Normalized maximum likelihood models in genomics Ioan Tabus - - PowerPoint PPT Presentation

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Normalized maximum likelihood models in genomics Ioan Tabus - - PowerPoint PPT Presentation

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1 Normalized maximum likelihood models in genomics Ioan Tabus Department of Signal Processing Tampere University of Technology Department of Signal Processing NML models in genomics 8.7.2008 2 Universal distributions Optimality of the

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1 NML models in genomics 8.7.2008

Department of Signal Processing

Normalized maximum likelihood models in genomics Ioan Tabus

Department of Signal Processing Tampere University of Technology

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2 NML models in genomics 8.7.2008

Department of Signal Processing

Universal distributions Optimality of the Normalized Maximum Likelihood model The universal distribution q(.) is solution of two minmax problems:

  • 1. Best distribution for the worst string:

) ,..., ( )) ,..., ( ˆ , ,..., ( max min

1 1 1 ,..., (.)

1

n n n x x q

x x q x x x x P

n

θ

  • 2. Best distribution q(.) for the average regret of the worst

generating distribution g(.) :

) ,..., ( )) ,..., ( ˆ , ,..., ( max min

1 1 1 (.) (.) n n n g g q

x x q x x x x P E θ

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3 NML models in genomics 8.7.2008

Department of Signal Processing

Two related goals: DNA compression and DNA modelling

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Department of Signal Processing

Finding the regressor in the past

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Department of Signal Processing

The NML model for approximate matching

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6 NML models in genomics 8.7.2008

Department of Signal Processing

Memoryless model

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Department of Signal Processing

The NML for memoryless discrete regression

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Department of Signal Processing

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Department of Signal Processing

Memory model

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Department of Signal Processing

NML for the class of memory models

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Department of Signal Processing

NML-1 Encoding algorithm

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Department of Signal Processing

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Department of Signal Processing

Compression of Human Genome (average 1.45 bit/base)

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Department of Signal Processing

Compression ratio in bits / base when compressing the human genome (only A,C,G,T alphabet) with a 10 MB window size. Blue: bzip2, Red:GeNML.

5 10 15 20 25 0.5 1 1.5 2 2.5 Chromosome Bits / base

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15 NML models in genomics 8.7.2008

Department of Signal Processing

Approximate matching in DNA analysis

Use a universal coding of the binary mask resulting from matching two candidate sequences Normalized maximum likelihood models

For memoryless sources (Bernoulli) For sources with memory

Example: the DNA locus HUMGHCSA (about 65000 bases)

The genes GH-1 and GH-2 are human growth hormone genes CS-5, CS-1, and CS-2 are chorionic somatomammotropin genes expressed either in pituitary gland or in placenta

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Department of Signal Processing

DNA duplications and their role in evolution and disease

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Department of Signal Processing

Traditional approach to gene duplication

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Department of Signal Processing

Hamming versus NML 0 (memoryless)

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Department of Signal Processing

Optimizing the overall cost for duplication analysis

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Department of Signal Processing

Encoding the pointers and the mask

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Department of Signal Processing

Dynamic programming problem

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Department of Signal Processing

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Department of Signal Processing

NML with memory (orders 1 and 2)

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Department of Signal Processing

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Department of Signal Processing

Summarizing the duplication scenarios

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Department of Signal Processing

NML Encoding

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Department of Signal Processing

Efficient search of regressor

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Department of Signal Processing

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Department of Signal Processing

Renormalization to account for contiguous matches

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Department of Signal Processing

Accounting for contiguous perfect matches

Unconstrained case: Constrained case

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Department of Signal Processing

Accounting for contiguous perfect matches: efficient approach

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Department of Signal Processing

Comparison of NML 1 and NML 2

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Department of Signal Processing

Open avenues

Universal models provide efficient representation tools for genomic sequences More refined model order selection procedures may better account for non-stationarity along sequences The techniques are easy to extend to more adaptive tools Derive the exact NML model for more structured classes of models with memory