
of Dollar, Dax, F.A.Z., Dow Jones and the mean values for all 7 variables 
obtained on 3 levels. The table 4 shows that the repeated application of self-organization 
gives more accurate approximation, which results in better predictions in the 
second level. The models obtained in the 3 level are overfitted, therefore the 
prediction error increases. 

Table 4: Multilevel application (model f). 

MAPE [%]  
MAD [%]  
Level  
1  
2  
3  
1  
2  
3  
Dollar  
0.68  
0.51  
0.11  
2.32  
2.17  
11.67  
Dax  
0.35  
0.24  
0.10  
2.20  
1.24  
5.21  
F.A.Z.  
0.22  
0.23  
0.03  
1.54  
1.27  
2.32  
Dow Jones  
0.27  
0.16  
0.06  
2.15  
0.84  
4.84  
Mean  
0.267  
0.184  
0.051  
1.43  
0.98  
3.67  

The efforts in using the GMDH type neural networks are much less than in neural 
networks, where the architecture must be chosen by trial and error. Only an 
adaptive synthesis of the network structure allows an automatic model generation 
and therefore applications in the fields where lots of decisions and forecasts (monitoring 
of complex systems with many controlled variables) repeating over short time 
periods are needed. 

7. Objective selection of the best model 

It is the aim of self-organising modelling to get in an objective way models of 
optimal complexity. But there are several freedoms in choice of class of systems 
to be model (linear/non-linear), time lag and in selection of appropriate 
parameters (number of best models, complexity etc.). To reduce such a 
subjectivity it is recommended to generate several alternative models (linear, 
non-linear, with several complexity and time lags) and in a second layer to 
select the best model outputs or to generate there combination. Table 5 shows 
obtained results. 

Table 5. Selection of best model results (model g): prediction error MAD [%]. 

Linear  
Non-linear  
Second  
Model  
1  
2  
3  
1  
2  
3  
layer  
Dollar  
2.88  
2.10  
0.89  
1.25  
1.41  
1.40  
1.55  
F.A.Z.  
1.22  
1.45  
1.01  
0.82  
1.12  
1.57  
0.88  
Dax  
1.36  
2.41  
1.51  
1.69  
2.43  
4.54  
1.94  
Dow Jones  
1.14  
1.26  
1.44  
3.75  
3.25  
3.79  
2.93  
Mean  
1.14  
1.29  
0.90  
1.21  
1.35  
1.81  
1.20  

8. Non-parametric inductive selection methods 

8.1. Modeling of fuzzy systems 

The physical model is the best tool for function approximation and random 
process forecasting of deterministic objects where inputs and outputs are 
measured accurately with absence of noises. In the case of insufficient a priori 
information, not very accurate measurements, noisy and short data sample, better 
results can be reached by the use of non-physical models. But in the case of so-called 
ill-defined objects, dispersion of noise is too big, even for the use of non-physical 
models. In this case application of clustering of data sample is to be 
recommended, which can be considered as discrete form of physical model of ill-defined 
objects. 

Almost all objects of recognition and control in economics, ecology, biology and 
medicine are undeterministic or fuzzy. Deterministic (robust) part and 
additional black boxes acting on each output of object can represent them. The 
only information about these boxes is that they have limited values of output 
variables, which are similar to the corresponding states of object. 

According to Ashby [33] diversity of control system is to be not smaller, than 
diversity of the object itself. The Law of Adequateness, given by S.Beer, 
establishes that for optimal control the objects are to be compensated by 
corresponding black boxes of the control system [13]. For optimal pattern 
recognition and clustering only partial compensation is necessary. More of what 
we are interested in is to minimise the degree of compensation by all means to 
get more accurate results. 

The methods of cluster analysis and selection of analogous patterns discussed 
below are denoted as non-parametric because there is no need to estimate 
parameters. The method of cluster analysis was described in [20] in more detail. 

8.2. Method of analogues complexing 

The equal fuzziness of the model and object is reached automatically if the 
object itself is used for forecasting. This is done by searching analogues from 
the given data sample which are equivalent to the physical model. Forecasts are 
not calculated in the classical sense but selected from the table of observation 
data. 

The main assumptions are the following: 

- the system to be modelled is described by a multidimensional process; 

- observations of data sample are enough (long time series); 

- the multidimensional process is sufficiently representative, i.e. the 
essential system variables are included in the observations; 

- it is possible that a part of past behaviour will be repeated. 

If we succeed in finding for the last part of behaviour trajectory (starting 
pattern), one or more analogous parts in the past (analogous pattern) the 
prediction can be achieved by applying the known continuation of these analogous 
patterns [8]. 

Using a sliding window which generates the set of possible patterns {Pi,k+1}, 
where and k+1 is the width of sliding window and also of the patterns, the 
output pattern is 

The algorithm of selection of the analogous pattern has the following task: 

For the given output pattern it is necessary to select the most similar patterns 
and to evaluate the forecast with the help of these patterns. 

Method of analogue complexing is recommended in the case when the input 
observations of data sample is long enough. Analogues substitute the physical 
model. It means that optimal analogue can be found by selection sorting-out 
procedure, using internal accuracy type criterion. To divide data sample into 
two parts is not necessary. There should be provided several optimization of 
algorithms parameters, to rise up the accuracy of processes short-term 
forecasting. The selection task is a four-dimensional problem with the following 
dimensions: 

- set of variables used; 

- number of analogues selected; 

- width of the patterns (number of lines, used in each); 

- values of weight coefficients with which patterns are complexed. 

As method of optimization the comparison of variants by internal criterion of 
accuracy is used. The criterion is calculated on the whole length of sample. 
This is the way of short-time forecasting problem solution on one step ahead. 
More difficult is the problem of long-term step-by-step random processes 
forecast. To select similar patterns from all possible patterns in the time 
series, the following steps are developed: 

A. Reducing variable set size 

The choice of an optimal set of variables can be realised by preselection. It is 
necessary to identify a subset of effective variables, which were defined as the 
nucleus [17]. 

One method of automatic generation of the nucleus is the automatic 
classification of variables by means of the algorithm of objective cluster 
analysis, described in [20,22]. Another method gives the GMDH algorithm for 
linear model construction. The models selected in the last layer indicate an 
ensemble of variables for which we have to seek the most consistent pattern 
analysis. 

B. Transformation of analogues 

Most processes in large-scale systems are evolutionary. In this case 
stationarity as one important condition of successful use of the method of 
analogues is not fulfilled. As time-series may be non-stationary patterns with 
similar shapes may have different mean values, standard deviations and trends. 
In the literature, it is recommended to evaluate the difference between the 
process and its trend, which is an unknown function of time. Another possibility 
gives the selection of differences where the criterion of stationarity is used 
as selection criterion. The results of the method of analogues depend on the 
selected trend function. 

It is advisable to determine transformed patterns , where 

The weights w, w for each pattern P, k>1 can be estimated by means of the least 
squares method, which gives not only the unknown weights but also the total sum 
of squares , which can be used in the following (step 3) as a measure of 
similarity. 

C. Selection of the most similar analogues 

The closest analogue is called the first analogue A, the next one in distance A 
is called the second analogue and so on until the last analogue A. Distances can 
be measured by means of the Euclidean distance of points of the output pattern 
and the analogue or by other measures of distance. In our case it is not 
necessary to find a proximity measure, but the total sum of the squares gives us 
information about the proximity between and . 

D. Combining forecasts 

Every selected analogous has its continuation which gives a forecast. In such a 
way we obtain F forecasts, which are to combine. In the literature there are 
several principles for combination of forecasts. 

The unknown predictions of the M systems variables can be assumed as a linear 
combination of the continuations of selected analogous patterns, i.e.: 

, 

The unknown parameters g, g, , will be estimated by means of parametric 
selection procedures e.g. using self-organising methods. The only problem is the 
small number of observations, i.e. the number of selected patterns. 

8.3. Prediction of characteristics of stock market by analogues complexing 

On the base of observations in the period of February 22 to May 30, 1995 (66 
days) the analogue complexing algorithm was used. Table 7 shows prediction error 
(MAD [%]) of four variables (Dollar, Dax, F.A.Z., Dow Jones) and the mean 
prediction error over all 7 variables. The width of the patterns varies from 6 
to 15 days. 

Table 7. Prediction error (MAD [%]) of analogues complexing 

Width  
6  
7  
8  
9  
10  
11  
12  
13  
14  
15  
Dollar  
2.61  
3.28  
2.62  
2.79  
2.91  
2.91  
1.86  
1.48  
2.16  
8.96  
F.A.Z.  
1.418  
2.609  
1.597  
1.485  
1.187  
1.391  
1.435  
1.869  
0.723  
1.118  
Dax  
1.427  
1.702  
1.7  
2.307  
2.962  
2.761  
2.761  
2.612  
2.372  
1.122  
Dow J  
1.36  
1.708  
1.622  
6.979  
5.393  
4.647  
4.966  
3.849  
3.363  
2.793  
Mean  
1.174  
1.575  
1.581  
2.356  
2.458  
2.08  
1.944  
1.789  
1.632  
1.877  

The forecasts are obtained by means of linear combination of the continuations 
of 5 selected analogous pattern, where the unknown weights gj are estimated by 
means of parametric selection procedures. 

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Ivakhnenko A. G.,  Muller J. "Recent Developments of Self-Organising Modeling in Prediction and Analysis of Stock Market" 

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