Estimation of Scale Transformation for Approximate Periodic Time Series with Long-Term Trend

Approximate periodic time series means it has an approximate periodic trend. The so-called approximate periodicity refers that it looks like having periodicity, however the length of each period is not constant such as sunspot data. Approximate periodic time series has a wide application prospect in modelling social economic phenomenon. As for approximate periodic time series, the key problem is to depict its approximate periodic trend because it can be dealt as an ordinary time series only if its approximate periodic trend has been depicted. However, there is little study on depicting approximate periodic trend. In the paper, the authors first establish some necessary theories, especially bring forward the concept of shape-retention transformation with lengthwise compression and obtain necessary and sufficient condition for linear shape-retention transformation with lengthwise compression, then basing on the theories the authors present a method to estimate scale transformation, which can model approximate periodic trend very clearly. At last, a simulated example is analyzed by this presented method. The results show that the presented method is very effective and very powerful.