Integral estimation based on Markovian design

TL;DR

This paper introduces a novel integral estimation method using a Markovian sensor trajectory that does not require knowledge of the trajectory distribution, achieving faster convergence rates than traditional methods.

Contribution

The paper proposes a new integral estimation approach based on Markovian trajectories that improves convergence rates without needing distributional assumptions.

Findings

Probabilistic bounds on estimator convergence are established.

The method outperforms classical Monte Carlo in convergence speed.

Simulations and ocean temperature application demonstrate effectiveness.

Abstract

Suppose that a mobile sensor describes a Markovian trajectory in the ambient space. At each time the sensor measures an attribute of interest, e.g., the temperature. Using only the location history of the sensor and the associated measurements, the aim is to estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g., Monte Carlo, the proposed approach does not require any knowledge on the distribution of the sensor trajectory. Probabilistic bounds on the convergence rates of the estimator are established. These rates are better than the traditional "root n"-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, the good behaviour of the procedure is demonstrated through simulations and an application to the evaluation of the average temperature of oceans is considered.