In this paper, we develop a method for estimating and clustering two-dimensional spectral density functions (2D-SDFs) for spatial data from multiple subregions. We use a common set of adaptive basis functions to explain the similarities among the 2D-SDFs in a low-dimensional space and estimate the basis coefficients by maximizing the Whittle likelihood with two penalties. We apply these penalties to impose the smoothness of the estimated 2D-SDFs and the spatial dependence of the spatially-correlated subregions. The proposed technique provides a score matrix, that is comprised of the estimated coefficients associated with the common set of basis functions representing the 2D-SDFs. Instead of clustering the estimated SDFs directly, we propose to employ the score matrix for clustering purposes, taking advantage of its low-dimensional property. In a simulation study, we demonstrate that our proposed method outperforms other competing estimation procedures used for clustering. Finally, to validate the described clustering method, we apply the procedure to soil moisture data from the Mississippi basin to produce homogeneous spatial clusters. We produce animations to dynamically show the estimation procedure, including the estimated 2D-SDFs and the score matrix, which provide an intuitive illustration of the proposed method.
In this paper, we develop a method for the simultaneous estimation of spectral density functions (SDFs) for a collection of stationary time series that share some common features. Due to the similarities among the SDFs, the log-SDF can be represented using a common set of basis functions. The basis shared by the collection of the log-SDFs is estimated as a low-dimensional manifold of a large space spanned by a prespecified rich basis. A collective estimation approach pools information and borrows strength across the SDFs to achieve better estimation efficiency. Moreover, each estimated spectral density has a concise representation using the coefficients of the basis expansion, and these coefficients can be used for visualization, clustering, and classification purposes.The Whittle pseudo-maximum likelihood approach is used to fit the model and an alternating blockwise Newton-type algorithm is developed for the computation. A web-based shiny App found at "https://ncsde.shinyapps.io/NCSDE" is developed for visualization, training, and learning the SDFs collectively using the proposed technique. Finally, we apply our method to cluster similar brain signals recorded by the for identifying synchronized brain regions according to their spectral densities. KEYWORDS roughness penalty, time series clustering, Whittle likelihood INTRODUCTIONNonparametric techniques for estimating functional structures have been developed in a variety of settings including regression, density estimation, and survival analysis. In time series analysis, the spectral density function plays an important role in characterizing the frequency content of a signal. Mainly, the estimated spectral density can be used to detect the periodicities of the signals in the frequency domain.In practice, it is common to utilize a discrete Fourier transform (DFT) of the input signal and provide a mathematical approximation of the full integral solution of the Fourier transformation. The squared-magnitude of a DFT of the data is called periodogram. However, the raw periodogram is not a consistent estimator for the spectral density of a stationary random process. One classical method to obtain a consistent estimator is to smooth the periodogram across frequencies. Yuen 1 analyzed the performance of three methods of periodogram smoothing for spectrum estimation. Wahba 2 developed an objective optimum smoothing procedure for estimating the log-spectral density using the spline to smooth the log-periodogram. A discrete spectral average estimator and lag window estimators were introduced in the work of Brockwell and Davis. 3 Both of the two methods are consistent. Brillinger 4 introduced periodogram kernel smoothing.Statistics in Medicine. 2018;37:4789-4806.wileyonlinelibrary.com/journal/sim
A new species of narrow-mouthed frog of Kaloula is described in the Nonggang National Nature Reserve, Sino-Vietnamese border region of southern China. Kaloula nonggangensis sp. nov. is distinguished from its congeners by a combination of the following characters: medium size (SVL 41.4-52.7 mm in 18 adult males, 52.2 mm in 1 female); smooth or slightly rough olive dorsum with irregular dark-green marks and brown spots; tips of the fingers widely dilated and truncated; males with nearly fully webbed toes; males with two side protuberant osseous tubercles on the upper surface of the tips of fingers and chest beige with small lemon-colored spots. K. nonggangensis sp. nov. is found in habitats ranging from cultivated fields adjacent to the forest to primary evergreen forest in karst habitats. Based upon a 16S ribosomal RNA mitochondrial gene fragment, K. nonggangensis sp. nov. is embedded within the K. verrucosa group (including K. borealis, K. rugifera and K. verrucosa), and displays a low genetic distance to these species (< 3%). Considering the distinct morphology and karyotype we nevertheless suggest a status as separate species for these allopatrically distributed lineages.
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