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  • 1.
    Abdalmoaty, Mohamed
    KTH, Reglerteknik.
    Learning Stochastic Nonlinear Dynamical Systems Using Non-stationary Linear Predictors2017Licentiate thesis, monograph (Other academic)
    Abstract [en]

    The estimation problem of stochastic nonlinear parametric models is recognized to be very challenging due to the intractability of the likelihood function. Recently, several methods have been developed to approximate the maximum likelihood estimator and the optimal mean-square error predictor using Monte Carlo methods. Albeit asymptotically optimal, these methods come with several computational challenges and fundamental limitations.

    The contributions of this thesis can be divided into two main parts. In the first part, approximate solutions to the maximum likelihood problem are explored. Both analytical and numerical approaches, based on the expectation-maximization algorithm and the quasi-Newton algorithm, are considered. While analytic approximations are difficult to analyze, asymptotic guarantees can be established for methods based on Monte Carlo approximations. Yet, Monte Carlo methods come with their own computational difficulties; sampling in high-dimensional spaces requires an efficient proposal distribution to reduce the number of required samples to a reasonable value.

    In the second part, relatively simple prediction error method estimators are proposed. They are based on non-stationary one-step ahead predictors which are linear in the observed outputs, but are nonlinear in the (assumed known) input. These predictors rely only on the first two moments of the model and the computation of the likelihood function is not required. Consequently, the resulting estimators are defined via analytically tractable objective functions in several relevant cases. It is shown that, under mild assumptions, the estimators are consistent and asymptotically normal. In cases where the first two moments are analytically intractable due to the complexity of the model, it is possible to resort to vanilla Monte Carlo approximations. Several numerical examples demonstrate a good performance of the suggested estimators in several cases that are usually considered challenging.

  • 2.
    Ablikim, M.
    et al.
    Inst High Energy Phys, Beijing 100049, Peoples R China.
    Adlarson, Patrik
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Nuclear Physics.
    Johansson, Tord
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Nuclear Physics.
    Kupsc, Andrzej
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Nuclear Physics. Natl Ctr Nucl Res, PL-02093 Warsaw, Poland.
    Schönning, Karin
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Nuclear Physics.
    Thorén, Viktor
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Nuclear Physics.
    Wolke, Magnus
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Nuclear Physics.
    Zu, J.
    State Key Lab Particle Detect & Elect, Beijing 100049, Peoples R China;State Key Lab Particle Detect & Elect, Hefei 230026, Peoples R China;Univ Sci & Technol China, Hefei 230026, Peoples R China.
    Measurement of the branching fraction for the decay ψ(3686)→ϕKS0KS02023In: Physical Review D: covering particles, fields, gravitation, and cosmology, ISSN 2470-0010, E-ISSN 2470-0029, Vol. 108, no 5, article id 052001Article in journal (Refereed)
    Abstract [en]

    Based on (448.1±2.9)×106 ψ(3686) events collected with the BESIII detector operating at the BEPCII collider, the decay ψ(3686)→ϕK0SK0S is observed for the first time. Taking the interference between ψ(3686) decay and continuum production into account, the branching fraction of this decay is measured to be B(ψ(3686)→ϕK0SK0S)=(3.53±0.20±0.21)×10−5, where the first uncertainty is statistical and the second is systematic. Combining with the world average value for B(J/ψ→ϕK0SK0S), the ratio B(ψ(3686)→ϕK0SK0S)/B(J/ψ→ϕK0SK0S) is determined to be (6.0±1.6)%, which is suppressed relative to the 12% rule.

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    FULLTEXT01
  • 3.
    Abodayeh, Kamaleldin
    et al.
    Prince Sultan Univ, Dept Math & Gen Sci, Riyadh, Saudi Arabia.
    Raza, Ali
    Air Univ, Dept Math, Stochat Anal & Optimizat Res Grp, PAF Complex E-9, Islamabad 44000, Pakistan.
    Arif, Muhammad Shoaib
    Air Univ, Dept Math, Stochat Anal & Optimizat Res Grp, PAF Complex E-9, Islamabad 44000, Pakistan.
    Rafiq, Muhammad
    Univ Cent Punjab, Fac Engn, Lahore, Pakistan.
    Bibi, Mairaj
    Comsats Univ, Dept Math, Chak Shahzad Campus Pk Rd, Islamabad, Pakistan.
    Mohsin, Muhammad
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Stochastic Numerical Analysis for Impact of Heavy Alcohol Consumption on Transmission Dynamics of Gonorrhoea Epidemic2020In: Computers, Materials and Continua, ISSN 1546-2218, E-ISSN 1546-2226, Vol. 62, no 3, p. 1125-1142Article in journal (Refereed)
    Abstract [en]

    This paper aims to perform a comparison of deterministic and stochastic models. The stochastic modelling is a more realistic way to study the dynamics of gonorrhoea infection as compared to its corresponding deterministic model. Also, the deterministic solution is itself mean of the stochastic solution of the model. For numerical analysis, first, we developed some explicit stochastic methods, but unfortunately, they do not remain consistent in certain situations. Then we proposed an implicitly driven explicit method for stochastic heavy alcohol epidemic model. The proposed method is independent of the choice of parameters and behaves well in all scenarios. So, some theorems and simulations are presented in support of the article.

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    fulltext
  • 4.
    Abrahamsson, Linda
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Statistical models of breast cancer tumour growth for mammography screening data2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Download full text (pdf)
    fulltext
  • 5.
    Abrahamsson, Per Anders
    et al.
    Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science. Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science.
    Adami, Hans Olov
    Taube, Adam
    Kim, KyungMann
    Zelen, Marvin
    Kulldorff, Martin
    Re: Long-term survival and mortality in prostate cancer treated with noncurative intent1995In: UROLGY, Vol. 154, p. 460-465Article in journal (Refereed)
  • 6.
    Achcar, JA
    et al.
    Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science.
    Agrawal, MC
    Anand, KN
    Ali, MM
    Ali, MM
    Bagui, SC
    Baker, RD
    Balamurali, S
    Balasooriya, U
    Bansal, AK
    Barry, J
    Bonett, DG
    Box, G
    Carling, K
    Caudill, SB
    Chakraborti, S
    Chatfield, C
    Chatterjee, S
    Cornell, JA
    Cox, D
    Draper, NR
    Ehrenberg, A
    Finney, DJ
    25 years of applied statistics1998In: JOURNAL OF APPLIED STATISTICS, ISSN 0266-4763, Vol. 25, no 1, p. 3-22Article in journal (Refereed)
  • 7. Adami, Hans-Olov
    et al.
    Bergström, Reinhold
    Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science.
    Weiderpass, Elisabete
    Persson, Ingemar
    Barlow, Lotti
    McLaughlin, Joseph K.
    Risk for endometrial cancer following breast cancer: A prospective study in Sweden1997In: Cancer Causes & Control, Vol. 8, p. 821-827Article in journal (Refereed)
  • 8. Adami, H-O
    et al.
    Bergström, R
    Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science.
    Engholm, G
    Nyrén, O
    Wolk, A
    Ekbom, A
    Englund, A
    Baron, J
    A prospective study of smoking and risk of prostate cancer1996In: Int J Cancer, Vol. 67, p. 764-768Article in journal (Refereed)
  • 9.
    Adami, J
    et al.
    Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science.
    Nyren, O
    Bergstrom, R
    Ekbom, A
    McLaughlin, JK
    Hogman, C
    Fraumeni, JF
    Glimelius, B
    Blood transfusion and non-Hodgkin lymphoma: Lack of association1997In: ANNALS OF INTERNAL MEDICINE, ISSN 0003-4819, Vol. 127, no 5, p. 365-&Article in journal (Refereed)
  • 10.
    Adlersson, Albert
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Is eXplainable AI suitable as a hypotheses generating tool for medical research? Comparing basic pathology annotation with heat maps to find out2023Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Hypothesis testing has long been a formal and standardized process. Hypothesis generation, on the other hand, remains largely informal. This thesis assess whether eXplainable AI (XAI) can aid in the standardization of hypothesis generation through its utilization as a hypothesis generating tool for medical research. We produce XAI heat maps for a Convolutional Neural Network (CNN) trained to classify Microsatellite Instability (MSI) in colon and gastric cancer with four different XAI methods: Guided Backpropagation, VarGrad, Grad-CAM and Sobol Attribution. We then compare these heat maps with pathology annotations in order to look for differences to turn into new hypotheses. Our CNN successfully generates non-random XAI heat maps whilst achieving a validation accuracy of 85% and a validation AUC of 93% – as compared to others who achieve a AUC of 87%. Our results conclude that Guided Backpropagation and VarGrad are better at explaining high-level image features whereas Grad-CAM and Sobol Attribution are better at explaining low-level ones. This makes the two groups of XAI methods good complements to each other. Images of Microsatellite Insta- bility (MSI) with high differentiation are more difficult to analyse regardless of which XAI is used, probably due to exhibiting less regularity. Regardless of this drawback, our assessment is that XAI can be used as a useful hypotheses generating tool for research in medicine. Our results indicate that our CNN utilizes the same features as our basic pathology annotations when classifying MSI – with some additional features of basic pathology missing – features which we successfully are able to generate new hypotheses with. 

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    XAI_BachelorThesis
  • 11.
    Adriansson, Nils
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Mattsson, Ingrid
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Forecasting GDP Growth, or How Can Random Forests Improve Predictions in Economics?2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    GDP is used to measure the economic state of a country and accurate forecasts of it is therefore important. Using the Economic Tendency Survey we investigate forecasting quarterly GDP growth using the data mining technique Random Forest. Comparisons are made with a benchmark AR(1) and an ad hoc linear model built on the most important variables suggested by the Random Forest. Evaluation by forecasting shows that the Random Forest makes the most accurate forecast supporting the theory that there are benefits to using Random Forests on economic time series. 

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    fulltext
  • 12.
    Ahlberg, Daniel
    et al.
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden.
    Deijfen, Maria
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Competing first passage percolation on random graphs with finite variance degrees2019In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, no 3, p. 545-559Article in journal (Refereed)
    Abstract [en]

    We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate lambda(1) (lambda(2)) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if lambda(1) = lambda(2), then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V is an element of (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If lambda(1) not equal lambda(2), on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

  • 13.
    Ahlberg, Daniel
    et al.
    Stockholm Univ, Dept Math, Stockholm, Sweden..
    Griffiths, Simon
    Pontificia Univ Catolica Rio de Janeiro, Dept Matemat, Rio de Janeiro, Brazil..
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, Uppsala, Sweden..
    TO FIXATE OR NOT TO FIXATE IN TWO-TYPE ANNIHILATING BRANCHING RANDOM WALKS2021In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 49, no 5, p. 2637-2667Article in journal (Refereed)
    Abstract [en]

    We study a model of competition between two types evolving as branching random walks on Z(d). The two types are represented by red and blue balls, respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of Z(d) contain one ball each which are independently coloured red with probability p and blue otherwise. We address the question of fixation, referring to the sites and eventually settling for a given colour or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for p not equal 1/2 and that every site will change colour infinitely often almost surely for the balanced initial condition p = 1/2.

  • 14.
    Ahlberg, Daniel
    et al.
    Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
    Griffiths, Simon
    PUC Rio, Dept Matemat, BR-22451900 Gavea, RJ, Brazil.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Morris, Robert
    Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
    Competition in growth and urns2019In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 54, no 2, p. 211-227Article in journal (Refereed)
    Abstract [en]

    We study survival among two competing types in two settings: a planar growth model related to two-neighbor bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or infinity, depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls ( but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one-for-one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two-type growth model on Z(2), one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

  • 15.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.;Uppsala Univ, Dept Math, SE-75106 Uppsala, Sweden..
    Steif, Jeffrey E.
    Univ Gothenburg, Chalmers Univ Technol, Math Sci, SE-41296 Gothenburg, Sweden..
    Pete, Gabor
    Hungarian Acad Sci, Renyi Inst, 13-15 Realtanoda U, H-1053 Budapest, Hungary.;Budapest Univ Technol & Econ, Inst Math, 1 Egry Jozsef U, H-1111 Budapest, Hungary..
    Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?2017In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 53, no 4, p. 2135-2161Article in journal (Refereed)
    Abstract [en]

    Consider a monotone Boolean function f : {0, 1}(n) -> {0, 1} and the canonical monotone coupling {eta(p) : p is an element of [0, 1]} of an element in {0, 1}(n) chosen according to product measure with intensity p is an element of [0, 1]. The random point p is an element of [0, 1] where f (eta(p)) flips from 0 to 1 is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large n, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majority and percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on R arises in this way for some sequence of Boolean functions.

  • 16.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Matematica Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
    Tassion, Vincent
    Univ Geneva, 2-4 Rue Lievre, CH-1211 Geneva, Switzerland.
    Teixeira, Augusto
    Inst Matematica Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.
    Sharpness of the phase transition for continuum percolation in R22018In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 172, no 1-2, p. 525-581Article in journal (Refereed)
    Abstract [en]

    We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter.c. Moreover, For. <.c, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution. At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius. For. >.c, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast. The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

  • 17.
    Ahlberg, Daniel
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Inst Nacl Matemat Pura & Aplicada, Rio De Janeiro, RJ, Brazil;Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden;Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
    Tykesson, Johan
    Chalmers Univ Technol, Dept Math, SE-41296 Gothenburg, Sweden;Univ Gothenburg, Gothenburg, Sweden.
    Gilbert´s disc model with geostatical marking2018In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 50, no 4, p. 1075-1094Article in journal (Refereed)
    Abstract [en]

    We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in R-2 with radii determined by an underlying stationary and ergodic random field phi: R-2 -> [0, infinity), independent of the Poisson process. This setting, in which the random field is independent of the point process, is often referred to as geostatistical marking. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of R(2 )does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.

  • 18.
    Ahlberg, Ellen
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Will Svenska Akademiens Ordlista Improve Swedish Word Embeddings?2022Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Unsupervised word embedding methods are frequently used for natural language processing applications. However, the unsupervised methods overlook known lexical relations that can be of value to capture accurate semantic word relations. This thesis aims to explore if Swedish word embeddings can benefit from prior known linguistic information. Four knowledge graphs extracted from Svenska Akademiens ordlista (SAOL) are incorporated during the training process using the Probabilistic Word Embeddings with Laplacian Priors (PELP) model. The four implemented PELP models are compared with baseline results to evaluate the use of side information. The results suggest that various lexical relations in SAOL are of interest to generate more accurate Swedish word embeddings.

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  • 19.
    Ahlqvist, David
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Probability Theory and Combinatorics.
    Elliptic Curves and Cryptography2022Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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    fulltext
  • 20.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    A homogeneity test of large dimensional covariance matrices under non-normality2018In: Kybernetika (Praha), ISSN 0023-5954, E-ISSN 1805-949X, Vol. 54, no 5, p. 908-920Article in journal (Refereed)
    Abstract [en]

    A test statistic for homogeneity of two or more covariance matrices is presented when the distributions may be non-normal and the dimension may exceed the sample size. Using the Frobenius norm of the difference of null and alternative hypotheses, the statistic is constructed as a linear combination of consistent, location-invariant, estimators of trace functions that constitute the norm. These estimators are defined as U-statistics and the corresponding theory is exploited to derive the normal limit of the statistic under a few mild assumptions as both sample size and dimension grow large. Simulations are used to assess the accuracy of the statistic.

  • 21.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    A significance test of the RV coefficient in high dimensions2019In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 131, p. 116-130Article in journal (Refereed)
    Abstract [en]

    The RV coefficient is an important measure of linear dependence between two multivariate data vectors. Using unbiased and computationally efficient estimators of its components, a modification to the RV coefficient is proposed, and used to construct a test of significance for the true coefficient. The modified estimator improves the accuracy of the original and, along with the test, can be applied to data with arbitrarily large dimensions, possibly exceeding the sample size, and the underlying distribution need only have finite fourth moment. Exact and asymptotic properties are studied under fairly general conditions. The accuracy of the modified estimator and the test is shown through simulations under a variety of parameter settings. In comparisons against several existing methods, both the proposed estimator and the test exhibit similar performance to the distance correlation. Several real data applications are also provided.

  • 22.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    A unified approach to testing mean vectors with large dimensions2019In: AStA Advances in Statistical Analysis, ISSN 1863-8171, E-ISSN 1863-818X, Vol. 103, no 4, p. 593-618Article in journal (Refereed)
    Abstract [en]

    A unified testing framework is presented for large-dimensional mean vectors of one or several populations which may be non-normal with unequal covariance matrices. Beginning with one-sample case, the construction of tests, underlying assumptions and asymptotic theory, is systematically extended to multi-sample case. Tests are defined in terms of U-statistics-based consistent estimators, and their limits are derived under a few mild assumptions. Accuracy of the tests is shown through simulations. Real data applications, including a five-sample unbalanced MANOVA analysis on count data, are also given.

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  • 23.
    Ahmad, M Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Generalized tests of correlation for vectors with large dimensions using modified RV coefficient2019Report (Other academic)
    Abstract [en]

    Tests of zero correlation between two or more vectors with large dimension, possibly largerthan the sample size, are considered when the data may not necessarily follow a normal distribution. A single sample case for several vectors is rst proposed, which is then extended tothe common covariance matrix under the assumption of homogeneity across several independentpopulations. The test statistics are constructed using a recently proposed modicationof the RV coecient for high-dimensional vectors. The accuracy of the tests is shown through simulations.

  • 24.
    Ahmad, M Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Location-invariant and non-invariant tests for large dimensional covariance matrices under normality and non-normality2014Report (Other academic)
    Abstract [en]

    Test statistics for homogeneity, sphericity and identity of high-dimensional covariance matrices are presented under a wide variety of very general conditions when the dimension of the vector, $p$, may exceed the sample size, $n_i$, $i = 1, \ldots, g$. First, location-invariant tests are presented under normality assumption, followed by their robustness to normality by replacing the normality assumption with a mild alternative multivariate model. The two types of tests are then presented in non-invariant form, again under normality and non-normality. Tests of homogeneity of covariance matrices in all cases are immediately supplemented by the tests for sphericity and identity of the common covariance matrix under the null hypothesis. Both location-invariant and non-invariant tests are composed of estimators that are defined as $U$-statistics with kernels of different degrees. Hence, the asymptotic theory of $U$-statistics is employed to arrive at the limiting null and alternative distributions of tests for all cases. These limit distributions are derived using a very mild and practically viable set of assumptions mainly on the traces of the unknown covariance matrices. Finally, corrections and improvements of a few other tests are also presented.

  • 25.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Location-invariant Multi-sample U-tests for Covariance Matrices with Large Dimension2017In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 44, no 2, p. 500-523Article in journal (Refereed)
    Abstract [en]

    For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location-invariant estimators defined as U-statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.

  • 26.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Location-invariant tests of homogeneity of large-dimensional covariance matrices2017In: Journal of Statistical Theory and Practice, ISSN 1559-8608, E-ISSN 1559-8616, Vol. 11, no 4, p. 731-745Article in journal (Refereed)
    Abstract [en]

    A test statistic for homogeneity of two or more covariance matrices of large dimensions is presented when the data are multivariate normal. The statistic is location-invariant and defined as a function of U-statistics of non-degenerate kernels so that the corresponding asymptotic theory is employed to derive the limiting normal distribution of the test under a few mild and practical assumptions. Accuracy of the test is shown through simulations with different parameter settings.

  • 27.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Multiple comparisons of mean vectors with large dimension under general conditions2019In: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 89, no 6, p. 1044-1059Article in journal (Refereed)
    Abstract [en]

    Multiple comparisons for two or more mean vectors are considered when the dimension of the vectors may exceed the sample size, the design may be unbalanced, populations need not be normal, and the true covariance matrices may be unequal. Pairwise comparisons, including comparisons with a control, and their linear combinations are considered. Under fairly general conditions, the asymptotic multivariate distribution of the vector of test statistics is derived whose quantiles can be used in multiple testing. Simulations are used to show the accuracy of the tests. Real data applications are also demonstrated.

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  • 28.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics. Uppsala Univ, Dept Stat, Uppsala, Sweden.
    On Testing Sphericity and Identity of a Covariance Matrix with Large Dimensions2016In: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 25, no 2, p. 121-132Article in journal (Refereed)
    Abstract [en]

    Tests for certain covariance structures, including sphericity, are presented when the data may be high-dimensional but not necessarily normal. The tests are formulated as functions of location-invariant estimators defined as U-statistics of higher order kernels. Under a few mild assumptions, the limit distributions of the tests are shown to be normal. The accuracy of the tests is demonstrated by simulations.

  • 29.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Testing homogeneity of several covariance matrices and multi-sample sphericity for high-dimensional data under non-normality2017In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 46, no 8, p. 3738-3753Article in journal (Refereed)
    Abstract [en]

    A test for homogeneity of g 2 covariance matrices is presented when the dimension, p, may exceed the sample size, n(i), i = 1, ..., g, and the populations may not be normal. Under some mild assumptions on covariance matrices, the asymptotic distribution of the test is shown to be normal when n(i), p . Under the null hypothesis, the test is extended for common covariance matrix to be of a specified structure, including sphericity. Theory of U-statistics is employed in constructing the tests and deriving their limits. Simulations are used to show the accuracy of tests.

  • 30.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Tests for independence of vectors with large dimension2017Report (Other academic)
    Abstract [en]

    Given a random sample of n iid vectors, each of dimension p and partitioned into b sub- vectors of sizes pi, i = 1;:::;b. Location-invariant and non-invariant test statistics for independence of sub-vectors are presented when pi may exceed n and the distribution need not be normal. The tests are composed of U -statistics based estimators of the Frobenius norm of the di erence between the null and alternative hypotheses. Asymptotic distributions of the tests are provided for n;pi! 1, where their nite-sample performance is demonstrated through simulations. Some related and subsequent tests are brie y described. Relations of the proposed tests to certain multivariate measures are discussed, which are of interest on their own.

  • 31.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Tests for proportionality of matrices with large dimension2022In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 189, article id 104865Article in journal (Refereed)
    Abstract [en]

    A test for proportionality of two covariance matrices with large dimension, possibly larger than the sample size, is proposed. The test statistic is simple, computationally efficient, and can be used for a large class of multivariate distributions including normality. The properties of the statistic, including asymptotic distribution, are given under high-dimensional set up. Through simulations, the statistic is shown to perform accurately, and outperform its recent competitors, constructed on the basis of similar principles. An extension to the multi-sample case is given.

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  • 32.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Tests of Zero Correlation Using Modified RV Coefficient for High-Dimensional Vectors2019In: Journal of Statistical Theory and Practice, ISSN 1559-8608, E-ISSN 1559-8616, Vol. 13, no 3, article id 43Article in journal (Refereed)
    Abstract [en]

    Tests of zero correlation between two or more vectors with large dimension, possibly larger than the sample size, are considered when the data may not necessarily follow a normal distribution. A single-sample case for several vectors is first proposed, which is then extended to the common covariance matrix under the assumption of homogeneity across several independent populations. The test statistics are constructed using a recently proposed modification of the RV coefficient (a correlation coefficient for vector-valued random variables) for high-dimensional vectors. The accuracy of the tests is shown through simulations.

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  • 33.
    Ahmad, M. Rauf
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics. Swedish Univ Agr Sci, Dept Energy & Technol, Uppsala, Sweden.
    U-Tests of General Linear Hypotheses for High-Dimensional Data Under Nonnormality and Heteroscedasticity2015In: Journal of Statistical Theory and Practice, ISSN 1559-8608, E-ISSN 1559-8616, Vol. 9, no 3, p. 544-570Article in journal (Refereed)
    Abstract [en]

    Test statistics are presented for general linear hypotheses, with special focus on the two-sample profile analysis. The statistics are a modification to the classical Hotelling’s T2 statistic, are basically designed for the case when the dimension, p, may exceed the sample sizes, ni, and are valid under the violation of any assumption associated with T2, such as normality, homoscedasticity, or equal sample sizes. Under a few mild assumptions replacing the classical ones, the test statistics are shown to follow a normal limit under both the null and alternative hypothesis. As the test statistics are defined as a linear combination of U-statistics, the limits are correspondingly obtained using the asymptotic theory of degenerate (for null) and nondegenerate (for alternative) U-statistics. Simulation results, under a variety of parameter settings, are used to show the accuracy and robustness of the test statistics. Practical application of the tests is also illustrated using a few real data sets.

  • 34.
    Ahmad, M. Rauf
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Ahmed, S. Ejaz
    Brock Univ, Dept Math & Stat, St Catharines, ON, Canada..
    On the distribution of the T-2 statistic, used in statistical process monitoring, for high-dimensional data2021In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 168, article id 108919Article in journal (Refereed)
    Abstract [en]

    A modification to the asymptotic distribution of the T-2-statistic used in multivariate process monitoring is provided when the dimension of the vectors may exceed the sample size. Under certain mild condition, a unified limit distribution is obtained that is applicable for both Phase I and II charts. Further the limit holds for charts based on individual observations as well as subgroup means. The limit is easily applicable and does not need any data preprocessing or dimension reduction. Simulations are used to demonstrate the accuracy of the proposed limit.

  • 35.
    Ahmad, M. Rauf
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Ahmed, S. Ejaz
    Brock Univ, Dept Math & Stat, St Catharines, ON, Canada.
    Some correlation tests for vectors of large dimension2023In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 52, no 7, p. 2144-2160Article in journal (Refereed)
    Abstract [en]

    For a random sample of n iid p-dimensional vectors, each partitioned into b sub-vectors of dimensions pi, i=1,…,b, tests for zero correlation of sub-vectors are presented when pi ≫ n and the distribution need not be normal. The test statistics are composed of U-statistics based estimators of the Frobenius norm measuring the distance between the null and alternative hypotheses. Asymptotic distributions of the tests are provided for n,pi → ∞, with their finite-sample performance demonstrated through simulations. Some related tests are discussed. A real data application is also given.

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  • 36.
    Ahmad, M. Rauf
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Pavlenko, Tatjana
    KTH, Stockholm, Sweden.
    A U-classifier for high-dimensional data under non-normality2018In: Journal of Multivariate Analysis, ISSN 0047-259X, E-ISSN 1095-7243, Vol. 167, p. 269-283Article in journal (Refereed)
    Abstract [en]

    A classifier for two or more samples is proposed when the data are high-dimensional and the distributions may be non-normal. The classifier is constructed as a linear combination of two easily computable and interpretable components, the U-component and the P-component. The U-component is a linear combination of U-statistics of bilinear forms of pairwise distinct vectors from independent samples. The P-component, the discriminant score, is a function of the projection of the U-component on the observation to be classified. Together, the two components constitute an inherently bias-adjusted classifier valid for high-dimensional data. The classifier is linear but its linearity does not rest on the assumption of homoscedasticity. Properties of the classifier and its normal limit are given under mild conditions. Misclassification errors and asymptotic properties of their empirical counterparts are discussed. Simulation results are used to show the accuracy of the proposed classifier for small or moderate sample sizes and large dimensions. Applications involving real data sets are also included.

  • 37.
    Ahmad, M. Rauf
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    von Rosen, D.
    Tests for high-dimensional covariance matrices using the theory of U-statistics2015In: Journal of Statistical Computation and Simulation, ISSN 0094-9655, E-ISSN 1563-5163, Vol. 85, no 13, p. 2619-2631Article in journal (Refereed)
    Abstract [en]

    Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p >> n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

  • 38.
    Ahmad, M. Rauf
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Von Rosen, Dietrich
    Tests of Covariance Matrices for High Dimensional Multivariate Data Under Non Normality2015In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 44, no 7, p. 1387-1398Article in journal (Refereed)
    Abstract [en]

    Ahmad et al. (in press) presented test statistics for sphericity and identity of the covariance matrix of a multivariate normal distribution when the dimension, p, exceeds the sample size, n. In this note, we show that their statistics are robust to normality assumption, when normality is replaced with certain mild assumptions on the traces of the covariance matrix. Under such assumptions, the test statistics are shown to follow the same asymptotic normal distribution as under normality for large p, also whenp >> n. The asymptotic normality is proved using the theory of U-statistics, and is based on very general conditions, particularly avoiding any relationship between n and p.

  • 39.
    Ahmady Phoulady, Hady
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Brownian Motions and Scaling Limits of Random Trees2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
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  • 40.
    Aho, André
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    A Statistical Analysis of Weighting Techniques for Portfolio Construction: Insights for Portfolio Managers and Investors2023Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This thesis investigates the relative merits and drawbacks of six portfolio weighting techniques, including two traditional (equal and value-weighting), two optimization-based (mean-variance and risk parity), and two statistical (principal component analysis and ridge regression) techniques. The focus is thus on selecting the weighting technique, an aspect of portfolio construction that market practitioners sometimes overlook. The analysis, implemented on the Swedish market, employs historical backtesting, Monte Carlo simulations, and stress tests to evaluate various techniques under diverse market conditions. The results reveal that no single portfolio consistently outperforms or underperforms across all metrics and scenarios, highlighting the importance of a comprehensive set of performance and risk measures for informed investment decisions. Furthermore, the statistical techniques, principal component analysis and ridge regression demonstrate competitive risk-adjusted returns relative to the traditional and optimization-based techniques implemented in this thesis. The results suggest that market practitioners should consider incorporating these somewhat uncommon techniques alongside more traditional techniques in portfolio management, depending on their investment objectives and risk tolerance.

  • 41.
    Aho, André
    et al.
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Löw, Simon
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Statistical arbitrage: Can a pairs trading strategy beat a buy-and-hold strategy?2022Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    In this thesis, the aim is to investigate whether a pairs trading strategy on Swedish stocks can generate a higher risk-adjusted return compared to a buy-and-hold strategy on a benchmark index. The benchmark index is the OMX Stockholm Benchmark-index (OMXSBPI), which is an index that should reflect the Swedish market in general. With a statistical focus, a trading algorithm is built which is then evaluated on data between the years 2018 to 2021. The statistical concepts this thesis is based on are stationarity and cointegration and it is the Augmented Dickey-Fuller test that forms the basis for being able to test these concepts. The risk-adjusted return for the strategy is evaluated using the popular measure Sharpe ratio, which is then compared to the Sharpe ratio for the OMXSBPI-index. The results obtained in this study can not confirm that the pairs trading strategy is better than a buy-and-hold strategy on the OMXSBPI-index in terms of risk-adjusted return. One indication, however, is that the strategy seems to perform better in conditions when the market is declining. In 2018, the index went down by 7.7060 while the strategy went up by 7.5100 percent. As it is data for only one year, it is not possible to determine whether it is due to chance or a potential edge of the strategy. 

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  • 42.
    Alam, Mahbub
    et al.
    Tata Inst Fundamental Res, Sch Math, Mumbai 400005, Maharashtra, India.
    Ghosh, Anish
    Tata Inst Fundamental Res, Sch Math, Mumbai 400005, Maharashtra, India.
    Yu, Shucheng
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Quantitative Diophantine approximation with congruence conditions2021In: Journal de Théorie des Nombres de Bordeaux, ISSN 1246-7405, E-ISSN 2118-8572, Vol. 33, no 1, p. 261-271Article in journal (Refereed)
    Abstract [en]

    In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.

  • 43.
    Albert, Michael
    et al.
    Univ Otago, Dept Comp Sci, Dunedin, New Zealand.
    Holmgren, Cecilia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Johansson, Tony
    Stockholm Univ, Dept Math, Stockholm, Sweden.
    Skerman, Fiona
    Masaryk Univ, Fac Informat, Brno, Czech Republic.
    Embedding Small Digraphs and Permutations in Binary Trees and Split Trees2020In: Algorithmica, ISSN 0178-4617, E-ISSN 1432-0541, Vol. 82, no 3, p. 589-615Article in journal (Refereed)
    Abstract [en]

    We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335-364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409-432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

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  • 44.
    Ali, Mohammad
    Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
    Performance of Three Classification Techniques in Classifying Credit Applications Into Good Loans and Bad Loans: A Comparison2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The use of statistical classification techniques in classifying loan applications into good loans and bad loans gained importance with the exponential increase in the demand for credit. It is paramount to use a classification technique with a high predictive capacity to ensure the profitability of the business venture.

     

    In this study we aim to compare the predictive capability of three classification techniques: 1) Logistic regression, 2) CART, and 3) random forests. We apply these techniques on German credit data using an 80:20 learning:test split, and compare the performance of the models fitted using the three classification techniques. The probability of default pi for each observation in the test set is calculated using the models fitted on the training dataset. Each test set sample xi is then classified into a good loan or a bad loan, based on a threshold , such that xi bad loan class if pi  . We chose several  thresholds in order to compare the performance of each of the three classification techniques on five model suitability statistics: Accuracy, precision, negative predictive value, recall, and specificity.

     

    None of the classifiers turned out to be best at all the five cross-validation statistics. However, logistic regression has the best performance at low probability of default thresholds. On the other hand, for higher thresholds, CART performs best in accuracy, precision, and specificity measures, while random forest performs best for negative predictive value and recall measures. 

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  • 45.
    Allander, Erik
    et al.
    Uppsala University, Humanistisk-samhällsvetenskapliga vetenskapsområdet, Faculty of Social Sciences, Department of Information Science.
    Bring, Johan
    Gudmundsson, Ludvig
    Mattson, Stefan
    Olafsson, Olafur
    Rigner, Karl-Gustav
    Sigurgeirsson, Bardur
    Taube, Adam
    What is the long term value of multiphasic screening and the initial judgement of benefit?1997In: Scandinavian Journal of Social Medicine, Vol. Suppl 51, p. 1-20Article in journal (Refereed)
  • 46.
    Alm, Sven Erick
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics. Matematisk statistik.
    Approximation and Simulation of the Distributions of Scan Statistics for Poisson Processes in Higher Dimensions1998In: Extremes, Vol. 1, p. 111-126Article in journal (Refereed)
    Abstract [en]

    Given a Poisson process in two or three dimensions we are interested in the scan statistic, i.e. the largest number of points contained in a translate of a fixed scanning set restricted to lie inside a rectangular area.

    The distribution of the scan statistic is accurately approximated for rectangular scanning sets, using a technique that is also extended to higher dimensions.

    The accuracy of the approximation is checked through simulation.

  • 47.
    Alm, Sven Erick
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics. Matematisk statistik.
    Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence2003In: Bernoulli, ISSN 1350-7265, Vol. 9, no 2, p. 351-371Article in journal (Refereed)
    Abstract [en]

    For $n\ge0$, let $\lambda_n$ be the median of the $\Gamma(n+1,1)$ distribution. We prove that the sequence $\{\alpha_n=\lambda_n-n\}$ decreases from $\log 2$ to $2/3$ as $n$ increases from 0 to $\infty$. The difference, $1-\alpha_n$, between the mean and the median thus increases from $1-\log 2$ to $1/3$.

    This result also proves the following conjecture by Chen \& Rubin about the Poisson distributions: Let $Y_{\mu}\sim\text{Poisson}(\mu)$, and \lambda_n$ be the largest $\mu$ such that $P(Y_{\mu}\le n)=1/2$, then $\lambda_n-n$ is decreasing in $n$.

    The sequence $\{\alpha_n\}$ is related to a sequence $\{\theta_n\}$, introduced by Ramanujan, which is known to be decreasing and of the form

    $\theta_n=\frac13+\frac4{135(n+k_n)}$, where $\frac2{21}<k_n\le\frac8{45}$. We also show that the sequence $\{k_n\}$ is decreasing.

  • 48.
    Alm, Sven Erick
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    On Measures of Average Degree for Lattices2006In: Combinatorics, Probability and Computing, Vol. 15, no 4, p. 477-488Article in journal (Refereed)
    Abstract [en]

    The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices.

    These measures are compared on a class of lattices and are analyzed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.

  • 49.
    Alm, Sven Erick
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Deijfen, Maria
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden..
    First Passage Percolation on \(\mathbb {Z}^2\): A Simulation Study2015In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 161, no 3, p. 657-678Article in journal (Refereed)
    Abstract [en]

    First passage percolation on is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges. In this paper, the speed of the growth and the shape of the infected set is studied by aid of large-scale computer simulations, with focus on continuous passage time distributions. It is found that the most important quantity for determining the value of the time constant, which indicates the inverse asymptotic speed of the growth, is , where are i.i.d. passage time variables. The relation is linear for a large class of passage time distributions. Furthermore, the directional time constants are seen to be increasing when moving from the axis towards the diagonal, so that the limiting shape is contained in a circle with radius defined by the speed along the axes. The shape comes closer to the circle for distributions with larger variability.

  • 50.
    Alm, Sven Erick
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Linusson, Svante
    First critical probability for a problem on random orientations in G(n,p)2014In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 19, p. 69-Article in journal (Refereed)
    Abstract [en]

    We study the random graph G (n,p) with a random orientation. For three fixed vertices s, a, b in G(n,p) we study the correlation of the events {a -> s} (there exists a directed path from a to s) and {s -> b}. We prove that asymptotically the correlation is negative for small p, p < C-1/n, where C-1 approximate to 0.3617, positive for C-1/n < p < 2/n and up to p = p(2)(n). Computer aided computations suggest that p(2)(n) = C-2/n, with C-2 approximate to 7.5. We conjecture that the correlation then stays negative for p up to the previously known zero at 1/2; for larger p it is positive.

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