The method of self-organizing maps (SOM) is a method of exploratory data analysis used for clustering and projecting multi-dimensional data into a lower-dimensional space to reveal hidden structure of the data. The algorithm used retains local similarity and neighborhood relations between the data items. In some cases we have to compare the structure of data items visualized on two or more self-organizing maps (i.e. the information about the same set of data items is gathered in different tasks, from different respondents or using time intervals (lags)). In this paper we introduce a method for systematic comparison of SOM maps in the form of similarity measurement. Based on the idea that the SOM retains local similarity relations of data items those maps can be compared in terms of corresponding neighborhood relations. We give two examples of case studies and discuss the method and its applicability as an additional and more precise measure of similarity of SOM maps.
The self-organizing map (SOM) is a method to visualize multidimensional data. The SOM performs mapping of multidimensional data onto a two-dimensional map while preserving proximity relationships as well as possible. The results of the SOM analysis are usually assessed visually. Interpretation of the SOM and discovered knowledge depends mostly on an interpreter. Subjective factors such as one’s attentiveness to both general patterns and local details of a large number of presented data items might diminish the objective value of data analysis.
When we use different sources of data that describe the same phenomenon but are collected somehow differently or the number of variables is varying then we have to assess whether the results of the two analyses are similar. As the SOM projects close units of the input space into nearby map units the local neighborhood should remain quite similar. In this paper we propose a simple method to compare the results of different self-organizing maps. The methodology is based on the measurement of similarities of the local neighborhood.
In the first part of the paper the used methods and techniques including similarity measurement methodology are introduced. In the second part of the paper two data sets as case studies are used to illustrate the similarity measurement methodology. Finally there is a discussion to analyze the results and the accuracy of the methodology. 2
The self-organizing map [
To reduce dimensionality of the data we use the principal component analysis (PCA).
The main idea of PCA is to reduce the dimensionality of a data set consisting of a
large number of interrelated variables, while retaining as much as possible of the
variation present in the data set [
The matrix reordering is a structuring method for graphs (and general data tables).
The method reorganizes the neighborhood graph data vertices according to specific
property – systems monotonicity [
While the SOM represents data on two-dimensional topological maps the local topological relations between data items can be used to assess whether the maps have similar structure. The local neighborhood is the basis of our approach to measure the similarity between maps and we expect the neighborhood relations to remain stable even when the overall orientation of the map changes.
The proposed methodology to measure similarity between the self-organizing maps consists of four main steps.
Firstly, to analyze general organization the resulting map is visually examined and clusters and their borders are identified, also the general orientation and locations of data items are identified. Thereafter the matrix of neighborhood relations is formed. Neighborhood assessment is based on the location of the best matching units (BMU a point on the map that is the closest to the input data vector) on the self-organizing map. Two data items are neighbors if they are marked to locate on the same node or in the neighboring nodes depending on the neighborhood range. The neighborhood on the hexagonal map is demonstrated on Fig. 1. The neighborhood matrix is an n-by-n square symmetric matrix N where n is the number of data items and the matrix can also be regarded as a graph. If there is neighborhood relation between ith and jth element then the value of the matrix element is marked 1 and 0 otherwise.
⎧1, ifneighbor nij = ⎨
⎩0, otherwise
Next stage of similarity analysis is the calculation and assessment of similarity
coefficients [
SMC =
number of matches total number of variables .
(1)
The SMC rates positive and negative similarity equally and can be used if positive and negative values have equal weight.
Jaccard Coefficient (J) is used if the negative and positive matches have different weights (are asymmetric).
J =
number of positive matches number of variables - negative matches . (2)
Jaccard Coefficient ignores negative matches and can be used if the variables have many 0 values.
If the value of the Jaccard coefficient and SMC is below 0.5 then the number of positive matches is less than half of the total matches.
Fourth part of the similarity measurement consists of finding how much the two neighboring matrixes are identical what is a maximum isomorphic subset. The task is not as complicated as the general isomorphic graph problem, because the order of the data items is known and to identify the maximum isomorphic subgraph we can use an AND operator. If aij & bij (elements of the neighborhood matrixes have both value 1), then the neighborhood relation is isomorphic. Here we can perform a new meta-level analysis and reorder and visualize the isomorphic sub-graph to see commonly shared information between two maps. For output the Graphviz1 software has been used. 6
We use two sets of data to illustrate the method of similarity measurement. The first is a research into the concepts of emotion in Estonian language. The survey consisted of two parts and as a result two different data matrixes describe the same set of emotion concepts. In our meta-analysis we attempt to analyze whether and to what extent the results of two tasks are comparable. The second data set is banking data. In this case the purpose of our meta-analysis is to detect whether and to what degree the dimensionality reduction method (PCA) applied to the data has preserved its structure. Those two data sets reveal different aspects of the comparison methodology. 6.1
The purpose of the study was to discover the hidden structure of the Estonian emotion concepts and test a hypothesis that the way the information about concepts is collected can influence its emergent structure. Two lexical tasks were carried out providing information about emotion concepts either through their relation to the episodes of emotional experience or through semantic interrelations of emotion terms (synonymy and antonymy).
The inquiry was carried out in written form during the summer months of 2003 in
Estonia. There were 24 emotion concepts selected for the study based on the results of
tests of free listings [
had to complete two tasks measuring the concepts by means of different levels of
knowledge (see [
The data of both tasks was analyzed by SOM (Fig. 2). In a visual comparison of the two maps we could see completely different structures, but there is a clear distinction of concepts of positive vs. negative emotions observable on both maps. The locations of these clusters are reversed, however. In addition, the upper part of Fig. 2b is divided into two subclusters as there is a group of concepts located in the uppermost right edge of the graph. It is hard to decide whether the obtained structures are different enough to claim the hypothesis that the way of approach (in form of our two tasks) can influence the emerging conceptual structure, proved.
a) b)
range neig. neig. neig. neig. coef. 1 96 100 52 144 0.8403 0.3611 2 198 216 150 264 0.8021 0.5682 3 276 346 230 392 0.7188 0.5867
The summary of the neighborhood relations between two tasks is given in Table 2. The number of relations is measured with different range of neighborhood, starting from 1 to 3. Increase in neighborhood range causes also increase in the number of relevant neighborhood relations. The SMC coefficient is decreasing if the neighborhood is increasing because of possible connections between the words that actually do not belong to the same neighborhood. We could use the SMC as an indicator of stability. Jaccard coefficient is increasing if the neighborhood range is widening and there is tendency to have more positive matches if the number of neighborhood relations increases.
As far as the neighborhood range remains open it is still difficult to decide, whether the two SOMs of our two tasks were different enough to claim our hypothesis of the case study proven.
The second step of meta-analysis is to find a maximum isomorphic sub-graph and to find a clue what the suitable range of the neighborhood could be. In the case the neighborhood range was provisionally set on 1, several separate fragments of conceptual networks were formed. The general structure of the data did not appear as a connected system. With the neighborhood range 2, the graph became connected. One can speculate that it represents the communal structure or a backbone of the conceptual data gathered from two tasks. The reordered data matrix and its graph are visible on Fig. 3. The lighter part of the reordered matrix is isomorphic part of the matrix.
A conclusion can be drawn, that the match of the two structures based on our two tasks is partial, and it is measurable in principle. The degree of measured structural isomorphism depends on the rigidity of the selected criteria of neighborhood. 6.2
The second data set is used to illustrate the impact of dimensionality reduction by PCA on the SOM maps. The aim of the study is to measure the similarity between the results of SOM mapping of original data and reduced data.
The second data set consists of banking data (1997—2000; http://www.bankofestonia.info). We have used 133 public quarterly reports by individual banks as a balance sheet and profit / loss statement (income statement). The 50 most important variables have been selected to form a short financial statement of a bank. All the variables are normalized by the variable of total assets to make the reports comparable.
We formed three sets of the banking data. The first set consisted of all 50 original variables (Original), for the second set 26 principal components describing 95% of variation were selected (PCA95) and for the third set 5 principal components describing 50% of variation were selected (PCA50). From those data sets three selforganizing maps were created (Fig. 4). Our aim has been to measure how similar those maps are and whether similar banks are projected into nearby map units in all cases.
a) b) c)
Analyzing the maps visually we can see that in general the maps have a similar structure. As we are interested in overall structure we marked only the first BMUs on the map. The labels are referring to the number of a report. Comparing the SOMs we could identify one bigger group on top, another on bottom and a darker area between them. The original and PCA50 map seem to be rather similar but in case of the PCA95 left-right sides are interchanged. On the bigger light area on top of the SOM there are located the bigger and main retail banks. At the bottom some smaller and niche banks are gathered. Experiment Neig. Orig PCA neig. Similar Total SMC Jaccard range neig. neig. neig. coef.
Orig vs. PCA 95% 1 1876 1772 1426 2222 0.9550 0.6418 Orig vs. PCA 50% 1 1876 2120 1480 2516 0.9414 0.5882 Orig vs. PCA 95% 2 4078 3972 3122 4928 0.8979 0.6335 Orig vs. PCA 50% 2 4078 4038 3046 5070 0.8856 0.6008
In Table 2 the similarity measurement coefficients of the banking data are given. The density of data items on the map is quite high and it is also visible in the number of neighborhood relations. There is a slight difference in the number of neighborhood relations between PCA95 and PCA50. It shows that the PCA retains the internal structure of the data items. The SMC value is very high in all cases, but is becoming lower if the neighborhood range is widening. The Jaccard coefficient shows about 0.6 similarities between the different representations of the data items.
As the banking data consisted of 133 data items the neighboring relations were much stronger than in case of linguistic data. In Fig. 5 there is given an isomorphic subgraph showing neighborhood relations between the SOM of original data and the SOM of PCA95. The neighborhood range is defined as 1. The graph illustrates quite well the structure within the data. The same grouping was visible on the graph representing only 50% of variations. When the neighborhood range was increased the isomorphic sub-graph became connected but at the same time the neighborhood relations became so dense that the structure was not clearly visible any more.
The analysis what is the impact of dimensionality reduction on the results gave us confirmation that even the dramatic dimensionality reduction by the PCA method retains the most important internal relations in the data. 7
The case studies give an overview of possibilities to measure similarity between
selforganizing maps that is based on the topology and neighborhood relations. We find
local neighborhood relations between the data items and measure the similarity of
relations by coefficients and by finding an isomorphic subgraph. There has been
proposed another method to evaluate two- or three-dimensional visualizations and to
measure distances between the two representations by Mandl and Eibl [
The similarity measurement coefficients together could give some additional information about the similarity. If the SMC value is high and at the same time the Jaccard coefficient has lower value then it indicates the presence of clustered structure. The bigger the difference is the smaller are the clusters. We can also use the SMC coefficient if the data items are exclusive like in the case of lexical study. There were two exclusive groups of data – positive and negative emotion concepts – that had weak neighborhood relations.
We expected to use the maximum isomorphic subgraph as a measure to identify the similarity between the SOMs, but it became a new meta-level tool to find hidden structure and to reveal the grouping structure of the data. The main parameter in similarity analysis is the range of neighborhood. The number of neighborhood relations increases if the number of data items or the range of the neighborhood widens. The size of a map has also an impact on the density of data items on the map. Depending on the density of the data items range 1 or 2 gives good insight into the hidden structure or the so-called backbone within the data. In both case studies the visualized isomorphic neighborhood matrix gave us a new perspective on relations between the data items.
In this paper, we have proposed a methodology to measure similarity between the self-organizing maps if the maps are describing the same phenomenon but use different sources of data or the number of variables are different. We illustrated the methodology by two sets of data. The results of the two case studies have shown us that the suggested method to measure similarity between two self-organizing map is applicable and it gives new insights into the data.
Acknowledgements. This study was supported by the Estonian Science Foundation, grant No 7149 and G5918.