Recurrent event time modeling: predictive and causal modelscausal.sund.ku.dk/CausalSearch.pdf ·...

un i v er s i ty of copenhagen

Faculty of Science

Recurrent event time modeling: predictive and causalmodels

Niels Richard Hansen, Patricia Reynaud-Bouret,Vincent Rivoirard and Alexander SokolDepartment of Mathematical Sciences

November 24, 2016

Slide 1/31

un i v er s i ty of copenhagen department of mathemat i ca l s c i ence s

Current interests in Hawkes processesHawkes processes ... have become very popular in empirical highfrequency finance this last decade.

Hawkes Processes in FinanceE. Bacry et al., Market Microstructure and Liquidity, 2015

... applications include the user viewing records in an IPTVsystem, and the patient records in hospitals, ...

Learning Granger Causality for Hawkes ProcessesH. Xu et al., ICML, 2016

In a variety of settings, we observe a series of events associatedwith a group of actors whose interactions trigger future events.The interactions between these actors can be modeled using anetwork. For example: Social networks; Biological neuralnetworks; Financial networks; Epidemiological networks;Seismological networks.

Tracking Dynamic Point Processes on NetworksE. C. Hall and R. M. Willet, IEEE Transactions on Information Theory, 2016

Slide 2/31— Niels Richard Hansen — Recurrent event time modeling: predictive and causal models — November 24, 2016


Motivating Examplescused on 14 manually chosen humangenomic regions and 30 randomly se-lected ones, which in total compose30 mega-bases (!1%) of the humangenome sequence. Of all possible func-tional elements in the ENCODE regions,epigenetic modifications and cis-regulatory elements, including promot-ers and TF-binding sites (TFBSs; togetherreferred to as TREs in this report), area major form of transcriptional regula-tion in eukaryotes. To identify the com-plex set of cis-acting transcriptional con-trol sequences and modification sites inthe ENCODE regions, a large number ofproteins (including POLR2A) that playvarious roles in transcription and severaltypes of histone modifications were as-sayed by different participating labora-tories.

The ENCODE experimental assaysof the transcriptional regulation, whichcollectively represent the first concertedeffort to systematically identify TREs inthe human genome on a large scale,have generated a large amount of data.With this information available (TheENCODE Project Consortium 2007), it isnow possible to conduct detailed surveysof different TFs and their TREs on vari-ous genomic levels (Fig. 1A). The pro-moter assay finds the promoter regionsimmediately upstream to genes’ tran-scription start sites (TSSs) on a 100-base-pair (bp) level, and the chromatin struc-ture analysis examines the correspon-dence between various TREs and aspectsof chromatin architecture that impli-cates mega-bases of DNA. In contrast,our analysis of the genomic distributionof TREs was conducted on an intermedi-ate genomic level, which involves10!100 kb of DNA encompassing severalgenes on average.

With such an unprecedented dataset, it is now also possible to examine TFcoassociation on a large genomic scale.It is highly desirable to present the problem and subsequentlyanalyze the data in a consistent and coherent statistical frame-work. To do this, we first coded the ChIP–chip experimentalresults as a binary 105 ! !30,000,000 data matrix (Fig. 1B)and then transformed it into a 105 ! 5669 count matrix usinga sliding window to both reduce the matrix size and incorpo-rate contextual information from neighboring nucleotide posi-tions (Fig. 1C). By presenting the data set in the matrix form,many well-studied, mathematically-sound statistical methodsand techniques such as the principal component analysis anddata randomization (Fig. 1D) can be adopted to tackle the prob-lem.

Below, we evaluate the genomic distribution of the newlyidentified TREs both by themselves and together with the genedistribution, determine TRE clusters and deserts in the ENCODE

regions, and study the relationship among the TFs that have beenassayed.

ResultsWe analyzed 105 lists of regulatory elements of 29 TFs in theENCODE regions. A list of TREs of a particular TF specifies thelocation in the genome of the regulatory elements of this factorunder certain cellular and experimental conditions. Disregardingoverlaps among sites, there are a total of 15,211 TREs identified.The numbers of TREs in each list, ranging from 1–1083 with anaverage of !145 per list, are plotted in Supplemental Figure 1 withlists from the same laboratory grouped together and labeled ac-cordingly. The overall landscape of all 44 ENCODE regions withidentified TREs is depicted in Figure 2 and clearly shows a posi-

Figure 1. Schematic introduction of the several concepts used in this study. (A) Studies of differenttranscription factors and their regulatory elements on various genomic levels. (TRE) Transcriptionalregulatory element; (TSS) transcription start site. (Modified from The ENCODE Project Consortium2004 and reprinted with permission from AAAS [www.sciencemag.org] © 2004.) (B) The binary datamatrix. Each row is the result track of a ChIP–chip experiment. Red dots are identified transcriptionalregulatory elements, in which each nucleotide position is coded as one. (C) The count matrix. A slidingwindow (the green boxes in B) was used to incorporate contextual information from neighboringpositions. Each gray dot represents the number of nucleotide positions in TREs in a sliding window. (D)Correlating two ChIP–chip tracks. The correlation can be done on either two binary vectors or twocorresponding count integer vectors (actually used, not shown). Two tracks can also be randomized togenerate a background distribution of the correlation.

Zhang et al.

788 Genome Researchwww.genome.org

Cold Spring Harbor Laboratory Press on January 6, 2010 - Published by genome.cshlp.orgDownloaded from

ACCGT CGTTGTGGCA GCAAC

cutoff

actualbindingsites

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Neuron spike data example

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Dynamics, feedback and latent variables

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Questions

We are interested in models that are not just predictive but alsohave a causal interpretation. That is, they are predictive aboutintervention effects.

• How can we handle feedback that doesn’t fit into theframework of DAGs?

• But it does if time is included explicitly.

• What are the consequences of latent variables for statisticalinference? Two solutions:

• model latent variables explicitly,• or characterize (parametrize) marginalized distributions.

We consider dynamic Markov properties and graphs for eventtimes.




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timeSlide 7/31— Niels Richard Hansen — Recurrent event time modeling: predictive and causal models — November 24, 2016



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timeSlide 8/31— Niels Richard Hansen — Recurrent event time modeling: predictive and causal models — November 24, 2016


Important observation

For a stationary dynamic process

dynamic Markov properties do not translateinto Markov properties for the invariantdistribution – the latter may have no

conditional independencies.

It is thus particularly difficult to draw causal inference aboutdynamical processes from cross-sectional samples.



Events and intensity models

The counting process for the kth event track will be modeled interms of an intensity:

P(Nkt+δ − Nk

t = 1 | Ft) ' λkt δ.

The process λkt is predictable w.r.t. the filtration Ft .

The filtration Ft holds information on all event tracks up to time t.

In the neuron science literature,

Nkt+δ − Nk

t | Ft ∼ Poi(λkt δ)

is applied to binned data for small enough δ (nonlinear,non-Gaussian time series).



Local Independence

Let V = {1, . . . , d} denote the set of track indices.

We say that k is conditionally locally independent of j and write

j 6→ k | V \{j , k} (1)

if λkt does not depend on track j .

Definition (Local Independence Graph)

A graph G = (V ,E ) is called a local independence graph if

(j , k) 6∈ E =⇒ j 6→ k | V \{j , k}.

A local independence graph directly encodes pairwise dynamicMarkov properties.



The global Markov property

Let A,B,C ⊆ V be disjoint.

We say that B is locally independent of A given C and write

A 6→ B | C (2)

if E (λkt | σ(NA∪B∪C0:t )) does not depend on tracks in A for k ∈ B.

Theorem (Didelez1)

If C δ-separates A from B in a local independence graph then (2)holds.

1Graphical models for marked point processes based on local independence.JRSS-B 70(1), 2008.



The global Markov property

• Local independence graphs can have cycles and can representfeedback.

• They can be given a causal interpretation e.g. viainterventions.

• δ-separation generalizes d-separation known for DAGs.

• Some technical conditions are required.

Definition

A graph G is faithful if

C δ-separates A from B ⇐⇒ A 6→ B | C .

Faithfulness + the global Markov property allows for inferenceabout G from marginal observations.



Linear filters and the Hawkes process

The intensity can be specified in terms of linear filters ofpreceeding events

λkt = βk0 +

p∑j=1

∑i :τ ji <t

g jk(t − τ ji )

This gives the (linear) Hawkes process.



The nonlinear Hawkes process

The intensity

λkt (g) = ϕ(βk0 +

p∑j=1

∑i :τ ji <t

g ik(t − τ ji )),

for ϕ : R→ [0,∞) gives the nonlinear Hawkes process1.

1Bremaud, P. and Massoulie, L. Stability of nonlinear Hawkes processes. Ann.

Probab. 24(3), 1996..



Linear filters

time0

track 1

track 2

track 3

intensity



Local independence graphs for Hawkes processes

Simple observation: The graph defined by

(j , k) ∈ E ⇐⇒ g jk 6= 0

is a local independence graph.



R code – using package ppstat

A penalized B-spline basis expansion is available via the ppSmooth

function.

pps <- ppSmooth(c(v8.1, v14.2, v4.1, v13.2, v2.2, v5.1) ~

s(v8.1) + s(v14.2) + s(v4.1) +

s(v13.2) + s(v2.2) + s(v5.1),

data = spikeData,

family = Hawkes("log"),

support = 0.5,

lambda = 10

)



Results – basis approximation with B-splines

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Testing

We can testH0 : g jk = 0

using that test statistic

(βjkλ )T(

Σjkλ

)−1βjkλ

with an approximating χ2-distribution.

Here, λ is the smoothing parameter and Σjkλ is the sandwich

estimator for the covariance matrix for the parameters belongingto the jkth filter function.



Test results

condTest(pps)

v8.1 v14.2 v4.1 v13.2 v2.2 v5.1

v8.1 0.51 (38, 39) 0.2 (47, 40) 0 (91, 53) 0.03 (46, 30) 0.03 (65, 45) 0.22 (59, 52)

v14.2 0 (88, 40) 0 (110, 42) 0.51 (53, 54) 0 (84, 28) 0.79 (38, 46) 0.36 (55, 52)

v4.1 0 (92, 53) 0 (90, 53) 0 (550, 63) 0.06 (51, 37) 0.19 (63, 54) 0.11 (72, 58)

v13.2 0.25 (35, 30) 0.07 (45, 32) 0.87 (29, 39) 0.36 (29, 27) 0.44 (40, 39) 0.46 (36, 36)

v2.2 0 (77, 46) 0 (84, 48) 0.09 (69, 54) 0.01 (61, 38) 0 (240, 52) 0.16 (63, 53)

v5.1 0.4 (54, 52) 0 (100, 52) 0.15 (70, 59) 0.99 (18, 35) 0.07 (68, 52) 0 (180, 55)



Refit after selection

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Local independence graph

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Does the graph have a causal interpretation?

• Yes, if the observable event tracks have a distribution with acausally valid graph.

• No, not in general if the causally valid graph containsunobserved processes.

Earlier work

NRH, P. Reynaud-Bouret and V. Rivoirard. Lasso and probabilistic inequalitiesfor multivariate point processes, Bernoulli 21(1), 2015.

NRH. Nonparametric likelihood based estimation of linear filters for point

processes. Statistics and Computing 25(3), 2015.

has focused on predictive models.



Markov properties for a faithful graph

X ZY

Y

X ZY

U1U2

Y

U1U2

Y

U1U2

X → Y | Z X → Y | ZY → X | Z Y → X | ZZ → Y | X Z → Y | XY → Z | X Y → Z | XX 6→ Z | Y X → Z | YZ 6→ Y | Y Z → X | YX → Z | ∅ X 6→ Z | ∅Z → X | ∅ Z 6→ X | ∅




X ZY

Y

X Z

Y

U1U2

Y

U1U2

Y

U1U2

X → Y | Z X → Y | ZY → X | Z Y → X | ZZ → Y | X Z → Y | XY → Z | X Y → Z | X

X 6→ Z | Y X → Z | YZ 6→ Y | Y Z → X | YX → Z | ∅ X 6→ Z | ∅Z → X | ∅ Z 6→ X | ∅




X ZY

Y

X Z

Y

U1U2

Y

U1U2

Y

U1U2

X → Y | Z X → Y | ZY → X | Z Y → X | ZZ → Y | X Z → Y | XY → Z | X Y → Z | XX 6→ Z | Y X → Z | YZ 6→ Y | Y Z → X | Y

X → Z | ∅ X 6→ Z | ∅Z → X | ∅ Z 6→ X | ∅




X Z

Y

Y

X Z

Y

U1U2

Y

U1U2

Y

U1U2

X → Y | Z X → Y | ZY → X | Z Y → X | ZZ → Y | X Z → Y | XY → Z | X Y → Z | XX 6→ Z | Y X → Z | YZ 6→ Y | Y Z → X | YX → Z | ∅ X 6→ Z | ∅Z → X | ∅ Z 6→ X | ∅



Causal search

A PC-type algorithm. Initiate with the complete directed graph onV . Let pa(j) denote the parents of vertex j .

• k = 0: Test j 6→ k | ∅. Remove the edge (j , k) if the test isnot rejected.

• k = 1: Test j 6→ k | m for (j , k), (m, k) ∈ E (j ,m ∈ pa(j)).Remove the edge (j , k) if the test is not rejected.

• . . .• k: Test j 6→ k | C for {j} ∪ C ⊆ pa(j) with |C | = k. Remove

the edge (j , k) if the test is not rejected.

The algorithm stops when pa(j) < k + 1 for all j .



Local independence graph

v8.1

v14.2

v4.1

v13.2

v2.2 v5.1

No MAGs, PAGs or (R)FCI-type algorithm yet / / /.



Problems

Proposed tests rely on joint as well as marginal Hawkes processmodels.

The class of Hawkes processes is not closed undermarginalization.

• Marginal independence can be tested using Ripley’sK -function under stationarity.

• Nonparametric tests for conditional local independence ofevent tracks are not abundant (that’s true for conditionalindependence tests as well without e.g. a Gaussianassumption).

• Causal effect estimates (intervention effects) have not beenconsidered.



Extensions

Conditional local independence was not defined in terms offactorization.

It was defined in terms of functional dependence of compensatorsfor counting processes.

A formal generalization to semimartingales is possible in terms ofthe characteristic triplet. We have resolved some issues for SDEs1,such as interventions being well-defined.

Open problem: Does generalizations beyond the counting / eventprocesses enjoy the global Markov property?

1A. Sokol and NRH. Causal interpretation of stochastic differential equations.Electron. J. Probab. 19, 2014.



Acknowledgement

Research sponsered by VILLUM FONDEN.

Causal inference with incomplete dataNRH, Steffen Lauritzen and Jonas Peters

http://www.math.ku.dk/~richard/causality/

Open postdoc and Ph.D.-positions with deadlines early December2016.

Thanks


http://www.math.ku.dk/~richard/causality/

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