Inge Svein Helland

• Helland, Inge Svein (2019). Symmetry in a space of conceptual variables. Journal of Mathematical Physics.  ISSN 0022-2488.  60(5), s 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 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.
• 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, s 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 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), s 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 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 (2013). A Basis for Statistical Theory and Quantum Theory, In Paul Bracken (ed.),  Advances in Quantum Mechanics.  IntechOpen.  ISBN 978-953-51-1089-7.  15.  s 335 - 360 Full text in Research Archive.
• Helland, Inge Svein; Sæbø, Solve & Tjelmeland, Håkon (2012). Near optimal prediction from relevant components. Scandinavian Journal of Statistics.  ISSN 0303-6898.  39(4), s 695- 713 . doi: 10.1111/j.1467-9469.2011.00770.x 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 (2009). A foundation of quantum mechanics based on focusing and symmetry. Journal of Physics: Conference Series (JPCS).  ISSN 1742-6588.  174 . doi: 10.1088/1742-6596/174/1/012031
• Helland, Inge Svein (2008). Quantum mechanics from focusing and symmetry. Foundations of Physics.  ISSN 0015-9018.  38, s 818- 842 . doi: 10.1007/s10701-008-9239-8
• Helland, Inge Svein (2007). Discussion of: Bayesian probability in quantum mechanics, by Rüdiger Schack, In J.M. Bernardo (ed.),  Bayesian Statistics 8: Proceedings of the Eighth Valencia International Meeting, June 1-6, Benidorm, Spain.  Oxford University Press.  ISBN 978-0-19-921465-5.  Kapittel 6.
• Helland, Inge Svein (2006). Extended statistical modeling under symmetry; The link toward quantum mechanics. Annals of Statistics.  ISSN 0090-5364.  34, s 42- 77
• Helland, Inge Svein (2006). Towards quantum mechanics from a theory of experiments, In V.K. Dobrev (ed.),  Quantum Theory and Symmetries IV.  Heron Press.  ISBN 954-580-195-6.  2.  s 721 - 732
• Helland, Inge Svein (2005). Quantum Theory as a Statistical Theory under Symmetry, In  Foundations of Probability and Physics - 3.  American Institute of Physics (AIP).  ISBN 0-7354-0235-3.  16.  s 127 - 149
• Helland, Inge Svein (2004). Statistical inference under a fixed symmetry group. International Statistical Review.  ISSN 0306-7734.  72, s 409- 422
• 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, s 53- 60
• Helland, Inge Svein (2001). Reduction of regression models under symmetry, In M. Viana & D. Rickards (ed.),  Algebraic Methods in Statistics.  American Mathematical Society (AMS). 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, s 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 0020-7748.  38(7), s 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 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.

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• Helland, Inge Svein (2018). Epistemic Processes. A Basis for Statistics and Quantum Theory. Springer Nature.  ISBN 978-3-319-95067-9.  170 s.
• Helland, Inge Svein (2010). Steps Towards a Unified Basis for Scientific Models and Methods. World Scientific.  ISBN 978-981-4280-85-3.  275 s.

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• Helland, Inge Svein (2010). ENVELOPE MODELS FOR PARSIMONIOUS AND EFFICIENT MULTIVARIATE LINEAR REGRESSION COMMENT. Statistica sinica.  ISSN 1017-0405.  20(3), s 978- 981
• Helland, Inge Svein (2009). On partial least squares regression.
• Helland, Inge Svein (2009). Partial least squares and mathematical statistics.
• Helland, Inge Svein (2009). Steps towards a unified basis.
• 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 (2004). Quantum mechanics from statistical theory under symmetry and complementarity. Statistical research report (Universitetet i Oslo. Matematisk institut. 12.
• Helland, Inge Svein (2003). Extended statistical modelling under symmetry; the link towards quantum mechanics.
• Helland, Inge Svein (2003). Partial least squares regression, In Samuel Kotz (ed.),  Encyclopedia of Statistical Sciences 2.  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). Discussion of: What is a statistical model?. Annals of Statistics.  ISSN 0090-5364.  30, s 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 (2000). Quantum theory from symmetry and reduction of statistical models. The compact case. Statistical research report (Universitetet i Oslo. Matematisk institut. 3. Show summary

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• Helland, Inge Svein (2000). Reduction of regression models under symmetry. Statistical research report (Universitetet i Oslo. Matematisk institut. 11. 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 (2000). Some theoretical aspects of partial least squares regression. Statistical research report (Universitetet i Oslo. Matematisk institut. 10. 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). Book Review: Statistical Regression with Measurement Error. Statistics in Medicine.  ISSN 0277-6715.  18
• Helland, Inge Svein (1999). Approaching regression methods through symmetry arguments. 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.

• Helland, Inge Svein (1999). Experiments, symmetries and quantum mechanics.
• Helland, Inge Svein (1999). Quantum theory from symmetries in a general parameter space. 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. 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

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