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1001 Solutions
Published in Jaakko Astola, Pauli Kuosmanen, Fundamentals of Nonlinear Digital Filtering, 2020
Jaakko Astola, Pauli Kuosmanen
Several nonlinear filters have been described in this chapter. The motivations behind these filters were explained, and their principles and basic properties were studied. Some filters result from efforts to find a good compromise between the median and the mean filters. In our opinion most of these compromises are relatively successful. Another group of filters can be categorized as ad-hoc techniques. They are typically characterized by an excellent behavior in a specific limited task and a failure to behave well outside it. Still, we believe that they have their own place and applications among nonlinear filters. A clear drawback of these filters is that it is often not possible to analyze their behavior because of their specialized structure. One important group of filters originates from robust statistics. These filters have many desirable properties, and we believe that in most of the filtering cases the application engineer should consider applying some of these filters. Especially, M-filters and their modifications should be studied more thoroughly as they have great potential. Some of the studied filters, like polynomial filters, perhaps better suit applications other than pure noise removal, like channel equalization or modeling of nonlinear systems.
Elements of Complexity Theory
Published in F. P. Tarasenko, Applied Systems Analysis, 2020
Different degrees of reliability of knowledge about probability distribution dictate the need to extract the necessary information from the same data set in different ways. Special algorithms of experimental data processing have been developed for different levels of a priori information about the distribution. Mathematical statistics consists of four sections: classical (parametric) statistics based on the assumption that the distribution function is known up to a finite number of parameters;nonparametric statistics, assuming that the observations are subject to distribution, the functional form of which is unknown;robust statistics considering cases where the distribution function is known approximately: the real function is located in some neighborhood of a given function;semi-parametric statistics, assuming that observations belong to a parametric family, with random parameters.
Introduction to Exploratory Data Analysis
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
We do cover some density estimation, such as model-based clustering (Chapter 6) and histograms (Chapter 9). The reader is referred to Scott (2015) for an excellent treatment of the theory and methods of multivariate density estimation in general or Silverman (1986) for kernel density estimation. For more information on MATLAB implementations of density estimation the reader can refer to Martinez and Martinez (2015) and Martinez and Cho (2014). Finally, we will likely encounter outlier detection as we go along in the text, but this topic, along with robust statistics, will not be covered as a stand-alone subject. There are several books on outlier detection and robust statistics. These include Hoaglin, Mosteller, and Tukey (1983), Huber (1981), and Rousseeuw and Leroy (1987). A rather dated paper on the topic is Hogg (1974).
Structural and non-structural statistical methods: implications for delineating geochemical anomalies
Published in Applied Earth Science, 2020
Bijan Roshanravan, Seyed Hasan Tabatabaei, Oliver Kreuzer, Hamid Moini, Mohammad Parsa
Owing to the fact that in standard statistics, the average and standard deviation and the average-related parameters are sensitive to outlier or boundary values, in this research, robust statistics were also used and all calculations were also repeated using the robust statistics. Robust statistical methods take into account the deviations when estimating the parameters of models in non-normal distributions, thus increasing the accuracy of the fitted models and associated inference. One of the best robust dispersion estimators is median absolute deviation (MAD) that is less influenced by the outliers in a data set than the standard deviation (Rousseeuw and Croux 1993). In this regard, the data median is the most crucial robust parameter. By calculating the median and the MAD, used instead of the standard deviation, it is possible to reach new thresholds and geochemical anomalies lacking any distribution type and outlier values. For a set of data MAD is defined as follows (Rousseeuw and Croux 1993):where is value of each data, is new median, and is primary data median. This means that all the values are deducted from the primary data median, and then, the median is calculated from the resulted dataset. Contrary to the standard deviation in which all deviations are powered by 2, in this method, the MAD is calculated. Consequently, it is not sensitive to the boundary values and the deviations do not exceed large values.
Mindfulness mediates the relationship between mental toughness and pain catastrophizing in cyclists
Published in European Journal of Sport Science, 2018
Martin I. Jones, John K. Parker
The study is also unique because of the application of robust statistics that overcome some of the problems associated with classical techniques. The assumptions of Pearson’s correlations and ordinary least squares regression were not met in the current data. Rather than risking the computation of inaccurate P values and confidence intervals or dropping outliers and then misinterpreting the findings, we adopted robust approaches recommended by Wilcox (2017).