Conditional compositional biplots: theory and application

The biplot has proved to be a powerful descriptive and analytical tool in many areas of applications of statistics. For compositional data the necessary theoretical adaptation has been provided, with illustrative applications, by Aitchison (1990) and Aitchison and Greenacre (2002). These papers were restricted to the interpretation of simple compositional data sets. In many situations the problem has to be described in some form of conditional modelling. For example, in a clinical trial where interest is in how patients’ steroid metabolite compositions may change as a result of different treatment regimes, interest is in relating the compositions after treatment to the compositions before treatment and the nature of the treatments applied. To study this through a biplot technique requires the development of some form of conditional compositional biplot. This is the purpose of this paper. We choose as a motivating application an analysis of the 1992 US President ial Election, where interest may be in how the three-part composition, the percentage division among the three candidates - Bush, Clinton and Perot - of the presidential vote in each state, depends on the ethnic composition and on the urban-rural composition of the state. The methodology of conditional compositional biplots is first developed and a detailed interpretation of the 1992 US Presidential Election provided. We use a second application involving the conditional variability of tektite mineral compositions with respect to major oxide compositions to demonstrate some hazards of simplistic interpretation of biplots. Finally we conjecture on further possible applications of conditional compositional biplots

Fronts from complex two-dimensional dispersal kernels: theory and application to Reid's paradox

Bimodal dispersal probability distributions with characteristic distances differing by several orders of magnitude have been derived and favorably compared to observations by Nathan [Nature (London) 418, 409 (2002)]. For such bimodal kernels, we show that two-dimensional molecular dynamics computer simulations are unable to yield accurate front speeds. Analytically, the usual continuous-space random walks (CSRWs) are applied to two dimensions. We also introduce discrete-space random walks and use them to check the CSRW results (because of the inefficiency of the numerical simulations). The physical results reported are shown to predict front speeds high enough to possibly explain Reid's paradox of rapid tree migration. We also show that, for a time-ordered evolution equation, fronts are always slower in two dimensions than in one dimension and that this difference is important both for unimodal and for bimodal kernels

Front spreading in population dynamic models. Theory and application to the Neolithic transition

This thesis presents population dynamics models that can be applied to predict the rate of spread of the Neolithic transition (change from hunter-gathering to farming economics) across the European continent, which took place about 9000 to 5000 years ago. The first models in this thesis provide predictions at a continental scale. We develop population dynamics models with explicit kernels and apply realistic data. We also derive a new time-delayed reaction-diffusion equation which yields speeds about a 10% slower than previous models. We also deal with a regional variability: the slowdown of the Neolithic front when reaching the North of Europe. We develop simple reaction-diffusion models that can predict the measured speeds in terms of the non-homogeneous distribution of pre-Neolithic (Mesolithic) population in Europe, which were present in higher densities at the North of the continent. Such models can explain the observed speeds.

The calculation of accurate normal coordinates I: general theory, and application to methane

The calculation of accurate and reliable vibrational potential functions and normal co-ordinates is discussed, for such simple polyatomic molecules as it may be possible. Such calculations should be corrected for the effects of anharmonicity and of resonance interactions between the vibrational states, and should be fitted to all the available information on all isotopic species: particularly the vibrational frequencies, Coriolis zeta constants and centrifugal distortion constants. The difficulties of making these corrections, and of making use of the observed data are reviewed. A programme for the Ferranti Mercury Computer is described by means of which harmonic vibration frequencies and normal co-ordinate vectors, zeta factors and centrifugal distortion constants can be calculated, from a given force field and from given G-matrix elements, etc. The programme has been used on up to 5 × 5 secular equations for which a single calculation and output of results takes approximately l min; it can readily be extended to larger determinants. The best methods of using such a programme and the possibility of reversing the direction of calculation are discussed. The methods are applied to calculating the best possible vibrational potential function for the methane molecule, making use of all the observed data.

Phylogenies reveal new interpretation of speciation and the Red Queen

The Red Queen metaphor has species accumulating small changes to keep up with a continually changing environment, with speciation occurring at a constant rate. This constant-rate claim is now tested against four competing models, using 101 phylogenies of animal, plant and fungal taxa. The results provide a new interpretation of the Red Queen; a view linking speciation to rare stochastic events that cause reproductive isolation.

Speciation and variation in the occurrence of haloacetic acids in three water supply systems in England

An investigation into the speciation and occurrence of nine haloacetic acids (HAAs) was conducted during the period of April 2007 to March 2008 and involved three drinking water supply systems in England, which were chosen to represent a range of source water conditions; these were an upland surface water, a lowland surface water and a groundwater. Samples were collected seasonally from the water treatment plants and at different locations in the distribution systems. The highest HAA concentrations occurred in the upland surface water system, with an average total HAA concentration of 21.3 μg/L. The lowest HAA levels were observed in the groundwater source, with a mean concentration of 0.6 μg/L. Seasonal variations were significant in the HAA concentrations; the highest total HAA concentrations were found during the autumn, when the concentrations were approximately two times higher than in winter and spring. HAA speciation varied among the water sources, with dichloroacetic acid and trichloroacetic acid dominant in the lowland surface water system and brominated species dominant in the upland surface water system. There was a strong correlation between trihalomethanes and HAAs when considering all samples from the three systems in the same data set (r2=0.88); however, the correlation was poor/moderate when considering each system independently.

Water quality trends in the Windrush catchment: Nitrogen speciation and sediment interactions

For the predominantly agricultural River Windrush catchment, spatial variations in concentrations of nitrogen species and suspended sediment were strongly related to geology and land use. Temporal patterns of NO3- and NO2- concentrations during the three year study were highly correlated with seasonal variations in baseflow. Suspended sediment concentrations were mainly controlled by storm discharge. Variations in total ammonium concentrations reflected both flow controls. Suspended sediment effects total ammonium and organic nitrogen transport to the aquatic system, and in-stream cycling processes. Organic nitrogen did not display consistent seasonal variations, but concentrations occasionally exceeding those of NO3-. Overall, NO3- and organic nitrogen were the most important at 60% and -40%, of total nitrogen load, respectively. Future assessments of agriculture impact on river water quality should consider the total nitrogen load, and not solely that of NO3-.

Nitrogen speciation and phosphorus fractionation dynamics in a lowland chalk catchment

A detailed analysis of temporal and spatial trends in nitrogen (N) speciation and phosphorus (P) fractionation in the Wylye, a lowland Chalk sub-catchment of the Hampshire Avon, UK is presented, identifying the sources contributing to nutrient enrichment, and temporal variability in the fractionation of nutrients in transit from headwaters to lower reaches of the river. Samples were collected weekly from ten monitoring stations with daily sampling at three further sites over one year, and monthly inorganic N and total reactive P (TRP) concentrations at three of the ten weekly monitoring stations over a ten year period are also presented. The data indicate significant daily and seasonal variation in nutrient fractionation in the water column, resulting from plant uptake of dissolved organic and inorganic nutrient fractions in the summer months, increased delivery of both N and P from diffuse sources in the autumn to winter period and during high flow events, and lack of dilution of point source discharges to the Wylye from septic tank, small package Sewage Treatment Works (STW) and urban Waste Water Treatment Works (WwTW) during the summer low flow period. Weekly data show that contributing source areas vary along the river with headwater N and P strongly influenced by diffuse inorganic N and particulate P fluxes, and SRP and organic-rich point source contributions from STW and WwTW having a greater influence in the lower reaches. Long-term data show a decrease in TRP concentrations at all three monitoring stations, with the most pronounced decrease occurring downstream from Warminster WwTW, following the introduction of P stripping at the works in 2001. Inorganic N demonstrates no statistically significant change over the ten year period of record in the rural headwaters, but an increase in the lower reaches downstream from the WwTW which may be due to urban expansion in the lower catchment.