Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Power transformations in correspondence analysis

Power transformations of positive data tables, prior to applying the correspondence analysis algorithm, are shown to open up a family of methods with direct connections to the analysis of log-ratios. Two variations of this idea are illustrated. The first approach is simply to power the original data and perform a correspondence analysis this method is shown to converge to unweighted log-ratio analysis as the power parameter tends to zero. The second approach is to apply the power transformation to thecontingency ratios, that is the values in the table relative to expected values based on the marginals this method converges to weighted log-ratio analysis, or the spectral map. Two applications are described: first, a matrix of population genetic data which is inherently two-dimensional, and second, a larger cross-tabulation with higher dimensionality, from a linguistic analysis of several books.

Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing oneor more parameters in their definition. Methods that can be linked in this way arecorrespondence analysis, unweighted or weighted logratio analysis (the latter alsoknown as "spectral mapping"), nonsymmetric correspondence analysis, principalcomponent analysis (with and without logarithmic transformation of the data) andmultidimensional scaling. In this presentation I will show how several of thesemethods, which are frequently used in compositional data analysis, may be linkedthrough parametrizations such as power transformations, linear transformations andconvex linear combinations. Since the methods of interest here all lead to visual mapsof data, a "movie" can be made where where the linking parameter is allowed to vary insmall steps: the results are recalculated "frame by frame" and one can see the smoothchange from one method to another. Several of these "movies" will be shown, giving adeeper insight into the similarities and differences between these methods.

Spectral properties of chaotic signals generated by the skew tent map

Chaotic signals have been considered potentially attractive in many signal processing applications ranging from wideband communication systems to cryptography and watermarking. Besides, some devices as nonlinear adaptive filters and phase-locked loops can present chaotic behavior. In this paper, we derive analytical expressions for the autocorrelation sequence, power spectral density and essential bandwidth of chaotic signals generated by the skew tent map. From these results, we suggest possible applications in communication systems. (C) 2009 Elsevier B.V. All rights reserved.

Benthic cover map of Heron Reef derived from a high-spatial-resolution multi-spectral satellite image using object based image analysis

Providing accurate maps of coral reefs where the spatial scale and labels of the mapped features correspond to map units appropriate for examining biological and geomorphic structures and processes is a major challenge for remote sensing. The objective of this work is to assess the accuracy and relevance of the process used to derive geomorphic zone and benthic community zone maps for three western Pacific coral reefs produced from multi-scale, object-based image analysis (OBIA) of high-spatial-resolution multi-spectral images, guided by field survey data. Three Quickbird-2 multi-spectral data sets from reefs in Australia, Palau and Fiji and georeferenced field photographs were used in a multi-scale segmentation and object-based image classification to map geomorphic zones and benthic community zones. A per-pixel approach was also tested for mapping benthic community zones. Validation of the maps and comparison to past approaches indicated the multi-scale OBIA process enabled field data, operator field experience and a conceptual hierarchical model of the coral reef environment to be linked to provide output maps at geomorphic zone and benthic community scales on coral reefs. The OBIA mapping accuracies were comparable with previously published work using other methods; however, the classes mapped were matched to a predetermined set of features on the reef.

Habitat map of Danajon Bank, Philippines, derived from a high-spatial-resolution multi-spectral satellite image and georeferenced point intercept transect and spot-check survey field data, using an object-based image classification method

A mosaic of two WorldView-2 high resolution multispectral images (Acquisition dates: October 2010 and April 2012), in conjunction with field survey data, was used to create a habitat map of the Danajon Bank, Philippines (10°15'0'' N, 124°08'0'' E) using an object-based approach. To create the habitat map, we conducted benthic cover (seafloor) field surveys using two methods. Firstly, we undertook georeferenced point intercept transects (English et al., 1997). For ten sites we recorded habitat cover types at 1 m intervals on 10 m long transects (n= 2,070 points). Second, we conducted geo-referenced spot check surveys, by placing a viewing bucket in the water to estimate the percent cover benthic cover types (n = 2,357 points). Survey locations were chosen to cover a diverse and representative subset of habitats found in the Danajon Bank. The combination of methods was a compromise between the higher accuracy of point intercept transects and the larger sample area achievable through spot check surveys (Roelfsema and Phinn, 2008, doi:10.1117/12.804806). Object-based image analysis, using the field data as calibration data, was used to classify the image mosaic at each of the reef, geomorphic and benthic community levels. The benthic community level segregated the image into a total of 17 pure and mixed benthic classes.

Spectral properties of chaotic signals with applications in communications

This paper investigates the characteristics of the Power Spectral Density (PSD) of chaotic signals generated by skew tent maps. The influence of the Lyapunov exponent on the autocorrelation sequence and on the PSD is evaluated via computational simulations. We conclude that the essential bandwidth of these signals is strongly related to this exponent and they can be low-pass or high-pass depending on the family`s parameter. This way, the PSD of a chaotic signal is a function of the generating map although this is not a one-to-one relationship. (C) 2009 Elsevier Ltd. All rights reserved.

Using a cartographic modeling language to manipulate spectral satellite imagery

Land related information about the Earth's surface is commonIJ found in two forms: (1) map infornlation and (2) satellite image da ta. Satellite imagery provides a good visual picture of what is on the ground but complex image processing is required to interpret features in an image scene. Increasingly, methods are being sought to integrate the knowledge embodied in mop information into the interpretation task, or, alternatively, to bypass interpretation and perform biophysical modeling directly on derived data sources. A cartographic modeling language, as a generic map analysis package, is suggested as a means to integrate geographical knowledge and imagery in a process-oriented view of the Earth. Specialized cartographic models may be developed by users, which incorporate mapping information in performing land classification. In addition, a cartographic modeling language may be enhanced with operators suited to processing remotely sensed imagery. We demonstrate the usefulness of a cartographic modeling language for pre-processing satellite imagery, and define two nerv cartographic operators that evaluate image neighborhoods as post-processing operations to interpret thematic map values. The language and operators are demonstrated with an example image classification task.