Application of Multivariate Statistical Analysis for the Detection of Structural Changes in the Series of Monitoring Data

A new approach to the study of time series by the projection pursuit methods is described. The ideas are illustrated on the time series of the monitoring of the environment and climate: (a) on time series of anomalies of global mean annual temperature -- the main climatological parameter; (b) on time series of atmospheric CO2 concentrations -- the main greenhouse gases; (c) on time series of vegetation index (NDVI) -- the main global characteristic of biota activity on the satellite data. With the aid of the shift operator for time signal, we construct a curve in n-dimensional Euclidian space (shift operator and integer n are the parameters of method). So an analysis of a time series is reduced to the analysis of the most informative projections [for example, by the criterion of factor analysis or spectral analysis (discrete Fourie analysis)] of the corresponding n-dimensional curve. We show that the comparison of such projections for model-test time series with the projection of the time series under investigation gives an effective way of finding the structural changes of the monitoring time series. For example, the case of the Hansen-Lebedeff time series of anomalies of the global mean annual temperature (see Rends 'go), shows that the structure of the series in the interval from 1920 until 1950 essentially differs from the structure on the intervals 1880-1920 and 1950-1987. For the series of CO2 on the Mauna Loa and Barrow monitoring stations, we obtained dynamics of the amplitudes of the year and semi-year cycles. We give the construction of a nonparametric estimation of a model of the initial time series using k-dimensional projection of n-dimensional curve. As a consequence, for example, we found the main components of the CO2 time series and obtained the models of the yearly behaviors of NDVI time series which permit one to carry out statistically stable classification of ecosystems by ecotypes and to describe dynamics of the separate ecosystems. Thus it is proposed a tool for the creation of the statistical description of the current state of the given monitoring series in the form of geometrical image. These geometrical images permit us to analyze the anomalies in the monitoring series in the terms of deviation of these images. As it follows from the examples given below such a method of the analysis of monitoring data is an effective method. Between the theoretical let us stress the following: we show how the methods of the analysis of the time series widely used in statistical treatment of monitoring data could also be used in our approach as the tools of the projections pursuit for comparing the images of the curves of the signal under investigation with the curves of the corresponding signals; it is shown that the proposed approach permits us to join in a united method the achievement of the theory of operators of a generalized shift and exploratory analysis on the basis of the projection pursuit

A Statistical Model of Background Air Pollution Frequency Distributions

The authors of this paper describe an approach for identifying statistically stable central tendencies in the frequency distributions of time series of observations of background atmospheric pollutants. The data were collected as daily mean values of concentrations of sulfur dioxide and suspended particulate matter at five monitoring stations -- three in the USSR, one in Norway, and one in Sweden. In their approach, the authors use well-developed statistical techniques and the usual method of constructing multimodal distributions. The problem is subdivided into two parts: first, a decomposition of the observations in order to obtain a description of each season separately and second, an investigation of this description in order to derive statistically stable characteristics of the entire data set. The main hypothesis of the investigation is that dispersion processes interact in such a way that in the zone of influence of one process (near its mode) the "tails" of the other process are not observed. This permits illumination of interrelations between the physics and the chemistry of the atmosphere. During the last 15-20 years, a wide range of monitoring programs has been initiated at national and international levels including, for example, the European Monitoring and Evaluation Program (EMEP) under the auspices of the ECE, and the Background Air Pollution Monitoring Network (BAPMoN) under the auspices of the WMO. The flow of data from the system of monitoring stations has led to national and international projects for the development of extensive environmental data bases such as NOAANET (NDAA), GRID/GEMS/UNEP/NASA, etc. The degree of information obtained should be sufficient for the goals of the analysis but often there is an overabundance of such data. The methods discussed in this paper therefore help in air pollution assessments, particularly with respect to distinguishing the baseline components, and their trends over decades

An Exploratory Analysis of Long-Term Trends in Atmospheric CO2 Concentrations

A new methodological approach for analysis of monitoring data is discussed. The main ideas are illustrated for the example of the CO2 problem. The analysis of CO2 concentrations obtained from a global network of monitoring stations permitted the authors to construct a nonparametric evaluation of the spatial-temporal distribution of this field. They propose a parabolic parametrization of the long-term tendency of this field as a function of time (in one-year time steps). A function of the predicitve ability of a model is defined on the basis of the technique of "supervised training". This function is computed for a parabolic model and it is shown that this model constructed for the first 15 years of observations evaluates the tendency for the next 15 years quite well. The main problem that is solved in this paper is how to correlate the projections of different models for the carbon cycle and different scenarios of the annual release of carbon into the atmosphere with the projection that reflect parametrization of the trends of CO2-monitoring data

Generalised Elliptic Functions

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed sees the functions defined as logarithmic derivatives of the sigma-function, a modified Riemann theta-function. We can make use of known properties of the sigma function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given. The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future.Comment: 16 page

Statistical Analysis of Long Term Trends in Atmospheric CO2 Concentration at Baseline Stations

Carbon dioxide is one of several greenhouse gases that can modify the earth's heat balance by absorbing outgoing radiation from the earth's surface, thereby increasing the amount of heat retained by the atmosphere (the so-called greenhouse effect). Changes in CO2 are therefore of considerable importance. In this paper, the long-term trends are assessed at four baseline stations -- Mauna Loa (Hawaii), Barrow (Alaska), American Samoa and South Pole. The authors conclude that a parabolic model provides the best fit for the observed rates of CO2 concentration growth over the last 20-30 years

Stratifying derived categories of cochains on certain spaces

In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a topological content.Comment: 27 page

Combinatorial Alexander Duality -- a Short and Elementary Proof

Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof.Comment: 7 pages, 2 figure; v3: the sign function was simplifie

On one integrable system with a cubic first integral

Recently one integrable model with a cubic first integral of motion has been studied by Valent using some special coordinate system. We describe the bi-Hamiltonian structures and variables of separation for this system.Comment: LaTeX with AMS fonts, 9 page

A constructive approach to the soliton solutions of integrable quadrilateral lattice equations

Scalar multidimensionally consistent quadrilateral lattice equations are studied. We explore a confluence between the superposition principle for solutions related by the Backlund transformation, and the method of solving a Riccati map by exploiting two kn own particular solutions. This leads to an expression for the N-soliton-type solutions of a generic equation within this class. As a particular instance we give an explicit N-soliton solution for the primary model, which is Adler's lattice equation (or Q4).Comment: 22 page

On maximally superintegrable systems

Locally any completely integrable system is maximally superintegrable system such as we have the necessary number of the action-angle variables. The main problem is the construction of the single-valued additional integrals of motion on the whole phase space by using these multi-valued action-angle variables. Some constructions of the additional integrals of motion for the St\"ackel systems and for the integrable systems related with two different quadratic $r$-matrix algebras are discussed. Among these system there are the open Heisenberg magnet and the open Toda lattices associated with the different root systems.Comment: 12 pages, LaTeX with AmsFont