In the following, operations are defined componentwise. Specifically, if v = (v i ) 1≤i≤r , w = (w i ) 1≤i≤r and (u (k) ), k ≥ 1, are in Z r , then

Analysis of near-surface measurements at several measuring points in Graz, Austria, reveals the impact of restrictive measures during the COVID-19 pandemic on the emission of atmospheric pollutants. We quantify the effects at traffic hotspots, industrial and residential areas. Using historical data collected over several years, we are able to account for meteorological and seasonal confounders. Our analysis is based on daily means as well as intraday pollution level curves. Nitrogen dioxide (NO 2 ) has decreased drastically while the levels of particulate matter PM 10 and carbon monoxide (CO) mostly exhibit little change. Traffic data shows that the decrease in traffic frequency is parallel to the decline in the levels of NO 2 and NO.

We consider the problem of estimating the conditional distribution P(Y ∈ A|X) of a functional data object Y = (Y (t) : t ∈ [0, 1]) in the space of continuous functions, given covariates X in a general space and assuming that Y and X are related by a functional linear regression model. Two estimation methods are proposed, based on either the empirical distribution of the estimated model residuals, or fitting functional parametric models to the model residuals. We show that consistent estimation can be achieved under relatively mild assumptions. We exemplify a general class of sets A specifying path properties of Y that are of interest in applications. The proposed methods are studied in several simulation experiments, and data analyses of electricity price and pollution curves.

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We consider the problem of consistently estimating the conditional distribution P (Y ∈ A|X) of a functional data object Y = (Y (t) : t ∈ [0, 1]) given covariates X in a general space, assuming that Y and X are related by a functional linear regression model. Two natural estimation methods are proposed, based on either bootstrapping the estimated model residuals, or fitting functional parametric models to the model residuals and estimating P (Y ∈ A|X) via simulation. Whether either of these methods lead to consistent estimation depends on the consistency properties of the regression operator estimator, and the space within which Y is viewed. We show that under general consistency conditions on the regression operator estimator, which hold for certain functional principal component based estimators, consistent estimation of the conditional distribution can be achieved, both when Y is an element of a separable Hilbert space, and when Y is an element of the Banach space of continuous functions. The latter results imply that sets A that specify path properties of Y , which are of interest in applications, can be considered. The proposed methods are studied in several simulation experiments, and data analyses of electricity price and pollution curves.