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• 1.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics.
A generalised PAV algorithm for monotonic regression in several variables2004In: COMPSTAT. Proceedings in Computational Statistics / [ed] J. Antoch, Heidelberg, NY: PhysicaVerlag/Springer , 2004, p. 761-767Conference paper (Refereed)

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.

• 2.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Hasse diagrams and the generalized PAV-algorithm for monotonic regression in several explanatory variables2005In: Computational Statistics and Data Analysis, ISSN 0167-9473Article in journal (Refereed)

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, …, xp).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.

• 3.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics.
An algorithm for isotonic regression problems2004In: European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS / [ed] P. Neittaanmäki, T. Rossi, K. Majava and O. Pironneau, Jyväskylä: University of Jyväskylä , 2004, p. 1-9Conference paper (Refereed)

We consider the problem of minimizing the distance from a given n-dimensional vector to a set defined by constraintsof the form   xi $\leq$ 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.

• 4.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics.
An O(n2) algorithm for isotonic regression2006In: Large-Scale Nonlinear Optimization / [ed] Pillo, Gianni; Roma, Massimo, New York: Springer Science+Business Media B.V., 2006, p. 25-33Conference paper (Other academic)

We consider the problem of minimizing the distance from a given n-dimensional vector to a set defined by constraints of the form xixj. Such constraints induce a partial order of the components xi, 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.

• 5.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, Faculty of Arts and Sciences.
An O(n2) algorithm for isotonic regression problems2006In: Large-Scale Nonlinear Optimization / [ed] G. Di Pillo and M. Roma, Springer-Verlag , 2006, p. 25-33Chapter in book (Refereed)

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

• 6.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Isotonic regression and normalisation of environmental quality data2003In: TIES The International Environmetrics Society 2003,2003, 2003Conference paper (Other academic)
• 7.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Unit of Applied Statistics and Mathematics, Swedish University of Agricultural Sciences, Box 7013, SE-750 07 Uppsala, Sweden.
Semiparametric smoothers for trend assessment of multiple time series of environmental quality data2008In: Environmetrics, ISSN 1180-4009, E-ISSN 1099-095XArticle in journal (Other academic)

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.

• 8.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Monotonic and Semiparametric Regression for the Detection of Trends in Environmental Quality Data2005Doctoral thesis, comprehensive summary (Other academic)

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.

1. Hasse diagrams and the generalized PAV-algorithm for monotonic regression in several explanatory variables
Open this publication in new window or tab >>Hasse diagrams and the generalized PAV-algorithm for monotonic regression in several explanatory variables
2005 (English)In: Computational Statistics and Data Analysis, ISSN 0167-9473Article in journal (Refereed) Submitted
##### Abstract [en]

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, …, xp).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.

Mathematics
##### Identifiers
urn:nbn:se:liu:diva-13600 (URN)
Available from: 2005-12-16 Created: 2005-12-16 Last updated: 2015-06-02
2. Monotonic regression for the detection of temporal trends in environmental quality data
Open this publication in new window or tab >>Monotonic regression for the detection of temporal trends in environmental quality data
2005 (English)In: Match, ISSN 0340-6253, Vol. 54, no 3, p. 535-550Article in journal (Refereed) Published
##### National Category
Social Sciences
##### Identifiers
urn:nbn:se:liu:diva-13601 (URN)
Available from: 2005-12-16 Created: 2005-12-16 Last updated: 2017-12-13
3. Trend analysis of mercury in fish using nonparametric regression
Open this publication in new window or tab >>Trend analysis of mercury in fish using nonparametric regression
2005 (English)Report (Other (popular science, discussion, etc.))
##### Abstract [en]

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.

##### Series
LIU-MAI-R, Department of Mathematics, Division of Statistics ; 2005-07
##### Keywords
additive models, thin plate splines, monotonic regression, trend assessment, normalisation, mercury, fish
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-13602 (URN)
Available from: 2005-12-16 Created: 2005-12-16 Last updated: 2009-03-03
4. Estimation of the human impact on nutrient loads carried by the Elbe River
Open this publication in new window or tab >>Estimation of the human impact on nutrient loads carried by the Elbe River
2004 (English)In: Environmental Monitoring and Assessment, ISSN 0167-6369, Vol. 96, no 1-3, p. 15-33Article in journal (Refereed) Published
##### Abstract [en]

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.

##### Keywords
Elbe River, nitrogen, normalization, phosphorus, trend detection
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-13603 (URN)10.1023/B:EMAS.0000031722.88972.62 (DOI)
Available from: 2005-12-16 Created: 2005-12-16 Last updated: 2009-05-19
• 9.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics .
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics .
A generic procedure for simultaneous estimation of monotone trends and seasonal patterns in time series of environmental data2003In: EnviroInfo 2003,2003, 2003, p. 629-634Conference paper (Other academic)
• 10.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics .
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics .
A generic procedure for simultaneous estimation of monotone trends and seasonal patterns in time series of environmental data2003In: SPRUCE VI,2003, 2003Conference paper (Other academic)
• 11.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Trend analysis of mercury in fish using nonparametric regression2005Report (Other (popular science, discussion, etc.))

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.

• 12.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Monotonic regression for assessment of trends in environmental quality data2004In: European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS / [ed] P. Neittaanmäki, T. Rossi, K. Majava and O. Pironneau, Jyväskylä: University of Jyväskylä, Department of Mathematical Information Technology , 2004, p. 1-12Conference paper (Refereed)

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.

• 13.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Monotonic Regression for Assessment of Trends in Environmental Quality Data2004Report (Other academic)
• 14.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Monotonic regression for the detection of temporal trends in environmental quality data2005In: Match, ISSN 0340-6253, Vol. 54, no 3, p. 535-550Article in journal (Refereed)
• 15.
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics .
Linköping University, Faculty of Arts and Sciences. Linköping University, Department of Mathematics, Statistics .
Estimation of the human impact on nutrient loads carried by the Elbe River2003Report (Other academic)
• 16.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Linköping University, Department of Mathematics, Statistics . Linköping University, The Institute of Technology. GKSS Research Centre, Institute for Coastal Research, Geesthacht, Germany.
Estimation of the human impact on nutrient loads carried by the Elbe River2004In: Environmental Monitoring and Assessment, ISSN 0167-6369, Vol. 96, no 1-3, p. 15-33Article in journal (Refereed)

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.

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