Some statistical aspects of LULU smoothers

[摘要] The smoothing of time series plays a very important role in various practical applications. Estimatingthe signal and removing the noise is the main goal of smoothing. Traditionally linear smoothers wereused, but nonlinear smoothers became more popular through the years.From the family of nonlinear smoothers, the class of median smoothers, based on order statistics, is themost popular. A new class of nonlinear smoothers, called LULU smoothers, was developed by usingthe minimum and maximum selectors. These smoothers have very attractive mathematical properties.In this thesis their statistical properties are investigated and compared to that of the class of mediansmoothers.Smoothing, together with related concepts, are discussed in general. Thereafter, the class of mediansmoothers, from the literature is discussed. The class of LULU smoothers is defined, their propertiesare explained and new contributions are made. The compound LULU smoother is introduced and itsproperty of variation decomposition is discussed. The probability distributions of some LULUsmootherswith independent data are derived. LULU smoothers and median smoothers are compared accordingto the properties of monotonicity, idempotency, co-idempotency, stability, edge preservation, outputdistributions and variation decomposition. A comparison is made of their respective abilities for signalrecovery by means of simulations. The success of the smoothers in recovering the signal is measuredby the integrated mean square error and the regression coefficient calculated from the least squaresregression of the smoothed sequence on the signal. Finally, LULU smoothers are practically applied.

[发布机构] Stellenbosch University