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  • 1.
    Abramowicz, Konrad
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Numerical analysis for random processes and fields and related design problems2011Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D).

    In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed.

    In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided.

    In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity.

    In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.

  • 2.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Arnqvist, Per
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sjöstedt de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Secchi, Piercesare
    Vantini, Simone
    Vitelli, Valeria
    Was it snowing on lake Kassjön in January 4486 BC? Functional data analysis of sediment data2014Conference paper (Other academic)
  • 3.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Häger, Charlotte
    Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy.
    Hérbert-Losier, Kim
    Swedish Winter Sports Research Centre Mid Sweden; University Department of Health Sciences, Östersund, Sweden.
    Pini, Alessia
    MOX – Department of Mathematics, Politecnico di Milano.
    Schelin, Lina
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy.
    Strandberg, Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vantini, Simone
    MOX – Department of Mathematics, Politecnico di Milano.
    An inferential framework for domain selection in functional anova2014In: Contributions in infinite-dimensional statistics and related topics / [ed] Bongiorno, E.G., Salinelli, E., Goia, A., Vieu, P, Esculapio , 2014Conference paper (Refereed)
    Abstract [en]

    We present a procedure for performing an ANOVA test on functional data, including pairwise group comparisons. in a Scheff´e-like perspective. The test is based on the Interval Testing Procedure, and it selects intervals where the groups significantly differ. The procedure is applied on the 3D kinematic motion of the knee joint collected during a functional task (one leg hop) performed by three groups of individuals.

  • 4.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Häger, Charlotte
    Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy.
    Pini, Alessia
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. Department of Statistical Sciences, Università Cattolica del Sacro Cuore, Milan, Italy.
    Schelin, Lina
    Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation. Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Sjöstedt de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vantini, Simone
    Nonparametric inference for functional-on-scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament2018In: Scandinavian Journal of Statistics, Vol. 45, no 4, p. 1036-1061Article in journal (Refereed)
    Abstract [en]

    Motivated by the analysis of the dependence of knee movement patterns during functional tasks on subject-specific covariates, we introduce a distribution-free procedure for testing a functional-on-scalar linear model with fixed effects. The procedure does not only test the global hypothesis on the entire domain but also selects the intervals where statistically significant effects are detected. We prove that the proposed tests are provided with an asymptotic control of the intervalwise error rate, that is, the probability of falsely rejecting any interval of true null hypotheses. The procedure is applied to one-leg hop data from a study on anterior cruciate ligament injury. We compare knee kinematics of three groups of individuals (two injured groups with different treatments and one group of healthy controls), taking individual-specific covariates into account.

  • 5.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Multivariate piecewise linear interpolation of a random field2011Manuscript (preprint) (Other academic)
    Abstract [en]

    We consider a multivariate piecewise linear interpolation of a continuous random field on a-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is defined by N field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field in mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields we provide the upper bound for the approximation accuracy in the uniform mean square norm.

  • 6.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On the error of the Monte Carlo pricing method for Asian option2008In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, p. 1-10Article in journal (Refereed)
    Abstract [en]

    We consider a Monte Carlo method to price a continuous arithmetic Asian option with a given precision. Piecewise constant approximation and plain simulation are used for a wide class of models based on L\'{e}vy processes. We give bounds of the possible discretization and simulation errors. The sufficient numbers of discretization points and simulations to obtain requested accuracy are derived. To demonstrate the general approach, the Black-Scholes model is studied in more detail. We undertake the case of continuous averaging and starting time zero,  but the obtained results can be applied to the discrete case  and generalized for any time before an execution date. Some numerical experiments and comparison to the PDE based method are also presented.

  • 7.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Piecewise multilinear interpolation of a random field2013In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 45, no 4, p. 945-959Article in journal (Refereed)
    Abstract [en]

    We consider a piecewise-multilinear interpolation of a continuous random field on a d-dimensional cube. The approximation performance is measured using the integrated mean square error. Piecewise-multilinear interpolator is defined by N-field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field, in the mean square sense, and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields, we provide the upper bound for the approximation accuracy in the uniform mean square norm.

  • 8.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Spline approximation of a random process with singularity2011In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 141, no 3, p. 1333-1342Article in journal (Refereed)
    Abstract [en]

    Let a continuous random process X defined on [0,1] be (m+β)-smooth, 0m, 0<β1, in quadratic mean for all t>0 and have an isolated singularity point at t=0. In addition, let X be locally like a m-fold integrated β-fractional Brownian motion for all nonsingular points. We consider approximation of X by piecewise Hermite interpolation splines with n free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate n^(m+β) for the whole interval.

  • 9.
    Abramowicz, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Seleznjev, Oleg
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stratified Monte Carlo quadrature for continuous random fields2015In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 17, no 1, p. 59-72Article in journal (Refereed)
    Abstract [en]

    We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H¨older class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

  • 10.
    Abramowizc, Konrad
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Arnqvist, Per
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Secchi, Piercesare
    Politecnico di Milano, Italy.
    Sjöstedt de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vantini, Simone
    Politecnico di Milano, Italy.
    Vitelli, Valeria
    Oslo University, Norway.
    Clustering misaligned dependent curves applied to varved lake sediment for climate reconstruction2017In: Stochastic environmental research and risk assessment (Print), ISSN 1436-3240, E-ISSN 1436-3259, Vol. 31, no 1, p. 71-85Article in journal (Refereed)
    Abstract [en]

    In this paper we introduce a novel functional clustering method, the Bagging Voronoi K-Medoid Aligment (BVKMA) algorithm, which simultaneously clusters and aligns spatially dependent curves. It is a nonparametric statistical method that does not rely on distributional or dependency structure assumptions. The method is motivated by and applied to varved (annually laminated) sediment data from lake Kassjön in northern Sweden, aiming to infer on past environmental and climate changes. The resulting clusters and their time dynamics show great potential for seasonal climate interpretation, in particular for winter climate changes.

  • 11. Chen, Zhi-Qiang
    et al.
    Abramowicz, Konrad
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Raczkowski, Rafal
    Ganea, Stefana
    Wu, Harry X.
    Lundqvist, Sven-Olof
    Mörling, Tommy
    Sjöstedt de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Gil, Maria Rosario Garcia
    Mellerowicz, Ewa J.
    Method for accurate fiber length determination from increment cores for large-scale population analyses in Norway spruce2016In: Holzforschung, ISSN 0018-3830, E-ISSN 1437-434X, Vol. 70, no 9, p. 829-838Article in journal (Refereed)
    Abstract [en]

    Fiber (tracheid) length is an important trait targeted for genetic and silvicultural improvement. Such studies require large-scale non-destructive sampling, and accurate length determination. The standard procedure for non-destructive sampling is to collect increment cores, singularize their cells by maceration, measure them with optical analyzer and apply various corrections to suppress influence of non-fiber particles and cut fibers, as fibers are cut by the corer. The recently developed expectation-maximization method (EM) not only addresses the problem of non-fibers and cut fibers, but also corrects for the sampling bias. Here, the performance of the EM method has been evaluated by comparing it with length-weighing and squared length-weighing, both implemented in fiber analyzers, and with microscopy data for intact fibers, corrected for sampling bias, as the reference. This was done for 12-mm increment cores from 16 Norway spruce (Picea abies (L.) Karst) trees on fibers from rings 8-11 (counted from pith), representing juvenile wood of interest in breeding programs. The EM-estimates provided mean-fiber-lengths with bias of only +2.7% and low scatter. Length-weighing and length2-weighing gave biases of -7.3% and +9.3%, respectively, and larger scatter. The suggested EM approach constitutes a more accurate non-destructive method for fiber length (FL) determination, expected to be applicable also to other conifers.

  • 12. Hébert-Losier, Kim
    et al.
    Pini, Alessia
    Vantini, Simone
    Strandberg, Johan
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Abramowicz, Konrad
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Schelin, Lina
    Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy.
    Häger, Charlotte
    Umeå University, Faculty of Medicine, Department of Community Medicine and Rehabilitation, Physiotherapy.
    One-leg hop kinematics 20years following anterior cruciate ligament rupture: Data revisited using functional data analysis2015In: Clinical Biomechanics, ISSN 0268-0033, E-ISSN 1879-1271, Vol. 30, no 10, p. 1153-1161Article in journal (Refereed)
    Abstract [en]

    BACKGROUND: Despite interventions, anterior cruciate ligament ruptures can cause long-term deficits. To assist in identifying and treating deficiencies, 3D-motion analysis is used for objectivizing data. Conventional statistics are commonly employed to analyze kinematics, reducing continuous data series to discrete variables. Conversely, functional data analysis considers the entire data series.

    METHODS: Here, we employ functional data analysis to examine and compare the entire time-domain of knee-kinematic curves from one-leg hops between and within three groups. All subjects (n=95) were part of a long-term follow-up study involving anterior cruciate ligament ruptures treated ~20years ago conservatively with physiotherapy only or with reconstructive surgery and physiotherapy, and matched knee-healthy controls.

    FINDINGS: Between-group differences (injured leg, treated groups; non-dominant leg, controls) were identified during the take-off and landing phases, and in the sagittal (flexion/extension) rather than coronal (abduction/adduction) and transverse (internal/external) planes. Overall, surgical and control groups demonstrated comparable knee-kinematic curves. However, compared to controls, the physiotherapy-only group exhibited less flexion during the take-off (0-55% of the normalized phase) and landing (44-73%) phase. Between-leg differences were absent in controls and the surgically treated group, but observed during the flight (4-22%, injured leg>flexion) and the landing (57-85%, injured leg<internal rotation) phases in the physiotherapy-only group.

    INTERPRETATION: Functional data analysis identified specific functional knee-joint deviations from controls persisting 20years post anterior cruciate ligament rupture, especially when treated conservatively. This approach is suggested as a means for comprehensively analyzing complex movements, adding to previous analyses.

  • 13.
    Rani, Raffaele
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Abramowicz, Konrad
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Falster, Daniel S.
    Sterck, Frank
    Brännström, Åke
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Evolution and Ecology Program, International Institute for Applied Systems Analysis, Laxenburg, Austria.
    Effects of bud-flushing strategies on tree growth2018In: Tree Physiology, ISSN 0829-318X, E-ISSN 1758-4469, Vol. 38, no 9, p. 1384-1393Article in journal (Refereed)
    Abstract [en]

    Allocation of carbohydrates between competing organs is fundamental to plant development, growth and productivity. Carbohydrates are synthesized in mature leaves and distributed via the phloem vasculature to developing buds where they are consumed to produce new biomass. The distribution and mass-allocation processes within the plant remain poorly understood and may involve complex feedbacks between different plant functions, with implications for the emergent structure of the plant. Here, we investigate how the order in which dormant buds are flushed affects the development of tree size and reproductive output during the first 20 years of growth in full light and shaded canopy environments. We report the following findings: (i) Bud-flushing strategies strongly affect the temporal dynamics of height, mass and the size of reproduction pool, as well as the resulting architectures. (ii) Bud-flushing strategies affect tree growth by altering the rate of growth and final size of trees. (iii) No single bud-flushing strategy performs best when both the size and allocation for reproduction of the resulting trees are compared. However, we observe that the strategy that optimizes the net carbon gain for the entire tree architecture always results in a high reproduction output. (iv) Branch turnover and meristem regeneration enhance the performance of certain strategies with respect to the measured quantities. These results highlight the importance of employing generic models of architecture (i.e., non-species-specific) to identify general mechanisms of carbon allocation and the spatial distribution of newly formed biomass in growing trees.

  • 14. Wenk, Elizabeth Hedi
    et al.
    Abramowicz, Konrad
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Westoby, Mark
    Falster, Daniel S.
    Investment in reproduction for 14 iteroparous perennials is large and associated with other life-history and functional traits2018In: Journal of Ecology, ISSN 0022-0477, E-ISSN 1365-2745, Vol. 106, no 4, p. 1338-1348Article in journal (Refereed)
    Abstract [en]

    1. While theoretical models predict reproductive allocation (RA) should approach 100% of available energy as a plant ages, available empirical data suggest much lower RA values in perennial plants. In this study, we have two aims. First, we assess whether the discrepancy between theory and data arises from methodological differences in how growth and RA are calculated. Specifically, we hypothesize RA in older plants is large when compared to growth in leaf area, that is, after excluding turnover of stem and leaf tissues. Second, we hypothesize that species with cheap tissues or those that are shorter reach RA = 0.5 at a younger age.

    2. We measured investment in leaf, stem and reproduction on individuals from 14 co-occurring woody perennial iteroparous species. A fire chronosequence allowed us to use a space-for-time substitution to estimate RA schedules for each species, simultaneously measuring reproductive and vegetative production on individuals differing in age.

    3. For most (11 of 14) species, we found RA eventually reached 100% of available energy, with another two species reaching at least 80%. Increases in RA were associated with a decline in growth of leaf area. Comparing species, we found that species with cheap leaves reached RA = 0.5 sooner (they could be called fast-living), whereas delayed maturation and slower increases in RA were associated with greater maximum height.

    4. Synthesis. Explicitly accounting for the cost of leaf replacement leads to the high estimates of reproductive allocation in perennial plants predicted by theoretical models, limiting or even halting leaf area expansion. For some species, so much energy is allocated to reproduction that leaf area declines year-upon-year for multiple growing seasons preceding death. Connecting lifetime reproductive allocation schedules with leaf area expansion, leaf life span, and plant maximum height demonstrates how reproductive allocation schedules synthesize a plant's life-history strategy, making them a valuable tool for connecting plant traits and demography.

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