De-noising a time series, that is a sequence of observations of a variable measured at equidistant points in time, or an image, that is a rectangular array of pixels, is a common task nowadays. The objective is to extract a varying level (a “signal”) representing the path followed by the time series or the true image which is overlaid by irrelevant noise.
Linear filters like moving averages are computationally simple and eliminate normal noise efficiently. However, their output is heavily affected by strongly deviating observations (called outliers, spikes or impulses), which can be caused for instance by measurement artifacts. Moreover, linear filters do not preserve abrupt changes (also called step changes or jumps) in the signal or edges in an image. Tukey (1977) suggests median filters, also called running medians, for these purposes.
We focus on the time series setting in the following. Let y 1, …, y N be observations of a variable at equidistant points in time. De-noising these...
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References and Further Reading
Davies L, Fried R, Gather U (2004) Robust signal extraction for online monitoring data. J Stat Plan Infer 122:65–78
Gather U, Fried R, Lanius V (2006) Robust detail-preserving signal extraction. In: Schelter B, Winterhalder M, Timmer J (eds) Handbook of time series analysis. Wiley, New York, pp. 131–158
Tukey JW (1977) Exploratory data analysis (preliminary edition 1971). Addison-Wesley, Reading MA
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Fried, R., George, A.C. (2011). Median Filters and Extensions. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_361
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DOI: https://doi.org/10.1007/978-3-642-04898-2_361
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