In the paper, we brie y describe a problem of identi cation of entrepreneurship determinants with respect to economic development of countries. In order to solve this problem, we need to identify correlations between entrepreneurship and macroeconomic indicators. The main attention in the paper is focused on selecting a proper computer tool for solving this problem. As a tool supporting identi cation, SelfOrganizing Feature Maps (SOMs) have been chosen. Some modi cation of the clustering process using SOMs is proposed by us to improve classi cation results and e ciency of the learning process. At the end, we indicate some challenges of further research.
The phenomenon of entrepreneurship is the subject of various levels of
observations such as the entrepreneur, industry, region or nation with respect to
many aspects re ecting the entrepreneurship level. The worldwide interest in
the entrepreneurship especially innovative entrepreneurship based on advanced
knowledge and technology shows the importance of the phenomenon particularly
for underdeveloped countries, for nations of aging societies, for those with youth
unemployment growing rate, and for the global economy as well. Our research
concerns designing e ective methods for computer support of identi cation of
entrepreneurship determinants as it is the key factor of countries economic
development. Therefore, in our research, we are going to build a specialized computer
aided system based on applying neural networks to determine the cross-countries
di erences referring to propensity for entrepreneurship and the country
framework in order to assess policy gaps and opportunities for future actions. The
multidimensional analysis enables us to form speci c recommendations to a country
government on how to lead a policy toward entrepreneurship development. The
question how to increase the development level by the entrepreneurship
stimulation policy is still open. Building an e ective and boosting entrepreneurship
system is challenge of the century, see e.g. [
The Clustering Procedure using Self Organizing
Feature Maps
The concept of a Self-Organizing Feature Map (SOM) was originally developed
by T. Kohonen [
In this section, we describe a clustering procedure used in experiments for nding correlations of groups of multidimensional objects using Self Organizing Feature Maps. We propose some modi cation to improve classi cation results and e ciency of the learning process, among others: { a modi ed coe cient for adjusting weights, { a modi ed way for adjusting weights of neighboring neurons (the modi cation coe cient is not constant, but it decreases along with the distance from the pattern neuron), { a modi ed way of the learning process (only neighboring neurons of the pattern neuron for a given input vector are trained).
Input for the procedure is a matrix of real numbers. Each row of the matrix represents a feature vector of one object (corresponding to one country) subjected to clustering. All rows (feature vectors) have the same dimension. An input matrix must have at least two rows (feature vectors). A fragment of exemplary data (for the indicator "New business density") subjected to clustering is shown in Table 1.
n = ceil(p2m + 0:5); where ceil is a function rounding up elements and m is a number of feature vectors. An initial size of the output matrix is 2 2. This size is increased, during a learning process, up to n n. A learning process is performed iteratively. In our research, a number of iterations has been set as 100. More iterations did not improve a quality of classi cation. Each feature vector is associated with an individual map. A map represents a matrix of neurons. We have as many maps as many feature vectors is present. We will treat this set of maps as a multilayer n. The parameter n is map labeled with M . An initial value of weights of the map is set on the basis of the following formula:
M [x][y][i] = random(min; max) 10 ; where x and y determine a position in the map and i is the index of the feature vector, i is integer included in the interval [1; k], where k is a number of all feature vectors subjected to clustering, random(min; max) is a pseudorandomnumber generator returning a number from the interval [min; max], where min and max determine minimal and maximal values of input feature vectors. A learning process includes the following steps: 1. Calculating a current coe cient for modi cation of weights of the map. 2. Calculating a new desired size of the map. 3. Random selection of the order of feature vectors for training the network. 4. Modi cation of weights of the map after calculation of the error on the basis of an input feature vector and current weights of the map.
Steps from 1 to 4 are performed iteratively up to the xed number of iterations (in our case, 100). After nishing the learning process, the testing process is run. In this process, assessment of classi cation results is made for each input feature vector used in the learning process. Assessment consists in calculation di erences between a given feature vector and weights of all neurons. A neuron with the smallest di erence is selected and identi ed as the pattern neuron for this feature vector. On the basis of pattern neurons, a map including all feature vectors and their assignments to centroids is created.
The current coe cient for modi cation of weights is calculated as: ec = e em ; where ec is a current epoch (its index changing from 1 to em), em is the maximal number of epochs.
The new desired size nd of the map is calculated as: nd = 2(ec 1)(n em ns) ; where ec is a current epoch, n is the maximal size of the map, ns is the initial size of the map, em is the maximal number of epochs. If the new desired size of the map is greater than the current one, the size nc of the map is increased in the following way:
n0c = nc + 1 if n0c is less than the maximal size of the map.
After changing the size of the map, weights need to be modi ed. For modication of weights in the map, we calculate auxiliary variables: xtemp = x ytemp = y (x (y Weights of the map for each input feature vector are changed in the following way:
M [x][y][i] = c1M [x 1][y][i] + c2M [x][y +c3M [x 1][y 1][i] + c4M [x][y][i] for i = 1; : : : ; k.
A di erence between a given feature vector input and weights of all neurons is calculated from the formula:
vu k d = tuX(M [x][y][i] i=1 input[i])2 for each neuron in the position x and y. Next, a neuron with the smallest d is selected and neighboring neurons ( 1 neurons in both directions, i.e., x and y) are modi ed according to:
M 0[x][y][i] = M [x][y][i]+ + (input[i]M [x][y][i])(1 0:4abs(x xtop))(1 0:4abs(y ytop)) for each i = 1; : : : ; k, where abs is the function of an absolute value, xtop = x + 1 and ytop = y + 1.
Results of the clustering process are presented in the form of minimal spanning trees with respect to distances between feature vectors and centroids. 3
Examined data consisted of entrepreneurship and macroeconomic indicators
called World Development Indicators (WDI) published by the World Bank [
Periodicity of data is annual. They cover developing and high-income economies. For each selected country, we have a time series consisting of annual values of a given indicator (cf. Table 1). Therefore, for the clustering process, we have as many feature vectors as many countries is selected. Each clustered object represents a time series. Examined indicators come from years of the rst decade of 21st Century. 4
The presented paper constitutes the rst attempt to dealing with the problem of identi cation of correlations between groups of time series obtained from the clustering process. Therefore, it has rather a rudimentary (introductory) character. In this section, we give some challenges of further research.
As the result of clustering process of the set of time series corresponding to a given indicator, we obtain a minimal spanning tree with respect to distances between feature vectors and centroids. An exemplary spanning tree is shown in Figure 1. It presents clusters of countries regarding the indicator called "New business density" showing new businesses registrations per thousand population 15-64 years old. According to the Figure 1 we can notice several groups of countries with similar values of the indicator, i.e., one cluster form countries: Vanuatu, Spain, Romania, Ireland, Latvia, Denmark, Singapore; whereas the second one covers: Bolivia, Philippines, Algeria, Argentina, Guatemala, Uganda, Jordan, Zambia, Morocco; and in the third we have: Malaysia, Kazakhstan, Uruguay, Croatia, Netherlands, France, Finland, Belgium, Portugal and Sweden.
In order to identify correlations between groups of time series formed in the clustering process, we need to apply some methods for comparison of topological structures of minimal spanning trees. In simple case, we can make one-to-one comparison, i.e., we compare a minimal spanning tree of one of the indicators with the one of another indicator. Results of comparison process should enable us to identify entrepreneurship determinants.
Moreover, we plan to test other clustering methods, among others, that
proposed by us (see [