Exploratory analysis of multivariate data: Applications of parallel coordinates in ecology

Book reviewed in this article:Handbook of Applicable Mathematics. Chief Editor Walter Lederman.Interpreting Multivariate Data: Proceedings of the Conference entitled ‘Looking at Multivariate Data’ held in Sheffield, UK, March 1980. Edited by Vic Barnett.Finite Mixture Distributions. By B. S. Everitt & D. J. Hand.Discrimination and Classification. By D. J. Hand.Classification: Methods for the Exploratory Analysis of Multivariate Data. By A. D. Gordon.Mathematical Psychology and Psychophysiology. Proceedings of the SIAM‐AMS Symposium in Philadelphia, 15 16 April 1980. Edited by Stephen Grossberg.Mete‐Analysis in Social Research. By Gene V. Glass, Barry McGaw & Mary Lee Smith.Direct Perception. By Claire F. Michaels & Claudia Carello.

CHEMOMETRICS III – REVISITING THE EXPLORATORY ANALYSIS OF MULTIVARIATE DATA. In this work, three methods for pattern recognition, used as exploratory data analysis, are revisited. A brief review of principal component analysis, PCA, an unsupervised method, is provided. Next, the Mahalanobis distance and the confidence ellipses usually drawn around the scores samples are discussed. Fisher’s canonical variate analysis (a supervised methodology) is the second method revisited in this work. The third exploratory data analysis methodology addressed is ANOVA-PCA, which uses the analysis of variance to separate variations into main effects, interaction and noise followed by principal component analysis. Unlike the other two, ANOVA-PCA was proposed recently and is still not yet explored in all its capabilities. One advantage of this method is the possibility to calculate the variance of each of the effects involved in the experimental design. The mathematical bases of the three methods are discussed as well as examples are presented.

The role of permutation tests in exploratory multivariate data analysis

A modern statistical technique for the exploratory analysis of multivariate data, called projection pursuit, is described. The method consists of the extraction and display of informative one- or two-dimensional projections of the multivariate dataset. Projection pursuit has much scope for application to ecological datasets. The example in this paper concerns element concentrations in soil samples from a coastal embayment on the east coast of Australia. The application attempts to search for patterns in the soils which could be related to vegetation patterns. The compositional nature of the dataset adds a novel element to the methodology.

In this work, we present an effective and scalable system for multivariate volume data visualization and analysis with a novel Transfer Function (TF) interface design that tightly couples parallel coordinates plots (PCP) and MDS-based dimension projection plots. In our system, the PCP visualizes the data distribution of each variate and the MDS plots project features. Together, they are integrated seamlessly to provide flexible feature classification without context switching between different data presentations during the user interaction. The proposed interface enables users to identify relevant correlation clusters and assign optical properties on them. To further support large scale multivariate volume data visualization and analysis, we develop three integrated parallel systems to accelerate the rendering of PCP, the layout of MDS, as well as parallel rendering of multivarite volume data. Our experiments show that the system is effective in multivariate volume data visualization and its performance is scalable for data sets with different sizes and number of variates.

Understanding complexity in the response of ornamental rosemary to different substrates: A multivariate analysis

Principal component analysis (PCA) is one of the most popular tools in multivariate exploratory data analysis. Its probabilistic version (PPCA) based on the maximum likelihood procedure provides a probabilistic manner to implement dimension reduction. Recently, the bilinear PPCA (BPPCA) model, which assumes that the noise terms follow matrix variate Gaussian distributions, has been introduced to directly deal with two-dimensional (2-D) data for preserving the matrix structure of 2-D data, such as images, and avoiding the curse of dimensionality. However, Gaussian distributions are not always available in real-life applications which may contain outliers within data sets. In order to make BPPCA robust for outliers, in this paper, we propose a robust BPPCA model under the assumption of matrix variate t distributions for the noise terms. The alternating expectation conditional maximization (AECM) algorithm is used to estimate the model parameters. Numerical examples on several synthetic and publicly available data sets are presented to demonstrate the superiority of our proposed model in feature extraction, classification and outlier detection.

The article presents and investigates new possibilities of multivariate data visualization and their analytical convenience. We demonstrate elementary principles, algorithms and graphical outputs of modern visualization methods with particular focus on Bertin matrices, RADVIZ, Projection pursuit and parallel coordinates. Illustrative examples show their practical implementation into the process of multivariate data analysis, hence providing the reader with an idea of the wide range of their application.

Lipids and volatile compounds of shea butter samples from different African countries (Burkina Faso, Chad, Mali, and Uganda) and different production technologies have been analyzed using an approach based on MALDI-TOF-MS and SPME-GC-MS combined with exploratory multivariate data analysis. By applying MALDI-MS analysis total DAGs were found in the range of 7.6- 22.6% and TAGs in the range of 77.7-92.4% of the shea butter samples. Thereby, major DAGs were found PO (7.3-21.9%), PL (0.0-26.3%), SO (16.3-42.1%), OO (15.1-57.4%), LO (9.1-17.9) and major TAGs were identified POO (2.7-7.8%), POS (4.9-8.3%), SOS (13.8-45.3%), SOO (18.9-39.5%), OOO (1.0-24.3%), AOO (1.4-2.5%) and AOS (1.4-3.7%), respectively. TAG/DAG ratios were found to be dependent on production conditions indicating for inactivation of the lipases by boiling the shea kernels before fat extraction. Volatiles pattern of shea samples detected by headspace-SPME-GC-MS contained more than 100 different compounds, among them many aldehydes, ketones and organic acids. PCA of MALDI data showed a dominating influence of geographic origin on TAG composition characterized mostly by a different saturation degree of the fatty acyl residues declining from West to East Africa. Similar geographic relationships could be visualised by exploratory PLS-DA of the volatiles data. Mainly lipid peroxidation products (in particular unsaturated aldehydes) were identified to be responsible for this sample distribution. Volatile patterns are also influenced by producing methods. For example, methylketones - probably of microbial origin - have been found as marker compounds for "heaping" of the shea kernels before fat extraction. Keywords: Butyrospermum parkii (C.F. Gaertn.), MALDI-TOF-MS, Partial least squares discriminant analysis (PLS-DA), Principal component analysis (PCA), Shea butter, Triacylglycerols, Vitellaria paradoxa (C.F. Gaertn.), Volatiles.

Interactive labelling of a multivariate dataset for supervised machine learning using linked visualisations, clustering, and active learning

Many factors may contribute to cyanobacterial bloom formation. This study examined possible relationships between the presence of zebra mussels (Dreissena polymorpha) and Microcystis spp. abundance. Experiments were conducted in twelve microcosms designed to mimic shallow lake ecosystems. Zebra mussels significantly reduced nitrate, dissolved organic nitrogen, and total dissolved nitrogen concentrations, and had no effect on ammonia, phosphate levels, or dissolved organic carbon. Consequently, the N:P ratio was reduced in microcosms with zebra mussels to 6:1, which is below the Redfield ration of 16:1. Zebra mussels also increased the abundance of Microcystis and Microcystis: Pseudokirchneriella biovolume. In experiments done without zebra mussels, nutrient ratios were manipulated and low N:P caused a similar increase in Microcystis and Microcystis: Pseudokirchneriella biovolume. The shift in N:P in the presence of zebra mussels were related to higher rates of nitrate flux into sediments and reduced flux of phosphate into sediments. It is this shift in N:P, and possibly some level of selective feeding, that is believed to have driven changes in the relative abundance of Microcystis. Finally, in order to compare the experimental results with changes caused by zebra mussel invasion in the natural environment, the data from 15 Wisconsin lakes before and after the zebra mussel invasions were analysed.

Preserving all multidimensional data in two-dimensional visualization is a long-standing problem in Visual Analytics, Machine Learning/Data Mining, and Multiobjective Pareto Optimization. While Parallel and Radial (Star) coordinates preserve all n-D data in two dimensions, they are not sufficient to address visualization challenges of all possible datasets such as occlusion. More such methods are needed. Recently, the concepts of lossless General Line Coordinates that generalize Parallel, Radial, Cartesian, and other coordinates were proposed with initial exploration and application of several subclasses of General Line Coordinates such as Collocated Paired Coordinates and Star Collocated Paired Coordinates. This article explores and enhances benefits of General Line Coordinates. It shows the ways to increase expressiveness of General Line Coordinates including decreasing occlusion and simplifying visual pattern while preserving all n-D data in two dimensions by adjusting General Line Coordinates for given n-D datasets. The adjustments include relocating, rescaling, and other transformations of General Line Coordinates. One of the major sources of benefits of General Line Coordinates relative to Parallel Coordinates is twice less number of point and lines in visual representation of each n-D points. This article demonstrates the benefits of different General Line Coordinates for real data visual analysis such as health monitoring and benchmark Iris data classification compared with results from Parallel Coordinates, Radvis, and Support Vector Machine. The experimental part of the article presents the results of the experiment with about 70 participants on efficiency of visual pattern discovery using Star Collocated Paired Coordinates, Parallel, and Radial Coordinates. It shows advantages of visual discovery of n-D patterns using General Line Coordinates subclass Star Collocated Paired Coordinates with n = 160 dimensions.

Parallel coordinate plots (PCPs) are commonly used in information visualization to provide insight into multi-variate data. These plots help to spot correlations between variables. PCPs have been successfully applied to unstructured datasets up to a few millions of points. In this paper, we present techniques to enhance the usability of PCPs for the exploration of large, multi-timepoint volumetric data sets, containing tens of millions of points per timestep. The main difficulties that arise when applying PCPs to large numbers of data points are visual clutter and slow performance, making interactive exploration infeasible. Moreover, the spatial context of the volumetric data is usually lost. We describe techniques for preprocessing using data quantization and compression, and for fast GPU-based rendering of PCPs using joint density distributions for each pair of consecutive variables, resulting in a smooth, continuous visualization. Also, fast brushing techniques are proposed for interactive data selection in multiple linked views, including a 3D spatial volume view. These techniques have been successfully applied to three large data sets: Hurricane Isabel (Vis'04 contest), the ionization front instability data set (Vis'08 design contest), and data from a large-eddy simulation of cumulus clouds. With these data, we show how PCPs can be extended to successfully visualize and interactively explore multi-timepoint volumetric datasets with an order of magnitude more data points.

In this paper, we present some graphical techniques for cluster analysis of high-dimensional data. Parallel coordinate plots and parallel coordinate density plots are graphical techniques which map multivariate data into a two-dimensional display. The method has some elegant duality properties with ordinary Cartesian plots so that higher-dimensional mathematical structures can be analyzed. Our high interaction software allows for rapid editing of data to remove outliers and isolate clusters by brushing. Our brushing techniques allow not only for hue adjustment, but also for saturation adjustment. Saturation adjustment allows for the handling of comparatively massive data sets by using the α-channel of the Silicon Graphics workstation to compensate for heavy overplotting.

Despite predictions that the zebra mussel (Dreissena polymorpha) invasion of North America would damage fisheries, analyses of actual effects on fish have been few and equivocal. We analyze 26 years of data on fish populations in the Hudson River to quantify changes associated with the zebra mussel invasion. Based on our measurements of changes in the lower food web, we predicted that populations of open-water fish species (e.g., Alosa spp.) would suffer and populations of littoral fish species (e.g., Centrarchidae) would prosper from the zebra mussel invasion. We found that the median decrease in abundance of open-water species was 28%, whereas the median increase in abundance of littoral species was 97%. Populations of open-water species shifted downriver away from the zebra mussel population, whereas those of littoral species shifted upriver. Median apparent growth rates fell by 17% among open-water species and rose by 12% in the single littoral species studied. Many of the observed changes were large and involved species of commercial or recreational importance (e.g., American shad (Alosa sapidissima), black basses (Micropterus spp.)). The influence of zebra mussels on fish should vary widely across ecosystems as a function of system morphology, factors that limit primary production, and diets of the fish species.