Multiple time series of environmental quality data with similar, but not necessarily identical, trends call for multivariate methods for trend detection and adjustment for covariates. Here, we show how an additive model in which the multivariate trend function is specified in a nonparametric fashion (and the adjustment for covariates is based on a parametric expression) can be used to estimate how the human impact on an ecosystem varies with time and across components of the observed vector time series. More specifically, we demonstrate how a roughness penalty approach can be utilized to impose different types of smoothness on the function surface that describes trends in environmental quality as a function of time and vector component. Compared to other tools used for this purpose, such as Gaussian smoothers and thin plate splines, an advantage of our approach is that the smoothing pattern can easily be tailored to different types of relationships between the vector components. We give explicit roughness penalty expressions for data collected over several seasons or representing several classes on a linear or circular scale. In addition, we define a general separable smoothing method. A new resampling technique that preserves statistical dependencies over time and across vector components enables realistic calculations of confidence and prediction intervals.

We consider the problem of minimizing the distance from a given n-dimensional vector to a set defined by constraints of the form x_i ≤ x_j. Such constraints induce a partial order of the components x_i, which can be illustrated by an acyclic directed graph. This problem is also known as the isotonic regression (IR) problem. IR has important applications in statistics, operations research and signal processing, with most of them characterized by a very large value of n. For such large-scale problems, it is of great practical importance to develop algorithms whose complexity does not rise with n too rapidly. The existing optimization-based algorithms and statistical IR algorithms have either too high computational complexity or too low accuracy of the approximation to the optimal solution they generate. We introduce a new IR algorithm, which can be viewed as a generalization of the Pool-Adjacent-Violator (PAV) algorithm from completely to partially ordered data. Our algorithm combines both low computational complexity O(n²) and high accuracy. This allows us to obtain sufficiently accurate solutions to IR problems with thousands of observations.

Large-Scale Nonlinear Optimization reviews and discusses recent advances in the development of methods and algorithms for nonlinear optimization and its applications, focusing on the large-dimensional case, the current forefront of much research.

The chapters of the book, authored by some of the most active and well-known researchers in nonlinear optimization, give an updated overview of the field from different and complementary standpoints, including theoretical analysis, algorithmic development, implementation issues and applications

Monotonic regression is a nonparametric method for estimation ofmodels in which the expected value of a response variable y increases ordecreases in all coordinates of a vector of explanatory variables x = (x₁, …, x_p).Here, we examine statistical and computational aspects of our recentlyproposed generalization of the pool-adjacent-violators (PAV) algorithm fromone to several explanatory variables. In particular, we show how the goodnessof-fit and accuracy of obtained solutions can be enhanced by presortingobserved data with respect to their level in a Hasse diagram of the partial orderof the observed x-vectors, and we also demonstrate how these calculations canbe carried out to save computer memory and computational time. Monte Carlosimulations illustrate how rapidly the mean square difference between fittedand expected response values tends to zero, and how quickly the mean squareresidual approaches the true variance of the random error, as the number of observations increases up to 10⁴.

Natural fluctuations in the state of the environment can long conceal or distort important trends in the human impact on our ecosystems. Accordingly, there is increasing interest in statistical normalisation techniques that can clarify the anthropogenic effects by removing meteorologically driven fluctuations and other natural variation in time series of environmental quality data. This thesis shows that semi- and nonparametric regression methods can provide effective tools for applying such normalisation to collected data. In particular, it is demonstrated how monotonic regression can be utilised in this context. A new numerical algorithm for this type of regression can accommodate two or more discrete or continuous explanatory variables, which enables simultaneous estimation of a monotonic temporal trend and correction for one or more covariates that have a monotonic relationship with the response variable under consideration. To illustrate the method, a case study of mercury levels in fish is presented, using body length and weight as covariates. Semiparametric regression techniques enable trend analyses in which a nonparametric representation of temporal trends is combined with parametrically modelled corrections for covariates. Here, it is described how such models can be employed to extract trends from data collected over several seasons, and this procedure is exemplified by discussing how temporal trends in the load of nutrients carried by the Elbe River can be detected while adjusting for water discharge and other factors. In addition, it is shown how semiparametric models can be used for joint normalisation of several time series of data.

The International Council for the Exploration of the Sea (ICES) has longcompiled extensive data on contaminants in biota. We investigated how trendassessment of mercury in muscle tissue from fish (flounder and Atlantic cod)might be facilitated by using nonparametric regression to normalise observedlevels of this contaminant with respect to body length and weight. Specifically,we examined response surfaces and annual normalised means obtained byemploying purely additive models (AM), thin plate splines (TPS), andmonotonic regression (MR) to model mercury levels as functions of samplingyear and one or two covariates. Our analysis showed that TPS and MR modelscan be more satisfactory than purely additive models, because the formertechniques enable estimation of time-dependent relationships between themercury concentration and the covariates. However, the major obstacle fortrend assessment of the collected mercury data was a substantial interannualvariation that was related to factors other than body length and weight.Nevertheless, several time series of flounder data that started in the 1970s and1980s showed obvious downward trends, and these trends were particularly2strong in the Elbe estuary. When the analysis was limited to data collected after1990, an overall Mann-Kendall test for all sampling sites revealed astatistically significant downward trend for flounder, whereas it was notsignificant for cod.

We present a new algorithm for monotonic regression in one or more explanatory variables. Formally, our method generalises the well-known PAV (pool-adjacent-violators) algorithm from fully to partially ordered data. The computational complexity of our algorithm is O(n2). The goodness-of-fit to observed data is much closer to optimal than for simple averaging techniques.

We consider the problem of minimizing the distance from a given n-dimensional vector to a set defined by constraintsof the form   xi  xj Such constraints induce a partial order of the components xi, which can be illustrated by an acyclic directed graph.This problem is known as the isotonic regression (IR) problem. It has important applications in statistics, operations research and signal processing. The most of the applied IR problems are characterized by a very large value of n. For such large-scale problems, it is of great practical importance to develop algorithms whose complexity does not rise with n too rapidly.The existing optimization-based algorithms and statistical IR algorithms have either too high computational complexity or too low accuracy of the approximation to the optimal solution they generate. We introduce a new IR algorithm, which can be viewed as a generalization of the Pool-Adjacent-Violator (PAV) algorithm from completely to partially ordered data. Our algorithm combines both low computational complexity O(n2) and high accuracy. This allows us to obtain sufficiently accurate solutions to the IR problems with thousands of observations.

The reunification of Germany led to dramatically reduced emissions of nitrogen (N) and phosphorus (P) to the environment. The aim of the present study was to examine how these exceptional decreases influenced the amounts of nutrients carried by the Elbe River to the North Sea. In particular, we attempted to extract anthropogenic signals from time series of riverine loads of nitrogen and phosphorus by developing a normalization technique that enabled removal of natural fluctuations caused by several weather-dependent variables. This analysis revealed several notable downward trends. The normalized loads of total-N and NO₃-N exhibited an almost linear trend, even though the nitrogen surplus in agriculture dropped dramatically in 1990 and then slowly increased. Furthermore, the decrease in total-P loads was found to be considerably smaller close to the mouth of the river than further upstream. Studying the predictive ability of different normalization models showed the following: (i) nutrient loads were influenced primarily by water discharge; (ii) models taking into account water temperature, load of suspended particulate matter, and salinity were superior for some combinations of sampling sites and nutrient species; semiparametric normalization models were almost invariably better than ordinary regression models.