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
(2025).
Quantum probability for statisticians; some new ideas.
Methodology and Computing in Applied Probability.
ISSN 1387-5841.
27.
doi:
10.1007/s11009-025-10214-1.
Full text in Research Archive
Show summary
It is argued from several points of view that quantum probabilities might play a role in statistical settings. New approaches toward quantum foundations have postulates that appear to be equally valid in macroscopic settings. One such approach is described here in detail, while one other is briefly sketched. In particular, arguments behind the Born rule, which gives the basis for quantum probabilities, are given. A list of ideas for possible statistical applications of quantum probabilities is provided and discussed. A particular area is machine learning, where there exists substantial literature on links to quantum probability. Here, an idea about model reduction is sketched and is motivated from a quantum probability model. Quantum models can play a role in model reduction, where the partial least squares regression model is a special case. It is shown that for certain experiments, a Bayesian prior given by a quantum probability can be motivated.
-
Helland, Inge Svein
(2025).
Some mathematical issues regarding a new approach towards quantum foundations.
Journal of Mathematical Physics.
ISSN 0022-2488.
66(9).
doi:
10.1063/5.0251682.
Full text in Research Archive
Show summary
In this article, the weakest possible theorem providing a foundation for the Hilbert space formalism of quantum theory is stated. The necessary postulates are formulated, and the mathematics is spelled out in detail. It is argued that, from this approach, a general epistemic interpretation of quantum mechanics is natural. Some applications to the Bell experiment and to decision theory are briefly discussed. The article represents the conclusion of a series of articles and books on quantum foundations.
-
Helland, Inge Svein
(2025).
On optimal linear prediction.
Scandinavian Journal of Statistics.
ISSN 0303-6898.
53(1),
p. 16–32.
doi:
10.1111/sjos.70006.
Full text in Research Archive
Show summary
The main purpose of this article is to show that, under certain assumptions in a linear prediction setting, near optimal methods based upon model reduction can be provided. The optimality is formulated in terms of the expected mean square prediction error. The optimal model reduction turns out, under a certain assumption, to correspond to the statistical model for partial least squares (PLS) regression discussed by the author elsewhere, and under a certain specific condition, a PLS-like predictor is proved to be good compared to other predictors. It is also proved in this article that the situation with two different model reductions can be fit into a quantum mechanical setting. Thus, the article contains a synthesis of three cultures: mathematical statistics as a basis, algorithms introduced by chemometricians and used very much by applied scientists as a background, and finally, notions from quantum foundation as an alternative point of view.
-
-
Helland, Inge Svein
(2024).
A new approach towards quantum foundation and some consequences.
Academia Quantum.
1(1).
doi:
10.20935/AcadQuant7282.
Full text in Research Archive
Show summary
A general theory based on six postulates is introduced. The basic notions are theoretical variables that are associated with an observer or with a group of communicating observers. These variables may be accessible or inaccessible. From these postulates, the ordinary formalism of quantum theory is derived. The mathematical derivations are not given in this article, but I refer to the recent articles. Three possible applications of the general theory can be given as follows: (1) the variables may be decision variables connected to the decisions of a person or a group of persons, (2) the variables may be statistical parameters or future data, and (3) most importantly, the variables are physical variables in some context. The last application gives a completely new foundation of quantum mechanics, a foundation which in my opinion is much easier to understand than ordinary formalism. So-called paradoxes like that of Schrödinger’s cat can be clarified under the theory. Explanations of the outcomes of David Bohm’s version of the EPR (Einstein–Podolsky–Rosen) experiment and the Bell experiment are provided. Finally, references to links toward relativity theory and quantum field theory are given. The concluding remarks point to further possible developments.
-
Helland, Inge Svein
(2024).
On probabilities in quantum mechanics.
APL Quantum.
1(3).
doi:
10.1063/5.0218982.
Full text in Research Archive
Show summary
This is an attempt to clarify certain concepts related to a debate on the interpretation of quantum mechanics, a debate between Khrennikov on the one hand and Stacey and Schack on the other hand. Central to this debate is the notion of quantum probabilities. I first take up the probability concept in the QBist school, as seen from my point of view, and then give my own arguments for the Born formula for calculating quantum probabilities. In that connection, I also sketch some consequences of my approach toward the foundation and interpretation of quantum theory. I discuss my general views on QBism as a possible alternative interpretation of quantum mechanics before I give some final remarks.
-
Helland, Inge Svein
(2023).
A simple quantum model linked to decisions.
Foundations of Physics.
ISSN 0015-9018.
53(1).
doi:
10.1007/s10701-022-00658-7.
Full text in Research Archive
Show summary
This article may be seen as a summary and a final discussion of the work that the author has done in recent years on the foundation of quantum theory. It is shown that quantum mechanics as a model follows under certain specific conditions from a quite different, much simpler model. This model is connected to the mind of an observer, or to the joint minds of a group of communicating observers. The model is based upon conceptual variables, and an important aspect is that an observer (a group of observers) must decide on which variable to measure. The model is then linked more generally to a theory of decisions. The results are discussed from several angles.
-
Helland, Inge Svein
(2023).
An alternative foundation of quantum theory.
Foundations of Physics.
ISSN 0015-9018.
54(1).
doi:
10.1007/s10701-023-00735-5.
Full text in Research Archive
Show summary
A new approach to quantum theory is proposed in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the discrete case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is shown here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. This simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. The interpretation inferred from the proposed foundation here may be called a general epistemic interpretation of quantum theory. A special case of this interpretation is QBism; it also has a relationship to several other interpretations.
-
Helland, Inge Svein
(2022).
On reconstructing parts of quantum theory from two related maximal conceptual variables.
International journal of theoretical physics.
ISSN 0020-7748.
61.
doi:
10.1007/s10773-022-05047-4.
Full text in Research Archive
Show summary
In the book Helland (2021) the general problem of reconstructing the Hilbert space formulation in quantum theory is discussed from the point of view of what I called conceptual variables, any variables defined by a person or by a group of persons. These variables may be inaccessible, i.e., impossible to assign numerical value to by experiments or by measurements, or accessible. One basic assumption in Helland (2021) and here is that group actions g ∈ G are defined on a space where some maximally accessible variable varies, and then accessible functions of these maximal variables are introduced. By using group representation theory the basic Hilbert space formalism is restored under the assumption that the observator or the set of observators has two related maximally accessible variables in his (their) mind(s). The notion of relationship is precisely defined here. Symmetric (self-adjoint) operators are connected to each variable, and in the discrete case the possible values of the variables are given by the eigenvalues of the operators. In this paper the main results from Helland (2021) are made more precise and more general. It turns out that the conditions of the main theorem there can be weakened in two essential ways: 1) No measurements need to be assumed, so the result is also applicable to general decision situations; 2) States can have arbitrary phase factors. Some consequences of this approach towards quantum theory are also discussed here.
-
Helland, Inge Svein
(2022).
The Bell Experiment and the Limitations of Actors.
Foundations of Physics.
ISSN 0015-9018.
52(3).
doi:
10.1007/s10701-022-00572-y.
Full text in Research Archive
Show summary
The well known Bell experiment with two actors Alice and Bob is considered. First the simple deduction leading to the CHSH inequality under local realism is reviewed, and some arguments from the literature are recapitulated. Then I take up certain background themes before I enter a discussion of Alice’s analysis of the situation. An important point is that her mind is limited by the fact that her Hilbert space in this context is two-dimensional. General statements about a mind’s limitation during a decision process are derived from recent results on the reconstruction of quantum theory from conceptual variables. These results apply to any decision situation. Let all the data from the Bell experiment be handed over to a new actor Charlie, who performs a data analysis. But his mind is also limited: He has a four-dimensional Hilbert space in the context determined by the experiment. I show that this implies that neither Alice nor Charlie can have the argument leading to the CHSH inequality as a background for making decisions related to the experiment. Charlie may be any data analyst, and he may communicate with any person. It is argued that no rational person can be convinced by the CHSH argument when making empirical decisions on the Bell situation.
-
Helland, Inge Svein
(2022).
On the diversity and similarity of mathematical models in science.
American Review of Mathematics and Statistics.
ISSN 2374-2348.
10(1),
p. 1–16.
doi:
10.15640/arms.v10n1a1.
Full text in Research Archive
Show summary
In this article, the notion of a mathematical model in science is attempted to be enlightened from several points of view. It is shown that mathematical models are introduced differently and used differently in different areas of science. In the present article the use of models in statistics is taken as a basis, but links are given to several other areas. Two such links are described in some detail here: First the link connected to the chemometricians’ Partial Least Squares algorithm; a link that now has been generalized to the more sophisticated envelope model. Secondly, statistics, as is well known, relies heavily on making decisions, and it may also in certain cases be connected to a process of taking measurements. A mathematical model for these two activities, connected to the mind of an observer, is introduced. This model is taken further and is shown to be important in a new proposal for a foundation of quantum theory. Quantum mechanics, as seen from this point of view, is described in some detail. The discussion here is close to the discussion in a recent book and in a recently published article by the author in a recognized physical journal.
-
Helland, Inge Svein
(2019).
Symmetry in a space of conceptual variables.
Journal of Mathematical Physics.
ISSN 0022-2488.
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 0886-9383.
32(9).
doi:
10.1002/cem.3044.
Full text in Research Archive
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.
-
Helland, Inge Svein
(2015).
The quantum formulation derived from assumptions of epistemic processes.
Journal of Physics: Conference Series (JPCS).
ISSN 1742-6588.
597(1).
doi:
10.1088/1742-6596/597/1/012041.
Full text in Research Archive
Show summary
Motivated by Quantum Bayesianism I give background for a general epistemic approach to quantum mechanics, where complementarity and symmetry are the only essential features. A general definition of a symmetric epistemic setting is introduced, and for this setting the basic Hilbert space formalism is arrived at under certain technical assumptions. Other aspects of ordinary quantum mechanics will be developed from the same basis elsewhere.
-
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 0169-7439.
146,
p. 128–135.
doi:
10.1016/j.chemolab.2015.05.012.
Full text in Research Archive
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 re-parametrization 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 input-parameters 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 R-package 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 B (Statistical Methodology).
ISSN 1369-7412.
75(5),
p. 851–877.
doi:
10.1111/rssb.12018.
Full text in Research Archive
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 likelihood-based 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 0303-6898.
39(4),
p. 695–713.
doi:
10.1111/j.1467-9469.2011.00770.x.
Full text in Research Archive
Show summary
The random x regression model is approached through the group of rotations of the eigenvectors for the x-covariance 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 non-defined posterior. Nevertheless we can devise an approximate scale group with a proper invariant prior leading to a well-defined 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 1-6, Benidorm, Spain.
Oxford University Press.
ISSN 9780199214655.
Full text in Research Archive
-
Helland, Inge Svein
(2006).
Extended statistical modeling under symmetry; The link toward quantum mechanics.
Annals of Statistics.
ISSN 0090-5364.
34,
p. 42–77.
Full text in Research Archive
-
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 9789545801952.
p. 721–732.
Full text in Research Archive
-
-
Helland, Inge Svein
(2004).
Statistical inference under a fixed symmetry group.
International Statistical Review.
ISSN 0306-7734.
72,
p. 409–422.
Full text in Research Archive
-
Helland, Inge Svein
(2003).
Rotational symmetry, model reduction and optimality of prediction from the PLS population model.
Chemometrics and Intelligent Laboratory Systems.
ISSN 0169-7439.
68,
p. 53–60.
Full text in Research Archive
-
Helland, Inge Svein
(2002).
Discussion of: What is a statistical model?
Annals of Statistics.
ISSN 0090-5364.
30,
p. 1286–1289.
Full text in Research Archive
-
Helland, Inge Svein
(2001).
Reduction of regression models under symmetry.
In Viana, M. & Rickards, D. (Ed.),
Algebraic Methods in Statistics.
American Mathematical Society (AMS).
Full text in Research Archive
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 high-dimensional. 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
(2001).
Some theoretical aspects of partial least squares regression.
Chemometrics and Intelligent Laboratory Systems.
ISSN 0169-7439.
58,
p. 97–107.
Full text in Research Archive
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 0020-7748.
38(7),
p. 1851–1881.
Full text in Research Archive
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
1-1 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.
View all works in NVA
-
Helland, Inge Svein
(2023).
Correction to: On reconstructing parts of quantum theory from two related maximal conceptual variables. (Intern. J. Theor. Phys. 61, 69, 2022.) (International Journal of Theoretical Physics, (2022), 61, 3, (69), 10.1007/s10773-022-05047-4).
International journal of theoretical physics.
ISSN 0020-7748.
62(3).
doi:
10.1007/s10773-023-05321-z.
Full text in Research Archive
-
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 0022-2488.
61(1).
doi:
10.1063/1.5127336.
Full text in Research Archive
-
Helland, Inge Svein
(2010).
ENVELOPE MODELS FOR PARSIMONIOUS AND EFFICIENT MULTIVARIATE LINEAR REGRESSION COMMENT.
Statistica sinica.
ISSN 1017-0405.
20(3),
p. 978–981.
Full text in Research Archive
-
-
-
-
-
-
-
-
-
Helland, Inge Svein
(2003).
Partial least squares regression.
In Kotz, Samuel (Eds.),
Encyclopedia of Statistical Sciences 2.
Full text in Research Archive
Show summary
Partial least squares regression was introduced as an algorithm in
the early 1980's, and it has gained much popularity in chemometrics.
PLSR - or PLSR1 - is a regression method for collinear data, and can
be seen as a competitor to principal component regression. The method
makes it possible to combine prediction with a study of a joint latent
structure in the $x, y$-variables. PLSR2 is a generalization to
several dependent variables. PLSR1 can be understood from a
statistical point of view via its parameter algorithm and the
corresponding population model.
-
Helland, Inge Svein
(2002).
Quantum mechanics from statistical parameter models.
Full text in Research Archive
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).
Rotational symmetry, model reduction and optimality of prediction from the PLS population model.
Full text in Research Archive
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 0277-6715.
18.
Full text in Research Archive
-
Helland, Inge Svein
(2004).
Quantum Theory as a Statistical Theory under Symmetry and Complementarity.
Universitetet i Oslo.
Full text in Research Archive
Show summary
The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantum mechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability/quantum inference theory in the simple case. Planck's constant, and the Schrödinger equation are also derived from this conceptual framework. The theory is illustrated by one and by two spin 1/2 particles; in particular, a statistical discussion of Bell's inequality is given.
-
Helland, Inge Svein
(2003).
Statistical inference under symmetry.
Universitetet i Oslo.
Full text in Research Archive
Show summary
We explore the consequences of adjoining a symmetry group to a statistical model. Group actions are first induced on the sample space, and then on the parameter space. It is argued that the right invariant measure induced by the group on the parameter space is a natural non-informative prior for the parameters of the model. The permissible sub-parameters are introduced, i.e., the subparameters upon which group actions can be defined. Equivariant estimators are similarly defined. Orbits of the group are defined on the sample space and on the parameter space; in particular the group action is called transitive when there is only one orbit. Credibility sets and confidence sets are shown (under right invariant prior and assuming transitivity on the parameter space) to be equal when defined by permissible sub-parameters and constructed from equivariant estimators. The effect of different choices of transformation group is illustrated by examples, and properties of the the orbits on the sample space and on the parameter space are discussed. It is argued that model reduction should be constrained to one or several orbits of the group. Using this and other natural criteria and concepts, among them concepts related to design of experiments under symmetry, leads to links towards chemometrical prediction methods and towards the foundation of quantum theory.
-
-
-
Helland, Inge Svein
(2000).
Some theoretical aspects of partial least squares regression.
Universitetet i Oslo.
Full text in Research Archive
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.
Universitetet i Oslo.
Full text in Research Archive
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 high-dimensional. 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 Statistical Parameter Space.
Universitetet i Oslo.
ISSN 9788255311898.
Full text in Research Archive
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 group-theoretical 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.
Universitetet i Oslo.
Full text in Research Archive
Show summary
If a natural non-transitive 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.
Universitetet i Oslo.
Full text in Research Archive
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.
-
-
-
-
-
-
-
View all works in NVA
Published
Nov. 30, 2010 11:20 PM
- Last modified
Sep. 20, 2022 12:04 PM
