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Classic wavelet methods were developed in the Euclidean spaces for multiscale representation and analysis of regularly sampled signals (time series) and images. This paper introduces a method of representing scattered spherical data by multiscale spherical wavelets. The method extends the recent pioneering work of Narcowich and Ward [Appl. Comput. Harmon. Anal., 3 (1996), pp. 324-336] by employing multiscale rather than single-scale spherical basis functions and by introducing a bottom-up procedure for network design and bandwidth selection. Decomposition and reconstruction algorithms are proposed for efficient computation. An analytical investigation confirms the localization property of the resulting spherical wavelets. The proposed method is illustrated by numerical examples. It is also employed to analyze and compress a real-data set consisting of the surface air temperatures observed on a global network of weather stations.

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Abstract-Internet service utilities host multiple server applications on a shared server cluster (server farm). One of the essential tasks of the hosting service provider is to allocate servers to each of the websites to maintain a certain level of quality of service for different classes of incoming requests at each point of time, and optimize the use of server resources, while maximizing its profits. Such a proactive management of resources requires accurate prediction of workload, which is generally measured as the amount of service requests per unit time. As a time series, the workload exhibits not only short time random fluctuations but also prominent periodic (daily) patterns that evolve randomly from one period to another. We propose a solution to the Web server load prediction problem based on a hierarchical framework with multiple time scales. This framework leads to adaptive procedures that provide both longterm (in days) and short-term (in minutes) predictions with simultaneous confidence bands which accommodate not only serial correlation but also heavy-tailedness, and nonstationarity of the data. The long-term load is modeled as a dynamic harmonic regression (DHR), the coefficients of which evolve according to a random walk, and are tracked using sequential Monte Carlo (SMC) algorithms; whereas the short-term load is predicted using an autoregressive model, whose parameters are also estimated using SMC techniques. We evaluate our method using real-world Web workload data.