Quasi-Monte Carlo cubature methods often sample the integrand using Sobol' (or other digital) sequences to obtain higher accuracy than IID sampling. An important question is how to conservatively estimate the error of a digital sequence cubature so that the sampling can be terminated when the desired tolerance is reached. We propose an error bound based on the discrete Walsh coefficients of the integrand and use this error bound to construct an adaptive digital sequence cubature algorithm. The error bound and the corresponding algorithm are guaranteed to work for integrands lying in a cone defined in terms of their true Walsh coefficients. Intuitively, the inequalities defining the cone imply that the ordered Walsh coefficients do not dip down for a long stretch and then jump back up. An upper bound on the cost of our new algorithm is given in terms of the unknown decay rate of the Walsh coefficients.
Quasi-Monte Carlo methods are used for numerically integrating multivariate functions. However, the error bounds for these methods typically rely on a priori knowledge of some semi-norm of the integrand, not on the sampled function values. In this article, we propose an error bound based on the discrete Fourier coefficients of the integrand. If these Fourier coefficients decay more quickly, the integrand has less fine scale structure, and the accuracy is higher. We focus on rank-1 lattices because they are a commonly used quasi-Monte Carlo design and because their algebraic structure facilitates an error analysis based on a Fourier decomposition of the integrand. This leads to a guaranteed adaptive cubature algorithm with computational cost O(mb m ), where b is some fixed prime number and b m is the number of data points.
Bayesian Decision Trees provide a probabilistic framework that reduces the instability of Decision Trees while maintaining their explainability. While Markov Chain Monte Carlo methods are typically used to construct Bayesian Decision Trees, here we provide a deterministic Bayesian Decision Tree algorithm that eliminates the sampling and does not require a pruning step. This algorithm generates the greedy-modal tree (GMT) which is applicable to both regression and classification problems. We tested the algorithm on various benchmark classification data sets and obtained similar accuracies to other known techniques. Furthermore, we show that we can statistically analyze how was the GMT derived from the data and demonstrate this analysis with a financial example. Notably, the GMT allows for a technique that provides explainable simpler models which is often a prerequisite for applications in finance or the medical industry.
In the field of sensitivity analysis, Sobol' indices are sensitivity measures widely used to assess the importance of inputs of a model to its output. The estimation of these indices is often performed trough Monte Carlo or quasi-Monte Carlo methods. A notable method is the replication procedure that estimates first-order indices at a reduced cost in terms of number of model evaluations.An inherent practical problem of this estimation is how to quantify the number of model evaluations needed to ensure that estimates satisfy a desired error tolerance. This paper addresses this challenge by proposing a reliable error bound for first-order and total effect Sobol' indices. Starting from the integral formula of the indices, the error bound is defined in terms of the discrete Walsh coefficients of the different integrands.We propose a sequential estimation procedure of Sobol' indices using the error bound as a stopping criterion. The sequential procedure combines Sobol' sequences with either Saltelli's strategy to estimate both first-order and total effect indices, or the replication procedure to estimate only firstorder indices.Keywords Sobol' index¨error bound¨sequential methodq uasi-Monte Carlo
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