Multivariate point process models
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1 Faculty of Science Multivariate point process models Niels Richard Hansen Department of Mathematical Sciences January 8, 200 Slide /20
2 Ideas and outline General aim: To build and implement a flexible (non-parametric), intensity based framework of multivariate point process models aka discretely marked point processes. Secondary aim: Development and implementation of model selection techniques LASSO, grouped LASSO, penalization based ideas with the multivariate point processes as a testbed. Joint with Patricia Bouret, Vincent Rivoirard. Main application: Organizational model of active transcription regulatory elements along genomes joint with Lisbeth Carstensen, Albin Sandelin, Ole Winther. Slide 2/20 Niels Richard Hansen Multivariate point process models January 8, 200
3 Genomic scales an e- se an c- s, ister re la- - n- in of ay ral s- ra- We focus on the distribution of point-like transcriptional regulatory elements at the meso-genomic scale. Figure from Zhang et al, Genome Res., 8-, 200 Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
4 Transcription regulator binding loci Attempt of a broad definition: Transcription regulators are proteins that modify, interact with or bind to the DNA, chromatin or other transcription regulators to either activate or repress the transcription of DNA. An active transcription regulatory loci is a loci on the genome where we observe the presence of a transcription regulator. Fact: Transcription regulators cluster in promoter regions and intergenic regions. Why? Is there a combined effect? Do they recruit each other...? With the different regulators as marks and with measurements of the active loci as points on the meso-genomic scale we use a multivariate point process model of the organization of active loci. Slide 4/20 Niels Richard Hansen Multivariate point process models January 8, 200
5 Obtaining and preprocessing data (ChIP-chip) TGGCA ACCGT GCAAC CGTTG cutoff actual binding sites Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
6 ENCODE affymetrix data sirt rara pu pol2 type brg cebpe ctcf type p00 h4kac4histone hk2mehistone hk2mehistone h4kac4histone p00 pol2 ctcf cebpe brg position pu rara sirt The 0 Affymetrix ChIP-chip measurements for the ENCODE pilot project segment ENm00. Histone modifications are not regarded as point-like, figure gives start positions. Slide 6/20 Niels Richard Hansen Multivariate point process models January 8, 200
7 Embryonic mouse stem cell data (ChIP-seq) Suz2 STAT type Sox2 Nanog Oct4 type Smad p00 Smad p00 Sox2 STAT Oct4 Suz2 Nanog.0e+0.0e+08.e+08 position Mouse, chromosome I: of active transcription factor binding loci measured by ChIP-seq for embryonic stem cells. 2 2 Chen et al, Cell, 06-, 2008 Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
8 The Hawkes process If t i <... < t ini denote the observed points for transcription regulator i we specify the intensity process for the occurrence of regulator j as λ j (t) = ϕ(β 0 + h i,j (t t ik )) i k:t ik <t for a fixed ϕ. We expand the h-function in a suitable basis; B i,j h i,j (s) = β i,j b B b(s) b= and the minus-log-likelihood in terms of β = (β 0, β ij b ) is directly expressible. Slide 8/20 Niels Richard Hansen Multivariate point process models January 8, 200
9 ppstat The current implementation in the R package ppstat offers Formula interface to model specification. Standard (e.g. spline) basis function expansions of linear filters. Inclusion of continuous time covariate effects and additive model specification. Currently only with ϕ = exp. Soon with a more flexible class of ϕ s including ϕ(x) = x +. Why is ϕ(x) = x + a problem? The minus-log-likelihood is a priory not twice differentiable, but it turns out to be so under certain regularity assumptions. Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
10 Some spline bases B(t) t B(t) t Slide 0/20 Niels Richard Hansen Multivariate point process models January 8, 200
11 university of copenhagen department of mathematical sciences Estimated multiplicative effects - ES cells E2f Zfx c.myc n.myc Nanog Sox2 Oct4 Suz2 STAT p00 Smad E2f Zfx c.myc n.myc Nanog Sox2 Oct4 Suz2 STAT p00 Smad position Slide /20 Niels Richard Hansen Multivariate point process models January 8,
12 Interaction terms The formula i k:t ik <t h i,j (t t ik ) that entered in the intensity is an additive model of linear, univariate filters. Question: Can we capture interaction effects? Suggestion: Introduce bivariate, linear filters h (i,r),j (t t ik, t t rl ) k:t ik <t l:t rl <t and expand them in a suitable bivariate basis. Slide 2/20 Niels Richard Hansen Multivariate point process models January 8, 200
13 Bivariate bases Many possible choices of 2d-bases: Thin plate splines h(s, t) = (s s 0 ) 2 + (t t 0 ) 2 log((s s 0 ) 2 + (t t 0 ) 2 ). Tensor products h(s, t) = h (s)h 2 (t). For illustration we consider an example with a spline based tensor product construction with 6 basis functions. Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
14 Bivariate spline tensor product bases value Sox Oct4 Slide 4/20 Niels Richard Hansen Multivariate point process models January 8, 200
15 Additive estimated Oct4-Sox2 effect on Nanog what joint oct4 sox2 Sox2 400 value Oct4 ĥ Oct4,Nanog (s) + ĥ Sox2,Nanog (t) Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
16 Estimated Oct4-Sox2 interaction effect on Nanog what joint oct4 sox2 Sox2 400 value Oct4 ĥ Oct4,Nanog (s) + ĥ Sox2,Nanog (t) + ĥ (Sox2,Oct4),Nanog (s, t) Slide 6/20 Niels Richard Hansen Multivariate point process models January 8, 200
17 Difference what joint oct4 sox2 Sox2 400 value Oct4 ĥ (Sox2,Oct4),Nanog (s, t) Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
18 What do we model? The point process model is a model of the organizational part of evolution. It is a model of which loci are active for a given cell line/organism/tissue under given experimental conditions. The driving stochastic mechanism is evolution we do not account for measurement noise or biological variation in an explicit way. Data are integrated over cells we do not observe single cell transcription factor bindings. Slide 8/20 Niels Richard Hansen Multivariate point process models January 8, 200
19 Which conclusions do the model facilitate? Working assumption: Organizational association Functional association It is a systemic analysis that facilitate phrasing and answering question of a systemic nature. Example: For two tissues find systemic differences in the organization of active regulatory loci. Slide /20 Niels Richard Hansen Multivariate point process models January 8, 200
20 Thanks to Lisbeth Carstensen, Albin Sandelin and Ole Winther and to all of you for attending this meeting. Slide 20/20 Niels Richard Hansen Multivariate point process models January 8, 200
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