This thesis describes new methodologies for wavelet analysis of high frequency data, and presents fast, GPU-accelerated implementations in a new R package, WaveCUDA. We address two main features of high frequency data, namely their large size and also the irregular spacing of their time values.

To deal with the size issue, we provide an R package with new parallel algorithms to perform wavelet transforms with the Lifting Scheme. We factorise the Coiflet 6 and Least Asymmetric 8 wavelet transforms into lifting steps and implement the Haar and Daubechies Extremal Phase 4 filters. We achieve considerable speedups, both in these transforms and in code that carries out thresholding schemes for nonparametric regression. We implement standard thresholding rules and provide a fast implementation of the potentially computationally expensive wavelet cross validation method of choosing the threshold. We then present a new regularised V-fold wavelet cross validation scheme for non-dyadic V.

We contribute to wavelet analysis of irregularly spaced data by using an entropy-based method for selecting an optimal grid at which to interpolate.