Detection of Singularity Locations in Time Series Using Multi-Scale Wavelet Decomposition

In this paper, we employ multi-scale wavelet decomposition to detect singularity locations in time series. Multi-scale wavelet decomposition serves as an effective analytical method that decomposes time series into sub-sequences at different scales, facilitating better understanding of features and variations within the sequence. Through multi-scale wavelet decomposition of time series, we can accurately pinpoint singularity positions and further investigate their impact on the sequence. This method employs wavelet transform algorithms (such as Discrete Wavelet Transform - DWT) that can be implemented using functions like wavedec in MATLAB or pywt.wavedec in Python's PyWavelets library. The implementation typically involves decomposing the signal into approximation and detail coefficients across multiple levels, where singularities manifest as significant modulus maxima in the detail coefficients across scales. This approach enables in-depth analysis of anomalies in time series, thereby improving data analysis and forecasting capabilities through systematic singularity detection and characterization.