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Adaptive estimation for Hawkes processes; application to genome analysis
Patricia Reynaud-Bouret, Sophie Schbath
TL;DR
Detecting favored or avoided genomic distances requires modeling event dependence beyond standard sequence models. This paper develops nonasymptotic penalized model selection for Hawkes processes, with data-driven penalty calibration and an Islands strategy. The estimator is adaptive minimax for certain function classes, while real-data results agree with and refine biological knowledge.
Problem
The paper addresses detecting favored or avoided distances between genomic events, a dependence structure not captured by standard sequence models and lacking nonasymptotic Hawkes model-selection theory.
Method
The paper combines nonasymptotic penalized model selection for Hawkes processes with data-driven penalty calibration and an Islands strategy for estimating interaction ranges.
Results
The estimator is adaptive minimax for certain function classes, the calibrated penalty works well in simulations, and real-data results agree with previous analyses.
Takeaways & Limitations
The Islands strategy coupled with the angle penalty appears suited to characterizing biological-event dependence and estimating its interaction range.
Takeaways & Limitations
Existing AIC-based selection requires a true fixed knot set asymptotically and prior knowledge of the interaction-function support, while the theoretical penalty constant is not computable in practice.
Abstract
from arXiv · showhide
The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes process. The biological problem is actually complex enough to need a nonasymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is shown to be adaptive minimax for Hölderian functions with regularity in $(1/2,1]$: those aspects have not yet been studied for the Hawkes' process. Moreover, we introduce an efficient strategy, named Islands, which is not classically used in model selection, but that happens to be particularly relevant to the biological question we want to answer. Since a multiplicative constant in the theoretical penalty is not computable in practice, we provide extensive simulations to find a data-driven calibration of this constant. The results obtained on real genomic data are coherent with biological knowledge and eventually refine them.
1. Introduction.
The paper models genomic event occurrences with Hawkes processes to detect favored or avoided distances, addressing limitations of existing genome-analysis procedures. It develops nonasymptotic penalized projection-estimation methods with theoretical guarantees while retaining broader applicability beyond genomic data.
- Motivation: The study directly models occurrences of diverse genomic events to identify favored or avoided distances beyond the memory scale of standard Markov-chain models.Examples include DNA patterns, genes, and other biological signals occurring along genomes.
- Limitations of existing methods: FADO can model general Hawkes interactions and produce smooth estimates, but its AIC-based knot selection has theoretical and practical weaknesses.Its theory assumes a true fixed knot set, while its practical behavior degrades with many possible knot sets.
- Contributions: The paper constructs a nonasymptotic penalized model-selection approach for Hawkes processes, including theoretical results such as oracle inequalities.The approach is intended to select irregular knot sets whose complexity can grow with the observed DNA-sequence length.
- Methodological choices: The theoretical procedure uses penalized projection, or least-squares, estimators rather than penalized maximum likelihood estimators and restricts estimation to piecewise constant functions.The framework focuses on self-exciting Hawkes processes with nonnegative h, while keeping the final estimator computable for self-inhibition.
- Scope: The formalism is kept general because Hawkes processes also model earthquakes, financial data, and economic data, enabling applications beyond genomics.The biological applications constitute a more specialized part of the development.
- Paper organization: Section 3 establishes a nonasymptotic result for projection estimators and derives a penalty whose penalized estimator satisfies an oracle inequality.The result supports selecting a good estimator from a family of projection estimators.
2. Framework.
The framework models a stationary Hawkes process with bounded-support interactions, estimates its intensity through least-squares projection, and selects among interval-based models using penalization. It includes nested, regular, irregular, and sparse Islands strategies, while noting that signed self-inhibition is computationally accommodated but not theoretically studied.
- Process assumptions: The Hawkes process is stationary under p < 1, with interaction function h supported on the known interval (0,A].For DNA applications, A represents the maximal distance at which linear genomic interaction remains reasonable.
- Intensity estimation: The process is observed on [−A,T], typically with T significantly larger than A, and intensity candidates are defined in L2 for least-squares estimation.The Hilbert-space formulation supports the least-square estimators used throughout the framework.
- Model selection: For a model m consisting of disjoint intervals, the projection estimator depends on m, motivating data-driven model selection through bias-variance penalization.The approximation error is identified with the bias, while the stochastic error scales as |m|/T up to a slowly varying quantity C_T.
- Model selection: A penalized criterion selects a model whose projection estimator satisfies an oracle inequality, making it nearly as effective as the best candidate in the model family.The guarantee may hold with high probability or in expectation, up to a residual term and a multiplicative factor that can increase slightly with T.
- Model strategies: The Islands strategy targets localized interactions by allowing every subset of a regular partition, producing sparse models when h is nonzero only on a small interval or union of intervals.The framework also considers nested, regular, and irregular model families with different numbers and structures of candidate models.
- Signed interactions: The projection and penalized projection estimators can be computed without using the sign of h, although signed self-inhibition is not theoretically studied because of major technical issues.Negative h represents self-inhibition, while positive h represents self-excitation.
3. Main results.
The main results control clipped projection estimators on bounded parameter subsets, showing near-sharp risk bounds and minimaxity up to a logarithmic factor. They also establish an expectation oracle inequality for penalized model selection with a parameter-dependent logarithmic penalty, while noting limitations from Hawkes-process concentration.
- Parameter restrictions: The theoretical results are restricted to a bounded subset of L2 because extreme ν or p→1 can make the number of process points explode or vanish.The subset is defined using H, η>ρ>0, and 0<P<1.
- Projection estimation: For a fixed model, the clipped projection estimator is consistent with convergence rate below log(T)/T when s belongs to the model.Proposition 1 controls the risk through approximation bias and a stochastic term.
- Projection estimation: When s belongs to the model, the variance term is bounded above by a constant times |m|log(T)/T, and a matching lower bound gives only a log(T) gap.The logarithmic loss is attributed to controlling the unbounded intensity, whose bound on [0,T] is of order log(T).
- Projection estimation: The clipped projection estimator is minimax on the bounded parameter class up to the logarithmic term.The relevant class is identified as Sm ∩ Lη,ρ,H,P.
- Model selection: The penalized projection estimator satisfies an expectation oracle inequality when the true function and candidate models meet the stated boundedness and partition assumptions.The theorem assumes s∈Lη,ρ,H,P and models written on a regular partition satisfying (3.2).
- Model selection: The penalty has the form a parameter-dependent constant times |m|log(T)^2/T, and Gaussian concentration inequalities do not apply to Hawkes processes.The multiplicative constant depends on H, η, ρ, P, and related parameters; additional logarithmic loss remains.
4. Practical data-driven calibration via simulations.
Because the theoretical penalty constant depends on unknown parameters and is too large for practical use, the simulations calibrate implementable penalties for Regular, Irregular, and Islands strategies. The angle-calibrated Islands estimator performs especially well, selecting dimensions accurately and detecting local features in h.
- Motivation: The theoretical penalty constant is impractical because it depends on unknown parameters, motivating data-driven calibration of the multiplicative constant.The theoretical penalty shape remains a guide, while its constant is calibrated empirically.
- Simulation design: The simulations compare Regular, Irregular, and Islands strategies on models with at most 15 intervals, excluding Nested because it would involve only five models.The estimators used are nontruncated because truncated estimators require unknown parameters and impose nonnegative estimates of h.
- Calibration methods: The angle method automatically identifies a contrast-curve angle near the true dimension and is implemented for the Irregular and Islands strategies.It retains a penalty proportional to model dimension while avoiding manual inspection of the contrast curve.
- Simulation results: As T increases, risk decreases for all methods, while methods 1, 2, and 4—Regular minimal penalty and Irregular or Islands angle penalties—are consistently best.The same three methods remain most precise as T or c increases, and their Oracle Ratio improves with larger T.
- Simulation results: Methods 1, 2, and 4 select the true dimension in most simulations, whereas the other methods tend to overestimate it.In an example simulation, method 4 places spikes correctly, while method 2 misses the smallest bump.
- Simulation results: The Islands strategy with angle penalty, method 4, is appropriate for detecting local spikes, bumps, and even negative jumps in h.This conclusion is based on the simulated estimator behavior and resulting reconstructions.
5. Applications on real data.
Applied to two E. coli genomic datasets, the penalized angle estimator with the Islands strategy identified biologically coherent favored and avoided distances. Compared with FADO, it better indicated the effective support and selected a smaller-dimensional model, while using piecewise-constant estimators.
- Gene occurrences: For 4290 gene occurrences, the estimator indicated no correlation down to 2600 bps, avoidance at 0–500 bps, and favored distances around 700–2000 bps.With support shortened to A = 5000 and A = 2000, the pattern sharpened to a negative effect below 250 bps and a positive effect around 1000 bps.
- Biological interpretation: The gene-distance pattern is coherent with genes generally not overlapping, averaging about 1000 bps in length, and occurring in compact bacterial genomes.These biological observations support the detected negative effect below 250 bps and positive effect around 1000 bps.
- DNA motif tataat: For 1036 tataat motif occurrences, the estimator found favored distances below 600 bps, around 0–1500 bps, and around 3000 bps.After shortening A to 5000, three favored-distance types emerged: less than 300 bps, around 1000 bps, and around 3500 bps.
- Comparison with FADO: The new approach agrees with FADO but shows no significant behavior after 3000 bps, whereas FADO fluctuates until the interval’s end.The comparison was made using piecewise-constant estimators, although FADO can also use splines of any fixed degree.
- Comparison with FADO: The Islands method selected a smaller model dimension than FADO: |m| = 4 versus |m| = 15.Its main limitation is that it considers only piecewise-constant estimators, which the authors state suffice to obtain a general trend.
6. Minimax properties.
The section establishes minimax guarantees for Hawkes-process estimation, including an adaptive minimax result for the Nested strategy over Hölder regularities in (1/2,1]. It also shows that Irregular and Islands strategies achieve rates matching minimax lower bounds up to logarithmic factors for sparse-function classes.
- Nested strategy: The Nested strategy yields an adaptive minimax estimator, a property previously unknown for the Hawkes model.The strategy was not implemented for technical reasons described earlier.
- Nested strategy: The clipped penalized projection estimator with the Nested strategy is adaptive minimax over {HL,a ∩ Lη,ρ: a ∈ (1/2,1]} up to a logarithmic term.Choosing the model collection appropriately and applying the theoretical penalty gives adaptation to a.
- Sparse strategies: The Irregular and Islands strategies provide performance guarantees for sparse functions, with Islands sparsity defined through the support of h for the biological application.Irregular sparsity concerns piecewise functions whose partitions have few intervals, whereas Islands targets sparse support.
- Sparse strategies: When N ≃ log(T) and D ≃ log(T)^a with a < 1, both strategies match the 1/T rate up to logarithmic terms: the lower bound is of order log(T)^a log log T/T and the estimator risk is bounded by log(T)^(a+2)/T.The comparison is stated for the sparse-function sets considered in the section.
7. Technical results.
The technical results establish oracle inequalities for Hawkes-process estimation on a high-probability event controlling local point counts and the random norm. They also identify model-complexity limitations, while extending the penalty result to more general self-inhibiting processes.
- Theorem 2: Theorem 2 provides simultaneous estimation inequalities for all models in a finite family under bounded ν and h assumptions and conditions controlling the model family.The result holds for the practical, unclipped estimator ˜s and on an event with probability at least 1 − 3#{MT}e−x.
- Control event: The key event B controls the number of points in intervals of length A, thereby bounding the intensity and stabilizing the random norm DT.This event makes the unbounded intensity and potentially poorly behaved random norm manageable within the analysis.
- Technical limitations: The choices of N, R, r, and MT are essential for controlling B, preventing the analysis from handling model families with very high complexity.The limitation is attributed to dependence and the process’s lack of boundedness.
- Generalization: For the more general process allowing self-inhibition, the oracle inequality remains valid after intersecting B with the positivity event B′, although B′ is not observable.The result indicates that a penalty proportional to model dimension still works in the self-inhibiting case.
8. Sketch of proofs for the technical and minimax results.
The proofs establish the norm and contrast properties underlying the estimator’s analysis, then derive oracle inequalities through concentration arguments and minimax lower bounds through Kullback–Leibler control and Birgé’s lemma.
- Technical results: Lemma 1 links the Hawkes process’s Bartlett spectrum to the quantity g2 defining the L2 space.Its proof uses known Hawkes-process identities, Plancherel’s identity, and an upper bound on the spectral density.
- Technical results: Lemma 2 proves that D2 defines a norm equivalent to the L2 norm, which is essential for the subsequent analysis.The associated quadratic form T has expectation given by the relevant L2 quantity, and the proof is omitted as available in [23].
- Technical results: The functional γT is a contrast because E(γT(f)) is minimized at f = s, using martingale properties and the norm property from Lemma 2.The argument relies on a concentration inequality for χ2-type statistics that applies to any counting process.
- Oracle inequalities: Theorem 2’s oracle-inequality proof controls two random terms separately, using χ2-type concentration and union bounds over the model family.The doubly random selected-model term is controlled through predictable-process representations, while the second term is handled on the event B.
- Oracle inequalities: The proof controls event B through interval point counts and deviations of D2_T from its mean, with the latter obtained from coupling-based Hawkes-process concentration.The remaining computations are described as lengthy and omitted, with details referenced in [23].
- Minimax lower bounds: For minimax lower bounds, Lemma 4 shows that Kullback–Leibler distance grows linearly with T and clarifies its link to the L2 norm; Lemma 5 then applies Birgé’s lemma.Lemma 5 supplies the finite-family construction used for the different lower-bound settings under stated parameter conditions.
9. Conclusion.
The paper proposes an adaptive-minimax model-selection method for Hawkes processes, with a data-driven penalty calibration and an Islands strategy coupled with an angle penalty for the biological problem. Future work includes modeling interactions with other event types and testing whether the interaction function is nonzero.
- The proposed Hawkes-process model-selection method is proved adaptive minimax for certain function classes.
- A data-driven calibration of the penalty’s multiplicative constant works well in simulations.
- The Islands strategy coupled with the angle penalty is designed for the paper’s biological problem.
- Future work should extend the Islands strategy to interactions with other event types, such as promoters or genes.
- A test procedure should determine whether h is nonzero, equivalently whether any interaction exists.