Editing Self-organizing map (section)

== Interpretation ==
[[File:SOMsPCA.PNG|thumb|One-dimensional SOM versus principal component analysis (PCA) for data approximation. SOM is a red [[broken line]] with squares, 20 nodes. The first principal component is presented by a blue line. Data points are the small grey circles. For PCA, the [[fraction of variance unexplained]] in this example is 23.23%, for SOM it is 6.86%.<ref>Illustration is prepared using free software: Mirkes, Evgeny M.; [http://www.math.le.ac.uk/people/ag153/homepage/PCA_SOM/PCA_SOM.html ''Principal Component Analysis and Self-Organizing Maps: applet''], University of Leicester, 2011</ref>]]

There are two ways to interpret a SOM. Because in the training phase weights of the whole neighborhood are moved in the same direction, similar items tend to excite adjacent neurons. Therefore, SOM forms a semantic map where similar samples are mapped close together and dissimilar ones apart. This may be visualized by a [[U-Matrix]] (Euclidean distance between weight vectors of neighboring cells) of the SOM.<ref name="UltschSiemon1990">{{cite book |first1= Alfred |last1= Ultsch |first2= H. Peter |last2= Siemon |chapter= Kohonen's Self Organizing Feature Maps for Exploratory Data Analysis |title= Proceedings of the International Neural Network Conference (INNC-90), Paris, France, July 9–13, 1990 |pages= [https://archive.org/details/innc90parisinter0001inte/page/305 305–308] |editor1-first= Bernard |editor1-last= Widrow |editor2-first= Bernard |editor2-last= Angeniol |publisher= Kluwer |location= Dordrecht, Netherlands |year= 1990 |volume= 1 |isbn= 978-0-7923-0831-7 |chapter-url= http://www.uni-marburg.de/fb12/datenbionik/pdf/pubs/1990/UltschSiemon90 |url= https://archive.org/details/innc90parisinter0001inte/page/305 }}</ref><ref name="Ultsch2003">{{cite tech report |last=Ultsch |first=Alfred |year=2003 |title=U*-Matrix: A tool to visualize clusters in high dimensional data |publisher=Department of Computer Science, University of Marburg |url=http://www.uni-marburg.de/fb12/datenbionik/pdf/pubs/2003/ultsch03ustar |id=36 |pages=1-12}}</ref><ref>{{cite conference |last1=Saadatdoost |first1=Robab |first2=Alex Tze Hiang |last2=Sim |last3=Jafarkarimi |first3=Hosein |title=Application of self organizing map for knowledge discovery based in higher education data |book-title=Research and Innovation in Information Systems (ICRIIS), 2011 International Conference on |publisher=IEEE |date=2011 |doi=10.1109/ICRIIS.2011.6125693 |isbn=978-1-61284-294-3}}</ref>

The other way is to think of neuronal weights as pointers to the input space. They form a discrete approximation of the distribution of training samples. More neurons point to regions with high training sample concentration and fewer where the samples are scarce.

SOM may be considered a nonlinear generalization of [[Principal components analysis]] (PCA).<ref>{{cite book |last=Yin |first=Hujun |chapter=Learning Nonlinear Principal Manifolds by Self-Organising Maps |title={{harvnb|Gorban|Kégl|Wunsch|Zinovyev|2008}}}}</ref> It has been shown, using both artificial and real geophysical data, that SOM has many advantages<ref>{{cite journal | last1 = Liu | first1 = Yonggang | last2 = Weisberg | first2 = Robert H | year = 2005 | title = Patterns of Ocean Current Variability on the West Florida Shelf Using the Self-Organizing Map | journal = Journal of Geophysical Research | volume = 110 | issue = C6 | page = C06003 | doi = 10.1029/2004JC002786 | bibcode = 2005JGRC..110.6003L | doi-access = free }}</ref><ref>{{cite journal | last1 = Liu | first1 = Yonggang | last2 = Weisberg | first2 = Robert H. | last3 = Mooers | first3 = Christopher N. K. | year = 2006 | title = Performance Evaluation of the Self-Organizing Map for Feature Extraction | journal = Journal of Geophysical Research | volume = 111 | issue = C5 | page = C05018 | doi = 10.1029/2005jc003117 | bibcode = 2006JGRC..111.5018L | doi-access = free }}</ref> over the conventional [[feature extraction]] methods such as Empirical Orthogonal Functions (EOF) or PCA. Additionally, researchers found that Clustering and PCA reflect different facets of the same local feedback circuit of human brain, with the SOM providing the shared learning rules that guide both processes. In other words, Clustering and PCA synergize via SOM. <ref>Liu, C., Bowen, E. F. W., & Granger, R. (2025). A formal relation between two disparate mathematical algorithms is ascertained from biological circuit analyses. bioRxiv. https://doi.org/10.1101/2025.03.28.645962</ref>

Originally, SOM was not formulated as a solution to an optimisation problem. Nevertheless, there have been several attempts to modify the definition of SOM and to formulate an optimisation problem which gives similar results.<ref>{{cite book |last=Heskes |first=Tom |chapter=Energy Functions for Self-Organizing Maps |editor-last=Oja |editor-first=Erkki |editor2-last=Kaski |editor2-first=Samuel |title=Kohonen Maps |publisher=Elsevier |date=1999 |pages=303–315 |doi=10.1016/B978-044450270-4/50024-3 |isbn=978-044450270-4}}</ref> For example, [[Elastic map]]s use the mechanical metaphor of elasticity to approximate [[Nonlinear dimensionality reduction#Principal curves and manifolds|principal manifolds]]:<ref>{{cite book |editor-link=Alexander Nikolaevich Gorban |editor-last=Gorban |editor-first=Alexander N.  |editor2-last=Kégl |editor2-first=Balázs |editor3-last=Wunsch |editor3-first=Donald C. |editor4-last=Zinovyev |editor4-first=Andrei |url=https://www.researchgate.net/publication/271642170  |title=Principal Manifolds for Data Visualization and Dimension Reduction |series=Lecture Notes in Computer Science and Engineering  |volume=58 |publisher=Springer |date=2008 |isbn=978-3-540-73749-0}}</ref> the analogy is an elastic membrane and plate.