| 1. |
- Malmberg, Filip, et al.
(författare)
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An efficient algorithm for exact evaluation of stochastic watersheds
- 2014
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Ingår i: Pattern Recognition Letters. - : Elsevier BV. - 0167-8655 .- 1872-7344. ; 47, s. 80-84
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Tidskriftsartikel (refereegranskat)abstract
- The stochastic watershed is a method for unsupervised image segmentation proposed by Angulo and Jeulin (2007). The method first computes a probability density function (PDF), assigning to each piece of contour in the image the probability to appear as a segmentation boundary in seeded watershed segmentation with randomly selected seeds. Contours that appear with high probability are assumed to be more important. This PDF is then post-processed to obtain a final segmentation. The main computational hurdle with the stochastic watershed method is the calculation of the PDF. In the original publication by Angulo and Jeulin, the PDF was estimated by Monte Carlo simulation, i.e., repeatedly selecting random markers and performing seeded watershed segmentation. Meyer and Stawiaski (2010) showed that the PDF can be calculated exactly, without performing any Monte Carlo simulations, but do not provide any implementation details. In a naive implementation, the computational cost of their method is too high to make it useful in practice. Here, we extend the work of Meyer and Stawiaski by presenting an efficient (quasi-linear) algorithm for exact computation of the PDF. We demonstrate that in practice, the proposed method is faster than any previously reported method by more than two orders of magnitude. The algorithm is formulated for general undirected graphs, and thus trivially generalizes to images with any number of dimensions.
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| 2. |
- Ahmad, M. Rauf, et al.
(författare)
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Tests for high-dimensional covariance matrices using the theory of U-statistics
- 2015
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Ingår i: Journal of Statistical Computation and Simulation. - : Informa UK Limited. - 0094-9655 .- 1563-5163. ; 85:13, s. 2619-2631
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Nzabanita, Joseph, et al.
(författare)
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Bilinear regression model with Kronecker and linear structures for the covariance matrix
- 2015
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Ingår i: Afrika Statistika. - : Statistics and Probability African Society (SPAS). - 2316-090X. ; 10:2, s. 827-837
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, the bilinear regression model based on normally distributed random matrix is studied. For these models, the dispersion matrix has the so called Kronecker product structure and they can be used for example to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations in a flip-flop relation are established and the consistency of estimators is studied.
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| 4. |
- Pielaszkiewicz, Jolanta Maria, et al.
(författare)
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On n/p-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrice
- 2018
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Ingår i: The Electronic Journal of Linear Algebra. - Pensacola, FL, United States : International Linear Algebra Society. - 1537-9582 .- 1081-3810. ; 33, s. 24-40
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Tidskriftsartikel (refereegranskat)abstract
- The joint distribution of standardized traces of 1/n XX' and of (1/n XX')(2), where the matrix X : p x n follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition n/p ->(n,p ->infinity) c> 0. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of n and p.
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| 5. |
- Ekström, Magnus, 1966-, et al.
(författare)
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A general measure of skewness
- 2012
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Ingår i: Statistics and Probability Letters. - : Elsevier. - 0167-7152 .- 1879-2103. ; 82:8, s. 1559-1568
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Tidskriftsartikel (refereegranskat)abstract
- A very general measure of skewness based on the quantiles is introduced, which includes several well-known measures as special cases. Sample versions of our measure may be used as test statistics for testing the hypothesis of symmetry about an unknown value. We provide large sample theory for such a statistic and discuss the asymptotic relative efficiencies of this against some competing test statistics for symmetry.
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| 6. |
- Li, Ying
(författare)
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A two step model for linear prediction, with connections to PLS
- 2011
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In the thesis, we consider prediction of a univariate response variable, especially when the explanatory variables are almost collinear. A two step approach has been proposed. The first step is to summarize the information in the explanatory variables via a bilinear model with a Krylov structured design matrix. The second step is the prediction step where a conditional predictor is applied. The two step approach gives us a new insight in partial least squares regression (PLS). Explicit maximum likelihood estimators of the variances and mean for the explanatory variables are derived. It is shown that the mean square error of the predictor in the two step model is always smaller than the one in PLS. Moreover, the two step model has been extended to handle grouped data. A real data set is analyzed to illustrate the performance of the two step approach and to compare it with other regularized methods.
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| 7. |
- Pielaszkiewicz, Jolanta Maria, 1985-, et al.
(författare)
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Cumulant-moment relation in free probability theory
- 2014
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Ingår i: Acta et Commentationes Universitatis Tartuensis de Mathematica. - : Tartu University Press. - 1406-2283 .- 2228-4699. ; 18:2, s. 265-278
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Tidskriftsartikel (refereegranskat)abstract
- The goal of this paper is to present and prove a cumulant-moment recurrent relation formula in free probability theory. It is convenient tool to determine underlying compactly supported distribution function. The existing recurrent relations between these objects require the combinatorial understanding of the idea of non-crossing partitions, which has been considered by Speicher and Nica. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts. Similarly like the non-recursive formulation of Mottelson our formula demands only summing over partitions of the set. The proof of non-recurrent result is given with use of Lagrange inversion formula, while in our proof the calculations of the Stieltjes transform of the underlying measure are essential.
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| 8. |
- von Rosen, Dietrich
(författare)
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A generalization of the exponential function to model growth
- 2018
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Ingår i: International Journal of Applied Mathematics. - 1992-9978 .- 1992-9986. ; 48, s. 152-167
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Tidskriftsartikel (refereegranskat)abstract
- We generalize the exponential function to model instantaneous relative growth. The modified function is defined by a linear relationship between a continuous quantity (rather than time) and logarithmic relative growth. The corresponding formula is ln[q'(t)/q(t)] = a+b·q(t) where q'(t)/q(t) is instantaneous relative growth of a quantity q, t refers to time, a denotes initial logarithmic relative growth, and b is a shape parameter in terms of its sign, as well as a scaling parameter in terms of its magnitude. For calculating q(t), the exponential integral Ei[-b·q] = ∫(exp[-b·q]/q)dq is needed. The problem of taking the inverse of Ei[x] = zEi is addressed. In order to distinguish two possible solutions for given zEi, we define the two inverse functions Ei(-1)x > 0[zEi] and Ei(-1)x < 0[zEi]. An indirect method for their numerical evaluation is developed. With the generalized exponential function, one can model sigmoid growth (b < 0), exponential growth (b = 0), and explosive growth (b > 0), where the term "explosive growth" refers to a relative growth rate that increases with time. The resulting formula of generalized exponential growth is where qC is a calibrating quantity at time tC. For b = 0, the two functions equal the (standard) exponential function q(t) = qC· exp[(t-tc)·exp[a]]. In the case of sigmoid growth, the inflection point quantity is -1/b, which depends only on one parameter (b). Negative growth can be modeled by substituting t-tC with tC - t. Any two points of logarithmic relative growth can be connected unambiguously with the generalized exponential function, to derive the corresponding function of q(t). Furthermore, we derive formulas for the conversion of a segmented curve of logarithmic relative growth as a function of time, into an equivalent growth curve of q(t).Finally, the generalized exponential function is compared with a 2nd-degree polynomial and the nonlinear Schnute function. In conclusion, the generalized exponential function is useful for modeling a path of changing relative growth continuously, and to translate it into a growth curve of quantity as a function of time.
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| 9. |
- Ahmad, M. Rauf, et al.
(författare)
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Tests of Covariance Matrices for High Dimensional Multivariate Data Under Non Normality
- 2015
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Ingår i: Communications in Statistics - Theory and Methods. - : Informa UK Limited. - 0361-0926 .- 1532-415X. ; 44:7, s. 1387-1398
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 10. |
- Ahmed, S. Ejaz, et al.
(författare)
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Estimation of Several Intraclass Correlation Coefficients
- 2015
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Ingår i: Communications in statistics. Simulation and computation. - : Informa UK Limited. - 0361-0918 .- 1532-4141. ; 44:9, s. 2315-2328
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Tidskriftsartikel (refereegranskat)abstract
- An intraclass correlation coefficient observed in several populations is estimated. The basis is a variance-stabilizing transformation. It is shown that the intraclass correlation coefficient from any elliptical distribution should be transformed in the same way. Four estimators are compared. An estimator where the components in a vector consisting of the transformed intraclass correlation coefficients are estimated separately, an estimator based on a weighted average of these components, a pretest estimator where the equality of the components is tested and then the outcome of the test is used in the estimation procedure, and a James-Stein estimator which shrinks toward the mean.
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