Discovery of non-obvious relationships between time series is an important problem in many domains, such as financial, sensory, and scientific data analysis. We consider data mining in aligned time series, which arise, e.g., in numerous online monitoring applications, and we are interested in finding time series which reflect the same external events. The time series can have different vertical positions, scales and overall trends, however still show related features at the same locations. The features can be short-term, such as small peaks and turns, or long-term, such as wider mountains and valleys. We propose using a wavelet transformation of a time series to produce a natural set of features for the sequence. Wavelet transformation yields features which describe properties of the sequence, both at various locations and at varying time granularities. In the proposed method, these features are processed so that they are insensitive to changes in the vertical position, scaling, and overall trend of the time series. We discuss the use of these features in data mining, in tasks such as clustering. We demonstrate how the features allow a flexible analysis of different aspects of the similarity: we show how one can examine how the similarity between time series changes as a function of time or as a function of time granularity considered. We present experimental results with real financial data sets. Experiments indicate that the proposed method can produce useful results. For instance, important similarities can be found in time series, which would be considered unrelated by visual inspection. Experiments with compression give encouraging results for the application of the method in mining massive time series data sets.