| 1. |
- Abramowicz, Konrad, et al.
(författare)
-
An inferential framework for domain selection in functional anova
- 2014
-
Ingår i: Contributions in infinite-dimensional statistics and related topics. - : Esculapio. - 9788874887637
-
Konferensbidrag (refereegranskat)abstract
- 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.
|
|
| 2. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
-
Domain selection and family-wise error rate for functional data : a unified framework
- 2023
-
Ingår i: Biometrics. - : John Wiley & Sons. - 0006-341X .- 1541-0420. ; 79:2, s. 1119-1132
-
Tidskriftsartikel (refereegranskat)abstract
- Functional data are smooth, often continuous, random curves, which can be seen as an extreme case of multivariate data with infinite dimensionality. Just as component-wise inference for multivariate data naturally performs feature selection, subset-wise inference for functional data performs domain selection. In this paper, we present a unified testing framework for domain selection on populations of functional data. In detail, p-values of hypothesis tests performed on point-wise evaluations of functional data are suitably adjusted for providing a control of the family-wise error rate (FWER) over a family of subsets of the domain. We show that several state-of-the-art domain selection methods fit within this framework and differ from each other by the choice of the family over which the control of the FWER is provided. In the existing literature, these families are always defined a priori. In this work, we also propose a novel approach, coined threshold-wise testing, in which the family of subsets is instead built in a data-driven fashion. The method seamlessly generalizes to multidimensional domains in contrast to methods based on a-priori defined families. We provide theoretical results with respect to consistency and control of the FWER for the methods within the unified framework. We illustrate the performance of the methods within the unified framework on simulated and real data examples, and compare their performance with other existing methods.
|
|
| 3. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
-
Multiresolution clustering of dependent functional data with application to climate reconstruction
- 2019
-
Ingår i: Stat. - : John Wiley & Sons. - 2049-1573. ; 8:1
-
Tidskriftsartikel (refereegranskat)abstract
- We propose a new nonparametric clustering method for dependent functional data, the double clustering bagging Voronoi method. It consists of two levels of clustering. Given a spatial lattice of points, a function is observed at each grid point. In the first‐level clustering, features of the functional data are clustered. The second‐level clustering takes dependence into account, by grouping local representatives, built from the resulting first‐level clusters, using a bagging Voronoi strategy. Depending on the distance measure used, features of the functions may be included in the second‐step clustering, making the method flexible and general. Combined with the clustering method, a multiresolution approach is proposed that searches for stable clusters at different spatial scales, aiming to capture latent structures. This provides a powerful and computationally efficient tool to cluster dependent functional data at different spatial scales, here illustrated by a simulation study. The introduced methodology is applied to varved lake sediment data, aiming to reconstruct winter climate regimes in northern Sweden at different time resolutions over the past 6,000 years.
|
|
| 4. |
- Abramowicz, Konrad, et al.
(författare)
-
Multivariate piecewise linear interpolation of a random field
- 2011
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
|
|
| 5. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
-
Nonparametric bagging clustering methods to identify latent structures from a sequence of dependent categorical data
- 2023
-
Ingår i: Computational Statistics & Data Analysis. - : Elsevier. - 0167-9473 .- 1872-7352. ; 177
-
Tidskriftsartikel (refereegranskat)abstract
- Nonparametric bagging clustering methods are studied and compared to identify latent structures from a sequence of dependent categorical data observed along a one-dimensional (discrete) time domain. The frequency of the observed categories is assumed to be generated by a (slowly varying) latent signal, according to latent state-specific probability distributions. The bagging clustering methods use random tessellations (partitions) of the time domain and clustering of the category frequencies of the observed data in the tessellation cells to recover the latent signal, within a bagging framework. New and existing ways of generating the tessellations and clustering are discussed and combined into different bagging clustering methods. Edge tessellations and adaptive tessellations are the new proposed ways of forming partitions. Composite methods are also introduced, that are using (automated) decision rules based on entropy measures to choose among the proposed bagging clustering methods. The performance of all the methods is compared in a simulation study. From the simulation study it can be concluded that local and global entropy measures are powerful tools in improving the recovery of the latent signal, both via the adaptive tessellation strategies (local entropy) and in designing composite methods (global entropy). The composite methods are robust and overall improve performance, in particular the composite method using adaptive (edge) tessellations.
|
|
| 6. |
|
|
| 7. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
-
Nonparametric inference for functional-on-scalar linear models applied to knee kinematic hop data after injury of the anterior cruciate ligament
- 2018
-
Ingår i: Scandinavian Journal of Statistics. - : John Wiley & Sons. - 0303-6898 .- 1467-9469. ; 45:4, s. 1036-1061
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 8. |
- Abramowicz, Konrad, 1983-
(författare)
-
Numerical analysis for random processes and fields and related design problems
- 2011
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
|
|
| 9. |
- Abramowicz, Konrad, 1983-, et al.
(författare)
-
On the error of the Monte Carlo pricing method for Asian option
- 2008
-
Ingår i: Journal of Numerical and Applied Mathematics. - 0868-6912. ; 96:1, s. 1-10
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 10. |
- Abramowicz, Konrad, et al.
(författare)
-
Piecewise multilinear interpolation of a random field
- 2013
-
Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 45:4, s. 945-959
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
