Contributions to the multivariate Analysis of Marine Environmental Monitoring

The thesis parts from the view that statistics starts with data, and starts by introducing the data sets studied: marine benthic species counts and chemical measurements made at a set of sites in the Norwegian Ekofisk oil field, with replicates and annually repeated. An introductory chapter details the sampling procedure and shows with reliability calculations that the (transformed) chemical variables have excellent reliability, whereas the biological variables have poor reliability, except for a small subset of abundant species. Transformed chemical variables are shown to be approximately normal. Bootstrap methods are used to assess whether the biological variables follow a Poisson distribution, and lead to the conclusion that the Poisson distribution must be rejected, except for rare species. A separate chapter details more work on the distribution of the species variables: truncated and zero-inflated Poisson distributions as well as Poisson mixtures are used in order to account for sparseness and overdispersion. Species are thought to respond to environmental variables, and regressions of the abundance of a few selected species onto chemical variables are reported. For rare species, logistic regression and Poisson regression are the tools considered, though there are problems of overdispersion. For abundant species, random coefficient models are needed in order to cope with intraclass correlation. The environmental variables, mainly heavy metals, are highly correlated, leading to multicollinearity problems. The next chapters use a multivariate approach, where all species data is now treated simultaneously. The theory of correspondence analysis is reviewed, and some theoretical results on this method are reported (bounds for singular values, centring matrices). An applied chapter discusses the correspondence analysis of the species data in detail, detects outliers, addresses stability issues, and considers different ways of stacking data matrices to obtain an integrated analysis of several years of data, and to decompose variation into a within-sites and between-sites component. More than 40 % of the total inertia is due to variation within stations. Principal components analysis is used to analyse the set of chemical variables. Attempts are made to integrate the analysis of the biological and chemical variables. A detailed theoretical development shows how continuous variables can be mapped in an optimal manner as supplementary vectors into a correspondence analysis biplot. Geometrical properties are worked out in detail, and measures for the quality of the display are given, whereas artificial data and data from the monitoring survey are used to illustrate the theory developed. The theory of display of supplementary variables in biplots is also worked out in detail for principal component analysis, with attention for the different types of scaling, and optimality of displayed correlations. A theoretical chapter follows that gives an in depth theoretical treatment of canonical correspondence analysis, (linearly constrained correspondence analysis, CCA for short) detailing many mathematical properties and aspects of this multivariate method, such as geometrical properties, biplots, use of generalized inverses, relationships with other methods, etc. Some applications of CCA to the survey data are dealt with in a separate chapter, with their interpretation and indication of the quality of the display of the different matrices involved in the analysis. Weighted principal component analysis of weighted averages is proposed as an alternative for CCA. This leads to a better display of the weighted averages of the species, and in the cases so far studied, also leads to biplots with a higher amount of explained variance for the environmental data. The thesis closes with a bibliography and outlines some suggestions for further research, such as a the generalization of canonical correlation analysis for working with singular covariance matrices, the use partial least squares methods to account for the excess of predictors, and data fusion problems to estimate missing biological data

Contributions to the multivariate Analysis of Marine Environmental Monitoring

The thesis parts from the view that statistics starts with data, and starts by introducing the data sets studied: marine benthic species counts and chemical measurements made at a set of sites in the Norwegian Ekofisk oil field, with replicates and annually repeated. An introductory chapter details the sampling procedure and shows with reliability calculations that the (transformed) chemical variables have excellent reliability, whereas the biological variables have poor reliability, except for a small subset of abundant species. Transformed chemical variables are shown to be approximately normal. Bootstrap methods are used to assess whether the biological variables follow a Poisson distribution, and lead to the conclusion that the Poisson distribution must be rejected, except for rare species. A separate chapter details more work on the distribution of the species variables: truncated and zero-inflated Poisson distributions as well as Poisson mixtures are used in order to account for sparseness and overdispersion. Species are thought to respond to environmental variables, and regressions of the abundance of a few selected species onto chemical variables are reported. For rare species, logistic regression and Poisson regression are the tools considered, though there are problems of overdispersion. For abundant species, random coefficient models are needed in order to cope with intraclass correlation. The environmental variables, mainly heavy metals, are highly correlated, leading to multicollinearity problems. The next chapters use a multivariate approach, where all species data is now treated simultaneously. The theory of correspondence analysis is reviewed, and some theoretical results on this method are reported (bounds for singular values, centring matrices). An applied chapter discusses the correspondence analysis of the species data in detail, detects outliers, addresses stability issues, and considers different ways of stacking data matrices to obtain an integrated analysis of several years of data, and to decompose variation into a within-sites and between-sites component. More than 40 % of the total inertia is due to variation within stations. Principal components analysis is used to analyse the set of chemical variables. Attempts are made to integrate the analysis of the biological and chemical variables. A detailed theoretical development shows how continuous variables can be mapped in an optimal manner as supplementary vectors into a correspondence analysis biplot. Geometrical properties are worked out in detail, and measures for the quality of the display are given, whereas artificial data and data from the monitoring survey are used to illustrate the theory developed. The theory of display of supplementary variables in biplots is also worked out in detail for principal component analysis, with attention for the different types of scaling, and optimality of displayed correlations. A theoretical chapter follows that gives an in depth theoretical treatment of canonical correspondence analysis, (linearly constrained correspondence analysis, CCA for short) detailing many mathematical properties and aspects of this multivariate method, such as geometrical properties, biplots, use of generalized inverses, relationships with other methods, etc. Some applications of CCA to the survey data are dealt with in a separate chapter, with their interpretation and indication of the quality of the display of the different matrices involved in the analysis. Weighted principal component analysis of weighted averages is proposed as an alternative for CCA. This leads to a better display of the weighted averages of the species, and in the cases so far studied, also leads to biplots with a higher amount of explained variance for the environmental data. The thesis closes with a bibliography and outlines some suggestions for further research, such as a the generalization of canonical correlation analysis for working with singular covariance matrices, the use partial least squares methods to account for the excess of predictors, and data fusion problems to estimate missing biological data.Postprint (published version

Researches in non-associative algebra

I have frequently been asked by biologists for mathematical help in connection with their problems. I was working on one such problem when an algebraist, observing my work without knowing what it was about, remarked that I was apparently using hypercomplex numbers. I was considering a certain type of inheritance specified by formulae which could be regarded as forming the multiplication table of a non -associative linear algebra; and my calculations could be regarded as manipulations of hypercomplex numbers in this algebra, or in another algebra derived from it by a process which I later called "duplication;I then realised that there are many such "genetic algebras ", representing different types of inheritance. They are in all cases non-associative as regards multiplication, though they can always be taken to be commutative. I found that a large class of genetic algebras (viz. those for "symmetrical inheritance" as defined in Paper VI, p. 2) possessed certain distinctive properties which seemed worthy of investigation for their own sake, and also for the sake of possible exploitation in genetics.Part Three, the main part of this thesis, consists of four papers in which this investigation is given - or rather is begun, for there are a good many problems left untackled.Part One consists of four papers (one written in collaboration with Dr A. Erdélyi) on some purely combinatory problems of non - associative algebra, suggested by the notations which I employed for products and powers in the genetic algebras. The combinatory per t 0.4-01.5 rt. theory is continued in theAconcluding postscipt which follows Paper X.Part Two shows how genetic algebras arise and are manipulate The multiplication table of a genetic algebra, the multiplication of hypercomplex numbers, and the above mentioned process of duplication, are simply a translation into symbols of the relevant essentials in the processes; of inheritance; and the symbolism as a whole is a convenient shorthand for reckoning with combinations and statistical distributions of genetic types, enabling one to dispense with some of the verbal arguments and the chessboard diagrams commonly used for the same purpose. In paper VI the treatment is made as general as possible with the object of showirg the relationship between different genetic algebras and something of their structure; and the concepts to be discussed in Part Three are here defined. In Paper V, which was published later but mostly written earlier than VI, the explanation is given in very much simpler mathematical language (for it was intended to be read by geneticists), and with more attention to practical applications. It can be explained very simply why multiplication in the genetic algebras is non- associative, that is to say(AB) C ≠ A (BC)This statement is interpreted:- "If the offspring of A and B mates with C, the probability distribution of genetic types in the progeny will not be the same as if A mates with the offspring of B and C."My symbolism was not essentially new: the novelty lay in is interpretationlin terms of hypercomplex numbers. In fact it could be said that genetic algebras had been used by geneticists in a primitive way for quite a long time without having been recognised. explicitly. Their explicit recognition is I believe more than a mere change of notation. Apart from greater brevity achieved in some applications, general theorems on linear algebras can be applied; transformations can be used which are quite meaningless genetically but which lead to genetically significant conclusions; and the use of an index notation and summation convention reduces the symbolism to manageable proportions when, with inheritance involving many genes, it threatens to become too heavy to handle.Biological considerations were thus the root of these researches, and I intend to return to the genetical applications later; for I believe that genetic algebras may throw light on some deeper problems of genetics. I cannot at present give solid justification for this belief in the sense of having successfully tackled problems otherwise unsolved, and I therefore wish that this thesis may not be judged as a finished achievement in biological investigation; but may be judged primarily as a contribution to algebra, suggested by biological problems, and perhaps having possibilities of application beyond the simple ones so far demonstrated

Reconstructions of science

'In vier Rekonstruktionen wird versucht, die natur- und sozialwissenschaftliche Diskussion der Basiskonzepte von Raum und Zeit zu vereinen. Dazu bedarf es einer neuen Diskursform, die bereits in Alfred North Whiteheads Naturphilosophie anklingt. Auf sozial-wissenschaftlicher Seite besinnen wir uns grundlegender Themen von Sozialphänomenologie, Strukturalismus und Interaktionismus. Fragestellungen von PrähistorikerInnen, ÄgyptologInnen und EthnomathematikerInnen werden wichtig, wo wir zeigen, daß unsere Konzepte von Raum und Zeit kulturelle Institutionen der Bedeutung sind, die ihrerseits Gesellschaft konstituieren und konstanter Rekonstruktion bedürfen. Die vierte Rekonstruktion greift die Frage der theoretischen Physik auf und stützt sich auf das integrative Instrument der Theorie der geometrischen Clifford Algebren. Wir leiten ab, daß und wie die inneren Symmetrien der Materie mit den äußeren Symmetrien der Raum-Zeit verbunden sind und daß die Metapher vom 'achtfachen Pfad', die Gell-Mann für einen Teil des Standardmodells verwendete, entgegen seiner Auffassung nicht als Witz zu verstehen ist. Der Faktor (D4)m in der Dirac-Gruppe jeder geometrischen Clifford Algebra C/p,q bildet eine Grundstruktur von Orientierung und Logik ab und korrespondiert daher mit einem Interface zwischen Geist und Materie.' (Autorenreferat)'In four reconstructions it is attempted to lead the natural and social science debate of the basis concepts of space and time in common. For this we need a new mode of science discourse which has already been initiated in Alfred North Whitehead's philosophy of nature. In social science we reconsider the basis themes of social phenomenology, structuralism and interactionism as far as those contribute to a space-time topic. Investigations of prehistorians, egyptologists and ethno-mathematicians are of importance where we demonstrate that our concepts of space and time represent cultural institutions of meaning which on their part constitute society and require that we constantly reconstruct them. The fourth reconstruction deals with the space-time of postmodern theoretical physics and is founded on the integrative instrument of the theory of geometric Clifford algebras. We show that and how the inner symmetrics of matter are connected with the outer symmetries of space-time and that Gell-Mann's metaphor of the 'eightfold path' that he used to denote part of the standard model of physics cannot be interpreted as quirk, in opposition to his own intention. The factor (D4)m in the Dirac group of any geometric Clifford Algebra C/p,q represents a ground template (or archetypal structure) for both orientation and logic and corresponds therefore with an interface between matter and mind.' (author's abstract)

Measuring Electron Correlation. The Impact of Symmetry and Orbital Transformations

In this perspective, the various measures of electron correlation used in wavefunction theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behaviour with respect to the choice of the $N$-electron and the one-electron basis. The impact of symmetry is discussed and we emphasize that the distinction between determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wavefunction expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wavefunctions are discussed and the effect of orbital rotations on the multireference character is reviewed by analysing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and $N$-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one

Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce.

Charles Peirce (1839-1914) was one of the most important logicians of the nineteenth century. This thesis traces the development of his algebraic logic from his early papers, with especial attention paid to the mathematical aspects. There are three main sources to consider. 1) Benjamin Peirce (1809-1880), Charles's father and also a leading American mathematician of his day, was an inspiration. His memoir Linear Associative Algebra (1870) is summarised and for the first time the algebraic structures behind its 169 algebras are analysed in depth. 2) Peirce's early papers on algebraic logic from the late 1860s were largely an attempt to expand and adapt George Boole's calculus, using a part/whole theory of classes and algebraic analogies concerning symbols, operations and equations to produce a method of deducing consequences from premises. 3) One of Peirce's main achievements was his work on the theory of relations, following in the pioneering footsteps of Augustus De Morgan. By linking the theory of relations to his post-Boolean algebraic logic, he solved many of the limitations that beset Boole's calculus. Peirce's seminal paper `Description of a Notation for the Logic of Relatives' (1870) is analysed in detail, with a new interpretation suggested for his mysterious process of logical differentiation. Charles Peirce's later work up to the mid 1880s is then surveyed, both for its extended algebraic character and for its novel theory of quantification. The contributions of two of his students at the Johns Hopkins University, Oscar Mitchell and Christine Ladd-Franklin are traced, specifically with an analysis of their problem solving methods. The work of Peirce's successor Ernst Schröder is also reviewed, contrasting the differences and similarities between their logics. During the 1890s and later, Charles Peirce turned to a diagrammatic representation and extension of his algebraic logic. The basic concepts of this topological twist are introduced. Although Peirce's work in logic has been studied by previous scholars, this thesis stresses to a new extent the mathematical aspects of his logic - in particular the algebraic background and methods, not only of Peirce but also of several of his contemporaries

BIALGEBRAIC STRUCTURES AND SMARANDACHE BIALGEBRAIC STRUCTURES

The study of bialgebraic structures started very recently. Till date there are no books solely dealing with bistructures. The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published in 1998. In the year 1999, bivector spaces was introduced. In 2001, concept of free De Morgan bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the construction of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic