| In statistics, smoothing is a technique that attempts to capture the key patterns or trends in data while leaving out the noise that is obscuring them. Nonparametric techniques are well-suited for smoothing as they do not rely on assumptions that the data arise from a given probability distribution.A common smoothing technique is Gaussian smoothing. Though common, the Gaussian kernel possesses infinite support, meaning that, in theory, every data point will be brought into consideration for each estimate thus, exact implementation is not possible. As an alternative, the truncated Gaussian kernel may be used. Here, the support of the kernel is truncated to some finite interval on the real line however, there is a discontinuity at the point of truncation, thereby leaking noise.To overcome these issues, a spatial smoothing algorithm (KZS), based on convolutions with the constant B-spline, that is, the uniform kernel, is introduced. This is an iterative algorithm corresponding to repeated applications of a moving average, in which the kernel approximates the Gaussian kernel having finite support.When applied in environments containing high levels of random noise and missing values, KZS achieved excellent recoveries of intricate signals and surfaces, having small relative errors. Based on several such illustrations, the KZS can be considered a comparable alternative to other commonly used smoothing methods.In a spatiotemporal application, using smoothing parameters determined from spectral information found in regional temperature, two global signals, 2-5 year periods and beyond 13 year periods, have been identified. Within monthly global temperature records, a long-term average temperature profile along latitude has been identified and parametrically approximated with great accuracy. A movie based on four-dimensional data of deviations from long-term local averages has been constructed to illustrate global warming. Maps of 2-5 year scales display deviations similarly to those observed during an El Nino event and provide the opportunity for explanation and prediction of weather anomalies in various global regions.In summary, the KZS smoothing algorithm is an effective tool for univariate and/or spatial analysis that is applicable to various areas of modern day research.Keywords: KZS, KZ filter, spline kernel, smoothing, moving average, B-spline, time series, global warming, El Nino. |
