Data depth for measurable noisy random functions

Abstract

In the literature on data depth applicable to random functions it is usually assumed that the trajectories of all the random curves are continuous, known at each point of the domain, and observed exactly. These assumptions turn out to be unrealistic in practice, as the functions are often observed only at a finite grid of time points, and in the presence of measurement errors. In this work, we provide the necessary theoretical background enabling the extension of the statistical methodology based on data depth to measurable (not necessarily continuous) random functions observed within the latter framework. It is shown that even if the random functions are discontinuous, observed discretely, and contaminated with additive noise, many common depth functionals maintain the fine consistency properties valid in the ideal case of completely observed noiseless functions. For the integrated depth for functions, we provide uniform rates of convergence over the space of integrable functions.

Flexible integrated functional depths

“…Results of this type are not available for infimal depths, or other non-integrated versions of depths for infinite-dimensional data (López-Pintado and Romo [29], López-Pintado and Romo [30], Chakraborty and Chaudhuri [5], Sguera, Galeano and Lillo [44]). It is important to mention that in the limit cases k = ±∞ the consistency may fail to hold true, as can be seen by modification of the example using the infimal depth in L 2 ([0, 1]) of Nagy and Ferraty [37], p. 99. Theorem 1.…”

Recognizing VSC DC Cable Fault Types Using Bayesian Functional Data Depth

“…Chenouri used it for improving the quality of nonparametric tests [34]. Nagy and Ferraty [35] use functional data analysis to represent discontinuous data. It was also used to analyze functional data [36,37].…”

Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis

“…These results are, of course, directly applicable to h-depth and functional projection depth. The next theorem is similar to [70,Theorem 7] but the proof leverages the KME representation of h-depth and uses the relationship between MMD and weak convergence to easily obtain the desired convergence result. Theorem 8.…”