SelfOrganizing Maps

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Self-Organizing Maps
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Basic Theory of SOMs

• Kinda a constrained K-means clustering
  (particularly matches Hasties ch. 13
  description)
• K-means where cluster centers are on smooth
  manifold
• A nearest-neighbor model designed to project
  multi-dimensional data points down to a
  low-dimensional space

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The Basic SOM Algo

• Take a bunch of high-dimensional data points and
  a low-dimensional graph (typically a 2-d grid)
• Each vertex of the graph has an associated vector
  that lives in the high-dimensional space
• The vector initially has random weights

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The Basic SOM Algo Cont

• Each iteration take a data point and move each
  vector towards that point based on how close the
  vector is to it
• OR just move vectors that are neighbors of the
  nearest vector
• At each iteration the learning rate decreases

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The Basic SOM Algo Cont

• Why?
• Well, there is a topographic relationship between
  the clusters
• What it looks like

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Using it for gene expression arrays

• Whitehead Institute likes SOMs
• In 1999, they pushed this method pretty hard
• Early success, but now not used that much for
  gene array clustering

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What they looked at

• Took yeast cells through out their cell cycle and
  looked at various RNA levels
• Clusters matched human analysis
• More interestingly, similar expression patterns
  for neighbors

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What else they looked at

• Blood stem cells and their differentiation
• Looked at how expression levels changed over a 24
  hour period as the stems cells differentiated
  into macrophages
• Underlying idea similar time courses, shared
  regulation
• Also looks at different gene regulations in
  different cell lines

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The good

• This algorithm runs in linear time
• Robust against noise
• A relation between clusters
• Bustas lament

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The bad

• No high-order relations
• Guess the number of vertices in the
  lower-dimensional space
• What the heck does an edge mean here?

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The ugly

• The child of Selma Hayek and Ed Norton

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Is there hope?

• Toronen uses a method to create multiple maps
  with different node numbers
• For example, maps of 3x4, 6x8, 12x16
• Pick the one which produces best separation
• Same connectivity, just different number

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Combining Hierarchical Clustering and SOMs

• Hierarchy of clusters
• Self-organizing process like SOM, but uses a
  binary tree topology
• Keeps splitting as long as researcher wants
• Problem with method is that it requires training

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Another Idea

• Pick an underlying feature space that means
  something biologically
• Easier said than done
• Perhaps try to map biological knowledge of gene
  regulatory networks into an appropriate feature
  space topology
• How unsupervised is that?

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A mathematical aside

• Can be thought of as a discretized version of
  principle curves
• Principle curves are a generalized version of
  principal components
• A smooth, 1-d curved approximation
• Which brings us to some next-level ish