Self-Organizing Maps (SOM) - Utrecht · PDF fileKV Spezielle Kapitel aus Informatik:...
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KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 1
Univ.-Ass. Dr. Markus Schedl
Department of Computational PerceptionJohannes Kepler University Linz
[email protected]://www.cp.jku.at
Self-Organizing Maps (SOM)
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 2
Overview
Basics
Sequential Training (On-Line Learning)
Batch SOM
Visualizing the SOM- SOM Grid- Music Description Map (MDM)- Bar Plots and Chernoff's Faces- U-Matrix and Distance Matrix- Smoothed Data Histogram (SDH)- Component Planes
Growing Hierarchical SOM
Aligned SOM
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 3
SOM:• "neural network" [Kohonen, 1982], [Kohonen, 2001]• SOM ~ k-means clustering + topology preservation• preservation of non-linear relationship between data items
Self-Organizing Map (SOM): Basics
Interpretation:• clustering with topology constraints (similar data items should be placed
close to each other on the map)• mapping from data/feature/input space to low-dim. visualization space
Basic Architecture:• Map: 2-dimensional array of interconnected units ("neurons")• connections define fixed topology "neighborhood"• units represent cluster centers
(prototypes, "model vectors", "weight vector", "reference vector")
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 4
+ tighter relationship between clusters+ more connections+ grid structure fits Gaussian structure in neighborhood kernel calculation (centroids of neighboring map units are equidistant)
Different Topologies / Grid Structures
+ easier to implement– diagonally neighboring map units do not perfectly fit to Gaussian neighborhood function
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 5
An Application of the Self-Organizing Map: The nepTune Interface
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© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 6
Input:• map of units ui with model vectors mi ("codebook") • training instances X = {xi}• a similarity measure sim(.,.) between data items (e.g., Euclidean distance)• parameters: α(t) (learning rate ∈ [0..1]) and
a neighborhood kernel function with parameter r(t) (‘neighborhood radius’),e.g., pseudo-Gaussian (dij = map distance btw. ui, uj)
On-line Learning: The Online Training Algorithm
Initialize each unit (model vector) mi to represent a randomly selected data item
Loop over time steps t, until convergence:1. Randomly select an example x2. Find the ‘winning unit’ (best matching unit) uc with mc = maxi(sim(mi,x))3. Adapt model vectors of all units as
mi(t +1) = mi(t) + α(t) · uic(t) · [x−mi (t)]
4. Update (decrease) training parameters α(t), r(t)
Online SOM Training Algorithm (one possible variant):
))(exp()( 22 trdtu ijij −=
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 7
Input:• map of units ui with model vectors mi• training instances X = {xi}• a similarity measure between examples (e.g., Euclidean distance)• a neighborhood kernel function with parameter r(t) (‘neighborhood radius’),
e.g., pseudo-Gaussian (dij = map distance btwn. ui, uj)
Off-line Learning: The Batch SOM Algorithm
Initialize each unit (model vector) mi to represent a randomly selected data item
Loop over time steps t, until convergence:1. Determine the best matching unit uc(i) for each data item xi (i.e., assign
each instance to its most similar model vector) → “Voronoi set”2. Update each model vector mi to better fit the data items assigned to it
and the data in its neighborhood:
3. Update (decrease) neighborhood radius r(t)
Batch SOM Training Algorithm (one possible variant):
))(exp()( 22 trdtu ijij −=
)()(
)1()(
)(
tutu
tk kic
k kkici ∑
∑ ⋅=+
xm
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 8
SOM: Illustration
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 9
• each data point (example) x uniquely ‘belongs to’ a unit (the BMU of x)• relationship between units: neighboring units cover similar data items• non-uniform distances between model vectors, uniform distances in visualization• "interpolation units" (units with no data associated) are possible
SOM: Illustration
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 10
• Random Initialization: - random values in same range as X (between min and max of each dimension)- randomly select data items from X and assign them to model vectors mi
+ fast– mapping not consistent for different runs
• Linear Initialization:perform Eigendecomposition of autocorrelation matrix of X → PCAtop 2 Eigenvectors (with largest Eigenvalues) span a 2-dimensional subspaceinitialize model vectors along these Eigenvectors→ predefined linear mapping to start with
+ mapping consistent for different runs (up to rotation / mirroring)– computationally more complex
Initialization of the Model Vectors
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© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 11
Example: WebSOM Project
Support in Browsing (Potentially Huge) Data Sets
[Kaski et al., 1998]
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 12
Example: Browsing Music Collections
„ViSMuC“ by Schedl, 2004
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© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 13
Example: Browsing Music Collections (II)
„PlaySOM“, IFS @
TU Wien, 2005
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 15
Visualizing attribute distributions on top of a learned SOM:
• Component Planes: visualize feature values of model vectorsassociated with the map units (or averaged feature values overall instances covered by a unit)
• Bar Charts or Chernoff's Faces: visualize all dimensions of model vectors for each map unit in one plot
[Vesanto, 1999], [Vesanto, 2002]
Visualizing the SOM
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 16
Visualizing the data distribution on top of a learned SOM:
• SOM-Grid: each data item is displayed within its BMU
• Music Description Map (MDM): aggregates similar map units and add descriptive labels
[Knees et al., 2006]
• U-Matrix: visualizes distances between units (via color)
• Distance Matrix: visualize aggregated distances of model vectors to all neighboring units
[Vesanto, 1999], [Vesanto, 2002]
• Smoothed Data Histogram (SDH):visualizes (smoothed) density of data items in an area
[Pampalk et al., 2002]
Visualizing the SOM
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 17
HorseZebraCow
DuckGoose
DoveChicken
TigerLion
OwlHawk
FoxDogWolf
Cat
Eagle
Small Medium Big 2-Legs
4-Legs Hair Hooves Mane
Feathers Hunt Run Fly
Swim
- explain mapping (labeling)- make correlations between attributes visible
Component Planes
Visualizing Attribute Distributions: Component Planes
Learned Map
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© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 19
each attribute value (dimension in data space)is displayed via a bar in a d-dimensional barchart visualization for each map unit
Visualizing Attribute Distributions: Bar Plots
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© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 20
psychologically motivated visualization method(people can quickly grasp a face's expression)
each attribute value (dimension in data space)is mapped to a specific property of the Chernoff face (e.g., mouth's contour, height/width of face, ear's slope, …)
Visualizing Attribute Distributions: Chernoff's Faces
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 21
Visualizing the data distribution on top of a learned SOM:
• SOM-Grid: each data item is displayed within its BMU
• Music Description Map (MDM): aggregates similar map units and add descriptive labels
[Knees et al., 2006]
• U-Matrix: visualizes distances between units (via color)
• Distance Matrix: visualize aggregated distances of model vectors to all neighboring units
[Vesanto, 1999], [Vesanto, 2002]
• Smoothed Data Histogram (SDH):visualizes (smoothed) density of data items in an area
[Pampalk et al., 2002]
Visualizing the SOM
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 22
SOM Grid for data set C103a: co-occurrences of artist names
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 23
SOM Grid for data set C103a: co-occurrences of artist names (II)KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 24
SOM Grid for larger data set
2572 songs, 7 genres, features: MFCCs
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 25
SOM Grid for larger data set: Aggregate data items using metadata
metadata available → summarize items w.r.t. properties (e.g., genre)
loss of information: Dance?
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 26
Music Description Map (MDM)[Knees et al., 2006]
- extension of the simple SOM grid- describes regions of the map by metadata- aggregates "similar" neighboring map units via region growing algorithm
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 27
uAk 0∈
determining the goodness G2t,u of a term t for map unit u
according to [Lagus, Kaski, 1999]:
if Manhattan distance between units u and k < threshold r0
if Manhattan distance between units u and i: r0 < d(u,i) < r1
filter all terms t with G2t,u < 0.01
cut-off of 30 keywords per map unit
MDM (II): Labeling Map Units
[Lagus, Kaski, 1999]
( )∑∑
∉
∈=u
u
Ai it
Ak ktut
F
FG
1
0
,
2,
,2
∑ ∑∑
⋅
⋅=
v ava ua
ata uaut tff
tffF
,,
,,,
uAi 1∈
#tracks of artist a on unit u
term frequency ofterm t for artist a
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 28
1. sort all units w.r.t. G2-values of contained terms → U
2. remove highest ranked unit u ∈ U,find similarly labeled units among u's neighbors→ if cosine similarity between label vectors of map unit u and its neighbors i < threshold θ, aggregate u and i
3. goto 2
MDM (II): Connecting Similar Map Units
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 29
U-Matrix: visualizes distances between units (model vectors)
Distance Matrix: visualizes difference of a unit's model vector to all neighboring units' model vectors
Visualizing Data Distributions: U-Matrix and Distance Matrix
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 30
Two methods for visualizing data on top of a learned SOM:• U-Matrix: visualizes distances between units (via color)• Smoothed Data Histogram (SDH):
visualizes (smoothed) density of examples in an area of the map
Visualizing Data Distributions: U-Matrix and SDH
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 31
• display smoothed density of data items associated with areas of the map• reveal clusters in the data• many pieces associated with a unit → cluster center
Smoothed Data Histograms (SDH)
Idea for smoothing / density estimation:
• voting matrix whose size equals size of SOM• data items “vote” for a number N of best-matching units• best-matching unit gets N points, 2nd best gets N-1 points, …• N-th best gets 1 point, all others get 0 points (N is parameter, ‘spread’)• the distribution of votes is visualized over the entire map, e.g.,
via a color map (interpolated voting matrix for smoothing)
[Pampalk et al., 2002]
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 32
Data S
paceVisualization S
pace
SOM and SDH: An Example
N=1 N=2 N=5
N=7 N=10 N=20
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 33
Be aware of influence of color scale on perception!
Smoothed Data Histograms
Matlab implementations of SOMs and SDHs (Toolboxes):http://www.cis.hut.fi/projects/somtoolbox/ (Google: “SOM Toolbox”)http://www.oefai.at/~elias/sdh/ (Google: “SDH Toolbox”)
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 34
SOM and SDH – A Sample Application nepTune
• Input: music collection (digital audio files)• calculate audio features for each track, e.g.
• rhythmic [Pampalk, Islands of Music: Analysis, Organization, andVisualization of Music Archives, Diploma Thesis 2001]
• timbral [Mandel & Ellis, Song-Level Features and Support Vector Machinesfor Music Classification, ISMIR 2005]
• train a SOM on audio features• calculate an SDH on the SOM• visualize SDH in 3D using smoothed voting matrix of SDH as height values• build a game-like user interface to explore the user’s
(or someone else’s) music collection
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nepTune (2)
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nepTune (3)
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nepTune (4)
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nepTune (5)
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Hierarchical Structuring:The Growing Hierarchical Self-Organizing Map (GHSOM)
Flat SOM:
Hierarchical SOM:
[Dittenbach et al., 2002]
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Start with 1 unit to expand (= mean of data), level 0Loop until no more units to expand
1. For each unit to expand create new 2x2 SOM (initialize orientation)2. Train SOM on data assigned to ‘parent unit’3. Decision 1: Insert new row or column?
If yes: insert new row/column and goto 2 4. Decision 2: hierarchically expand units of map?
If yes: add units to expand list
Decision 1: Insert new row or column if mean quantization error > threshold(i.e., map does not represent the data well); insert new row or column between unit with highest quantizationerror and adjacent unit with largest distance
Decision 2: Expand unit if quantization error of unit > threshold(i.e., unit does not represent its associated data items well)
Parameters: same as SOM (except no. of units) + 2 thresholds τ1, τ2
The GHSOM Algorithm
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 42
The GHSOM Algorithm:Decisions 1 and 2
∑∈
−=iVk
iki
i ||||||
1 mxV
MQE }|{ )( ikci k uuV ==
Mean quantization error of unit ui:
where
quantifies how well a unit i approximates ‘its’ data items
Mean quantization error of a SOM:
ii
i
nV MQEMMQE ⋅= ∑ ||
quantifies how well a SOM approximates the data items
Voronoi set Vi of unit ui:
all data items whose BMU is ui
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 43
Decision 2 : Expand unit if
MQEi > τ2MQE0*
where MQE0* is the mean quantization error of the whole dataset
with respect to the virtual unit located in the center of the whole dataset
(in contrast to MQE0, which is the mean quantization error of thedata items in the respective sub-branch of the GHSOM)
Decision 1 : Insert new row or column if MMQE > τ1MQE0where MQE0 is the MQE of a virtual unit m0 representing themean of all instances covered by the parent unit:
The GHSOM Algorithm:Decisions 1 (enlarge map) and 2 (insert new map)
Generally: τ1, τ2 are chosen such that 1 > τ1 >> τ2 > 0.
∑=i
i nxm0
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 44
The GHSOM Algorithm:Preservation of Orientation
Problem:• maps of descendants of a unit ui could have arbitrary orientation
→ no visible relationship between different sub-branches (other thancommon parent map)
Solution:• enforce/encourage a specific orientation of the sub-layer SOMs via
initialization
• initialize the model vectors of the 2x2 SOMs such that they correspondto the orientation of the parent map
• for example: initialize the 4 model vectors with the means of theparent vector and each of its 4 immediate neighbors
• for border units: extrapolate "virtual" units.Example: if ui is located on the left border and the unit to its right is ur,create virtual left neighbor ul with ml = mi + (mi – mr)
Exercise: How could the initialization function for the codebook of a new sublevel SOM expressed as weighted parent unit(s' neighbors) look like?
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 45
GHSOM on Animals
Hierarchical Component PlanesHierarchical Map
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GHSOM + SDH: deepTune
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GHSOM + SDH: deepTune (II)
Different HierarchyLevels
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Basic concepts:
• Goal: understand relationship between different ways of representingthe same data
• layers of mutually constrained SOMs (i.e., a stack of SOMs)• each layer trained on slightly different data space / view of the data
(i.e., different dimensions or distance definitions), but same data items• trained so that all layers have same orientation• constraints between layers to enforce smooth transitions between views
Visualizing Effects of Changes in Data Definition:Aligned SOMs
Use:• exploratory analysis of alternative data representations• visualize changes in the inherent structure of the data in response to
changes in features, relative feature weights, different ways of normalizing features, different similarity functions, ...
→ navigation through alternative data spaces
[Pampalk et al., 2003]
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
© 2010 Markus Schedl, partly based on material by Gerhard Widmer and Peter Knees 50
pmaxpmin
Stack of SOMs
Parameter Values(define different views of the data)
Distance between layers (relative to distance betweenunits in same layer)E.g., intra-SOM distance between neighboring units = 1inter-SOM distance "between" same map unit = 1/5
…
…
…
Aligned SOMs: The Basic Architecture
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
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• Initialize all layers• Loop
– Randomly select training instance x and layer l– Find best matching unit for x in l– Adapt neighborhood of best matching unit (intra- and inter-layer
neighborhood)
within layer
between layers
Neighborhood:
Aligned SOM: Training(Online version, simplified)
KV Spezielle Kapitel aus Informatik: Exploratory Data Analysis Self-Organizing Maps
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Input:• map of units uli with model vectors mli ("codebook"), l…layer• training instances X = {xi}• a similarity measure sim(.,.) between data items (e.g., Euclidean distance)• parameters: α(t) (learning rate ∈ [0..1]) and
a neighborhood kernel function with parameter r(t) (‘neighborhood radius’),e.g., pseudo-Gaussian (dij = map distance btw. uli, ukj)
Aligned SOM: On-line Learning
Initialize each unit (model vector) mli to represent a randomly selected data item (features weighted according to layer-specific weights, e.g., from 1:0 to 0:1)
Loop over time steps t, until convergence:1. Randomly select an example x and a layer l; apply weighting according
to view/data space of l → xl2. Find the ‘winning unit’ (best matching unit) uc with mc = maxi(sim(mli,xl))3. Adapt model vectors of all units in all layers as
mli(t +1) = mli(t) + α(t) · uic(t) · [xl−mli (t)]
4. Update (decrease) training parameters α(t), r(t)
Online SOM Training Algorithm:))(exp()( 22 trdtu ijij −=
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Aligned SOM on Animals
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http://www.ofai.at/~elias.pampalk/kdd03/IoMv2
http://www.ofai.at/~elias.pampalk/kdd03/animals
Aligned SOM Demos
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LiteratureSOM:[Kohonen, 1982]: Kohonen, T. Self-Organizing Formation of Topologically Correct Feature Maps. Biological Cybernetics, 43:59–69.[Kohonen, 2001]: Kohonen, T. Self-Organizing Maps, volume 30 of Springer Series in Information Sciences. Springer, Berlin, Germany, 3rd edition.[Vesanto, 1999]: Vesanto, J. SOM-Based Data Visualization Methods. Intelligent Data Analysis 3(2):111–126.[Vesanto, 2002]: Vesanto, J. Data Exploration Process Based on the Self-Organizing Map. PhD thesis, Helsinki University of Technology, Espoo, Finland.[Pampalk et al., 2002]: Pampalk, E., Rauber, A., and Merkl, D. Using Smoothed Data Histograms for Cluster Visualization in Self-Organizing Maps. In Proceedings of the International Conference on Artificial Neural Networks (ICANN 2002), Madrid, Spain. Springer.[Knees et al., 2006]: Knees, P., Pohle, T., Schedl, M., and Widmer, G. Automatically Describing Music on a Map. In Proceedings of the 2nd Workshop on Learning the Semantics of Audio Signals (LSAS 2008), Paris, France, June 2008.[Kaski et al., 1998]: WEBSOM – Self-Organizing Maps of Document Collections, Neurocomputing 21, 1998.
[Lagus, Kaski, 1999]: Keyword Selection Method for Characterising Text Document Maps, In Proceedings of the International Conference on Artificial Neural Networks (ICANN 1999), London, UK.
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Literature (II)GHSOM:[Dittenbach et al., 2002]: Dittenbach, M., Rauber, A., and Merkl, D. Uncovering Hierarchical Structure in Data Using the Growing Hierarchical Self-Organizing Map. Neurocomputing, 48(1–4):199–216.
Aligned SOM:[Pampalk et al. 2003]: Pampalk, E., Goebl, W., Widmer, G. Visualizing Changes in the Structure of Data for Exploratory Feature Selection, In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2003).