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.

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

Emissions from road traffic are hard to measure and therefore usually estimated in models. In this paper the construction of the widely used COPERT III model is examined, and the model is rewritten in mathematical notation. The original COPERT III software is easily handled but is not suitable as an emissiondata generating tool for fast calculations over a broad variety of driving conditions, which is required for sensitivity analysis. An alternative program has been developed to meet the desired properties of such a tool. The construction of the alternative program is discussed together with its new abilities and restrictions. Some differences between the results from the original COPERT III software and the alternative program are analyzed and discussed.

Many complex systems are simulated in models, and these models are hard to evaluate completely. For example, the sensitivity of the model output for different inputs is seldom analyzed. This paper provides an introduction to sensitivity analysis and proposes some basic sensitivity analysis methods. The results are important, for example, in choosing a sampling scheme. Themethods are applied to the COPERT III model, and the sensitivity results are compared and discussed. Certain theoretical results simplify the sensitivity analysis in a special case, and the paper ends by recommending a method.

Sensitivity analyses may be local or global, one at a time or all at a time, or classified in other ways. This paper examines some response surface methods for instant calculations of many sensitivities at the same time. The appropriateness of replacing the original model with a simpler response surface is discussed.The methods are also compared with one at a time methods. Sensitivity results are calculated by applying the methods to the COPERT III model. The conclusion is that a simple response surface method should be preferred.

Many systems, and some of their properties, are studied in simulation models. The models may be very complex and difficult to understand if the systems they are meant to simulate are complex in themselves. Even small, simple and well-described models can have properties that cannot easily be seen in the model formulations. Sensitivity and uncertainty in the model results are examples of such properties.

One important field where simulation models are used is road traffic, for emissions or other traffic-related topics. To <late, little has been written about sensitivity and uncertainty in road traffic emission models. More must be known about the construction of such models before sensitivity methods can be proposed.

This thesis describes one model for road traffic emissions, both for construction and for ability to be an emission data generating tool for the analyses that follow. Some suitable sensitivity methods are studied; they are based on different approaches such as sensitivity on the margins and measure over all situations by using response surface methods. All of the methods in the study are also applied to the described emission model.

Different approaches are compared on theoretical grounds in several ways, and the sensitivity results allow many other comparisons.

The output of a model or an experiment is often fairly uncertain because the underlying conditions have to some extent been guessed at or estimated. Such uncertainty is usually described by examining the distribution of the output given an input of known distribution. In this paper, we discuss why this approachmay not be suitable for some problems, and consider an alternate approach in cases in which we have only vague information concerning the distributions of the inputs, regardless as to whether these inputs are continm:ms or categorical. Our approach is well suited for a mean output that represents the weighted sum of the outputs of categories, if there is a large dataset with an uncertain input distribution. In this study, the method is applied to a road traffic emission scenario.

This paper discusses sensitivity analysis based on methods that divide the total variation of a particular response into components associated with the explaining variables or factors. Two methods for estimating variance components in no-interaction multiway designs are discussed, and their dependence on balancein the sampling design is studied. The methods are applied to datasets generated from a road traffic emission model.

The SwedishNational Road Administration SNRA administrates the constructionand maintenance of Swedish state roads. In addition, the SNRA hasan overall responsibility for road traffic safety in Sweden. Thesetasks are supported by a system of road traffic surveys by which wemean here surveys based on data collection from pre-selected siteson the roads. The data collection is conducted by observers or byusing equipment such as traffic analyzers and pneumatic tubes. Onepart of the system is the so-called Traffic Measurement Systemwhich includes the surveys of Annual average daily traffic AADT andChange of total vehicle mileage. Another part of the system isdirectly connected to road traffic safety and includes surveys ofvehicle speed, seat belt use, and bicycle helmet use, for example.The surveys of traffic safety are based on a master sample ofroads, designed for a broad use of traffic surveys. We present theSNRA traffic surveys and discuss research activities includingsampling and estimation issues, non-sampling errors and the use ofmodel-based inference from these surveys.

We examined under what circumstances the results of a large number of runs of the one-dimensional, physics-based SOIL/SOILN nitrate transport model can be combined into a reduced (or meta) model. We considered the total flow of nitrate from a given area and investigated when and how hidden linear structures can be extracted from the underlying model. The presence of such structures can justify the use of spatially aggregated inputs to compute spatially aggregated outputs. Extensive Monte-Carlo simulations showed that some linear structures emerged when the outputs for a long period of time were summed. Other linear structures appeared as relationships between two different components of the model outputs. However, different cropping systems respond differently to changes in anthropogenic or meteorological forcings. Therefore, we derived a reduced model of long-term leaching of nitrogen from the root zone in an agricultural area by combining each combination of soil type and cropping system. Reduced models can help make process-oriented models more transparent, and they are particularly suitable for incorporation into decision support systems.

Decision support systems (DSSs) for evaluation of different policy measures have two important functions: To assess how considered policy measures may influence the behavior of actors, and to predict the effects of a given set of actions generated from the anticipated behavior. So far, almost all attempts to construct DSSs for environmental management have focused on assessing the impact of a set of actions on the environment. Here, we describe the generic structure of a DSS that enables more complete evaluation of regional or national policies to reduce nitrogen inputs to water. In particular, we expound the principles for linking models of farm economic behavior to catchment-scale models of the transport and transformation of nitrogen in soil and water. First, we define system boundaries for nitrogen fluxes through the agricultural sector and the ambient environment to create a basis for model integration. Thereafter, we show how different modules operating on different temporal and spatial scales can be interlinked. Finally, we demonstrate how statistical emulators or meta-models can be derived to reduce the computational burden and increase the transparency of the DSS. In particular, we show when and how the temporal or spatial resolution of model inputs can be reduced without significantly influencing the estimates of annual nitrogen fluxes on a catchment scale.

Empirical data regarding the time scales of nutrient losses from soil to water and land to sea were reviewed. The appearance of strongly elevated concentrations of nitrogen and phosphorus in major European rivers was found to be primarily a post-war phenomenon. However, the relatively rapid water quality response to increased point source emissions and intensified agriculture does not imply that the reaction to decreased emissions will be equally rapid. Long-term fertilisation experiments have shown that important processes in the large-scale turnover of nitrogen operate on a time scale of decades up to at least a century, and in several major Eastern European rivers there is a remarkable lack of response to the dramatic decrease in the use of commercial fertilisers that started in the late 1980s. In Western Europe, studies of decreased phosphorus emissions have shown that riverine loads of this element can be rapidly reduced from high to moderate levels, whereas a further reduction, if achieved at all, may take decades. Together, the reviewed studies showed that the inertia of the systems that control the loss of nutrients from land to sea was underestimated when the present goal of a 50% reduction of the input of nutrients to the Baltic Sea and the North Sea was adopted. (C) 2000 Elsevier Science B.V.

There is a need to examine long-term changes in nitrogen leaching from arable soils. The purpose of this study was to analyse variations in specific leaching rates (kg ha-1 per year) and gross load (Mg per year) of N from arable land to watercourses in Sweden from a historical perspective. The start of the study was set to 1865 because information on crop distribution, yield and livestock has been compiled yearly since then. The SOIL/SOILN model was used to calculate nitrogen leaching. Calculations were done for cereals, grass and bare fallow for three different soil types in nine agricultural regions covering a range of climatic conditions. Results indicate that both specific leaching rates and gross load of nitrogen in the middle of 19th century were approximately the same as they are today for the whole of south and central Sweden. Three main explanations for this were (1) large areas of bare fallow typical for the farming practice at the time, (2) enhanced mineralisation from newly cultivated land, and (3) low yield. From 1865, i.e. the start of the calculations, N leaching rates decreased and were at their lowest around 1930. During the same period, gross load was also at its lowest despite the fact that the acreage of arable land was at its most extensive. After 1930, average leaching increased by 60% and gross load by 30%, both reaching a peak in the mid-1970s to be followed by a declining trend. The greatest increase in leaching was in regions where the increase in animal density was largest and these regions were also those where the natural conditions for leaching such as mild winters and coarse-textured soils were found. Extensive draining projects occurred during the period of investigation, in particular an intensive exploitation of lakes and wetlands. This caused a substantial drop in nitrogen retention and the probable increase in net load to the sea might thus have been more affected by this decrease in retention than the actual increase in gross load. (C) 2000 Elsevier Science B.V.

Monotonic regression is a non-parametric method that is designed especially for applications in which the expected value of a response variable increases or decreases in one or more explanatory variables. Here, we show how the recently developed generalised pool-adjacent-violators (GPAV) algorithm can greatly facilitate the assessment of trends in time series of environmental quality data. In particular, we present new methods for simultaneous extraction of a monotonic trend and seasonal components, and for normalisation of environmental quality data that are influenced by random variation in weather conditions or other forms of natural variability. The general aim of normalisation is to clarify the human impact on the environment by suppressing irrelevant variation in the collected data. Our method is designed for applications that satisfy the following conditions: (i) the response variable under consideration is a monotonic function of one or more covariates; (ii) the anthropogenic temporal trend is either increasing or decreasing; (iii) the seasonal variation over a year can be defined by one increasing and one decreasing function. Theoretical descriptions of our methodology are accompanied by examples of trend assessments of water quality data and normalisation of the mercury concentration in cod muscle in relation to the length of the analysed fish.

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.

In a Swedish vehicle speed survey for a multi-stage sample of road sites, data are collected by use of a measurement device installed on the road. Typically, some of the vehicles passing a chosen site will remain unobserved. Therefore, we suggest dividing the traffic into weighting classes. The main difficulty is to adjust the observed number of vehicles for missing data. Within class, one proposal is to add vehicles imputed by the device; another to use registration probability weighting. Models for the errors in the number of imputed vehicles, and in the estimated registration probabilities, enable theoretical evaluations of the proposals. From some empirical data, the models are evaluated.

The measured road lengths in a frame of roads, used for surveying vehicle speeds, may differ from the true ones. An error model is formulated and used to evaluate the impact of such errors on the bias and variance of the estimator of average speed. The theoretical investigation is supplemented by results from an empirical study of the errors in the frame.

The impact of some errors, associated with a road traffic survey, is examined. The survey aims at evaluating efforts to reduce the speeds on Swedish roads. It covers both state and urban roads, but we consider only the latter. From these roads, observational sites are selected by a three-stage sampling procedure. A measurement device installed on the road is used to collect data, from which the average speed on the roads is estimated.

We focus on errors in the frames used in the final sampling stage, and on errors due to rnissing data. The impact of these errors on the total error of the survey estimators is investigated. Also, possibilities to reduce the total error by weighting adjustments for missing data, and by re-allocation of the sample over sampling stages, are explored. We approach the problems partly theoretically, by use of various error models; partly empirically, by collecting data on the errors. Throughout. the sampling design of the survey is taken properly into account. We conclude that the frame error under consideration probably does not bias the estimator of average speed, and only implies a minor increase of its variance. It remains unclear if the estimator needs to be adjusted for missing data - a theoretical frame for further investigations is however provided. For unchanged total sample size, the precision of the estimator is likely to improve if the sample sizes in stage three are increased, and the sampling sizes in stage one decreased correspondingly.

Riverine transport is the, most important pathway for input of nutrients to the Gulf of Riga. The present study focused on updating existing estimates of the riverine nutrient contributions and on improving the favailable information on temporal and spatial variation in such input. The results show that the gulf received an average of 113,300 tons of nitrogen, 2050 tons of phosphorus and 64,900 tons of dissolved silica (DSi) annually during the time period 1977-1995. There was large interannual variation in loads, e.g., a factor two difference was found between the two most extreme years (1984 and 1990), this was attributed mainly to natural variation in water discharge. The seasonal distribution of nutrient loads exhibited a distinct pattern for practically all studied constituents, especially nitrate. Loads were high during the spring-flow and relatively low during the low-flow summer period. Examination of the spatial variation of nutrient loads showed that the Daugava River alone accounted for approximately 60% of the total riverine load. The highest area-specific loads of nitrate and phosphate were observed in the agriculturally dominated Lielupe River, and the highest loads of organic-nitrogen (org-N) and total phophorus (tot-P) were found in the Parnu River. However, the values for all the studied rivers and constituents were rather low (phosphorus) or moderate (nitrogen and silica) compared to those reported for many other drainage areas of the Baltic Sea. This was true despite the inefficient sewage treatment and intensive agriculture in the studied basins in the 1970s and 1980s. (C) 1999 Elsevier Science B.V. All rights reserved.

To correctlyassesstrends in water quality data, influencing variables suchasdischarge or temperature must betaken into account. This canbedone by using i one-step procedures like the PartialMann-KendallPMK test ormultiple regression, or ii two-steptechniques thatinclude a normalisation followed by a trend test ontheresiduals.Which approach is most appropriate depends strongly ontherelationship between the response variableunder considerationandthe influencing variables. For example, PMK tests can besuperiorif there are long andvarying time lags in the waterqualityresponse. Two-step procedures are particularly useful whentheshape of thetemporal trend is the primary interest, but they canbemisleading if one of the influencing variables itselfexhibitsatrend or long-term tendency. The present study discussestheadvantages and disadvantages of some trendtesting techniques,usingSwedish water quality data to illustrate the properties ofthemethods.

Time series of environmental data are collected to monitor the effectiveness of new emission reduction policies or to determine the general state of the environment. However, small gradual changes in such variables can easily be concealed by large fluctuations caused by prevailing weather conditions. Hence, there is a real need for procedures that facilitate separation of such natural variation from anthropogenic effects.

Taking meteorological or hydrological variables into consideration in a trend analysis can be done in several ways. The technique chosen to accomplish this objective depends on characteristics of the data set, for example the length of the time series and sampling frequencies, and the kind of relationships that exist between the response variable and the covariates. Two different approaches were examined in the studies underlying this thesis: multivariate non-parametric tests and parametric normalisation procedures. The non-parametric trend test proposed here was newly desinged, thus it was also necessary to conduct simulation studies to examine the performance of this method. By comparison, normalisation techniques have been used over the past few decades mainly to adjust for the impact of meteorological effects on air quality data. The choice of explanatory variables for such procedures was studied: first by examining variable selection procedures based on cross-validation, paying special attention to serially correlated response data; and secondly by considering variables derived from complex physics-based models as alternatives to measured variables. A number of other aspects that might influence the ability to detect trends were also explored, including level shifts due to instrument malfunctions.

Meteorological normalisation of time series of air quality data aims to extract anthropogenic signals by removing natural fluctuations in the collected data. We showed that the currently used procedures to select normalisation models can cause over-fitting to observed data and undesirable smoothing of anthropogenic signals. A simulation study revealed that the risk of such effects is particularly large when: (i) the observed data are serially correlated, (ii) the normalisation model is selected by leave-one-out cross-validation, and (iii) complex models, such as artificial neural networks, are fitted to data. When the size of the test sets used in the cross-validation was increased, and only moderately complex linear models were fitted to data, the over-fitting was less pronounced. An empirical study of the predictive ability of different normalisation models for tropospheric ozone in Finland confirmed the importance of using appropriate model selection strategies. Moderately complex regional models involving contemporaneous meteorological data from a network of stations were found to be superior to single-site models as well as more complex regional models involving both contemporaneous and time-lagged meteorological data from a network of stations.

Trend analyses of time series of environmental data are often carried out to assess the human impact on the environment under the influence of natural fluctuations in temperature, precipitation, and other factors that may affect the studied response variable. We examine the performance of partial Mann–Kendall (PMK) tests, i.e. trend tests in which the critical region is determined by the conditional distribution of one Mann-Kendall (MK) statistic for monotone trend, given a set of other MK statistics. In particular, we examine the impact of incorporating information regarding covariates in the Hirsch–Slack test for trends in serially correlated data collected over several seasons. Monte Carlo simulation of the performance of PMK tests demonstrates that the gain in power due to incorporation of relevant covariates can be large compared to the loss in power caused by irrelevant covariates. Furthermore, we have found that the asymptotic normality of the test statistics in such tests enables rapid and reliable determination of critical regions, unless the sample size is very small (n < 10) or the different MK statistics are very strongly correlated. A case study of water quality trends shows that PMK tests can detect and correct for rather complex relationships between river water quality and water discharge. The generic character of the PMK tests makes them particularly useful for scanning large sets of data for temporal trends.

Power analysis is an integral part of statistical hypothesis testing, and, when neither exactpower computations nor reasonable approximations are feasible, Monte Carlo simulations providea viable alternative. However, generating data for such simulations is often an intricate task, especially when the hypothesis testing is based on non-normal multivariate data withcomplex dependencies. Here, we show how process-based deterministic models can be employed to generate data with adequate statistical dependencies for realistic power simulations. In particular, we usedthe Integrated Nitrogen in Catchments INCA model to produce bivariate time series ofnitrogen concentration and water discharge data that included plausible temporal trends andrealistic cross-correlations, seasonal patterns, and memory effects. The random variation in thegenerated data was achieved by running the INCA model with various sets of weather data that were obtained by block resampling from a given time series of observed air temperatureand precipitation data. The assortment of temporal trends was created by altering the anthropogenic input of nitrogen to the catchment under consideration.Two tests for temporal trends in nitrogen concentration were compared: i a partial Mann-Kendall test in which water discharge was treated as a covariate; ii a two-stage procedurein which we first used a semi-parametric regression technique to remove the impact of natural fluctuations in water discharge, and we subsequently applied an ordinary Mann-Kendall testto the obtained residuals. Our simulations demonstrated that the two tests had comparablepower, but also that they involved empirical significance levels that were much higher than the nominal levels, possibly due to substantial serial dependence in the data.

The objective of meteorological normalisation of air quality measurements is to extract anthropogenic signals by removing meteorologically driven fluctuations in the collected data. We found that standard normalisation procedures involving regression of air quality on local meteorological data can be improved by incorporating information on four-day back trajectories of the sampled air masses. A case study of tropospheric ozone data revealed that the most efficient normalisation was achieved by including selected trajectory coordinates directly in multivariate regression models. Summarising the trajectories into clusters or sector values prior to the normalisation indicated that there was a slight loss of information.