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Kolmogorov–Zurbenko filters in spatiotemporal analysis

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This paper is an extension of WIRE publication Kolmogorov–Zurbenko filters, 2010. It addresses computational aspects of multidimensional KZ filtering for unevenly spaced data. Some real examples are provided to illustrate some of the details of such data analysis. The identification and separation of different spatial or temporal scales in spatiotemporal data can provide essential improvements to the accuracy of explanations. In particular, long term scales can be made to display clear patterns which are absolutely indistinguishable using standard multivariate analysis methods. WIREs Comput Stat 2018, 10:e1419. doi: 10.1002/wics.1419

This article is categorized under:

  • Applications of Computational Statistics > Signal and Image Processing and Coding
  • Data: Types and Structure > Image and Spatial Data
  • Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data
2D image restoration using 2D KZA filter. (a), (b), and (c) present, for a 2‐dimensional image, the 3‐dimensional coordinates (x, y, value), so that image restoration may be better visualized. (a) Displays the signal only, three height 1 cylinders with base in the xy plane. (b) Displays the signal plus noise independently distributed N(μ = 0, σ = 1). (c) Displays the KZA recovered signal from (b).
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Directional periodogram along 15°, applied to the simulated signals sin (2π·0.4·t) along 30° and 1.5sin (2π·0.05·t) along −30°, with added noise distributed N (0, 100).
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Composition of 2 waves as original signal and original signal covered by noise N (0, 102).
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Two separate signals as waves of amplitudes 1.5 and 1 and directions 30° and −30° in a 400 × 400 wave field.
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Long term trend component of Melanoma rate per hundred thousand for January 2000.
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Long term trend component of Melanoma rate per hundred thousand for January 1990.
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Smoothed periodogram of melanoma spectra for grid cell (42, −93) using KZP algorithm with DZ smoothing parameter set to 1%. Frequencies A and L correspond to 1 and 11 years, respectively.
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Log skin cancer monthly rate, United States.
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The global, long term component of specific humidity’s deviation from the cosine squared law, in 1972 and 2013. The global, long term component of specific humidity is constructed from the product of the filter KZ29,5 applied over monthly observations of time and the filter KZ3°,3°,4 applied over space to the spatiotemporal data SH(x i, y j, t k).
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Monthly mean precipitation deviations from the cosine squared law along latitudes.
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Mean sea level air pressure over latitudes.
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Global specific humidity distribution on January 2001, smoothed by a KZ3°,3°,4 four filter applied over a 3° × 3° grid of specific humidity data.
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The band‐pass filter [1‐KZ (1, 17)]*KZ (3, 11) (whose transfer function is the dotted black curve in the plot) has cut‐offs at 4.8 and 1.5 years, and works well to recover the El Niño‐like component. To implement this KZ filter, one needs to apply a moving average with a 17 months window, then take the original monthly data minus the result; then, iterate three times a moving average with 11 months window on the resulting data. The result must be enlarged by some scale coefficient (about 4) to counterbalance the amplitude attenuation (plotted above in blue). Finally, the red curve above provides the transfer function for the KZ filter KZ (5, 25), a low‐pass filter with cutoff at 7.3 years.
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DZ smoothed periodogram of the global component of specific humidity.
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The cosine squared law describes over 99% of the variation over latitudes of the global component of temperature.
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