Diffusion adaptive filtering algorithm based on the Fair cost function

To better perform distributed estimation, this paper, by combining the Fair cost function and adapt-then-combine scheme at all distributed network nodes, a novel diffusion adaptive estimation algorithm is proposed from an M-estimator perspective, which is called the diffusion Fair (DFair) adaptive filtering algorithm. The stability of the mean estimation error and the computational complexity of the DFair are theoretically analyzed. Compared with the robust diffusion LMS (RDLMS), diffusion Normalized Least Mean M-estimate (DNLMM), diffusion generalized correntropy logarithmic difference (DGCLD), and diffusion probabilistic least mean square (DPLMS) algorithms, the simulation experiment results show that the DFair algorithm is more robust to input signals and impulsive interference. In conclusion, Theoretical analysis and simulation results show that the DFair algorithm performs better when estimating an unknown linear system in the changeable impulsive interference environments.

APPLYING M-ESTIMATES TO DEFINE START VALUE OF EXPONENTIAL AVERAGE IN BRAUN’S FORECAST MODEL OF ZERO LEVEL

В экономическом прогнозировании коротких временных рядов часто применяется модель Брауна нулевого порядка. К одной из проблем использования этой модели на первых шагах прогнозирования относится оценка начального значения экспоненциальной средней. Как правило, в качестве такой оценки используется простое среднее арифметическое значение первых уровней ряда, которое является неустойчивой статистической оценкой. Поэтому в данном исследовании предложено для оценки начального значения экспоненциальной средней использовать робастные М-оценки Тьюки, Хампеля, Хьюбера и Эндрюса. Цель исследования заключается в определении целесообразности применения М-оценок для определения начального значения экспоненциальной средней в модели Брауна при прогнозировании коротких временных рядов экономических показателей. В результате проведенного экспериментального исследования установлено: а) к наиболее значимым факторам, влияющим на точность прогноза с использованием модели Брауна, относятся вид временного ряда, значение постоянной сглаживания, отбраковка аномальных уровней и вид весов; б) вид оценки начального значения экспоненциальной средней и число итераций при вычислении М-оценки являются менее значимыми факторами (в связи с этим обоснована целесообразность применения одношаговых М-оценок); в) на начальных шагах прогнозирования при ограниченном количестве уровней временного ряда, когда невозможно достоверно определить вид ряда и когда отсутствуют основания для отбраковки аномальных уровней, предпочтительнее использовать модель Брауна с весами Вейда и определять начальное значение экспоненциальной средней на основе одношаговых робастных М-оценок (в остальных случаях целесообразно применять простое среднее арифметическое значение). In economic forecasting of short-term time series Braun’s model of zero level is often applied. One of issues of usage of this model from the very beginning of forecasting is estimation of start value of exponential average. As usual, simple arithmetic mean of first levels of series, used as such estimate, is volatile statistical estimate. That’s why in this investigation it’s suggested to use Tukey’s, Hampel’s, Huber’s and Andrews’ robust M-estimates for estimation of start value of exponential average. Purpose of research is definition of reasonability of M-estimates application to define start value of exponential average in Braun’s model during forecasting of short-term time series of economic indicators. The results of conducted experimental research are as follows: a) the most important factors, that have significant impact on forecast accuracy with usage of Braun’s model, are type of time series, value of smoothing constant, removal of abnormal levels and type of weights; b) type of estimate of start value of exponential average and quantity of iterations in process of calculation of M-estimate are less significant factors; c) consequently, reasonability of usage of one-step M-estimates is justified; d) on the first steps of forecasting with limited quantity of levels of time series, when it’s impossible to define with certainty type of series and when there is no reasons for removal of abnormal levels, it’s preferable to use Braun’s model with Wade’s weights and define start value of exponential average based on one-step robust M-estimates (in other cases it’s better to use simple arithmetical mean).

Influence Function and Bootstrap Methods of Estimating the Standard Errors of the Estimators of Mixture Exponential Distribution Parameter

This work estimated the standard error of the maximum likelihood estimator (MLE) and the robust estimators of the exponential mixture parameter (θ) using the influence function and the bootstrap approaches. Mixture exponential random samples of sizes 10, 15, 20, 25, 50, and 100 were generated using 3 mixture exponential models at 2%, 5% and 10% contamination levels. The selected estimators namely: mean, median, alpha-trimmed mean, Huber M-estimate and their standard errors (Tn ) were estimated using the two approaches at the indicated sample sizes and contamination levels. The results were compared using the coefficient of variation, confidence interval and the asymptotic relative efficiency of Tn in order to find out which approach yields the more reliable, precise and efficient estimate of Tn. The results of the analysis show that the two approaches do not equally perform at all conditions. From the results, the bootstrap method was found to be more reliable and efficient method of estimating the standard error of the arithmetic mean at all sample sizes and contamination levels. In estimating the standard error of the median, the influence function method was found to be more effective especially when the sample size is small and yet contamination is high. The influence function based approach yielded more reliable, precise and efficient estimates of the standard errors of the alpha-trimmed mean and the Huber M-estimate for all sample sizes and levels of contamination although the reliability of the bootstrap method improved better as sample size increased to 50 and above. All simulations and analysis were carried out in R programming language.