Prediction and Testing for Non-Parametric Random Function Signals in a Complex System

Methods employed in the construction of prediction bands for continuous curves require a dierent approach to those used for a data point. In many cases, the underlying function is unknown and thus a distribution-free approach which preserves sufficient coverage for the entire signal is necessary in the signal analysis. This paper discusses three methods for the formation of (1-alpha)100% bootstrap prediction bands and their performances are compared through the coverage probabilities obtained for each technique. Bootstrap samples are first obtained for the signal and then three dierent criteria are provided for the removal of 100% of the curves resulting in the (1-alpha)100% prediction band. The first method uses the L1 distance between the upper and lower curves as a gauge to extract the widest bands in the dataset of signals. Also investigated are extractions using the Hausdorffdistance between the bounds as well as an adaption to the bootstrap intervals discussed in Lenhoffet al (1999). The bootstrap prediction bands each have good coverage probabilities for the continuous signals in the dataset. For a 95% prediction band, the coverage obtained were 90.59%, 93.72% and 95% for the L1 Distance, Hausdorff Distance and the adjusted Bootstrap methods respectively. The methods discussed in this paper have been applied to constructing prediction bands for spring discharge in a successful manner giving good coverage in each case. Spring Discharge measured over time can be considered as a continuous signal and the ability to predict the future signals of spring discharge is useful for monitoring flow and other issues related to the spring. While in some cases, rainfall has been tted with the gamma distribution, the discharge of the spring represented as continuous curves, is better approached not assuming any specific distribution. The Bootstrap aspect occurs not in sampling the output discharge curves but rather in simulating the input recharge that enters the spring. Bootstrapping the rainfall as described in this paper, allows for adequately creating new samples over different periods of time as well as specic rain events such as hurricanes or drought. The Bootstrap prediction methods put forth in this paper provide an approach that supplies adequate coverage for prediction bands for signals represented as continuous curves. The pathway outlined by the flow of the discharge through the springshed is described as a tree. A non-parametric pairwise test, motivated by the idea of K-means clustering, is proposed to decipher whether there is equality between two trees in terms of their discharges. A large sample approximation is devised for this lower-tail significance test and test statistics for different numbers of input signals are compared to a generated table of critical values.