We study the properties of a stochastic heat equation with a generalized mixed fractional Brownian noise. We obtain the covariance structure, stationarity and obtain bounds for the asymptotic behaviour of the solution. We suggest estimators for the unknown parameters based on discrete time observations and study their asymptotic properties.
Network lasso is a method for solving a multi-task learning problem through the regularized maximum likelihood method. A characteristic of network lasso is setting a different model for each sample. The relationships among the models are represented by relational coefficients. A crucial issue in network lasso is to provide appropriate values for these relational coefficients. In this paper, we propose a Bayesian approach to solve multi-task learning problems by network lasso. This approach allows us to objectively determine the relational coefficients by Bayesian estimation. The effectiveness of the proposed method is shown in a simulation study and a real data analysis.
Kawasaki (2023) introduced Borsuk- Ulam's theorem to nonlinear optimization. It applied Borsuk-Ulam's theorem to an n-tuple of parametric optimization problems with parameter u ∈ S^n, and presented an antipodal theorem for them. Further, it showed that given n convex sets in ℝ^n, it is possible to divide each width in half with a hyperplane. In this paper, we take another approach, which weakens the assumptions, simplifies the proofs, and includes a result beyond the scope of Kawasaki (2023). Further, we add a new insight to the ham-sandwich theorem.
pairwise comparison matrix can be used to measure the degree of consistency of the decision maker's judgment. Since the eigenvalues of a matrix are the roots of a characteristic equation, they can be obtained by numerical iterative methods such as Newton's method. However, the location of the initial point does not always guarantee that the maximal eigenvalue will be obtained. For the third-order pairwise comparison matrix, it has already been shown that the characteristic equations always have a single real root and that Newton's method using three as the initial point can always achieve the maximal eigenvalue. In this paper, we focus on the characteristic equation of a fourth-order pairwise comparison matrix and show that they have exactly two real roots. Furthermore, we present an initial point of Newton's method such that it always converges to the larger root, i.e., the maximal eigenvalue. Numerical experiments are also provided to compare convergence rate with other numerical iterative methods.
In model theory, function symbols for formal systems are usually assumed to be totally defined.This paper reports that category theory is defined and discussed as a formal system with partial operations in first order predicate logic.We also review a formal definition of relational calculus with a local boolean structure and satisfying de Morgan-Schröder equivalences.
We investigate the asymptotic properties of the minimum $L_1$-norm estimator of the drift parameter for fractional Ornstein-Uhlenbeck type process driven by a general Gaussian process.
Subgroup analysis is an exploratory analysis often conducted with the expectation of providing useful information to establish new research hypotheses in medicine.However, use of subgroup analysis is currently limited, as the number of subgroups is often small, and thus subgroups are selected arbitrarily and typically deviate from subgroups in which the readers of a paper are interested in.In this study, we propose a method for representing the results of subgroup analysis in graphical form by focusing on a comparative clinical study of treatment versus control groups, with survival time as the primary endpoint, and in which subgroups are created by dividing the range of continuous biomarkers.This method unifies and strengthens fragmented data, enabling findings obtained in each subgroup by assuming the Cox proportional hazard model.Moreover, this method renders it possible to obtain information about any subgroup that an outside researcher is interested in.
The study aims to evaluate the efficacy of each category in physical therapy for improving motor function of patients with stroke, focusing on physical therapy alone.106 patients who received the stroke physical therapy program in seven hospitals from April 2020 to March 2021 participated in the study.The contents of the program were classified into the following five categories; having no physical therapy (PT0), preparatory exercise in a static position (PT1), exercises to improve basic activities (PT2), gait exercises without special tools (PT3), and therapy with special tools (PT4).The endpoint of the study was the relative shortening ratio (RSR) that compared hours of supine-to-sitting before and after the program.The hours of supine-to-sitting before the program was identified as a confounder.Adjusting for the confounder, the relationship of PT1, PT2, PT3, and PT4 with RSR was evaluated by the method of multiple linear regression.Only PT2 was significantly related to the RSR; PT1, PT3, and PT4 were not.An equation was developed to predict the RSR based on PT2 adjusting for the confounder.The study indicates that the program putting higher percentage of hours for PT2 in stroke physical therapy could be more effective for improving the motor function of patients with stroke.
We study the problem of estimation of the parameter in a parabolic stochastic parabolic equation driven by an infinite dimensional mixed fractional Brownian motion.
Recently, many Markov chain Monte Carlo methods have been developed with deterministic reversible transform proposals inspired by the Hamiltonian Monte Carlo method. The deterministic transform is relatively easy to reconcile with the local information (gradient etc.) of the target distribution. However, as the ergodic theory suggests, these deterministic proposal methods seem to be incompatible with robustness and lead to poor convergence, especially in the case of target distributions with heavy tails. On the other hand, the Markov kernel using the Haar measure is relatively robust since it learns global information about the target distribution introducing global parameters. However, it requires a density preserving condition, and many deterministic proposals break this condition. In this paper, we carefully select deterministic transforms that preserve the structure and create a Markov kernel, the Weave-Metropolis kernel, using the deterministic transforms. By combining with the Haar measure, we also introduce the Haar-Weave-Metropolis kernel. In this way, the Markov kernel can employ the local information of the target distribution using the deterministic proposal, and thanks to the Haar measure, it can employ the global information of the target distribution. Finally, we show through numerical experiments that the performance of the proposed method is superior to other methods in terms of effective sample size and mean square jump distance per second.
For a sample from GEM distribution considered as a random discrete distribution of positive integers, there are probably unobserved less positive integers than the maximum sample.The number of these integers follows a mixed Poisson distribution.Here, we provide a simple proof of the convergence in distribution of this number.Similarly, its asymptotic distribution is a mixed Poisson.We derived the upper bounds for the total variation between the finite and asymptotic distributions.These distributions are illustrated by examples; the related total values are provided in table 1.
For analyzing clustered survival data, a flexible partially linear additive hazards model is proposed.To accommodate the nonlinear effects, the unknown regression function is approximated by B-splines.All regression coefficients are estimated through a system of pseudo-score functions.Under certain conditions, the proposed estimators are shown to be asymptotically normal, where a consistent estimator of the covariance matrix is given.Simulation studies are also conducted to evaluate the finite sample performance of the proposed method, which is illustrated using a real data set from an AIDS clinical trial.
The aim of this study is to propose stepwise multiple comparison procedures for comparing sizes of normal means.Specifically, we construct Tukey-Welsh's step down procedure and a closed testing procedure called Ryan-Einot-Gabriel-Welsch's procedure based on Imada (2020)'s single step procedure and compare them in terms of numerical results regarding power of the test.Furthermore, we illustrate our procedures by an example.
Parametric and nonparametric inference for stochastic processes driven by a fractional Brownian motion were investigated in Mishura (2008) and Prakasa Rao(2010) among others. Similar problems for processes driven by an infinite dimensional fractional Brownian motion were studied in Prakasa Rao (2004,2013), Cialenco (2009) and others. Parametric estimation for processes driven by infinite dimensional mixed fractional Brownian motion is discussed in this article.
Pairwise comparison matrix (PCM) and its maximum eigenvalue play key roles in analytic hierarchy process (AHP). We shed light on the characteristic polynomial of a PCM of 4th order. By computational simulation, we can confirm that the value of the characteristic polynomial for 4 is non-positive in case which Saaty’s discrete scale will be used. Thus we can show the real-number solution of the characteristic equation exists and is greater than 4 in the practical use of AHP.
The local structure of a cellular automaton is a set of its neighborhood that represent interacting cells and its local function that represent its interaction.In this paper we discuss the behavior, especially the reversibility of of CA-150 extended the local structure (neighborhood) symmetrically and show a relationship of the cell size and the neighborhood radius for CA-150 to be reversible.
