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Beta Tucker Decomposition for DNA Methylation Data
Aaron Schein UMass Amherst
Mingyuan Zhou Univ. Texas at Austin
Hanna Wallach Microsoft Research
Joint work with:
Pat Flaherty UMass Amherst
Dan Sheldon UMass Amherst
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DNA methylation
CATTCCGCCTTCTCTCCCGAGG
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DNA methylation
CpG dinucleotides
CATTCCGCCTTCTCTCCCGAGG
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DNA methylation
M
methylated unmethylated
CATTCCGCCTTCTCTCCCGAGG
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DNA methylationCGAGGCATTCCGCCTTCTCTCCCGAGGCATTCCGCCT
TCGACGCGCCTTCTCTCCCGCGCGACGCGCCTTCTCT
CCCGCGCGACGCGCCTTCTCTCCCGCGCTCGACGCG
CCTTCTCTCCCGCGCGACGCGCCTTCTCTCCCGCGCG
ACGCGCCTTCTCTCCCGCGCCGACGCGCCTTCTCTCC
CGCGCGACGCGCCTTCTCTCCCGCGCGACGCGCCTT
CTCTCCCGCGTCCCGCGACGCGCCTTCTCTCCCGCGA
GGCATTCCGCCTTCTTTTTTTTTTTTCGACGCGCCTTCT
CTCCCGCGCGACGCGCCTTCTCTCCCGCGTTTTTCTC
CCGAGGCATTCCGCCTTCTCCGACGCGCCTTCTCTCC
CGCGTTCTCTAGCGCCTTCTCTCCCGACGACGCGCCT
TCTCTCCCGCGCGACGCGACGCGCCTTCTCTCCCGC
GCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTTC
TCTCCCGACGCCTTCTCTCCCGACGCGCCTTCTCTCC
CGCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTT
CATTCCGCCTTCTGCTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
ACCTATCTCCCGAGGCATTCCGCCTTCTCTCCCGAGG
CATTCCGCCTTCTCTCCCGAGGCATTCCGCCTTCTTTT
TTTTTTTTTTTTTCTCCCGAGGCATTCCGCCTTCTCTTCT
CTAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTC
CCCCAGGCTGGATTGCTACACCTTCTCTAGTCCCCCA
GGCTGGATTGCTACACCTCCCGAGGCATGCATTCCG
CCTTTCTCTAGTCCCCCAGGCTGGATTGCTACACCTTC
TCTAGTCCCCCAGGCTGGATTGCTACACCTCTCTCCG
AGGCATTCCGCCTTCCTCTCCTCTCTCTCCCGAGTCTC
TAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTCC
CCCAGGCTGGATTGCTACACCTGCATTCCGCCTTCTC
TTTTTCCCGAGGCATTTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
Gene
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DNA methylationCGAGGCATTCCGCCTTCTCTCCCGAGGCATTCCGCCT
TCGACGCGCCTTCTCTCCCGCGCGACGCGCCTTCTCT
CCCGCGCGACGCGCCTTCTCTCCCGCGCTCGACGCG
CCTTCTCTCCCGCGCGACGCGCCTTCTCTCCCGCGCG
ACGCGCCTTCTCTCCCGCGCCGACGCGCCTTCTCTCC
CGCGCGACGCGCCTTCTCTCCCGCGCGACGCGCCTT
CTCTCCCGCGTCCCGCGACGCGCCTTCTCTCCCGCGA
GGCATTCCGCCTTCTTTTTTTTTTTTCGACGCGCCTTCT
CTCCCGCGCGACGCGCCTTCTCTCCCGCGTTTTTCTC
CCGAGGCATTCCGCCTTCTCCGACGCGCCTTCTCTCC
CGCGTTCTCTAGCGCCTTCTCTCCCGACGACGCGCCT
TCTCTCCCGCGCGACGCGACGCGCCTTCTCTCCCGC
GCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTTC
TCTCCCGACGCCTTCTCTCCCGACGCGCCTTCTCTCC
CGCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTT
CATTCCGCCTTCTGCTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
ACCTATCTCCCGAGGCATTCCGCCTTCTCTCCCGAGG
CATTCCGCCTTCTCTCCCGAGGCATTCCGCCTTCTTTT
TTTTTTTTTTTTTCTCCCGAGGCATTCCGCCTTCTCTTCT
CTAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTC
CCCCAGGCTGGATTGCTACACCTTCTCTAGTCCCCCA
GGCTGGATTGCTACACCTCCCGAGGCATGCATTCCG
CCTTTCTCTAGTCCCCCAGGCTGGATTGCTACACCTTC
TCTAGTCCCCCAGGCTGGATTGCTACACCTCTCTCCG
AGGCATTCCGCCTTCCTCTCCTCTCTCTCCCGAGTCTC
TAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTCC
CCCAGGCTGGATTGCTACACCTGCATTCCGCCTTCTC
TTTTTCCCGAGGCATTTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
CpG island
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DNA methylationCGAGGCATTCCGCCTTCTCTCCCGAGGCATTCCGCCT
TCGACGCGCCTTCTCTCCCGCGCGACGCGCCTTCTCT
CCCGCGCGACGCGCCTTCTCTCCCGCGCTCGACGCG
CCTTCTCTCCCGCGCGACGCGCCTTCTCTCCCGCGCG
ACGCGCCTTCTCTCCCGCGCCGACGCGCCTTCTCTCC
CGCGCGACGCGCCTTCTCTCCCGCGCGACGCGCCTT
CTCTCCCGCGTCCCGCGACGCGCCTTCTCTCCCGCGA
GGCATTCCGCCTTCTTTTTTTTTTTTCGACGCGCCTTCT
CTCCCGCGCGACGCGCCTTCTCTCCCGCGTTTTTCTC
CCGAGGCATTCCGCCTTCTCCGACGCGCCTTCTCTCC
CGCGTTCTCTAGCGCCTTCTCTCCCGACGACGCGCCT
TCTCTCCCGCGCGACGCGACGCGCCTTCTCTCCCGC
GCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTTC
TCTCCCGACGCCTTCTCTCCCGACGCGCCTTCTCTCC
CGCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTT
CATTCCGCCTTCTGCTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
ACCTATCTCCCGAGGCATTCCGCCTTCTCTCCCGAGG
CATTCCGCCTTCTCTCCCGAGGCATTCCGCCTTCTTTT
TTTTTTTTTTTTTCTCCCGAGGCATTCCGCCTTCTCTTCT
CTAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTC
CCCCAGGCTGGATTGCTACACCTTCTCTAGTCCCCCA
GGCTGGATTGCTACACCTCCCGAGGCATGCATTCCG
CCTTTCTCTAGTCCCCCAGGCTGGATTGCTACACCTTC
TCTAGTCCCCCAGGCTGGATTGCTACACCTCTCTCCG
AGGCATTCCGCCTTCCTCTCCTCTCTCTCCCGAGTCTC
TAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTCC
CCCAGGCTGGATTGCTACACCTGCATTCCGCCTTCTC
TTTTTCCCGAGGCATTTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
CpG island (often in the promoter region)
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DNA methylation
CGAGGCATTCCGCCTTCTCTCCCGAGGCATTCCGCCT
TCGACGCGCCTTCTCTCCCGCGCGACGCGCCTTCTCT
CCCGCGCGACGCGCCTTCTCTCCCGCGCTCGACGCG
CCTTCTCTCCCGCGCGACGCGCCTTCTCTCCCGCGCG
ACGCGCCTTCTCTCCCGCGCCGACGCGCCTTCTCTCC
CGCGCGACGCGCCTTCTCTCCCGCGCGACGCGCCTT
CTCTCCCGCGTCCCGCGACGCGCCTTCTCTCCCGCGA
GGCATTCCGCCTTCTTTTTTTTTTTTCGACGCGCCTTCT
CTCCCGCGCGACGCGCCTTCTCTCCCGCGTTTTTCTC
CCGAGGCATTCCGCCTTCTCCGACGCGCCTTCTCTCC
CGCGTTCTCTAGCGCCTTCTCTCCCGACGACGCGCCT
TCTCTCCCGCGCGACGCGACGCGCCTTCTCTCCCGC
GCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTTC
TCTCCCGACGCCTTCTCTCCCGACGCGCCTTCTCTCC
CGCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTT
M M M
M
MMMMMM
MM
MMM
MM MM
MM M M
MM
MMMM M M
MMMM M
MMM
MM M
M MM
MMM
MM
MMMM
MMMM
MMCATTCCGCCTTCTGCTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
ACCTATCTCCCGAGGCATTCCGCCTTCTCTCCCGAGG
CATTCCGCCTTCTCTCCCGAGGCATTCCGCCTTCTTTT
TTTTTTTTTTTTTCTCCCGAGGCATTCCGCCTTCTCTTCT
CTAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTC
CCCCAGGCTGGATTGCTACACCTTCTCTAGTCCCCCA
GGCTGGATTGCTACACCTCCCGAGGCATGCATTCCG
CCTTTCTCTAGTCCCCCAGGCTGGATTGCTACACCTTC
TCTAGTCCCCCAGGCTGGATTGCTACACCTCTCTCCG
AGGCATTCCGCCTTCCTCTCCTCTCTCTCCCGAGTCTC
TAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTCC
CCCAGGCTGGATTGCTACACCTGCATTCCGCCTTCTC
TTTTTCCCGAGGCATTTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
M
M
M
M M
M
M
M
M
M
M
Gene is silenced
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DNA methylation
CGAGGCATTCCGCCTTCTCTCCCGAGGCATTCCGCCT
TCGACGCGCCTTCTCTCCCGCGCGACGCGCCTTCTCT
CCCGCGCGACGCGCCTTCTCTCCCGCGCTCGACGCG
CCTTCTCTCCCGCGCGACGCGCCTTCTCTCCCGCGCG
ACGCGCCTTCTCTCCCGCGCCGACGCGCCTTCTCTCC
CGCGCGACGCGCCTTCTCTCCCGCGCGACGCGCCTT
CTCTCCCGCGTCCCGCGACGCGCCTTCTCTCCCGCGA
GGCATTCCGCCTTCTTTTTTTTTTTTCGACGCGCCTTCT
CTCCCGCGCGACGCGCCTTCTCTCCCGCGTTTTTCTC
CCGAGGCATTCCGCCTTCTCCGACGCGCCTTCTCTCC
CGCGTTCTCTAGCGCCTTCTCTCCCGACGACGCGCCT
TCTCTCCCGCGCGACGCGACGCGCCTTCTCTCCCGC
GCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTTC
TCTCCCGACGCCTTCTCTCCCGACGCGCCTTCTCTCC
CGCGCCTTCTCTCCCGCGCCTTCTCTCCCGACGCCTT
M
M
M
M
M M
M
M M
M
M
M
MMCATTCCGCCTTCTGCTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
ACCTATCTCCCGAGGCATTCCGCCTTCTCTCCCGAGG
CATTCCGCCTTCTCTCCCGAGGCATTCCGCCTTCTTTT
TTTTTTTTTTTTTCTCCCGAGGCATTCCGCCTTCTCTTCT
CTAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTC
CCCCAGGCTGGATTGCTACACCTTCTCTAGTCCCCCA
GGCTGGATTGCTACACCTCCCGAGGCATGCATTCCG
CCTTTCTCTAGTCCCCCAGGCTGGATTGCTACACCTTC
TCTAGTCCCCCAGGCTGGATTGCTACACCTCTCTCCG
AGGCATTCCGCCTTCCTCTCCTCTCTCTCCCGAGTCTC
TAGTCCCCCAGGCTGGATTGCTACACCTTCTCTAGTCC
CCCAGGCTGGATTGCTACACCTGCATTCCGCCTTCTC
TTTTTCCCGAGGCATTTCTCTAGTCCCCCAGGCTGGAT
TGCTACACCTTCTCTAGTCCCCCAGGCTGGATTGCTAC
M
M
M
M M
M
M
M
M
M
M
Gene is expressed
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Abnormal DNA methylation
It causes cancer• Hypomethylation of oncogenes
• Hypermethylation of tumor suppressor genes
[Baylin & Ohm (2006)]
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Cancer taxonomies
Sample 1 Sample 2
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Cancer taxonomies
Sample 1 Sample 2
``Breast cancer”
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Cancer taxonomies
Sample 3 Sample 4
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Cancer taxonomies
Sample 3 Sample 4
``Ovarian cancer”
``Breast cancer”
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Cancer taxonomies
Anatomically similar cancer cells may be genetically different
Anatomically different cancer cells may be genetically similar
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Cancer taxonomies
Goal: Develop new taxonomies based on genetic information
ML solution: Unsupervised dimensionality reductionPCA, NMF, ICA,…
[Flusberg et al. (2010)]
[Teschendorff et al. (2007)][Wang et al. (2006)]
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DNA methylation data
�ijhow methylated locus j is in sample i=
�ij 2 [0, 1]
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
�Lo
cus
6
![Page 18: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/18.jpg)
CP decomposition
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
�Lo
cus
6k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
k=1
k=2
k=3
'
K ``components”
⇥ �
�ij 'KX
k=1
✓ik�kj
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�ij 'KX
k=1
✓ik�kj⇡k
CP decomposition
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
�Lo
cus
6k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
k=1
k=2
k=3
'
K ``components”
⇥ �
k=1
k=2
k=3
k=1
k=2
k=3
⇧
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Tucker decomposition
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
�Lo
cus
6k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
c=1
c=2
'
C ``clusters’’ and K ``components”
⇥ �
c=1
c=2
k=1
k=2
k=3
⇧
�ij 'CX
c=1
✓ic
KX
k=1
⇡ck �kj
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Our contributions:• Novel generative model
• Based on the Tucker decomposition • Matches the true data-generating process
✓ Beta likelihood ✓ Latent variables match real ones ✓ Priors match known sources of noise
• Gibbs sampler with closed form conditionals
Is it better than PCA/NMF/ICA/etc in practice?
Beta Tucker decomposition
• Comparable performance on (contrived) prediction tasks
Is it better than PCA/NMF/ICA/etc in theory?• Yes
• ??
[Ma et al. (2015)]
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DNA methylation data
�ijhow methylated locus j is in sample i=
�ij 2 [0, 1]
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
�Lo
cus
6
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CGTTTTTCTCM
CGCCTTCTCTCCCG CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Sample i
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
y(m)ij num. of methylated CpG sites
in locus j of sample =
y(u)ijnum. of unmethylated CpG sites in locus j of sample
=
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Sample i
Locus j
[Wang & Petronis (2008)]
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Sample i
Locus j
[Wang & Petronis (2008)]
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
�(m)ij �(u)
ij
Two real-valued fluorescent intensities
[Wang & Petronis (2008)]
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CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
�(m)ij �(u)
ij
�ij :=�(m)ij
�(m)ij + �(u)
ij
``Beta value”
![Page 31: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/31.jpg)
CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
�(m)ij �(u)
ij
n
�(m)ij ,�(u)
ij
oJ
j=1
Histogram of intensities for given sample i
![Page 32: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/32.jpg)
CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
DNA methylation data
Locus j
Sample i
�(m)ij �(u)
ij
n
�(m)ij ,�(u)
ij
oJ
j=1
Histogram of intensities for given sample i
�(m)ij ⇠ Gam(· · · , ci)
�(u)ij ⇠ Gam(· · · , ci)
![Page 33: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/33.jpg)
Gamma-Beta relationship
�1 ⇠ Gam(↵1, c) �2 ⇠ Gam(↵2, c)
✓�1
�1 + �2
◆⇠ Beta(↵1, ↵2)
![Page 34: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/34.jpg)
Gamma-Beta relationship
�(m)ij ⇠ Gam(· · · , ci) �(u)
ij ⇠ Gam(· · · , ci)
�ij :=�(m)ij
�(m)ij + �(u)
ij
�ij ⇠ Beta(· · · , · · · )
![Page 35: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/35.jpg)
CGTTTTTCTC
MCGCCTTCTCTCCCG
CTCCCGCGTCCCGCGAM M M
ACGCGCCTTCTCTM
CGCCTTCTTTTTM
Locus j
Sample i
�(m)ij �(u)
ij
Beta Tucker decomposition
![Page 36: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/36.jpg)
Locus j
Sample i
CTCCCGCGTCCCGCGAM M M
�(m)ij �(u)
ij
Beta Tucker decomposition
![Page 37: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/37.jpg)
Locus j
Sample i
CTCCCGCGTCCCGCGAM M M
�(m)ij �(u)
ij
1 2 31
Beta Tucker decomposition
![Page 38: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/38.jpg)
Locus j
Sample i
CTCCCGCGTCCCGCGAM M M
�(m)ij �(u)
ij
1 2 31
1 2= + + 3 = 1
Beta Tucker decomposition
![Page 39: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/39.jpg)
Locus j
Sample i
CTCCCGCGTCCCGCGAM M M
�(m)ij �(u)
ij
1 2 31
1 2= + + 3 = 1
Beta Tucker decomposition
![Page 40: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/40.jpg)
Locus j
Sample i
CTCCCGCGTCCCGCGAM M M
�(m)ij �(u)
ij
1 2 31
1 2= + + 3 = 1+ +
Beta Tucker decomposition
![Page 41: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/41.jpg)
�(m)ij
�(u)ij
1 2= + + 3
= 1
+
+
Beta Tucker decomposition
Locus jSample i
![Page 42: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/42.jpg)
Locus jSample i
�(m)ij
�(u)ij
= s
= s
+
+
Beta Tucker decomposition
y(m)ijX
s=1
y(u)ijX
s=1
![Page 43: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/43.jpg)
Locus jSample i
�(m)ij
�(u)ij
= s
= s
+
+
Beta Tucker decomposition
y(m)ijX
s=1
y(u)ijX
s=1
⇠ Gam(
1,c i)
⇠ Gam(
b0,c i)
![Page 44: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/44.jpg)
⇠ Gam⇣b0 + y(m)
ij , ci⌘
⇠ Gam⇣b0 + y(u)ij , ci
⌘
Beta Tucker decomposition
Locus jSample i
�(m)ij
�(u)ij
![Page 45: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/45.jpg)
�(m)ij ⇠ Gam
⇣b0 + y(m)
ij , ci⌘
�(u)ij ⇠ Gam
⇣b0 + y(u)ij , ci
⌘
�ij :=�(m)ij
�(m)ij + �(u)
ij
�ij ⇠ Beta⇣b0 + y(m)
ij , b0 + y(u)ij
⌘
Equivalent to:
Beta Tucker decomposition
![Page 46: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/46.jpg)
�(m)ij ⇠ Gam
⇣b0 + y(m)
ij , ci⌘
�(u)ij ⇠ Gam
⇣b0 + y(u)ij , ci
⌘
�ij :=�(m)ij
�(m)ij + �(u)
ij
Beta Tucker decomposition
y(m)ij ⇠ Pois(· · · ) y(u)ij ⇠ Pois(· · · )
![Page 47: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/47.jpg)
Beta Tucker decomposition
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!
![Page 48: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/48.jpg)
Beta Tucker decomposition
the probability that sample i is in cluster c
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!
![Page 49: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/49.jpg)
Beta Tucker decomposition
the probability that samples in cluster c
methylate loci in component k
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!
![Page 50: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/50.jpg)
Beta Tucker decomposition
the probability that locus j is in component k
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!
![Page 51: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/51.jpg)
Beta Tucker decomposition
✓i ⇠ Dir(⌘1, . . . , ⌘C)
⇡ck ⇠ Beta(⌘(m)0 , ⌘(u)0 )
�j ⇠ Dir(⌫1, . . . , ⌫K)
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!
![Page 52: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/52.jpg)
Beta Tucker decomposition
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!{= pij
![Page 53: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/53.jpg)
Beta Tucker decomposition
y(m)ij ⇠ Pois
�
CX
c=1
✓ic
KX
k=1
⇡ck �kj
!{= pij
![Page 54: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/54.jpg)
Beta Tucker decomposition
y(m)ij ⇠ Pois(� pij)
the probability that sample i methylates CpG sites in locus j
![Page 55: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/55.jpg)
Beta Tucker decomposition
y(m)ij ⇠ Pois(� pij)
the occurrence rate of CpG sites
![Page 56: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/56.jpg)
Beta Tucker decomposition
y(u)ij ⇠ Pois
�� (1� pij)
�y(m)ij ⇠ Pois
�� pij
�
pij :=CX
c=1
✓ic
KX
k=1
⇡ck�kj
�(m)ij ⇠ Gam
⇣b0 + y(m)
ij , ci⌘
�(u)ij ⇠ Gam
⇣b0 + y(u)ij , ci
⌘
�ij :=�(m)ij
�(m)ij + �(u)
ij
![Page 57: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/57.jpg)
Beta Tucker decomposition
y(u)ij ⇠ Pois
�� (1� pij)
�y(m)ij ⇠ Pois
�� pij
�
pij :=CX
c=1
✓ic
KX
k=1
⇡ck�kj
�ij ⇠ Beta⇣b0 + y(m)
ij , b0 + y(u)ij
⌘
![Page 58: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/58.jpg)
k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
c=1
c=2
⇥ �
c=1
c=2
k=1
k=2
k=3
⇧
Beta Tucker decomposition
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (m)Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (u)
Sample 1
Sample 2
Sample 3
Sample 4
�
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
![Page 59: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/59.jpg)
Sample 1
Sample 2
Sample 3
Sample 4
�
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
c=1
c=2
⇥ �
c=1
c=2
k=1
k=2
k=3
⇧Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (m)Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (u)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(m)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(u)
Beta Tucker decomposition
![Page 60: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/60.jpg)
Sample 1
Sample 2
Sample 3
Sample 4
�
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
c=1
c=2
⇥ �
c=1
c=2
k=1
k=2
k=3
⇧Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (m)Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (u)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(m)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(u)
Inference
![Page 61: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/61.jpg)
Sample 1
Sample 2
Sample 3
Sample 4
�
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
c=1
c=2
⇥ �
c=1
c=2
k=1
k=2
k=3
⇧Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (m)Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (u)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(m)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(u)
Inference
P⇣⇥,⇧,� |Y (m), Y (u), · · ·
⌘
= Poisson Tucker decomposition[Schein et al. (2016)]
![Page 62: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/62.jpg)
Sample 1
Sample 2
Sample 3
Sample 4
�
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
k=1
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Sample 1
Sample 2
Sample 3
Sample 4
k=2
k=3
c=1
c=2
⇥ �
c=1
c=2
k=1
k=2
k=3
⇧Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (m)Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
Y (u)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(m)
Sample 1
Sample 2
Sample 3
Sample 4
Locu
s 1
Locu
s 2
Locu
s 3
Locu
s 4
Locu
s 5
Locu
s 6
⇤(u)
Inference
P⇣Y (m), Y (u) |⇤(m),⇤(u), · · ·
⌘
![Page 63: Beta Tucker Decomposition for DNA Methylation Dataas5530/ScheinFlahertyZhouSheldonWall... · 2019. 6. 6. · Locus j Sample i (m) ij (u) ij Two real-valued fluorescent intensities](https://reader036.vdocument.in/reader036/viewer/2022062318/610c15263e190f3bc8035b5a/html5/thumbnails/63.jpg)
Inference
y(m)ij ⇠ Pois
�� pij
�
⇠ Gam⇣b0 + y(m)
ij , ci⌘
�(m)ij
P (y(m)ij |�(m)
ij , · · · ) =?
Poisson is not conjugate to the gamma…
…but maybe the posterior still has a closed form…
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Inference
y(m)ij ⇠ Pois
�� pij
�
⇠ Gam⇣b0 + y(m)
ij , ci⌘
�(m)ij
The Bessel distribution!
P (y(m)ij |�(m)
ij , · · · ) = Bes
✓b0�1, 2
qci�
(m)ij � pij
◆ [Yuan & Kalbfleisch (2000)]
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The Bessel distribution
Bes(y; v, a) / 1
y!�(y + v)
⇣a2
⌘2y+v
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Sampling the Bessel
[Devroye (2002)] [Yuan & Kalbfleisch (2000)]
[Amos (1974)]
[Zhou (2015)]
Stable computation of Bessel functions
Exact rejection sampling (four methods)
Table sampling
Basic properties of Bessel distribution
It’s easy and fast
https://github.com/aschein/fatwalrus
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MCMC algorithm
P⇣⇥,⇧,� |Y (m), Y (u), · · ·
⌘
Poisson Tucker decompositionO(CK|Y>0|)
O(2IJ)P⇣Y (m), Y (u) |⇤(m),⇤(u), · · ·
⌘
Sample Bessel counts
� controls sparsity!
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Example results
⇥⇧
Top locus in component 8 is in
the promoter region of FLJ1030207
Hypomethylation of FLJ1030207 is
a strong indicator of ovarian cancer
[Model & Rujan (2009)]