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 so that it can be easily adopted 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 facilitate the use of the KZ filter and its extensions in different areas. I.Zurbenko Postdoctoral position at UC Berkeley with Jerzy Neyman and Elizabeth Scott provided a lot of ideas of applications supported in contacts with Murray Rosenblatt, Robert Shumway, Harald Cramér, David Brillinger, Herbert Robbins, Wilfrid Dixon, Emanuel Parzen.
Definition
KZ Filter
Let be a real-valued time series, the KZ filter with parameters and is defined as
where coefficients
are given by the polynomial coefficients obtained from equation
From another point of view, the KZ filter with parameters and can be defined as time iterations of a moving average (MA) filter of points. It can be obtained through iterations.
First iteration is to apply a MA filter over process
The second iteration is to apply the MA operation to the result of the first iteration,
Generally the kth iteration is an application of the MA filter to the (k − 1)th iteration. The iteration process of a simple operation of MA is very convenient computationally.
Properties
The impulse response function of the product of filters is the convolution of impulse responses. The coefficients of the KZ filter , can be interpreted as a distribution obtained by the convolution of k uniform discrete distributions on the interval where m is an odd integer. Therefore, the coefficient forms a tapering window which has finite support . The KZ filter has main weight concentrated on a length of with weights vanishing to zero outside. The impulse response function of the KZ filter has 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.
