An Introduction to Kohonen Self Organizing · PDF fileKohonen Self Organizing Maps Utility...
Transcript of An Introduction to Kohonen Self Organizing · PDF fileKohonen Self Organizing Maps Utility...
An Introduction to Kohonen SelfOrganizing Maps
Rajarshi Guha
Penn State University
An Introduction to Kohonen Self Organizing Maps – p.1/12
Kohonen Self OrganizingMaps
What are SOM’s ?Competitive Learning ANNElastic net of points which is made to fit theinput vector
Mode of FunctioningEach unit of the map recieves identicalinputsThe units compete for selectionThe selected neuron and surroundingneighbours get modifiedResults in groups of units becomingsensitized to features of the input vectors
An Introduction to Kohonen Self Organizing Maps – p.2/12
Kohonen Self OrganizingMaps
UtilityVisualizing N dimensional data in 2DDetecing similarity and degree’s ofsimilarity
An Introduction to Kohonen Self Organizing Maps – p.3/12
Structure of the Map
Square grid
Each grid point is avector containing thedescriptor values
The grid wraps roundthe edges
The grid is initializedwith random vectors
An Introduction to Kohonen Self Organizing Maps – p.4/12
Training the Map
Each descriptor vector in the training set ispresented to all grid points.
Select the closest matching grid point basedon minimum Euclidean distance
dsj =
√
√
√
√
m∑
i=1
(ssi − wji)
Modify the selected grid point and itsneighbours.
Degree of modification reduces with eachtraining iteration
Once all the training vectors have beenpresented, repeat the cycle.
An Introduction to Kohonen Self Organizing Maps – p.5/12
Modification of GridPoints, aka Learning
Selected grid point and topologically closegridpoints are modified
Leads to smoothing which leads to globalordering
The general form of modification is
mi(t + 1) = mi(t) + hci(t)[x(t) − mi(t)]
hci(t) is termed as the neighborhood function
As t → ∞, hci(t) → 0
An Introduction to Kohonen Self Organizing Maps – p.6/12
The NeighborhoodFunction
The neighborhood function can be
of various forms.
Neighborhood Sethci(t) = α(t) ifi ∈ Nc
hci(t) = 0 if i /∈ Nc
The size of the setdetermines howlarge an area willbe affected by theselected neuronIf it is initially toosmall, globalordering will notemerge.
An Introduction to Kohonen Self Organizing Maps – p.7/12
The NeighborhoodFunction
Gaussian Neighborhood Function
hci(t) = α(t) exp(
−‖rc−ri‖2
2σ2(t)
)
σ is the width of the smootheningCorresponds to the radius of theneighborhood set
An Introduction to Kohonen Self Organizing Maps – p.8/12
The Learning Factor
α(t) is termed the learning factor. Someproperties of the function are:
0 < α < 1
Decreases monotonically with time
When α = 0, learning stops
α can be varied in several waysConstant decrementFunction of training iterationRecursive - αi(t + 1) = αi(t)/(1 + hciαi(t))
An Introduction to Kohonen Self Organizing Maps – p.9/12
Classification
Assign an arbitrary class to the first grid point
For each grid point calculate distances toeach nearest neighbor
If the distance to a neighbor is less than auser specified threshold then the neighbor isin the same class as the grid point
Finally, find the closest matching neuron inthe training set for the initial grid point.Correspondingly modify class assignments ofall the other grid points
An Introduction to Kohonen Self Organizing Maps – p.10/12
References[1] Lingran, C.; Gasteiger, J. J. Am. Chem. Soc. 1997, 119, 4033.
[2] Gasteiger, J.; Zupan, J. Angew. Chem. Intl. Ed. Engl. 1993, 32, 503.
[3] Kohonen, T. Self Organizing Maps; Springer Series in Information SciencesSpringer: Espoo, Finland, 1994.
An Introduction to Kohonen Self Organizing Maps – p.12/12