page_title: Kolmogorov–Zurbenko filter
section_title: Properties
hierarchical_section_title: Kolmogorov–Zurbenko filter / Properties
caption_attribution_description: Transfer Function for Kolmogorov–Zurbenko_filter with k=1.
context_page_description: The Kolmogorov–Zurbenko filter was first proposed by A. N. Kolmogorov and formally defined by Zurbenko. It is a series of iterations of a moving average filter of length m, where m is a positive, odd integer. The KZ filter belongs to the class of low-pass filters. The KZ filter has two parameters, the length m of the moving average window and the number of iterations k of the moving average itself. It also can be considered as a special window function designed to eliminate spectral leakage.
context_section_description: The impulse response function of the product of filters is the convolution of impulse responses. The coefficients of the KZ filter aᵐ,ᵏ
  ₛ, can be interpreted as a distribution obtained by the convolution of k uniform discrete distributions on the interval [ −(m − 1)/2 , (m − 1)/2 ] where m is an odd integer. Therefore, the coefficient a
   forms a tapering window which has  finite support [ (m − 1)k + 1]. The KZ filter a
   has main weight concentrated on a length of m√k with weights vanishing to zero outside. The impulse response function of the KZ filter has k − 2 continuous derivatives and is asymptotically Gaussian distributed. Zero derivatives at the edges for the impulse response function make from it a sharply declining function, what is resolving in high frequency resolution. The energy transfer function of the KZ filter is
  It is a low-pass filter with a cut-off frequency of
  Compared to a MA filter, the KZ filter has much better performance in terms of attenuating the frequency components above the cutoff frequency. The KZ filter is essentially a repetitive MA filter. It is easy to compute and allows for a straightforward way to deal with missing data. The main piece of this procedure is a simple average of available information within the interval of m points disregarding the missing observations within the interval. The same idea can be easily extended to spatial data analysis. It has been shown that missing values have very little effect on the transfer function of the KZ filter.
  Arbitrary k will provide k power of this transfer function and will reduce side lobe value to  0.05ᵏ. It will be a perfect low pass filter. For practical purposes a choice of k within a range 3 to 5 is usually sufficient, when regular MA (k = 1) is providing strong spectral leakage of about 5%.
caption: Figure 1.Transfer function of  filter for k = 1.
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