The present work provides an application of Global Sensitivity Analysis to supervised machine learning methods such as Random Forests. These methods act as black boxes, selecting features in high--dimensional data sets as to provide accurate classifiers in terms of prediction when new data are fed into the system. In supervised machine learning, predictors are generally ranked by importance based on their contribution to the final prediction. Global Sensitivity Analysis is primarily used in mathematical modelling to investigate the effect of the uncertainties of the input variables on the output. We apply it here as a novel way to rank the input features by their importance to the explainability of the data generating process, shedding light on how the response is determined by the dependence structure of its predictors. A simulation study shows that our proposal can be used to explore what advances can be achieved either in terms of efficiency, explanatory ability, or simply by way of confirming existing results.

Bootstrap is a popular methodology for simulating input uncertainty. However, it can be computationally expensive when the number of samples is large. We propose a new approach called \textbf{Orthogonal Bootstrap} that reduces the number of required Monte Carlo replications. We decomposes the target being simulated into two parts: the \textit{non-orthogonal part} which has a closed-form result known as Infinitesimal Jackknife and the \textit{orthogonal part} which is easier to be simulated. We theoretically and numerically show that Orthogonal Bootstrap significantly reduces the computational cost of Bootstrap while improving empirical accuracy and maintaining the same width of the constructed interval.

We study a plug in least squares estimator for the change point parameter where change is in the mean of a high dimensional random vector under subgaussian or subexponential distributions. We obtain sufficient conditions under which this estimator possesses sufficient adaptivity against plug in estimates of mean parameters in order to yield an optimal rate of convergence $O_p(\xi^{-2})$ in the integer scale. This rate is preserved while allowing high dimensionality as well as a potentially diminishing jump size $\xi,$ provided $s\log (p\vee T)=o(\surd(Tl_T))$ or $s\log^{3/2}(p\vee T)=o(\surd(Tl_T))$ in the subgaussian and subexponential cases, respectively. Here $s,p,T$ and $l_T$ represent a sparsity parameter, model dimension, sampling period and the separation of the change point from its parametric boundary. Moreover, since the rate of convergence is free of $s,p$ and logarithmic terms of $T,$ it allows the existence of limiting distributions. These distributions are then derived as the {\it argmax} of a two sided negative drift Brownian motion or a two sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively. Thereby allowing inference of the change point parameter in the high dimensional setting. Feasible algorithms for implementation of the proposed methodology are provided. Theoretical results are supported with monte-carlo simulations.