arXiv subjects

Lauri Viitasaari

Publications and source records attributed to Lauri Viitasaari.

At least 37 records · Page 2Linked to original sources

arXiv2024

On optimal prediction of missing functional data with memory

This paper considers the problem of reconstructing missing parts of functions based on their observed segments. It provides, for Gaussian processes and arbitrary bijective transformations thereof, theoretical expressions for the $L^2$-optimal reconstruction of the missing parts. These functions are obtained as solutions of explicit integral equations. In the discrete case, approximations of the solutions provide consistent expressions of all missing values of the processes. Rates of convergence of these approximations, under extra assumptions on the transformation function, are provided. In the case of Gaussian processes with a parametric covariance structure, the estimation can be conducted separately for each function, and yields nonlinear solutions in presence of memory. Simulated examples show that the proposed reconstruction indeed fares better than the conventional interpolation methods in various situations.

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arXiv2024

On parameter estimation for $N(μ,σ^2 I_3)$ based on projected data into $\mathbb{S}^2$

We consider the projected normal distribution, with isotropic variance, on the 2-sphere using intrinsic statistics. We show that in this case, the expectation commutes with the projection and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows to estimate, after model identification, the parameters of the underlying normal distribution that generates the data.

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arXiv2024

On the Impact of Approximation Errors on Extreme Quantile Estimation with Applications to Functional Data Analysis

We study the effect of approximation errors in assessing the extreme behavior of heavy-tailed random objects. We give conditions for the approximation error such that the standard asymptotic results hold for the classical Hill estimator and the corresponding extreme quantile estimator. As an application, we consider the effect of discretization errors in the computation of the $L^p$-norms related to functional data. We approximate the norms both with Riemann sums and with Monte Carlo integration. We quantify connections between the number of observed functions, the number of discretization points, and the regularity of the underlying functions. In addition, we derive a new concentration inequality for order statistics. This, to the best of our knowledge, is the first Chernoff-type concentration inequality for order statistics presented in the literature that provides an explicit rate at which the ratio between order statistics and tail quantile function converges to one. In our application, the bound is used to provide concentration inequalities measuring the distance between the Hill estimator based on approximated norms and the one based on the true ones.

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arXiv2024

Zombie Epidemic -- on Modeling the Effect of Interventions

The recent COVID-19 pandemic has highlighted the need of studying extreme, life-threatening phenomena in advance. In this article, a zombie epidemic in Uusimaa region in Finland is modeled. A stochastic agent based simulation model is proposed and extensive simulations are conducted for this purpose. The model utilizes knowledge on defensive human behavior during crises. Studying the effects of a hypothetical zombie attack resembles examining the spread of deadly diseases and of rumors. A zombie attack is simulated in the most densely populated region in Finland. The region's exact population densities over its rasterized geographical map are utilized. Furthermore, the simulations are used to study the effect of implementing a (strict or partial) quarantine area in the epicenter. Computationally efficient Scala codes and video animations of the simulated epidemics are provided. The main findings emphasize the importance of implementing very strict measures, without delay, to stop the outbreak.

stat.AP↗

arXiv2024

On explosion time in stochastic differential equations driven by fractional Brownian motion

In this article, we study the explosion time of the solution to autonomous stochastic differential equations driven by the fractional Brownian motion with Hurst parameter $H>1/2$. With the help of the Lamperti transformation, we are able to tackle the case of non-constant diffusion coefficients not covered in the literature. In addition, we provide an adaptive Euler-type numerical scheme for approximating the explosion time.

math.PR↗

arXiv2024

On the existence and regularity of local times

We study the existence and regularity of local times for general $d$-dimensional stochastic processes. We give a general condition for their existence and regularity properties. To emphasize the contribution of our results, we show that they include various prominent examples, among others solutions to stochastic differential equations driven by fractional Brownian motion, where the behavior of the local time was not fully understood up to now and remained as an open problem in the stochastic analysis literature. In particular this completes the picture regarding the local time behavior of such equations, above all includes high dimensions and both large and small Hurst parameters. As other main examples, we also show that by using our general approach, one can quite easily cover and extend some recently obtained results on the local times of the Rosenblatt process and Gaussian quasi-helices.

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arXiv2023

Necessary and sufficient conditions for continuity of hypercontractive processes and fields

Sample path properties of random processes are an interesting and extensively studied topic, especially in the case of Gaussian processes. In this article, we study the continuity properties of hypercontractive fields, providing natural extensions for some known Gaussian results beyond Gaussianity. Our results apply to both random processes and random fields alike.

math.PR↗

arXiv2023

Transfer principle for fractional Ornstein-Uhlenbeck processes

We prove the transfer principle for fractional Ornstein-Uhlenbeck processes, i.e., we construct a Brownian motion that has the same filtration as the fractional Ornstein-Uhlenbeck process and then represent the fractional Ornstein-Uhlenbeck process by using the constructed Brownian motion. As applications of the transfer principle we consider prediction.

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arXiv2023

Concentration inequalities for leave-one-out cross validation

In this article we prove that estimator stability is enough to show that leave-one-out cross validation is a sound procedure, by providing concentration bounds in a general framework. In particular, we provide concentration bounds beyond Lipschitz continuity assumptions on the loss or on the estimator. We obtain our results by relying on random variables with distribution satisfying the logarithmic Sobolev inequality, providing us a relatively rich class of distributions. We illustrate our method by considering several interesting examples, including linear regression, kernel density estimation, and stabilized/truncated estimators such as stabilized kernel regression.

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arXiv2023

Discretisation error for stochastic integrals with respect to the fractional Brownian motion with discontinuous integrands and local times

We consider equidistant Riemann approximations of stochastic integrals $\int_0^T f(B^H_s)dB^H_s$ with respect to the fractional Brownian motion with $H>\frac12$, where $f$ is an arbitrary function of locally bounded variation, hence possibly possessing discontinuities. We prove that properly normalised approximation error converge in the $L^2$-topology to a functional of the local time, and we provide rate of convergence for this approximation. As such, our results complements some recent advances on the topic as well as provides new methods for simulation of local times.

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arXiv2023

On Lamperti transformation and characterisations of discrete random fields

In this article we characterise discrete time stationary fields by difference equations involving stationary increment fields and self-similar fields. This gives connections between stationary fields, stationary increment fields and, through Lamperti transformation, self-similar fields. Our contribution is a natural generalisation of recently proved results covering the case of stationary processes.

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arXiv2022

On sharp rate of convergence for discretisation of integrals driven by fractional Brownian motions and related processes with discontinuous integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $H>\frac12$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $L^1$-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $n^{1-2H}$ that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

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arXiv2022

Flexible transition probability model for assessing cost-effectiveness of breast cancer screening

Breast cancer is the most common cancer among Western women. Fortunately, organized screening has reduced breast cancer mortality and, consequently, the European Union has recommended screening with mammography for 50-69-year-old women. This recommendation is followed well in Europe. Widening the screening target age further is supported by conditional recommendations for 45-49- and 70-74-year-old women. However, before extending screening to new age groups, it's essential to carefully consider the benefits and costs locally as circumstances vary between different regions and/or countries. We propose a new approach to assess cost-effectiveness of breast cancer screening for a long-ongoing program with incomplete historical screening data. The new model is called flexible stage distribution model. It is based on estimating the stage distributions of breast cancer cases under different screening strategies. In the model, an ongoing screening strategy may be used as a baseline and other screening strategies may be incorporated by changes in the incidence rates. The model is flexible, as it enables to apply different approaches for estimating the altered stage distributions. Thus, if randomized data is available, one may rely on that. On the other hand, if randomized data is not available, altered stage distributions may be estimated by extrapolating the stage distributions of the youngest and oldest screened/non-screened age groups. We apply the proposed flexible stage distribution model for assessing incremental cost of extending the current biennial breast cancer screening to younger and older target ages in Finland.

stat.AP↗

arXiv2022

Variability of paths and differential equations with $BV$-coefficients

We define compositions $φ(X)$ of Hölder paths $X$ in $\mathbb{R}^n$ and functions of bounded variation $φ$ under a relative condition involving the path and the gradient measure of $φ$. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions $φ(X)$ with respect to a given Hölder path $Y$. These results are then used, together with Doss' transform, to obtain existence and, in a certain sense, uniqueness results for differential equations in $\mathbb{R}^n$ driven by Hölder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of two-dimensional fractional Brownian motions.

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arXiv2022

Geometric Characterization of the Eyring-Kramers Formula

In this paper we consider the mean transition time of an over-damped Brownian particle between local minima of a smooth potential. When the minima and saddles are non-degenerate this is in the low noise regime exactly characterized by the so called Eyring-Kramers law and gives the mean transition time as a quantity depending on the curvature of the minima and the saddle. In this paper we find an extension of the Eyring-Kramers law giving an upper bound on the mean transition time when both the minima/saddles are degenerate (flat) while at the same time covering multiple saddles at the same height. Our main contribution is a new sharp characterization of the capacity of two local minimas as a ratio of two geometric quantities, i.e., the smallest separating surface and the geodesic distance.

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arXiv2021

Sobolev regularity of occupation measures and paths, variability and compositions

We prove a result on the fractional Sobolev regularity of composition of paths of low fractional Sobolev regularity with functions of bounded variation. The result relies on the notion of variability, proposed by us in the previous article [43, arXiv:2003.11698]. Here we work under relaxed hypotheses, formulated in terms of Sobolev norms, and we can allow discontinuous paths, which is new. The result applies to typical realizations of certain Gaussian or Lévy processes, and we use it to show the existence of Stieltjes type integrals involving compositions.

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arXiv2021

Least square estimators in linear regression models under negatively superadditive dependent random observations

In this article we study the asymptotic behaviour of the least square estimator in a linear regression model based on random observation instances. We provide mild assumptions on the moments and dependence structure on the randomly spaced observations and the residuals under which the estimator is strongly consistent. In particular, we consider observation instances that are negatively superadditive dependent within each other, while for the residuals we merely assume that they are generated by some continuous function. In addition, we prove that the rate of convergence is proportional to the sampling rate $N$, and we complement our findings with a simulation study providing insights on finite sample properties.

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arXiv2021

Long-range dependent completely correlated mixed fractional Brownian motion

In this paper we introduce the long-range dependent completely correlated mixed fractional Brownian motion (ccmfBm). This is a process that is driven by a mixture of Brownian motion (Bm) and a long-range dependent completely correlated fractional Brownian motion (fBm, ccfBm) that is constructed from the Brownian motion via the Molchan--Golosov representation. Thus, there is a single Bm driving the mixed process. In the short time-scales the ccmfBm behaves like the Bm (it has Brownian Hölder index and quadratic variation). However, in the long time-scales it behaves like the fBm (it has long-range dependence governed by the fBm's Hurst index). We provide a transfer principle for the ccmfBm and use it to construct the Cameron--Martin--Girsanov--Hitsuda theorem and prediction formulas. Finally, we illustrate the ccmfBm by simulations.

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