Theoretical aspects of the SOM algorithm

Exploring Soil Pollution Patterns Using Self-Organizing Maps

The geochemical composition of bedrock is the key feature determining elemental concentrations in soil, followed by anthropogenic factors that have less impact. Concerning the latter, harmful effects on the trophic chain are increasingly affecting people living in and around urban areas. In the study area of the present survey, the municipalities of Cosenza and Rende (Calabria, southern Italy), topsoil were collected and analysed for 25 elements by inductively coupled plasma mass spectrometry (ICP-MS) in order to discriminate the different possible sources of elemental concentrations and define soil quality status. Statistical and geostatistical methods were applied to monitoring the concentrations of major oxides and minor elements, while the Self-Organizing Maps (SOM) algorithm was used for unsupervised grouping. Results show that seven clusters were identified-(I) Cr, Co, Fe, V, Ti, Al; (II) Ni, Na; (III) Y, Zr, Rb; (IV) Si, Mg, Ba; (V) Nb, Ce, La; (VI) Sr, P, Ca; (VII) As, Zn, Pb-according to soil elemental associations, which are controlled by chemical and mineralogical factors of the study area parent material and by soil-forming processes, but with some exceptions linked to anthropogenic input.

Global collaboration through local interaction in competitive learning

Feature maps, that preserve the global topology of arbitrary datasets, can be formed by self-organizing competing agents. So far, it has been presumed that global interaction of agents is necessary for this process. We establish that this is not the case, and that global topology can be uncovered through strictly local interactions. Enforcing uniformity of map quality across all agents results in an algorithm that is able to consistently uncover the global topology of diversely challenging datasets. The applicability and scalability of this approach is further tested on a large point cloud dataset, revealing a linear relation between map training time and size. The presented work not only reduces algorithmic complexity but also constitutes first step towards a distributed self organizing map.

Implications of Winter NAO Flavors on Present and Future European Climate

The North Atlantic Oscillation (NAO), a basic variability mode in the Northern Hemisphere, undergoes changes in its temporal and spatial characteristics, with significant implications on European climate. In this paper, different NAO flavors are distinguished for winter in simulations of a Coupled Atmosphere-Ocean GCM, using Self-Organizing Maps, a topology preserving clustering algorithm. These flavors refer to various sub-forms of the NAO pattern, reflecting the range of positions occupied by its action centers, the Icelandic Low and the Azores High. After having defined the NAO flavors, composites of winter temperature and precipitation over Europe are created for each one of them. The results reveal significant differences between NAO flavors in terms of their effects on the European climate. Generally, the eastwardly shifted NAO patterns induce a stronger than average influence on European temperatures. In contrast, the effects of NAO flavors on European precipitation anomalies are less coherent, with various areas responding differently. These results confirm that not only the temporal, but also the spatial variability of NAO is important in regulating European climate.

Analysis of Spatial Characteristics of Digital Signage in Beijing with Multi-Source Data

Digital signage is an important medium for urban outdoor advertising. Understanding the spatial distribution characteristics and factors that influence the site of digital signage are conducive to the efficient, standardized, and sustainable development of digital signage. The outdoor commercial digital signage within the Sixth Ring Road in Beijing is taken as the research object, and social network check-ins, housing prices, traffic network centrality and the mount of commercial facilities are considered factors that influence digital signage. The spatial distribution characteristics of digital signage are studied by using point pattern analysis methods. Moreover, we use three spatial clustering algorithms to study the hierarchical spatial characteristics of digital signage and test the effectiveness of the results. In addition, the factors that influence the distribution of digital signage are analyzed by Spearman correlation analysis. The results indicate that (1) the digital signage in Beijing generally presents a relatively concentrated distribution with centrality and forms an obvious gathering area and the agglomeration centers are mainly concentrated in the core parts of the central business district (CBD). (2) Digital signage is categorized into three groups, the traffic-oriented, the population-oriented, and the market-oriented. In addition, the spatial distribution of digital signage is consistent with the historical urban development of Beijing. (3) The social network check-ins with dynamic population characteristics had the highest correlation with the operation cost of digital signage. The spatial characteristics of digital signage evaluated in this study can effectively enhance the sustainable management of digital signage and provide a reference for research of the sustainable allocation of digital signage resources.

Urban BTEX Spatiotemporal Exposure Assessment by Chemometric Expertise

Normative regulations on benzene in fuels and urban management strategies are expected to improve air quality. The present study deals with the application of self-organizing maps (SOMs) in order to explore the spatiotemporal variations of benzene, toluene, ethylbenzene, and xylene levels in an urban atmosphere. Temperature, wind speed, and concentration values of these four volatile organic compounds were measured after passive sampling at 21 different sampling sites located in the city of Trieste (Italy) in the framework of a multi-year long-term monitoring program. SOM helps in defining pollution patterns and changes in the urban context, showing clear improvements for what concerns benzene, toluene, ethylbenzene, and xylene concentrations in air for the 2001-2008 timeframe.

Spatiotemporal dynamics of spring and stream water chemistry in a high-mountain area

The present study deals with the application of the self-organizing map (SOM) technique in the exploration of spatiotemporal dynamics of spring and stream water samples collected in the Chochołowski Stream Basin located in the Tatra Mountains (Poland). The SOM-based classification helped to uncover relationships between physical and chemical parameters of water samples and factors determining the quality of water in the studied high-mountain area. In the upper part of the Chochołowski Stream Basin, located on the top of the crystalline core of the Tatras, concentrations of the majority of ionic substances were the lowest due to limited leaching. Significantly higher concentration of ionic substances was detected in spring and stream samples draining sedimentary rocks. The influence of karst-type springs on the quality of stream water was also demonstrated.

Parameterless-Growing-SOM and Its Application to a Voice Instruction Learning System

An improved self-organizing map (SOM), parameterless-growing-SOM (PL-G-SOM), is proposed in this paper. To overcome problems existed in traditional SOM (Kohonen, 1982), kinds of structure-growing-SOMs or parameter-adjusting-SOMs have been invented and usually separately. Here, we combine the idea of growing SOMs (Bauer and Villmann, 1997; Dittenbach et al. 2000) and a parameterless SOM (Berglund and Sitte, 2006) together to be a novel SOM named PL-G-SOM to realize additional learning, optimal neighborhood preservation, and automatic tuning of parameters. The improved SOM is applied to construct a voice instruction learning system for partner robots adopting a simple reinforcement learning algorithm. User's instructions of voices are classified by the PL-G-SOM at first, then robots choose an expected action according to a stochastic policy. The policy is adjusted by the reward/punishment given by the user of the robot. A feeling map is also designed to express learning degrees of voice instructions. Learning and additional learning experiments used instructions in multiple languages including Japanese, English, Chinese, and Malaysian confirmed the effectiveness of our proposed system.

Local matrix learning in clustering and applications for manifold visualization

Electronic data sets are increasing rapidly with respect to both, size of the data sets and data resolution, i.e. dimensionality, such that adequate data inspection and data visualization have become central issues of data mining. In this article, we present an extension of classical clustering schemes by local matrix adaptation, which allows a better representation of data by means of clusters with an arbitrary spherical shape. Unlike previous proposals, the method is derived from a global cost function. The focus of this article is to demonstrate the applicability of this matrix clustering scheme to low-dimensional data embedding for data inspection. The proposed method is based on matrix learning for neural gas and manifold charting. This provides an explicit mapping of a given high-dimensional data space to low dimensionality. We demonstrate the usefulness of this method for data inspection and manifold visualization.

Seasonal and spatial variability of (210)Po, (238)U and (239+240)Pu levels in the river catchment area assessed by application of neural-network based classification

The present study deals with the application of self-organizing maps (SOM) in order to model, classify and interpret seasonal and spatial variability of (210)Po, (238)U and (239+240)Pu levels in the Vistula river basin. The data set represents concentration values for 3 alpha emitters ((210)Po, (238)U and (239+240)Pu) measured in surface water samples collected at 19 different sampling locations (8 in major Vistula stream while 11 in right or left Vistula tributaries) during four seasons (winter, spring, summer and autumn) in the framework of a one-year quality monitoring study. The advantages of an SOM algorithm, its classification and visualization ability for environmental data sets, are stressed. The neural-network based classification made it possible to reveal specific patterns related to both seasonal and spatial variability. In the middle and upper part of Vistula catchment as well as in the right-shore tributaries, concentrations of (210)Po and (238)U during summer and winter are the lowest. Concentrations of (210)Po and (238)U increase significantly during spring and autumn in the Vistula river catchment, especially in the delta of Vistula river. High concentration of anthropogenic originated (239+240)Pu indicates "site-specific" character of pollution in two large left-shore tributaries located in the middle part of the Vistula drainage area. Efficient classification of sampling locations could lead to an optimization of river radiochemical sampling networks and to a better tracing of natural and anthropogenic changes along Vistula river stream.

Identification and control of dynamical systems using the self-organizing map

In this paper, we introduce a general modeling technique, called vector-quantized temporal associative memory (VQTAM), which uses Kohonen's self-organizing map (SOM) as an alternative to multilayer perceptron (MLP) and radial basis function (RBF) neural models for dynamical system identification and control. We demonstrate that the estimation errors decrease as the SOM training proceeds, allowing the VQTAM scheme to be understood as a self-supervised gradient-based error reduction method. The performance of the proposed approach is evaluated on a variety of complex tasks, namely: i) time series prediction; ii) identification of SISO/MIMO systems; and iii) nonlinear predictive control. For all tasks, the simulation results produced by the SOM are as accurate as those produced by the MLP network, and better than those produced by the RBF network. The SOM has also shown to be less sensitive to weight initialization than MLP networks. We conclude the paper by discussing the main properties of the VQTAM and their relationships to other well established methods for dynamical system identification. We also suggest directions for further work.

1H NMR metabonomics of plasma lipoprotein subclasses: elucidation of metabolic clustering by self-organising maps

(1)H NMR spectra of plasma are known to provide specific information on lipoprotein subclasses in the form of complex overlapping resonances. A combination of (1)H NMR and self-organising map (SOM) analysis was applied to investigate if automated characterisation of subclass-related metabolic interactions can be achieved. To reliably assess the intrinsic capability of (1)H NMR for resolving lipoprotein subclass profiles, sum spectra representing the pure lipoprotein subclass part of actual plasma were simulated with the aid of experimentally derived model signals for 11 distinct lipoprotein subclasses. Two biochemically characteristic categories of spectra, representing normolipidaemic and metabolic syndrome status, were generated with corresponding lipoprotein subclass profiles. A set of spectra representing a metabolic pathway between the two categories was also generated. The SOM analysis, based solely on the aliphatic resonances of these simulated spectra, clearly revealed the lipoprotein subclass profiles and their changes. Comparable SOM analysis in a group of 69 experimental (1)H NMR spectra of serum samples, which according to biochemical analyses represented a wide range of lipoprotein lipid concentrations, corroborated the findings based on the simulated data. Interestingly, the choline-N(CH(3))(3) region seems to provide more resolved clustering of lipoprotein subclasses in the SOM analyses than the methyl-CH(3) region commonly used for subclass quantification. The results illustrate the inherent suitability of (1)H NMR metabonomics for automated studies of lipoprotein subclass-related metabolism and demonstrate the power of SOM analysis in an extensive and representative case of (1)H NMR metabonomics.

Concerning the differentiability of the energy function in vector quantization algorithms

The adaptation rule of Vector Quantization algorithms, and consequently the convergence of the generated sequence, depends on the existence and properties of a function called the energy function, defined on a topological manifold. Our aim is to investigate the conditions of existence of such a function for a class of algorithms including the well-known 'K-means' and 'Self-Organizing Map' algorithms. The results presented here extend several previous studies and show that the energy function is not always a potential but at least the uniform limit of a series of potential functions which we call a pseudo-potential. It also shows that a large number of existing vector quantization algorithms developed by the Artificial Neural Networks community fall into this class. The framework we define opens the way to studying the convergence of all the corresponding adaptation rules at once, and a theorem gives promising insights in that direction.

Segmentation and grading of brain tumors on apparent diffusion coefficient images using self-organizing maps

An accurate computer-assisted method to perform segmentation of brain tumor on apparent diffusion coefficient (ADC) images and evaluate its grade (malignancy state) has been designed using a mixture of unsupervised artificial neural networks (ANN) and hierarchical multiresolution wavelet. Firstly, the ADC images are decomposed by multiresolution wavelets, which are subsequently selectively reconstructed to form wavelet filtered images. These wavelet filtered images along with FLAIR and T2 weighted images have been utilized as the features to unsupervised neural network - self organizing maps (SOM) - to segment the tumor, edema, necrosis, CSF and normal tissue and grade the malignant state of the tumor. A novel segmentation algorithm based on the number of hits experienced by Best Matching Units (BMU) on SOM maps is proposed. The results shows that the SOM performs well in differentiating the tumor, edema, necrosis, CSF and normal tissue pattern vectors on ADC images. Using the trained SOM and proposed segmentation algorithm, we are able to identify high or low grade tumor, edema, necrosis, CSF and normal tissue. The results are validated against manually segmented images and sensitivity and the specificity are observed to be 0.86 and 0.93, respectively.

Understanding and reducing variability of SOM neighbourhood structure

The self-organizing map (SOM) is a nonlinear unsupervised method for vector quantization. In the context of classification and data analysis, the SOM technique highlights the neighbourhood structure between clusters. The correspondence between this clustering and the input proximity is called the topology preservation. We present here a stochastic method based on bootstrapping in order to increase the reliability of the induced neighbourhood structure. Considering the property of topology preservation, a local approach of variability (at an individual level) is preferred to a global one. The resulting (robust) map, called R-map, is more stable relatively to the choice of the sampling method and to the learning options of the SOM algorithm (initialization and order of data presentation). The method consists of selecting one map from a group of several solutions resulting from the same self-organizing map algorithm, but obtained with various inputs. The R-map can be thought of as the map, among the group of solutions, corresponding to the most common interpretation of the data set structure. The R-map is then the representative of a given SOM network, and the R-map ability to adjust the data structure indicates the relevance of the chosen network.

Self-organizing map algorithm and distortion measure

We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of the distortion measure. If we assume that the observations and the map lie in a compact Euclidean space, we prove the strong consistency of the map which almost minimizes the empirical distortion. Moreover, after calculating the derivatives of the theoretical distortion measure, we show that the points minimizing this measure and the equilibria of the Kohonen map do not match in general. We illustrate, with a simple example, how this occurs.

Unfolding preprocessing for meaningful time series clustering

Clustering methods are commonly applied to time series, either as a preprocessing stage for other methods or in their own right. In this paper it is explained why time series clustering may sometimes be considered as meaningless. This problematic situation is illustrated for various raw time series. The unfolding preprocessing methodology is then introduced. The usefulness of unfolding preprocessing is illustrated for various time series. The experimental results show the meaningfulness of the clustering when applied on adequately unfolded time series.

Batch and median neural gas

Neural Gas (NG) constitutes a very robust clustering algorithm given Euclidean data which does not suffer from the problem of local minima like simple vector quantization, or topological restrictions like the self-organizing map. Based on the cost function of NG, we introduce a batch variant of NG which shows much faster convergence and which can be interpreted as an optimization of the cost function by the Newton method. This formulation has the additional benefit that, based on the notion of the generalized median in analogy to Median SOM, a variant for non-vectorial proximity data can be introduced. We prove convergence of batch and median versions of NG, SOM, and k-means in a unified formulation, and we investigate the behavior of the algorithms in several experiments.

Magnification control in self-organizing maps and neural gas

We consider different ways to control the magnification in self-organizing maps (SOM) and neural gas (NG). Starting from early approaches of magnification control in vector quantization, we then concentrate on different approaches for SOM and NG. We show that three structurally similar approaches can be applied to both algorithms that are localized learning, concave-convex learning, and winner-relaxing learning. Thereby, the approach of concave-convex learning in SOM is extended to a more general description, whereas the concave-convex learning for NG is new. In general, the control mechanisms generate only slightly different behavior comparing both neural algorithms. However, we emphasize that the NG results are valid for any data dimension, whereas in the SOM case, the results hold only for the one-dimensional case.

The Parameterless Self-Organizing Map Algorithm

The parameterless self-organizing map (PLSOM) is a new neural network algorithm based on the self-organizing map (SOM). It eliminates the need for a learning rate and annealing schemes for learning rate and neighborhood size. We discuss the relative performance of the PLSOM and the SOM and demonstrate some tasks in which the SOM fails but the PLSOM performs satisfactory. Finally we discuss some example applications of the PLSOM and present a proof of ordering under certain limited conditions.

Optimization via Intermittency with a Self-Organizing Neural Network

One of the major obstacles in using neural networks to solve combinatorial optimization problems is the convergence toward one of the many local minima instead of the global minima. In this letter, we propose a technique that enables a self-organizing neural network to escape from local minima by virtue of the intermittency phenomenon. It gives rise to novel search dynamics that allow the system to visit multiple global minima as meta-stable states. Numerical experiments performed suggest that the phenomenon is a combined effect of Kohonen-type competitive learning and the iterated softmax function operating near bifurcation. The resultant intermittent search exhibits fractal characteristics when the optimization performance is at its peak in the form of 1/f signals in the time evolution of the cost, as well as power law distributions in the meta-stable solution states. The N-Queens problem is used as an example to illustrate the meta-stable convergence process that sequentially generates, in a single run, 92 solutions to the 8-Queens problem and 4024 solutions to the 17-Queens problem.

Stability of generalized topographic mappings between cell layers through correlational learning

We propose a simple topographic mapping formation model from a cell layer to a cell layer. Our model is a discrete one in that the state value of input and output cells takes 0 or 1 and input and output layers are represented by undirected graphs. A binary input pattern can be given to the network consisting of input and output cell layers. Such an input pattern can be represented by a subset of input cells. That is, a state value of an input cell takes 1 if a cell belongs to the subset, otherwise, a state value of an input cell is 0. Such a definition of an input pattern does not necessarily assume a short-range excitatory mechanism in an input layer. Thus, a topographic mapping described in this model is a map, which preserves the input pattern relation. By using the concept of input pattern separability, we showed an existence condition of certain learning rules, which are correlational. We have paid special attention to such correlational type learning rules, and have shown under the rules that topographic mappings are the only stable ones. As to the non-correlational learning rules, we also investigate the stability of generated mappings.

Double quantization of the regressor space for long-term time series prediction: method and proof of stability

The Kohonen self-organization map is usually considered as a classification or clustering tool, with only a few applications in time series prediction. In this paper, a particular time series forecasting method based on Kohonen maps is described. This method has been specifically designed for the prediction of long-term trends. The proof of the stability of the method for long-term forecasting is given, as well as illustrations of the utilization of the method both in the scalar and vectorial cases.

Self-organizing maps and clustering methods for matrix data

In this contribution we present extensions of the Self Organizing Map and clustering methods for the categorization and visualization of data which are described by matrices rather than feature vectors. Rows and Columns of these matrices correspond to objects which may or may not belong to the same set, and the entries in the matrix describe the relationships between them. The clustering task is formulated as an optimization problem: Model complexity is minimized under the constraint, that the error one makes when reconstructing objects from class information is fixed, usually to a small value. The data is then visualized with help of modified Self Organizing Maps methods, i.e. by constructing a neighborhood preserving non-linear projection into a low-dimensional "map-space". Grouping of data objects is done using an improved optimization technique, which combines deterministic annealing with "growing" techniques. Performance of the new methods is evaluated by applying them to two kinds of matrix data: (i) pairwise data, where row and column objects are from the same set and where matrix elements denote dissimilarity values and (ii) co-occurrence data, where row and column objects are from different sets and where the matrix elements describe how often object pairs occur.