Group method of data handling (GMDH) is a family of inductive algorithms for computer-based mathematical modeling of multi-parametric datasets that features fully automatic structural and parametric optimization of models.
GMDH is used in such fields as data mining, knowledge discovery, prediction, complex systems modeling, optimization and pattern recognition.[1] GMDH algorithms are characterized by inductive procedure that performs sorting-out of gradually complicated polynomial models and selecting the best solution by means of the external criterion. The last section of [2] contains a summary of the applications of GMDH in the 1970s.
Other names include "polynomial feedforward neural network",[3] or "self-organization of models". It was one of the first deep learning methods, used to train an eight-layer neural net in 1971.[4] [5]
This section is based on.
This is the general problem of statistical modelling of data: Consider a dataset
\{(x1,...,xk;y)s\}s=1:n
n
x1,...,xk
y
First, we split the full dataset into two parts: a training set and a validation set. The training set would be used to fit more and more model parameters, and the validation set would be used to decide which parameters to include, and when to stop fitting completely.
The GMDH starts by considering degree-2 polynomial in 2 variables. Suppose we want to predict the target using just the
i,j
a,b,c,d,e,f
a,b,c,d,e,f
i,j
i,j
| 12 | |
| k(k-1) |
| 12 | |
| k(k-1) |
We do not want to accept all the polynomial models, since it would contain too many models. To only select the best subset of these models, we run each model
f(i,
minMSE1
Suppose that after this process, we have obtained a set of
k1
z1,z2,...,
| z | |
| k1 |
The algorithm continues, giving us
minMSE1,minMSE2,...
minMSE
minMSEL+1>minMSEL
L+1
More sophisticated methods for deciding when to terminate are possible. For example, one might keep running the algorithm for several more steps, in the hope of passing a temporary rise in
minMSE
Instead of a degree-2 polynomial in 2 variables, each unit may use higher-degree polynomials in more variables:
Y(x1,...,xn)=a0+\sum\limits
| n | |
| i=1 |
{ai}xi+\sum\limits
| n | |
| i=1 |
| n | |
| {\sum\limits | |
| j=i |
{ai}}xixj+\sum\limits
| n | |
| i=1 |
| n | |
| {\sum\limits | |
| k=j |
{ai}}}xixjxk+ …
And more generally, a GMDH model with multiple inputs and one output is a subset of components of the base function (1):
Y(x1,...,xn)=a0+\sum\limits
| m | |
| i=1 |
aifi
where fi are elementary functions dependent on different sets of inputs, ai are coefficients and m is the number of the base function components.
External criteria are optimization objectives for the model, such as minimizing mean-squared error on the validation set, as given above. The most common criteria are:
Like linear regression, which fits a linear equation over data, GMDH fits arbitrarily high orders of polynomial equations over data.[6] [7]
To choose between models, two or more subsets of a data sample are used, similar to the train-validation-test split.
GMDH combined ideas from:[8] black box modeling, successive genetic selection of pairwise features,[9] the Gabor's principle of "freedom of decisions choice",[10] and the Beer's principle of external additions.[11]
Inspired by an analogy between constructing a model out of noisy data, and sending messages through a noisy channel,[12] they proposed "noise-immune modelling": the higher the noise, the less parameters must the optimal model have, since the noisy channel does not allow more bits to be sent through.
The model is structured as a feedforward neural network, but without restrictions on the depth, they had a procedure for automatic models structure generation, which imitates the process of biological selection with pairwise genetic features.
The method was originated in 1968 by Prof. Alexey G. Ivakhnenko in the Institute of Cybernetics in Kyiv.
Period 1968–1971 is characterized by application of only regularity criterion for solving of the problems of identification, pattern recognition and short-term forecasting. As reference functions polynomials, logical nets, fuzzy Zadeh sets and Bayes probability formulas were used. Authors were stimulated by very high accuracy of forecasting with the new approach. Noise immunity was not investigated.
Period 1972–1975. The problem of modeling of noised data and incomplete information basis was solved. Multicriteria selection and utilization of additional priory information for noiseimmunity increasing were proposed. Best experiments showed that with extended definition of the optimal model by additional criterion noise level can be ten times more than signal. Then it was improved using Shannon's Theorem of General Communication theory.
Period 1976–1979. The convergence of multilayered GMDH algorithms was investigated. It was shown that some multilayered algorithms have "multilayerness error" – analogous to static error of control systems. In 1977 a solution of objective systems analysis problems by multilayered GMDH algorithms was proposed. It turned out that sorting-out by criteria ensemble finds the only optimal system of equations and therefore to show complex object elements, their main input and output variables.
Period 1980–1988. Many important theoretical results were received. It became clear that full physical models cannot be used for long-term forecasting. It was proved, that non-physical models of GMDH are more accurate for approximation and forecast than physical models of regression analysis. Two-level algorithms which use two different time scales for modeling were developed.
Since 1989 the new algorithms (AC, OCC, PF) for non-parametric modeling of fuzzy objects and SLP for expert systems were developed and investigated.[13] Present stage of GMDH development can be described as blossom out of deep learning neuronets and parallel inductive algorithms for multiprocessor computers. Such procedure is currently used in deep learning networks.[14]
There are many different ways to choose an order for partial models consideration. The very first consideration order used in GMDH and originally called multilayered inductive procedure is the most popular one. It is a sorting-out of gradually complicated models generated from base function. The best model is indicated by the minimum of the external criterion characteristic. Multilayered procedure is equivalent to the Artificial Neural Network with polynomial activation function of neurons. Therefore, the algorithm with such an approach usually referred as GMDH-type Neural Network or Polynomial Neural Network. Li showed that GMDH-type neural network performed better than the classical forecasting algorithms such as Single Exponential Smooth, Double Exponential Smooth, ARIMA and back-propagation neural network.[15]
Another important approach to partial models consideration that becomes more and more popular is a combinatorial search that is either limited or full. This approach has some advantages against Polynomial Neural Networks, but requires considerable computational power and thus is not effective for objects with a large number of inputs. An important achievement of Combinatorial GMDH is that it fully outperforms linear regression approach if noise level in the input data is greater than zero. It guarantees that the most optimal model will be founded during exhaustive sorting.
Basic Combinatorial algorithm makes the following steps:
In contrast to GMDH-type neural networks Combinatorial algorithm usually does not stop at the certain level of complexity because a point of increase of criterion value can be simply a local minimum, see Fig.1.