Multiway clustering of microarray data using Probabilistic Sparse Matrix Factorization

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Multi-way clustering of microarray data
usingProbabilistic Sparse Matrix Factorization

• Delbert Dueck, Quaid Morris, Brendan Frey
• Probabilistic Statistical Inference Group
• Department of Electrical and Computer Engineering
• University of Toronto

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Outline

• Introduction
• Biological Background
• Mouse genome data set
• Probabilistic Sparse Matrix Factorization
• Generative Model
• Approximate Inference
• Results
• Visualizations
• Statistical Significance
• Summary

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Introduction

• Genes encode basic information about an organism
• Gene expression is influenced by the presence of
  transcription factors
• The activity of each gene can be explained by the
  activities of a small number of transcription
  factors
• Expression data is from Zhang et al. (2004)
• Contains clear expression profiles for over
  20,000 known and predicted genes across 55 mouse
  tissue types

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Introduction Dataset
? G22709 genes ?
Entire data set X GT matrix (G22709, T55)
? 100 genes ?
? T55 tissues ?
T55tissues
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Generative Model

• We introduce an unsupervised technique that
  renders a multi-way clustering of the data
• Each genes expression profile (xg) is
• a linear combination (weights ygc, c?sg)
• of a small number (rgltN)
• of C possible factor profiles (zc, c?sg)

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Generative Model (entire matrix view)

• Y is constrained structurally
• rows must have ltN non-zero elements

( indicates a non-zero entry)
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Generative Model (likelihoods)

• Form a joint distribution, assuming
• varying levels of Gaussian noise in the data
• nothing about factor weights
• normally-distributed factor profiles
• uniformly-distributed factor assignments
• multinomially-distributed factor counts

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Approximate Inference Techniques

• Exact inference of Ps hidden variables is
  intractable
• We examined two approximate techniques
• Sparse Matrix Factorization (Srebro Jaakkola,
  2001)
• Search for a configuration of the hidden
  variables that maximizes P (iterated conditional
  modes)
• Probabilistic Sparse Matrix Factorization (PSMF)
• Search for a distribution over configurations of
  the hidden variables that accurately approximates
  P (variational EM)

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Factorized Variational Inference

• Parameterize Q
• Accounts for noise in factor profiles and
  uncertainty in factor selection
• Minimize KL-divergence between P and Q

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Variational EM algorithm for PSMFconvert into
image

• Use coordinate descent on free energy, F

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PSMF visualization (1 of 3)

• Probabilistic Sparse Matrix Factorization
• C50 possible factors
• maximum of N3 factors per gene
• Expression profiles (rows) are grouped by primary
  factor (sg1), then secondary factor (sg2), etc.

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PSMF visualization (2 of 3)

• Zoom in where primary factor is 3
• High expression in colon, small intestine, large
  intestine tissues
• enriched for GO-BP category lipid metabolism
  GO0006629 (p-value lt 10-10)

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PSMF visualization (3 of 3)

• Zoom in where primary factor is 3 and secondary
  factor is 33

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Results p-value histograms

• Genes are clustered by primary factor,
  secondary factor, etc.
• Compare these clusters with annotated GO
  categories by computing hypergeometric p-values
  of all possible cluster-GO category pairings
• Statistical significance of each cluster is the
  p-value of most enriched-for GO category

Histograms of cluster p-values
in? significant insignificant ? (a0.05 plus
Bonferoni correction)
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Results significant factors (N1)
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Results significant factors (N1)
It would be nice to get rid of random clustering
for the next three slides
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Results significant factors (N2)
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Results significant factors (N3)
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Results complete
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PSMF vs. SMF maximizing likelihoods

• SMF makes hard decisions during maximization
• Immediately converges to poor local maximum
• PSMF makes soft decisions (better approximation
  of posterior)

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Summary

• We introduce probabilistic sparse matrix
  factorization
• Useful tool for data vectors that are most
  naturally explained as a linear combination of a
  selection of prototype vectors
• Computes probability distributions instead of
  making hard decisions during factor selection
• PSMF finds more highly-enriched functional
  clusters than SMF and standard techniques
• Also outputs secondary and higher-order labels
  for each expression vector
• Useful for more refined visualization and
  functional prediction

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Questions?

• For more details and software,see
  www.psi.toronto.edu/delbert

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Future Directions different Q?
Iterated conditional modes (point estimates)