William E. Full

FCM: The fuzzy c-means clustering algorithm

Abstract This paper transmits a FORTRAN-IV coding of the fuzzy c -means (FCM) clustering program. The FCM program is applicable to a wide variety of geostatistical data analysis problems. This program generates fuzzy partitions and prototypes for any set of numerical data. These partitions are useful for corroborating known substructures or suggesting substructure in unexplored data. The clustering criterion used to aggregate subsets is a generalized least-squares objective function. Features of this program include a choice of three norms (Euclidean, Diagonal, or Mahalonobis), an adjustable weighting factor that essentially controls sensitivity to noise, acceptance of variable numbers of clusters, and outputs that include several measures of cluster validity.

Principal Components Analysis and Receptor Models in Environmental Forensics

Abstract This chapter provides an overview of quantitative, multivariate exploratory data analysis methods in use in environmental forensics and chemical fingerprinting. Given multiple contaminant sources and empirical chemical data from a study area, environmental scientists often wish to unravel the contributions of multiple sources with overlapping spatial and temporal distributions. Such problems may be addressed through a multivariate approach, such as principal components analysis (PCA) and self-training receptor-modeling techniques. These approaches are related in that they are exploratory data analysis methods—techniques applied when one wishes not to assume a priori knowledge of contributing sources and their fingerprints. PCA is widely used in environmental chemistry as well as many scientific disciplines, and can be implemented using commercial software packages. Four self-training receptor model methods are discussed, compared, and applied herein. There are advantages and disadvantages to each of these methods, but regardless of the method, success depends on the data analyst’s experience, sensitivity to chemical data structure, and sampling plan design. Finally, one must be aware that the compositions of original source profiles may be altered after a release (even for recalcitrant chemicals such as PCBs). In interpreting these models, the data analyst should have some knowledge of alteration, weathering, and degradation mechanisms.

Evaluation of multichannel Wiener Filters applied to fine resolution passive microwave images of first-year sea ice

Abstract Over the past two decades passive microwave imaging systems have proven to be effective reconnaissance tools in polar environments. However, the mechanical scan mechanism and high gain electronics characteristic of this class of sensors commonly impart noise and unwanted artifacts to image data they produce, complicating visual analysis and automated classification procedures. The fact that data in individual scan lines are characterized by statistical stationarity and that information in adjacent pixels is highly correlated due to oversampling of these image data suggests that Wiener multichannel filtering techniques may prove effective in this application. Wiener filters applied to passive microwave images of first-year sea ice were constructed. Four major parameters that define the filter (lag or pixel offset between the original and desired scenes, filter length, number of lines in the filter, and weight applied to the empirical correlation functions) were varied. Results were compared visually to assess the effect of each variable on image quality. Effective filters that limit high frequency noise and enhance ice characteristics use a lag of one pixel, consist of two or three channels, are five pixels in length, and weigh the auto- and cross-correlation functions equally.

Fundamental problems associated with “eigenshape analysis” and similar “factor” analysis procedures

Recent advances in image analysis techniques have allowed rapid generation of peripheral points of two-dimensional objects. Such points, defining overall shape of an object, have been analyzed using several techniques including geometric shape analysis, Fourier analysis, and, more recently, by “eigenshape” analysis. The latter technique purports to represent a general, optimal, and universally best technique to analyze shapes of fossils. Such a technique was devised allegedly due to failure of exisiting techniques. We seriously question the validity of the eigenshape technique and discuss limitations of such an approach. Objections presented in this paper are applicable to a variety of other multivariable data reduction techniques as well.

Grain-size variations in North Atlantic non-carbonate sediments and sources of terrigenous components

Abstract Fine-silt to medium-sand grain-size spectra of the carbonate-free fractions of 115 sea bed samples from the northern North Atlantic, the Labrador Sea and Baffin Bay can be explained as mixtures of five distinct end-members. The five end-members resemble combinations of grain-size modes which are characteristic of till-matrix (Dreimanis and Vagners, 1972). Therefore the ultimate source of most of the terrigenous deep-sea sediments in the study area is probably the veneer of glacial comminution products on the surrounding continents. Three sand-rich end-members exhibit patterns of distribution that are largely compatible with the modern transport of ice-rafted debris as inferred from iceberg drift observations. The best sorted of those sandy end-members however also appears to be concentrated in regions characterized by high-energy turbidity and contour currents and so may also represent hydraulically winnowed sediment. The distribution of two lutite end-members appears in general to reflect the transport of fines by moderate to low energy thermohaline currents; although, in Baffin Bay and the northeastern North Atlantic the lutite could have accumulated primarily in turbidites. Ice-rafting by icebergs calved from Greenland glaciers, the reworking of glacigenic sediments on the Iceland—Faeroes Ridge and injection of turbid glacial meltwater into deep Baffin Bay from West Greenland fiords are suggested as the principal means of introduction of clastic particles into the sediment distribution (redistribution) budget of the deep North Atlantic.

Polynomial Trend Surface Analysis Applied to AVHRR Images to Improve Definition of Arctic Leads

Abstract Polynomial trend surface analysis was applied to three AVHRR images to determine whether regional trends in image radiance can be removed with this procedure. Results suggest that trend surface techniques can be effective in removing region-scale variation in image radiances that are related to uneven illumination, intermittent cloud cover, and variation in the surface temperature field. The dominant effects of illumination in Channel 2 (visible) data, caused by variable sun angle and proximity of the scene to the terminator, can be minimized by removing (subtracting) the first- and second-order trend surfaces from the raw image. These low-order surfaces also remove regional variation in the surface temperature field, which leads to marginal improvement in binary images derived from Channel 4 (infrared) data. Optimum results for both Channel 2 and Channel 4 data are achieved when the third- and fourth-order surfaces are subtracted to remove local temperature and illumination anomalies that occur at smaller spatial scales, primarily in the vicinity of clouds. Application of higher order surfaces fails to improve image quality. There is some indication that the topography of these higher-order surfaces in part maps regional variation in lead density. Use of a best-fit criterion based on a strict variance technique (such as the least-squares method) to define the trend surface limits the effectiveness of the technique in this application. Criteria that allow for data to be weighted based on their distance from the plane about which they cluster are more appropriate to the structure of AVHRR radiance data typical of images that show sea ice. A criterion that incorporates a rule system based on fuzzy logic offers an alternative means of assessing goodness-of-fit that might prove appropriate in this application.

Forma: A program in C to trace object peripheries for two-dimensional shape analysis

FORMA is a computer algorithm and C language program for definition of the external outline of an object. The algorithm is applicable to a multitude of shape analysis studies, especially those concerned with Fourier shape analysis of sedimentary grains and biological specimens. The output of the program is a data file consisting of header information and a series of objects each described by packed edge points and object identification information. FORMA has been developed to be a general image analysis program designed to be interfaced with relatively inexpensive imaging systems.

An improved program for the calculation of high-resolution Fourier coefficents used for shape analysis

Abstract Fourier analysis in closed form represents a frequently used and powerful tool for shape analysis. Inherent in the proper application of this approach is the satisfaction of several underlying statistical assumptions. Past problems associated with the development of an accurate algorithm and subsequent computer code that satisfy these statistical assumptions have limited the widespread use of Fourier analysis in closed form. This manuscript addresses this need with a computer program (C_FOURIER) based on code developed over a decade. The accuracy of the Fourier coefficients depends on the accuracy of finding an objects center. This program uses a rapidly converging center-finding algorithm, appropriate for the majority of shapes encountered, and implements several methods to aid the convergence of more difficult objects. Object size is also estimated and can be used along with the Fourier coefficients in subsequent analysis. The program allows a great deal of flexibility for controlling the Fourier calculation accuracy and implementation parameters and is applicable to a wide variety of shape analysis studies. Additionally, the C_FOURIER program allows for the input of data from several existing video digitizing routines and is relatively straightforwardly adapted to other digitizing schemes. The product of C_FOURIER is a high-resolution Fourier series and a size term that is superior to any of the Fourier series in closed form in the past.