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Anastasia Papavasiliou

Publications and source records attributed to Anastasia Papavasiliou.

11 recordsLinked to original sources

Statistical Inference for Fractional Diffusions

This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them. The second section reviews existing theory of statistical inference for fractional diffusions, identifies remaining challenges and introduces a novel approach. The final section discusses results for the case where fractional diffusions result as a homogenisation limit.

math.PR↗

The Inverse Problem for Single Trajectories of Rough Differential Equations

Motivated by the need to develop a general framework for performing statistical inference for discretely observed random rough differential equations, our aim is to construct a geometric $p$-rough path ${\bf X}$ whose response $Y$, when driving a rough differential equation, matches the observed trajectory $y$. We call this the \textit{continuous inverse problem} and start by rigorously defining its solution. We then develop a framework where the solution can be constructed as a limit of solutions to appropriately designed \textit{discrete inverse problems}, so that convergence holds in $p$-variation. Our approach is based on calibrating the bounded variation paths whose limit defines the rough path `lift' of path $X$ to rough path ${\bf X}$ to the observed trajectory $y$. Moreover, we develop a general numerical algorithm for constructing the solution to the discrete inverse problem. The core idea of the algorithm is to use the signature representation of the path, iterating between the response and the control, each time correcting according to the required properties. We apply our framework to the case where the geometric $p$-rough path ${\bf X}$ is defined as the limit of piecewise linear paths in the $p$-variation topology. We express the discrete inverse problem for a fixed observation rate as a solution to a system of equations driven by piecewise linear paths and prove convergence to the solution of the continuous inverse problem for observation time $δ\to 0$. Finally, we show that, in this context, the numerical algorithm for solving the discrete inverse problem simplifies to an iterative simultaneous update of the local gradients and we prove that it converges in $p$-variation uniformly with respect to $δ$.

math.CA↗

Path Signatures Enable Model-Free Mapping of RNA Modifications

Detecting chemical modifications on RNA molecules remains a key challenge in epitranscriptomics. Traditional reverse transcription-based sequencing methods introduce enzyme- and sequence-dependent biases and fragment RNA molecules, confounding the accurate mapping of modifications across the transcriptome. Nanopore direct RNA sequencing offers a powerful alternative by preserving native RNA molecules, enabling the detection of modifications at single-molecule resolution. However, current computational tools can identify only a limited subset of modification types within well-characterized sequence contexts for which ample training data exists. Here, we introduce a model-free computational method that reframes modification detection as an anomaly detection problem, requiring only canonical (unmodified) RNA reads without any other annotated data. For each nanopore read, our approach extracts robust, modification-sensitive features from the raw ionic current signal at a site using the signature transform, then computes an anomaly score by comparing the resulting feature vector to its nearest neighbors in an unmodified reference dataset. We convert anomaly scores into statistical p-values to enable anomaly detection at both individual read and site levels. Validation on densely-modified \textit{E. coli} rRNA demonstrates that our approach detects known sites harboring diverse modification types, without prior training on these modifications. We further applyied this framework to dengue virus (DENV) transcripts and mammalian mRNAs. For DENV sfRNA, it led to revealing a novel 2'-O-methylated site, which we validate orthogonally by qRT-PCR assays. These results demonstrate that our model-free approach operates robustly across different types of RNAs and datasets generated with different nanopore sequencing chemistries.

q-bio.GN↗

Statistical Inference for the Rough Homogenization Limit of Multiscale Fractional Ornstein-Uhlenbeck Processes

We study the problem of parameter estimation for the homogenization limit of multiscale systems involving fractional dynamics. In the case of stochastic multiscale systems driven by Brownian motion, it has been shown that in order for the Maximum Likelihood Estimators of the parameters of the limiting dynamics to be consistent, data needs to be subsampled at an appropriate rate. We extend these results to a class of fractional multiscale systems, often described as scaled fractional kinetic Brownian motions. We provide convergence results for the MLE of the diffusion coefficient of the limiting dynamics, computed using multiscale data. This requires the development of a different methodology to that used in the standard Brownian motion case, which is based on controlling the spectral norm of the inverse covariance matrix of a discretized fractional Gaussian noise on an interval.

math.ST↗

Approximate Likelihood Construction for Rough Differential Equations

The paper is split in two parts: in the first part, we construct the exact likelihood for a discretely observed rough differential equation, driven by a piecewise linear path. In the second part, we use this likelihood in order to construct an approximation of the likelihood for a discretely observed differential equation driven by a general class of rough paths. Finally, we study the behaviour of the approximate likelihood when the sampling frequency tends to infinity.

math.ST↗

Diffusion Parameter Estimation for the Homogenized Equation

We construct a novel estimator for the diffusion coefficient of the limiting homogenized equation, when observing the slow dynamics of a multiscale model, in the case when the slow dynamics are of bounded variation. Previous research suggests subsampling the data on fixed intervals and computing the corresponding quadratic variation. However, to achieve optimality, this approach requires knowledge of scale separation variable $ε$. Instead, we suggest computing the quadratic variation corresponding to the local extrema of the slow process. Our approach results to a natural subsampling and avoids the issue of choosing a subsampling rate. We prove that the estimator is asymptotically unbiased and we numerically demonstrate that its $L_2$-error is of order ${\mathcal O}(ε^2)$.

math.ST↗

Parameter estimation for rough differential equations

We construct the "expected signature matching" estimator for differential equations driven by rough paths and we prove its consistency and asymptotic normality. We use it to estimate parameters of a diffusion and a fractional diffusions, that is, a differential equation driven by fractional Brownian motion.

math.PR↗

A Distributed Procedure for Computing Stochastic Expansions with Mathematica

The solution of a (stochastic) differential equation can be locally approximated by a (stochastic) expansion. If the vector field of the differential equation is a polynomial, the corresponding expansion is a linear combination of iterated integrals of the drivers and can be calculated using Picard Iterations. However, such expansions grow exponentially fast in their number of terms, due to their specific algebra, rendering their practical use limited. We present a Mathematica procedure that addresses this issue by re-parametrising the polynomials and distributing the load in as small as possible parts that can be processed and manipulated independently, thus alleviating large memory requirements and being perfectly suited for parallelized computation. We also present an iterative implementation of the shuffle product (as opposed to a recursive one, more usually implemented) as well as a fast way for calculating the expectation of iterated Stratonovich integrals for Brownian Motion.

math.PR↗

Coarse-grained modeling of multiscale diffusions: the p-variation estimates

We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We suggest a novel approach that makes use of the total $p$-variation, as defined in the theory of rough paths and avoids the subsampling step. The method is applied to a multiscale OU process.

math.ST↗

Particle Filters for Multiscale Diffusions

We consider multiscale stochastic systems that are partially observed at discrete points of the slow time scale. We introduce a particle filter that takes advantage of the multiscale structure of the system to efficiently approximate the optimal filter.

stat.CO↗

Asymptotic Stability of the optimal filter for non-ergodic signals

In this paper, we study the problem of estimating a Markov chain $X$(signal) from its noisy partial information $Y$, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process ${\mathbb P}\{X_n|Y_n,...,Y_1\}$, referred to hereafter as the {\it optimal filter}. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.

math.PR↗