Automatic decomposition of time series into step, ramp, and impulse primitives

Time series data that can be modeled as linear combinations of weighted and shifted primitive functions such as ramps, steps and impulses are representative of many industrial, manufacturing, and business processes. Data of this type also are found in statistical process control, structural health monitoring, and other system diagnosis applications. Often, the existence of one or more of these primitive functions may be indicative of the occurrence of a specific process event, making their detection and interpretation of great interest. The human eye is an exceptional tool at this kind of pattern recognition. However, for processes that generate large amounts of data the human eye encounters difficulties related to speed and consistency necessitating an automated approach. In this paper, we consider the problem of decomposing a time series into its steps, ramps, and impulses constituents and expressing it as a linear combination of weighted and shifted versions of these primitives. We express the problem as a least squares error minimization coupled with a combinatorial search to arrive at an acceptable decomposition. We show that under certain conditions, such decomposition is possible and can be obtained efficiently using a sliding window approach. We illustrate the results with several examples.