Computational and mathematical fundamentals of continuous wavelet transform in the analysis of time-series with gaps

Abstract

The time-scale analysis, characterized by the application of the continuous wavelet transform (CWT) and other tools based in the CWT (as scalogram, global wavelet spectrum, and wavelet entropy) has been widespread in several knowledge areas as Astronomy, Physics, Chemistry, Medicine and Complexity and other, by your capacity into process experimental/observational signals in which we want to extract relevant information typical of each phenomenon considered and more significate than the simple detection of periodicities and/or fundamental frequencies. Generally, your correct application presupposes that let be given the sampling frequency of analyzed data but these requirements carried limitations that are particularly challenging in the treatment of experimental data where is common the existence of gaps. These gaps are characterized by irregular sampling over time caused, for example, by technical difficulties with equipment or by observation difficulties related to external conditions. In these situations we can not apply the wavelet tools and, in these cases, the literature proposes the application of an adaptive form of time-scale analysis based on numerical corrections. The adaptive CWT presents theoretical challenges related to open problems as the existence of inverse transform and computational challenges related to the inexistence codes open source or in scientific languages that require payments. In this context, in this research project, we proposed a theoretical and mathematical investigation of adaptive CWT and the computational implementation in the open language (as Python) with full availability, after the ends of this research work, for the academic community interested in your application. (AU)