Academic Interests

The statistical interpretation of quantum theory.

Regression methods and classification methods for cases with a large number of explanatory variables (classification variables).

The use of symmetry considerations in statistics.

The choice of conditioning in statistical models.
Higher education and employment history
Curriculum
Tags:
Statistics,
Statistics and biostatistics
Publications

Helland, Inge Svein
(2022).
On the diversity and similarity of mathematical models in science.
American Review of Mathematics and Statistics.
ISSN 23742348.
10(1),
p. 1–16.
Full text in Research Archive



Helland, Inge Svein
(2019).
Symmetry in a space of conceptual variables.
Journal of Mathematical Physics.
ISSN 00222488.
60(5),
p. 1–8.
doi:
10.1063/1.5082694.
Full text in Research Archive
Show summary
A conceptual variable is any variable defined by a person or by a group of persons. Such variables may be inaccessible,
meaning that they cannot be measured with arbitrary accuracy on the physical system under consideration at any given
time. An example may be the spin vector of a particle; another example may be the vector (position, momentum). In this
paper, a space of inaccessible conceptual variables is defined, and group actions are defined on this space. Accessible
functions are then defined on the same space. Assuming this structure, the basic Hilbert space structure of quantum
theory is derived: Operators on a Hilbert space corresponding to the accessible variables are introduced; when these
operators have a discrete spectrum, a natural model reduction implies a new model in which the values of the accessible
variables are the eigenvalues of the operator. The principle behind this model reduction demands that a group action
may also be defined also on the accessible variables; this is possible if the corresponding functions are permissible, a
term that is precisely defined. The following recent principle from statistics is assumed: every model reduction should
be to an orbit or to a set of orbits of the group. From this derivation, a new interpretation of quantum theory is briefly
discussed: I argue that a state vector may be interpreted as connected to a focused question posed to nature together
with a definite answer to this question. Further discussion of these topics is provided in a recent book published by the
author of this paper.

Helland, Inge Svein; Sæbø, Solve; Almøy, Trygve & Rimal, Raju
(2018).
Model and estimators for partial least squares regression.
Journal of Chemometrics.
ISSN 08869383.
32(9).
doi:
10.1002/cem.3044.
Show summary
Partial least squares (PLS) regression has been a very popular method for prediction. The method can in a natural way be connected to a statistical model, which now has been extended and further developed in terms of an envelope model. Concentrating on the univariate case, several estimators of the regression vector in this model are defined, including the ordinary PLS estimator, the maximum likelihood envelope estimator, and a recently proposed Bayes PLS estimator. These are compared with respect to prediction error by systematic simulations. The simulations indicate that Bayes PLS performs well compared with the other methods.


Sæbø, Solve; Almøy, Trygve & Helland, Inge Svein
(2015).
simrel  A versatile tool for linear model data simulation based on the concept of a relevant subspace and relevant predictors.
Chemometrics and Intelligent Laboratory Systems.
ISSN 01697439.
146,
p. 128–135.
doi:
10.1016/j.chemolab.2015.05.012.
Show summary
In the field of chemometrics and other areas of data analysis the development of new methods for statistical inference and prediction is the focus of many studies. The requirement to document the properties of new methods is inevitable, and often simulated data are used for this purpose. However, when it comes to simulating data there are few standard approaches. In this paper we propose a very transparent and versatile method for simulating response and predictor data from a multiple linear regression model which hopefully may serve as a
standard tool simulating linear model data. The approach uses the principle of a relevant subspace for prediction, which is known both from Partial Least Squares and envelope models, and is essentially based on a reparametrization of the random x regression model. The approach also allows for defining a subset of relevant observable predictor variables spanning the relevant latent subspace, which is handy for exploring methods for variable selection. The data properties are defined by a small set of inputparameters defined by the analyst. The versatile approach can be used to simulate a great variety of data with varying properties in order to compare statistical methods. The method has been implemented in an Rpackage and its use is illustrated by examples.

Cook, R. Dennis; Helland, Inge Svein & Su, Z.
(2013).
Envelopes and partial least squares regression.
Journal of The Royal Statistical Society Series Bstatistical Methodology.
ISSN 13697412.
75(5),
p. 851–877.
doi:
10.1111/rssb.12018.
Show summary
We build connections between envelopes, a recently proposed context for efficient estimation in multivariate statistics, and multivariate partial least squares (PLS) regression. In particular we establish an envelope as the nucleus of both univariate and multivariate PLS, which opens the door to pursuing the same goal as PLS but using different envelope estimators. It is argued that a likelihoodbased estimator is less sensitive to the number of PLS components selected and that it outperforms PLS in prediction and estimation.


Helland, Inge Svein; Sæbø, Solve & Tjelmeland, Håkon
(2012).
Near optimal prediction from relevant components.
Scandinavian Journal of Statistics.
ISSN 03036898.
39(4),
p. 695–713.
doi:
10.1111/j.14679469.2011.00770.x.
Show summary
The random x regression model is approached through the group of rotations of the eigenvectors for the xcovariance matrix together with scale transformations for each of the corresponding regression coefficients. The partial least squares model can be constructed from the orbits of this group. A generalization of Pitman’s Theorem says that the best equivariant estimator under a group is given by the Bayes estimator with the group’s invariant measure as the prior. A straightforward application of this theorem turns out to be impossible since the relevant invariant prior leads to a nondefined posterior. Nevertheless we can devise an approximate scale group with a proper invariant prior leading to a welldefined posterior distribution with a finite mean. This Bayes estimator is explored using Markov chain Monte Carlo technique. The estimator seems to require
heavy computations, but can be argued to have several nice properties. It is also a valid estimator when p>n.



Helland, Inge Svein
(2007).
Discussion of: Bayesian probability in quantum mechanics, by Rüdiger Schack.
In Bernardo, J.M. (Eds.),
Bayesian Statistics 8: Proceedings of the Eighth Valencia International Meeting, June 16, Benidorm, Spain.
Oxford University Press.
ISSN 9780199214655.

Helland, Inge Svein
(2006).
Towards quantum mechanics from a theory of experiments.
In Dobrev, V.K. (Eds.),
Quantum Theory and Symmetries IV.
Heron Press.
ISSN 9545801956.
p. 721–732.

Helland, Inge Svein
(2006).
Extended statistical modeling under symmetry; The link toward quantum mechanics.
Annals of Statistics.
ISSN 00905364.
34,
p. 42–77.


Helland, Inge Svein
(2004).
Statistical inference under a fixed symmetry group.
International Statistical Review.
ISSN 03067734.
72,
p. 409–422.


Helland, Inge Svein
(2001).
Reduction of regression models under symmetry.
In Viana, M. & Rickards, D. (Ed.),
Algebraic Methods in Statistics.
American Mathematical Society (AMS).

Helland, Inge Svein
(2001).
Some theoretical aspects of partial least squares regression.
Chemometrics and Intelligent Laboratory Systems.
ISSN 01697439.
58,
p. 97–107.
Show summary
We give a survey of partial least squares regression with one $y$variable from a theoretical point of view. Some general comments are
made on the motivation as seen by a statistician for this kind of studies, and the concept of soft modelling is criticized from the same
angle. Various aspects of the PLS algorithm are considered, and the population PLS model is defined. Asymptotic properties of the
prediction error are briefly discussed, and the relation to other regression methods are commented upon. Results indicating positive and negative properties of PLSR are mentioned, in particular the recent result of Butler, Denham and others which seem to show that PLSR can not be an optimal regression method in any reasonable way. The only possible path left towards some kind of optimality, seems then to be through first trying to find a good motivation for the population model and then possibly finding an optimal estimator under this model. Some results on this are sketched.

Helland, Inge Svein
(1999).
Quantum mechanics from symmetry and statistical modeling.
International journal of theoretical physics.
ISSN 00207748.
38(7),
p. 1851–1881.
Show summary
A version of quantum theory is derived from a set of plausible assumptions related
to the following general setting: For a given system there is a
set of experiments that can be performed, and for each such
experiment an ordinary statistical model is defined. The parameters
of the single experiments are functions of a hyperparameter, which
defines the state of the system. There is a symmetry group acting on the
hyperparameters, and for the induced action on the parameters of the
single experiment a simple consistency property is assumed, called
permissibility of the parametric function. The other assumptions
needed are rather weak. The derivation relies
partly on quantum logic, partly on a group representation of the
hyperparameter group, where the invariant spaces are shown to be in
11 correspondence with the equivalence classes of permissible
parametric functions. Planck's constant only plays a r\^{o}le
connected to generators of unitary group representations.
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Helland, Inge Svein
(2018).
Epistemic Processes. A Basis for Statistics and Quantum Theory.
Springer Nature.
ISBN 9783319950679.
170 p.

Helland, Inge Svein
(2010).
Steps Towards a Unified Basis for Scientific Models and Methods.
World Scientific.
ISBN 9789814280853.
275 p.
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Helland, Inge Svein
(2020).
Erratum: Symmetry in a space of conceptual variables (Journal of Mathematical Physics (2019) 60:5 (052101) DOI : 10.1063/1.5082694).
Journal of Mathematical Physics.
ISSN 00222488.
61(1).
doi:
10.1063/1.5127336.

Helland, Inge Svein
(2010).
ENVELOPE MODELS FOR PARSIMONIOUS AND EFFICIENT MULTIVARIATE LINEAR REGRESSION COMMENT.
Statistica sinica.
ISSN 10170405.
20(3),
p. 978–981.

Helland, Inge Svein
(2009).
Partial least squares and mathematical statistics.

Helland, Inge Svein
(2009).
Steps towards a unified basis.

Helland, Inge Svein
(2009).
On partial least squares regression.

Helland, Inge Svein
(2007).
PLS from model reduction under symmetry.

Helland, Inge Svein
(2006).
Discussion of Ruediger Schack: Bayesian probability in quantum mechanics.

Helland, Inge Svein
(2005).
Quantum mechanics as a statistical theory.

Helland, Inge Svein
(2005).
Towards quantum mechanics from a theory of experiments.

Helland, Inge Svein
(2004).
Quantum mechanics from statistical theory under symmetry.

Helland, Inge Svein
(2003).
Partial least squares regression.
In Kotz, Samuel (Eds.),
Encyclopedia of Statistical Sciences 2.

Helland, Inge Svein
(2002).
Discussion of: What is a statistical model?
Annals of Statistics.
ISSN 00905364.
30,
p. 1286–1289.

Helland, Inge Svein
(2002).
Quantum mechanics from statistical parameter models.
Show summary
The aim of the paper is to derive quantum mechanics from a
parametric structure extending slightly that of mathematical statistics. The
basic setting assumed is an unavailable hyperparameter space
$\Phi$, and subparameters $\theta^{a}(\cdot)$ defined as
functions on $\Phi$. There is a group of transformations $G$ acting on $\Phi$ and
subgroups $G^{a}$ such that $\theta^{a}(\cdot)$ is natural, i.e.,
allowing a group of transformations $\tilde{G}^{a}$ to be induced on its range.
Model reduction is assumed in
general to be via a natural function with respect to $\tilde{G}^{a}$. The possible models are
constrained through the corresponding experimental basis group in a precise
way, which is shown to give the usual Hilbert space formulation. The theory is
illustrated by one and by two particles with spin.
Several paradoxes and related themes of conventional quantum mechanics
are briefly discussed in this setting.

Helland, Inge Svein
(2001).
Quantum probability and statistics.

Helland, Inge Svein
(2001).
Rotational symmetry, model reduction and optimality of prediction from the PLS population model.
Show summary
The PLS model can be defined thrugh a simple restriction in the PLS population algorithm. Since the corresponding restriction does not hold for the sample estimates, sample PLS cannot be optimal in any strict sense. One purpose of this talk is to show that the PLS model itself can be connected to a certain optimality property related to rotational symmetry. Almost all known prediction methods behave as we expect under the group of rotations, so this is a natural group to look at in general. The theoretically best possible prediction methods under rotational symmetry can be defined in principle. Considering the freedom of choice we are left with when doing model reduction, we are led in a natural way to the PLS population model. Prediction methods arising from this are complicated, but can be studied.

Helland, Inge Svein
(1999).
Book Review: Statistical Regression with Measurement Error.
Statistics in Medicine.
ISSN 02776715.
18.

Helland, Inge Svein
(1999).
Experiments, symmetries and quantum mechanics.

Helland, Inge Svein
(2004).
Quantum mechanics from statistical theory under symmetry and complementarity.
Matematisk Institutt, UiO.
ISSN 08063842.
2004(12).

Helland, Inge Svein
(2003).
Extended statistical modelling under symmetry; the link towards quantum mechanics.
Department of Mathematics, University of Oslo.

Helland, Inge Svein
(2000).
Quantum theory from symmetry and reduction of statistical models. The compact case.
Department of Mathematics, UiO.
ISSN 8255312404.
Show summary
.

Helland, Inge Svein
(2000).
Some theoretical aspects of partial least squares regression.
Department of matematics, UiO.
ISSN 08063842.
Show summary
We give a survey of partial least squares regression with one
$y$variable from a theoretical point of view. Some general comments
are made on the motivation as seen by a statistician for this kind of
studies, and the concept of soft modelling is criticized from the same
angle. Various aspects of the PLS algorithm are considered, and the
population PLS model is defined. Asymptotic properties of the
prediction error are briefly discussed, and the relation to other
regression methods are commented upon. Results indicating positive and
negative properties of PLSR are mentioned, in particular the recent
result of Butler, Denham and others which seem to show that PLSR can
not be an optimal regression method in any reasonable way. The only
possible path left towards some kind of optimality, seems then to be
through first trying to find a good motivation for the population model
and then possibly finding an optimal estimator under this model. Some
results on this are sketched.

Helland, Inge Svein
(2000).
Reduction of regression models under symmetry.
Department of Mathematics, UiO.
ISSN 08063842.
Show summary
For collinear data nearly all regression methods that have been
proposed, are equivariant under the rotation group in the $x$space. It
is argued that the regression parameter along orbits of the rotation
group in principle always can be estimated in an optimal way as a
Pitman type estimator. On the other hand it is argued that it may pay
in general to reduce the parameter space of a statistical model when
this space is highdimensional. It follows that any reduction in the
regression model then must take place via the orbit index of the
rotation group. Further information can be found using the form of the
loss function. This is used to discuss the choice of regression model
and thereby the choice of regression method. The solution which seems
to emerge from this, is closely related to the population version of
the chemometricians' partial least squares regression. Estimation under
the reduced model is briefly discussed, as is model reduction in the
corresponding classification problem.

Helland, Inge Svein
(1999).
Quantum theory from symmetries in a general parameter space.
Department of Mathematics, University of Oslo.
ISSN 8255311890.
12.
Show summary
The aim of this paper is to show a connection between an extended
theory of statistical experiments on the one hand and the foundation
of quantum theory on the other hand. The main aspects of this
extension are: One assumes a hyperparameter space $\Phi$ common to several
potential experiments, and a basic symmetry group $G$ associated with
this space. The parameter $\theta_{a}$ of a single experiment, looked
upon as a parametric function $\theta_{a}(\cdot)$ on $\Phi$, is said to
be permissible if $G$ induces in a natural way a new group on the image
space of the function. If this is not the case, it is arranged for by
changing from $G$ to a subgroup $G_{a}$. The Haar measure of this subgroup
(confined to the spectrum; see below) is the prefered prior when the parameter is
unknown. It is assumed that the hyperparameter itself can never be
estimated, only a set of parametric functions. Model reduction is made
by restricting the space of complex `wave' functions, also regarded as
functions on $\Phi$, to an irreducible invariant subspace
$\mathcal{M}$ under $G$. The spectrum of a parametric function is
defined from natural grouptheoretical and statistical considerations.
We establish that a unique operator can be associated with every parametric
functions $\theta_{a}(\cdot)$, and essentially all of the
ordinary quantum theory formalism can be
retrieved from these and a few related assumptions. In particular the
concept of spectrum is consistent. The scope of the theory is illustrated on
the one hand by the example of a spin $1/2$ particle and a related
EPR discussion, on the other hand by a simple macroscopic example.

Helland, Inge Svein
(1999).
Restricted maximum likelihood from symmetry.
Department of Mathematics, University of Oslo.
ISSN 08063842.
14.
Show summary
If a natural nontransitive group is attached to a statistical
model, minimum risk equivariant estimators could be used on
orbits, and for the orbit index, maximum likelihood
estimation from the sample orbit index. This is used to motivate
REML

Helland, Inge Svein
(1999).
Approaching regression methods through symmetry arguments.
Department of Mathematics, University of Oslo.
ISSN 08063842.
15.
Show summary
For collinear data we consider regression methods which are equivariant
under the rotation group in the $x$space; in fact, this seems to
cover nearly all methods that have been proposed. It is argued that
the regression parameter along orbits of the rotation group always
can be selected in an optimal way, so any freedom in the choice
of method should be confined to the orbit index. Via a Pitman type
estimator a first order approximation for the estimated parameter
along orbits is found, and principal component regression, partial
least squares regression and ridge regression appear as the natural
methods under various assumptions. Some light is thrown on the
connection between these methods and on the possibility for
improvement.
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Published Nov. 30, 2010 11:20 PM
 Last modified Sep. 20, 2022 12:04 PM