Within statistics, the Kolmogorov–Zurbenko (KZ) 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.
Background A. N. Kolmogorov had the original idea for the KZ filter during a study of turbulence in the Pacific Ocean. Kolmogorov had just received the International Balzan Prize for his law of 5/3 in the energy spectra of turbulence. Surprisingly the 5/3 law was not obeyed in the Pacific Ocean, causing great concern. Standard fast Fourier transform (FFT) was completely fooled by the noisy and non-stationary ocean environment. KZ filtration resolved the problem and enabled proof of Kolmogorov's law in that domain. Filter construction relied on the main concepts of the continuous Fourier transform and their discrete analogues. The algorithm of the KZ filter came from the definition of higher-order derivatives for discrete functions as higher-order differences. Believing that infinite smoothness in the Gaussian window was a beautiful but unrealistic approximation of a truly discrete world, Kolmogorov chose a finitely differentiable tapering window with finite support, and created this mathematical construction for the discrete case. The KZ filter is robust and nearly optimal. Because its operation is a simple Moving Average (MA), the KZ filter performs well in a missing data environment, especially in multidimensional time series where missing data problem arises from spatial sparseness. Another nice feature of the KZ filter is that the two parameters have clear interpretation, enabling easy adoption by specialists in different areas. A few software packages for time series, longitudinal and spatial data have been developed in the popular statistical software R, which facilitates the use of the KZ filter and its extensions in different areas. The I. Zurbenko Postdoctoral position at UC Berkeley with Jerzy Neyman and Elizabeth Scott provided many applications, supported by contacts with Murray Rosenblatt, Robert Shumway, Harald Cramér, David Brillinger, Herbert Robbins, Wilfrid Dixon, Emanuel Parzen.
Definition KZ Filter Let { X ( t ) } , t = 0 , ± 1 , ± 2 , … {\displaystyle \{X(t)\},t=0,\pm 1,\pm 2,\dots } be a real-valued time series, the KZ filter with parameters m {\displaystyle m} and k {\displaystyle k} is defined as
K Z m , k [ X ( t ) ] = ∑ s = − k ( m − 1 ) / 2 k ( m − 1 ) / 2 X ( t + s ) × a s m , k {\displaystyle KZ_{m,k}[X(t)]=\sum _{s=-k(m-1)/2}^{k(m-1)/2}{X(t+s)\times {a_{s}^{m,k}}}}
where coefficients
a s m , k = c s k , m m k , s = − k ( m − 1 ) 2 , … , k ( m − 1 ) 2 {\displaystyle a_{s}^{m,k}={\frac {c_{s}^{k,m}}{m^{k}}},s={\frac {-k(m-1)}{2}},\dots ,{\frac {k(m-1)}{2}}}
are given by the polynomial coefficients obtained from equation
∑ r = 0 k ( m − 1 ) z r c r − k ( m − 1 ) / 2 k , m = ( 1 + z + ⋯ + z m − 1 ) k {\displaystyle \sum _{r=0}^{k(m-1)}{z^{r}c_{r-k(m-1)/2}^{k,m}=(1+z+\dots +z^{m-1})^{k}}}
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