We investigate here a kernel estimate of the spatial regression function r(x) = E(Y _u | X _u = x), x ∈ ℝ^d, of a stationary multidimensional spatial process { Z _u = (X _u, Y _u), u ∈ ℝ ^N }. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝ^N. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.

Classification of streamflow hydrographs plays an important role in a large number of hydrological and hydraulic studies. For instance, it allows decisions to be made regarding the implementation of hydraulic structures and characterization of different flood types, leading to a better understanding of extreme flow behavior. The employed hydrograph classification methods are generally based on a finite number of hydrograph characteristics and do not include all the available information contained in a discharge time series. In this paper, two statistical techniques from the theory of functional data classification are adapted and applied for the analysis of flood hydrographs. Functional classification directly employs all data of a discharge time series and thus contains all available information on shape, peak, and timing. This potentially allows a better understanding and treatment of floods as well as other hydrological phenomena. The considered functional methodology is applied to streamflow datasets from the province of Quebec, Canada. It is shown that classes obtained using functional approaches have merit and can lead to better representation than those obtained using a multidimensional hierarchical classification method. The considered methodology has the advantage of using all of the information contained in the hydrograph, thus reducing the subjectivity that is inherent in multidimensional analysis of the type and number of characteristics to be used and consequently diminishing the associated uncertainty.