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Continuous time Gaussian process dynamical models in gene regulatory network inference Aalto, Atte ; ; et al E-print/Working paper (2018) One of the focus areas of modern scientific research is to reveal mysteries related to genes and their interactions. The dynamic interactions between genes can be encoded into a gene regulatory network ... [more ▼] One of the focus areas of modern scientific research is to reveal mysteries related to genes and their interactions. The dynamic interactions between genes can be encoded into a gene regulatory network (GRN), which can be used to gain understanding on the genetic mechanisms behind observable phenotypes. GRN inference from time series data has recently been a focus area of systems biology. Due to low sampling frequency of the data, this is a notoriously difficult problem. We tackle the challenge by introducing the so-called continuous-time Gaussian process dynamical model (GPDM), based on Gaussian process framework that has gained popularity in nonlinear regression problems arising in machine learning. The model dynamics are governed by a stochastic differential equation, where the dynamics function is modelled as a Gaussian process. We prove the existence and uniqueness of solutions of the stochastic differential equation. We derive the probability distribution for the Euler discretised trajectories and establish the convergence of the discretisation. We develop a GRN inference method based on the developed framework. The method is based on MCMC sampling of trajectories of the GPDM and estimating the hyperparameters of the covariance function of the Gaussian process. Using benchmark data examples, we show that our method is computationally feasible and superior in dealing with poor time resolution. [less ▲] Detailed reference viewed: 32 (4 UL)Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian-fractional Brownian model Azmoodeh, Ehsan ; ; in Modern Stochastics: Theory and Applications (2015), 2(1), 2949 Detailed reference viewed: 31 (5 UL)Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind Azmoodeh, Ehsan ; in Statistical Inference for Stochastic Processes (2015), 18(3), 205227 Detailed reference viewed: 36 (0 UL)Necessary and sufficient conditions for Hölder continuity of Gaussian processes Azmoodeh, Ehsan ; ; et al in Statistics & Probability Letters (2014), 94 The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note ... [more ▼] The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov–Čentsov condition. [less ▲] Detailed reference viewed: 40 (0 UL)Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian-fractional Brownian model Azmoodeh, Ehsan ; ; E-print/Working paper (2014) We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst ... [more ▼] We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter H of the fractional part satisfies H∈(3/4,1), the central limit theorem holds. In the nonsemimartingale case, that is, where H∈(1/2,3/4], the convergence toward the normal distribution with a nonzero mean still holds if H=3/4, whereas for the other values, that is, H∈(1/2,3/4), the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator. [less ▲] Detailed reference viewed: 40 (0 UL)A general approach to small deviation via concentration of measures Azmoodeh, Ehsan ; E-print/Working paper (2014) Detailed reference viewed: 13 (0 UL)Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind Azmoodeh, Ehsan ; E-print/Working paper (2013) Detailed reference viewed: 34 (1 UL)Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion Azmoodeh, Ehsan ; in Journal of Theoretical Probability (2013) In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, where $B_t$ denotes the fractional Brownian motion with Hurst parameter $H \in (\frac{1}{2},1)$, for a ... [more ▼] In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, where $B_t$ denotes the fractional Brownian motion with Hurst parameter $H \in (\frac{1}{2},1)$, for a large class of convex functions $f$ is considered. In $\big[$\cite{a-m-v}, Statistics \& Decisions, \textbf{27}, 129-143$\big]$, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought arbitrary close to $H - \frac{1}{2}$. [less ▲] Detailed reference viewed: 61 (2 UL) |
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