| Field |
Value |
Language |
|
dc.contributor.author |
Colbrook, MJ |
|
|
dc.contributor.author |
Botev, ZI |
|
|
dc.contributor.author |
Kuritz, K |
|
|
dc.contributor.author |
MacNamara, S |
|
|
dc.date.accessioned |
2020-12-17T01:45:30Z |
|
|
dc.date.available |
2020-12-17T01:45:30Z |
|
|
dc.date.issued |
2020-10-01 |
|
|
dc.identifier.citation |
Studies in Applied Mathematics, 2020, 145, (3), pp. 357-396 |
|
|
dc.identifier.issn |
0022-2526 |
|
|
dc.identifier.issn |
1467-9590 |
|
|
dc.identifier.uri |
http://hdl.handle.net/10453/144791
|
|
|
dc.description.abstract |
© 2020 The Authors. Studies in Applied Mathematics published by Wiley Periodicals LLC Kernel density estimation on a finite interval poses an outstanding challenge because of the well-recognized bias at the boundaries of the interval. Motivated by an application in cancer research, we consider a boundary constraint linking the values of the unknown target density function at the boundaries. We provide a kernel density estimator (KDE) that successfully incorporates this linked boundary condition, leading to a non-self-adjoint diffusion process and expansions in nonseparable generalized eigenfunctions. The solution is rigorously analyzed through an integral representation given by the unified transform (or Fokas method). The new KDE possesses many desirable properties, such as consistency, asymptotically negligible bias at the boundaries, and an increased rate of approximation, as measured by the AMISE. We apply our method to the motivating example in biology and provide numerical experiments with synthetic data, including comparisons with state-of-the-art KDEs (which currently cannot handle linked boundary constraints). Results suggest that the new method is fast and accurate. Furthermore, we demonstrate how to build statistical estimators of the boundary conditions satisfied by the target function without a priori knowledge. Our analysis can also be extended to more general boundary conditions that may be encountered in applications. |
|
|
dc.language |
en |
|
|
dc.publisher |
Wiley |
|
|
dc.relation.ispartof |
Studies in Applied Mathematics |
|
|
dc.relation.isbasedon |
10.1111/sapm.12322 |
|
|
dc.rights |
Received: March Revised: June
DOI: ./sapm.
ORIGINAL ARTICLE
Kernel density estimation with linked boundary
conditions
Matthew J. Colbrook
Zdravko I. Botev
Karsten Kuritz
Shev MacNamara
Department of Applied Mathematics
and Mathematical Physics, University of
Cambridge, Cambridge, UK
School of Mathematics and Statistics,
The University of New South Wales,
NSW, Sydney, Australia
Institute for Systems Theory and
Automatic Control, University of
Stuttgart, Stuttgart, Germany
ARC Centre of Excellence for
Mathematical and Statistical Frontiers,
School of Mathematical and Physical
Sciences, University of Technology
Sydney, NSW, Australia
Correspondence
Matthew J. Colbrook,Department of
Applied Mathematics and Mathemat-
ical Physics, University of Cambridge,
Wilberforce Road, Cambridge CB WA,
UK.
Email: m.colbrook@damtp.cam.ac.uk
Funding information
ACEMS, Grant/AwardNumber:
DE; Deutsche Forschungs-
gemeinschaft, Grant/Award Number:
AL/-;Engineering and Physical
Sciences Research Council, Grant/Award
Number: EP/L/
Abstract
Kernel density estimation on a finite interval poses an
outstanding challenge because of the well-recognized
bias at the boundaries of the interval. Motivated by an
application in cancer research, we consider a boundary
constraint linking the values of the unknown target den-
sity function at the boundaries. We provide a kernel den-
sity estimator (KDE) that successfully incorporates this
linked boundary condition, leading to a non-self-adjoint
diffusion process and expansions in nonseparable gen-
eralized eigenfunctions. The solution is rigorously ana-
lyzed through an integral representation given by the
unified transform (or Fokas method). The new KDE pos-
sesses many desirable properties, such as consistency,
asymptotically negligible bias at the boundaries, and an
increased rate of approximation, as measured by the
AMISE. We apply our method to the motivating exam-
ple in biology and provide numerical experiments with
synthetic data, including comparisons with state-of-the-
art KDEs (which currently cannot handle linked bound-
ary constraints). Results suggest that the new method is
fast and accurate. Furthermore, we demonstrate how to
build statistical estimators of the boundary conditions
satisfied by the target function without a priori knowl-
edge. Our analysis can also be extended to more general
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduc-
tion in any medium, provided the original work is properly cited.
© The Authors. Studies in Applied Mathematics published by Wiley Periodicals LLC |
|
|
dc.rights |
info:eu-repo/semantics/openAccess |
|
|
dc.rights |
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduc-
tion in any medium, provided the original work is properly cited.
© The Authors. Studies in Applied Mathematics published by Wiley Periodicals LLC |
|
|
dc.subject |
0102 Applied Mathematics |
|
|
dc.subject.classification |
Mathematical Physics |
|
|
dc.title |
Kernel density estimation with linked boundary conditions |
|
|
dc.type |
Journal Article |
|
|
utslib.citation.volume |
145 |
|
|
utslib.for |
0102 Applied Mathematics |
|
|
pubs.organisational-group |
/University of Technology Sydney/Faculty of Science |
|
|
pubs.organisational-group |
/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences |
|
|
pubs.organisational-group |
/University of Technology Sydney |
|
|
utslib.copyright.status |
open_access |
* |
|
pubs.consider-herdc |
false |
|
|
dc.date.updated |
2020-12-17T01:45:26Z |
|
|
pubs.issue |
3 |
|
|
pubs.publication-status |
Published |
|
|
pubs.volume |
145 |
|
|
utslib.citation.issue |
3 |
|
