The Fourier transform occupies a central place in applied mathematics, statistics, computer sciences, and engineering. In this work, we introduce the one-dimensional quaternion Fourier transform, which is a generalization of the Fourier transform. We derive the conjugate symmetry of the one-dimensional quaternion Fourier transform for a real signal. We also collect other properties, such as the derivative and Parseval’s formula. We finally study the application of this transformation in probability theory.
A boxplot is an exploratory data analysis (EDA) tool for a compact distributional summary of a data set. It is designed to captures all typical observations and displays the location, spread, skewness and the tail of the data. The precision of some of this functionality is considered to be more reliable for symmetric data type and thus less appropriate for skewed data such as the extreme data. Many observations from extreme data were erroneously marked as outliers by the Tukeys standard boxplot. We proposed a modified boxplot fence adjustment using the Bowley coefficient, a robust skewness measure. The adjustment will enable us to detect inconsistent observations without any parametric assumption about the distribution of the data. The new boxplot is capable of displaying some additional features such as the location parameter region of the Gumbel fitted extreme data. A simulated and real life data were used to show the advantages of this development over those found in the literature.
In this study, we model extreme rainfall to study the high rainfall events in the province of South Sulawesi, Indonesia. We investigated the effect of the El Nino South Oscillation (ENSO), Indian Ocean Dipole Mode (IOD), and Madden–Julian Oscillation (MJO) on extreme rainfall events. We also assume that events in a location are affected by events in other nearby locations. Using rainfall data from the province of South Sulawesi, the results showed that extreme rainfall events are related to IOD and MJO.
There have been several efforts in the literature to extend the traditional Fourier transformation by using the quaternion algebra. This paper presents the one-dimensional quaternion Fourier transform. We derive its properties which are the extensions of corresponding properties of the one-dimensional Fourier transformation. Finally, the convolution theorem related to the one-dimensional quaternion Fourier transform is discussed.
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