Matematisk Statistik, Publikationer

In this paper, the multilinear normal distribution is introduced as an extension of the matrix-variate normal distribution. Basic properties such as marginal and conditional distributions, moments, and the characteristic function, are also presented.

The estimation of parameters using a flip-flop algorithm is also briefly discussed.

The problemof estimating parameters of amultivariate normal p-dimensional random vector is considered for a banded covariance structure reflecting mdependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.

In this paper, we consider profile analysis of several groups where the groups have partly equal means. This leads to a profile analysis for a growth curve model. The likelihood ratio statistics are given for the three hypotheses known in literature as parallelism, level hypothesis and flatness. Furthermore, exact and asymptotic distributions are given in the relevant cases.

We consider the problem of estimating a collection of integrals with respect to an unknown finite measure μ from noisy observations of some of the integrals. A new method to carry out Bayesian inference for the integrals is proposed. We use a Dirichlet or Gamma process as a prior for μ, and construct an approximation to the posterior distribution of the integrals using the sampling importance resampling algorithm and samples from a new multidimensional version of a Markov chain by Feigin and Tweedie. We prove that the Markov chain is positive Harris recurrent, and that the approximating distribution converges weakly to the posterior as the sample size increases, under a mild integrability condition. Applications to polymer chemistry and mathematical finance are given. © 2008 Board of the Foundation of the Scandinavian Journal of Statistics.

 We introducea statistical model for learning the genetic structure ofpopulations. A novel MCMC type method isintroduced. 

  Interacting parallell Markov chains are shown to converge to a maximum of the posterior on a set of partitions of a finite set of discrete data. This is demonstrated on an example of population genetics data.

When there is auxiliary information in survey sampling, the design based optimal regression estimator of a finite population mean is known to be at least asymptocially more effective than the corresponding GREG estimator . We show that the optimal estimator can be seen as a calibration estimator.

A new measure based on comparison of empirical distributions for sub sequences or parallel runs and the full sequence of Markov chain Monte Carlo simulations, is proposed as a criterion of stability or convergence. The measure is also put forward as a loss function when the design of a Markov chain is optimized. The comparison is based on a Kullback-Leibler (KL) type distance over value sets defined by the output data. The leading term in a series expansion gives an interpretation in terms of the relative uncertainty of cell frequencies. The validity of this term is studied by simulation in two analytically tractable cases with Markov dependency. The agreement between the leading term and the KL-measure is close, in particular when the simulations are extensive enough for stable results. Comparisons with established criteria turn out favourably in examples studied. © Board of the Foundation of the Scandinavian Journal of Statistics 2005.

Minimization of stochastic complexity (SC) was used as a method for classification of genotypic fingerprints. The method was applied to fluorescent amplified fragment length polymorphism (fAFLP) fingerprint patterns of 507 Vibrionaceae representatives. As the current BinClass implementation of the optimization algorithm for classification only works on binary vectors, the original fingerprints were discretized in a preliminary step using the sliding-window band-matching method, in order to maximally preserve the information content of the original band patterns. The novel classification generated using the BinClass software package was subjected to an in-depth comparison with a hierarchical classification of the same dataset, in order to acknowledge the applicability of the new classification method as a more objective algorithm for the classification of genotyping fingerprint patterns. Recent DNA-DNA hybridization and 16S rRNA gene sequence experiments proved that the classification based on SC-minimization forms separate clusters that contain the fAFLP patterns for all representatives of the species Enterovibrio norvegicus, Vibrio fortis, Vibrio diazotrophicus or Vibrio campbellii, while previous hierarchical cluster analysis had suggested more heterogeneity within the fAFLP patterns by splitting the representatives of the above-mentioned species into multiple distant clusters. As a result, the new classification methodology has highlighted some previously unseen relationships within the biodiversity of the family Vibrionaceae.

Let $W$ be the number of points in $(0,t]$ of a stationary finite-state Markov ren ewal point process. We derive a bound for the total variation distance between the distribution of $W$ and a compound Poisson distribution. For any nonnegative rand om variable $\zeta$ we construct a ``strong memoryless time'' $\hat\zeta$ such tha t $\zeta-t$ is exponentially distributed conditional on $\{\hat\zeta\leq t,\zeta>t \}$, for each $t$. This is used to embed the Markov renewal point process into ano ther such process whose state space contains a frequently observed state which rep resents loss of memory in the original process. We then write $W$ as the accumulat ed reward of an embedded renewal reward process, and use a compound Poisson approx imation error bound for this quantity by Erhardsson. For a renewal process, the bo und depends in a simple way on the first two moments of the interrenewal time dist ribution, and on two constants obtained from the Radon-Nikodym derivative of the i nterrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.

Microbiologists have traditionally applied hierarchical clustering algorithms as their mathematical tool of choice to unravel the taxonomic relationships between micro-organisms. However, the interpretation of such hierarchical classifications suffers from being subjective, in that a variety of ad hoc choices must be made during their construction. On the other hand, the application of more profound and objective mathematical methods - such as the minimization of stochastic complexity - for the classification of bacterial genotyping fingerprints data is hampered by the prerequisite that such methods only act upon vectorized data. In this paper we introduce a new method, coined sliding window discretization, for the transformation of genotypic fingerprint patterns into binary vector format. In the context of an extensive amplified fragment length polymorphism (AFLP) data set of 507 strains from the Vibrionaceae family that has previously been analysed, we demonstrate by comparison with a number of other discretization methods that this new discretization method results in minimal loss of the original information content captured in the banding patterns. Finally, we investigate the implications of the different discretization methods on the classification of bacterial genotyping fingerprints by minimization of stochastic complexity, as it is implemented in the BinClass software package for probabilistic clustering of binary vectors. The new taxonomic insights learned from the resulting classification of the AFLP patterns will prove the value of combining sliding window discretization with minimization of stochastic complexity, as an alternative classification algorithm for bacterial genotyping fingerprints.

We describe a library of molecular fragments designed to model and predict non-bonded interactions between atoms. We apply the Bayesian approach, whereby prior knowledge and uncertainty of the mathematical model are incorporated into the estimated model and its parameters. The molecular interaction data are strengthened by narrowing the atom classification to 14 atom types, focusing on independent molecular contacts that lie within a short cutoff distance, and symmetrizing the interaction data for the molecular fragments. Furthermore, the location of atoms in contact with a molecular fragment are modeled by Gaussian mixture densities whose maximum a posteriori estimates are obtained by applying a version of the expectation-maximization algorithm that incorporates hyperparameters for the components of the Gaussian mixtures. A routine is introduced providing the hyperparameters and the initial values of the parameters of the Gaussian mixture densities. A model selection criterion, based on the concept of a 'minimum message length' is used to automatically select the optimal complexity of a mixture model and the most suitable orientation of a reference frame for a fragment in a coordinate system. The type of atom interacting with a molecular fragment is predicted by values of the posterior probability function and the accuracy of these predictions is evaluated by comparing the predicted atom type with the actual atom type seen in crystal structures. The fact that an atom will simultaneously interact with several molecular fragments forming a cohesive network of interactions is exploited by introducing two strategies that combine the predictions of atom types given by multiple fragments. The accuracy of these combined predictions is compared with those based on an individual fragment. Exhaustive validation analyses and qualitative examples ( e. g., the ligand-binding domain of glutamate receptors) demonstrate that these improvements lead to effective modeling and prediction of molecular interactions.

Objective The reproducibility of clinical records of the occlusion was assessed in three dimensions using mounted casts. Three distinct areas were examined: 1) mandibular positions (intercuspal position (IP) or retruded contact position (RCP)), 2) materials used in recording the occlusion, 3) clinical variation. Design Interocclusal records were made in a random order of three patients: one fixed prosthodontics case, one removable partial denture case and one complete denture case, with two different types of waxes, record rims, two different brands of vinyl polysiloxanes and one irreversible hydrocolloid. Setting Private practice and Karolinska Institute, Huddinge, Sweden, Subjects One general dental practitioner and three voluntary patients. Results Point estimation of variance components indicate that 70-93% of the variation of the positions of the mounted casts are caused by: 1) clinical variation for all three cases and in three directions, 2) the influence of recording materials 0-29%, and 3) mandibular positions (IP/RCP) 0-11%. The ranges of the positions of the mounted casts were lower for the dentate case (0.04-1.39 mm) than for the partially dentate case (0.17-2.65 mm), which in turn was lower than those for the edentulous case (1.42-5.59 mm). Conclusion Clinical variation seems to dominate the variation in positions of mounting casts when making interocclusal records, rather than mandibular position or the recording materials used. Therefore a dentist who makes one single interocclusal record cannot presume that it will reproduce the interocclusal relationship intended, which in the present study was most obvious for the edentulous case. The results showed that impression materials stabilised by a tray did not differ significantly from waxes and record rims concerning the reproducibility. Therefore the stabilised impression materials are an alternative, which also give additional advantages like reduction of appointments as well as superior accuracy.

We give a survey of different partitioning methods that have been applied to bacterial taxonomy. We introduce a theoretical framework, which makes it possible to treat the various models in a unified way. The key concepts of our approach are prediction and storing of microbiological information in a Bayesian forecasting setting. We show that there is a close connection between classification and probabilistic identification and that, in fact, our approach ties these two concepts together in a coherent way.

[No abstract available]

In this paper, we consider a parallel profile model for several groups. Given the parallel profile model we construct tests based on the likelihood ratio, without any restrictions on the parameter space, testing the covariance matrix for random-effects structure or sphericity. Furthermore, given both the parallel profile and random- effects covariance structure the level hypothesis is tested. The attained significance levels and the empirical powers for the given tests in this paper are compared with the tests given by Yokoyama and Fujikoshi (1993) and Yokoyama (1995). 

In this talk, we consider profile analysis of several groups where subvectors of the mean vectors are equal. This leads to a profile analysis in a growth curve model. The likelihood ratio statistics are given for the three hypotheses known in literature as parallelism, level hypothesis and flatness. Furthermore, exact and asymptotic distributions are given in the relevant cases. 

The Kronecker structured covariance matrix in multivariate normal distribution will be studied. Particularly, the mapping and parametrization which are induced by the Kronecker product are considered.

Furthermore, estiamtion and the uniqueness of the estimators will be discussed in the case of a covariance matrix which is a Kronecker product of several matrices.

A well known fact is that when testing hypotheses for covariance matrices, distributions of quadratic forms arise. A generalization of the distribution of the multivariate quadratic form XAX', where X is a p times n normally distributed matrix and A is a n times n symmetric real matrix, is presented. It is shown that the distribution of the quadratic form is the same as the distribution of a weighted sum of noncentral Wishart distributed matrices.

Using this characterization of the distribution several properties of the quadratic form XAX' will be shown.

When the true class conditional model and class probabilities are approximated in a pattern recognition/classification problem the performance of the optimal classifier is expected to deteriorate. But calculating this reduction is far from trivial in the general case. We present one generalization, and easily computable formulas for estimating the degradation in performance with respect to the optimal classifier. An example of an approximation is the Naive Bayes classifier. We generalize and sharpen results for evaluating this classifier.

Supporting high availability by checkpointing and switching to a backup upon failure of a primary has a cost. Trade-off studies help system architects to decide whether higher availability at the cost of higher response time is to strive for. The decision will lead to configuring a fault-tolerant server for best performance. This paper provides a mathematical model employing queuing theory that helps to compute the optimal checkpointing interval for a primary-backup replicated server. The optimization criterion is system availability. The model guides towards the checkpointing interval that is short enough to give low failover time, but long enough to utilize most of the system resources for servicing client requests. The novelty of the work is the detailed modelling of service times, wait times for earlier calls in the queue, and priority of checkpointing calls over client calls within the queues. Studies on the model in Mathematica and validation of a modelling assumption through simulations are included.

We present a theory of classification and predictive identification of bacteria. Bacterial strains are characterized by a binary vector and the taxonomy is specified by attaching a label to each vector. The theory is developed from only two basic assumptions, viz. that the sequence of pairs of feature vectors and the attached labels is judged (infinitely) exchangeable and predictively sufficient. We derive expressions for the training error and the probability of identification error and show that latter is an affine function of the former. We prove the law of large numbers for identification matrices, which contain the fundamental information of bacterial data. We prove the Bayesian risk consistency of the predictive identification rule given by the theory and show that the training error is a consistent estimate of the generalization error. © 2002 Published by Elsevier Science Inc.

Processes in Nature may be considered as deterministic or/and random. We are observing global problems such as climate changes (e.g. warming and extreme weather conditions), pollutions (e.g. acidification, fertilization, the spread of many types of pollutants through air and water) and whole ecosystems that are under pressure (e.g. the Baltic sea and the Arctic region). To understand the processes in Nature and (predict) understand what might occur it is not enough with empirical studies. One needs theoretical fundaments including models and theories to perform correct actions against different threats or at least to carry out appropriate simulation studies. For example, extreme value theory can explain some of the observed phenomena, classical risk analysis may be of help, different types of multivariate and high-dimensional analysis can explain data, time series analysis is essential, for forthcoming studies the theory of experimental designs is of interest, data assimilation together with inverse problem technique is useful for adjustment of data into mathematical models and the list can be made much longer. Behind all these approaches mathematics is hidden, sometimes at a very advanced level. Chemical and physical processes influence all observations but the challenge is to do appropriate approximations so that mathematical/statistical models can be applied. The main aim of this project is to present state of the art knowledge concerning the modelling of Nature with focus on mathematical modelling, in particular "inverse and ill-posed problems", as well as spatiotemporal models. Inverse and ill-posed problems are characterized by the property that the solutions are extremely sensitive to measurement and modelling errors. There are established connections between inverse problems and Bayesian inference but very little has been carried out with focus on parametric inference such as the likelihood approach. Concerning spatio-temporal models these are usually extensions of classical time series models or/and classical multivariate analysis models.

From the Nordic Council of Ministers, within the program Nordic - Russian Cooperation in Education and Research we asked for funding of 3 preparatory meetings where the plan was to create a series of events taking place during 2011-2013. Partner organizations were

• Institute of Problems of Mechanical Engineering, St. Petersburg
• St. Petersburg State University
• Helsinki University
• Swedish Agricultural University
• Stockholm University
• Linköping University

However, there were also some other participants from other universities.

The planned events should be connected to the following fields: applied mathematics, biophysics and mathematical statistics. Within applied mathematics: mathematical modelling and partial differential equations, inverse and ill-pose problems, data assimilation, dynamical systems, linear algebra, matrix theory; within biophysics; neural networks and inverse modelling of objects; within mathematical statistical; analyses of stochastic processes, spatio-temporal modelling, experimental design, where considered. There exists a wide overlap between these areas and it is challenging to systemize this overlap and transmit this knowledge to students and stakeholders. However, due to unsure funding it was decided to discuss what can be presented during a one-year program. Moreover, due to practical reasons only 2 meetings/workshops were held:

1. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology: May 20-21, 2010 at Department of Mathematics, Linköping University.
2. Workshop on Inverse Problems, Data, Mathematical Statistics and Ecology, Part II: August 25-27, 2010 at Department of Mathematics, Helsinki University.

The output from the above events can be summarized as follows:

• We have identified a number of different areas which can be taught on from different perspectives depending on students background of mathematics.
• We have learned to know many interesting researchers who are willing to share there experiences when for example creating a summer school.
• There is no doubt that we can organize cross-disciplinary summer/winter schools with focus on either the Baltic or Archtic regions.

This booklet is also part of the deliverables. It comprizes extended abstracts of the majority of the talks of the participants showing their great interest. It is in some way a unique cross-disciplinary document which has joined researchers from different areas from Russia, Finland and Sweden.

We are extremely grateful for the support given by the Nordic Council of Ministers (NCM-RU-PA-2009/10382) and all the enthusiastic contributions by the participants, including our host in Helsinki, professor Lassi Päivärinta.

Vladimir Kozlo, Linköping University

Martin Ohlso, Linköping University

Dietrich von Rosen, Linköping University/SLU

Many testing, estimation and confidence interval procedures discussed in the multivariate statistical literature are based on the assumption that the observation vectors are independent and normally distributed. The main reason for this is that often sets of multivariate observations are, at least approximately, normally distributed. Normally distributed data can be modeled entirely in terms of their means and variances/covariances. Estimating the mean and the covariance matrix is therefore a problem of great interest in statistics and it is of great significance to consider the correct statistical model. The estimator for the covariance matrix is important since inference on the mean parameters strongly depends on the estimated covariance matrix and the dispersion matrix for the estimator of the mean is a function of it.

In this thesis the problem of estimating parameters for a matrix normal distribution with different patterned covariance matrices, i.e., different statistical models, is studied.

A p-dimensional random vector is considered for a banded covariance structure reflecting m-dependence. A simple non-iterative estimation procedure is suggested which gives an explicit, unbiased and consistent estimator of the mean and an explicit and consistent estimator of the covariance matrix for arbitrary p and m.

Estimation of parameters in the classical Growth Curve model when the covariance matrix has some specific linear structure is considered. In our examples maximum likelihood estimators can not be obtained explicitly and must rely on numerical optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linearly structured covariance matrix.

This thesis also deals with the problem of estimating the Kronecker product structure. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. The proposed estimators are used to derive a likelihood ratio test for spatial independence. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is given. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm and the null distribution for the likelihood ratio statistic is discussed.

Model based probabilistic classification is heavily used in data mining and machine learning. For computational learning these models may need approximation steps however. One popular approximation in classification is to model the class conditional densities by factorization, which in the independence case is usually called the ’Naïve Bayes’ classifier. In general probabilistic independence cannot model all distributions exactly, and not much has been published on how much a discrete distribution can differ from the independence assumption. In this dissertation the approximation quality of factorizations is analyzed in two articles.

A specific class of factorizations is the factorizations represented by graphical models. Several challenges arise from the use of statistical methods for learning graphical models from data. Examples of problems include the increase in the number of graphical model structures as a function of the number of nodes, and the equivalence of statistical models determined by different graphical models. In one article an algorithm for learning graphical models is presented. In the final article an algorithm for clustering parts of DNA strings is developed, and a graphical representation for the remaining DNA part is learned.

In this article, we consider prediction of a univariate response from background data. The data may have a near-collinear structure and additionally group effects are assumed to exist. A two step estimation procedure is proposed. The first step is to summarize the information in the predictors via a bilinear model. The bilinear model has a Krylov structured within individual design matrix, which is the link to classical partial least squares (PLS) analysis and a between individual design matrix which handles group effects. The second step is the prediction step where a conditional expectation approach is used. The two step approach gives new insight in PLS. Explicit maximum likelihood estimator of the dispersion matrix and mean for the predictors are derived under the assumption that the covariance between the response and explanatory variable is known. It is shown that the mean square error of the two step approach is always smaller than PLS.

Ahmad, Ohlson, and von Rosen (2011a) present asymptotic distribution of a one-sample test statistic under non-normality, when the data are high dimensional, i.e., when the dimension of the vector, p, may exceed the sample size, n. This paper extends the case to a two-sample statistic to test the difference of mean vectors of two independent multivariate distributions, again under high-dimensional set up. Using the asymptotic theory of U-statistics, and under mild assumptions on the traces of the unknown covariance matrices, the statistic is shown to follow an approximate normal distribution when n and p are large. However, no relationship between n and p is assumed. An extension to the paired case is given, which, being essentially a one-sample statistic, supplements the asymptotic results obtained in Ahmad, Ohlson, and von Rosen (2011a).

A test statistic is considered for testing a hypothesis for the mean vector

for multivariate data, when the dimension of the vector,

p, may exceed the number of vectors, n, and the underlying distribution need

not necessarily be normal. With n, p large, and under mild assumptions,

the statistic is shown to asymptotically follow a normal distribution.

A by product of the paper is the approximate distribution

of a quadratic form, based on the reformulation of well-known Box's

approximation, under high-dimensional set up.

In this paper the multivariate normal distribution with a Kronecker product structured covariance matrix is studied. Particularly, estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The estimation procedure, suggested in this paper, is a generalization of the procedure derived by Srivastava et al. (2008), for a separable covariance matrix.

Furthermore, the restrictions imposed by separability and double separability are discussed.

This paper deals with the problem of testing spatial independence for dependent observations. The sample observationmatrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is given. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algori

The distribution of particles in a rarefied gas in a vessel can be described by the Boltzmann equation. As an approximation of the solution to this equation, Caprino, Pulvirenti and Wagner [3] constructed a random N-particle system.

In the equilibrium case, they prove in [3] that the L¹-distance between the density function of k particles in the N-particle process and the k-fold product of the solution to the stationary Boltzmann equation is of order 1/N. They do this in order to show that the N-particle system converges to the system described by the stationary Boltzmann equation as the number of particles tends to infinity.

This is different from the standard approach of describing convergence of an N-particle system. Usually, convergence in distribution of random measures or weak convergence of measures over the space of probability measures is used. The purpose of the present thesis is to compare different modes of convergence of the N-particle system as N tends to infinity assuming stationarity.