Seriation, i.e., nding a suitable linear order for a set of objects given data and a loss or merit function, is a basic problem in data analysis. Caused by the problem’s combinatorial nature, it is hard to solve for all but very small sets. Nevertheless, both exact solution methods and heuristics are available. In this paper we present the package seriation which provides an infrastructure for seriation with R. The infrastructure comprises data structures to represent linear orders as permutation vectors, a wide array of seriation methods using a consistent interface, a method to calculate the value of various loss and merit functions, and several visualization techniques which build on seriation. To illustrate how easily the package can be applied for a variety of applications, a comprehensive collection of examples is presented.

/pdf/getting-things-in-order-an-introduction-to-the-r-package-3auf64xblb.pdf

Getting Things in Order: An Introduction to the R Package seriation

Summary


We introduce a new estimator, the simultaneous multiscale change point estimator SMUCE, for the change point problem in exponential family regression. An unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level α. The probability of overestimating the true number of change points K is controlled by the asymptotic null distribution of the multiscale test statistic. Further, we derive exponential bounds for the probability of underestimating K. By balancing these quantities, α will be chosen such that the probability of correctly estimating K is maximized. All results are even non-asymptotic for the normal case. On the basis of these bounds, we construct (asymptotically) honest confidence sets for the unknown step function and its change points. At the same time, we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log-term. Finally, the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n∞, even for an unbounded number of change points. We illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions. The performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy.

Multiscale change point inference

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in L 2 ([0,1)) our results cover other metrics like Skorokhod metric on the space of cadlag functions and uniform metrics on C([0, 1]). We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) Holder continuous functions of order 0 < α ≤ 1 and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.

/pdf/consistencies-and-rates-of-convergence-of-jump-penalized-4hz1jrfbuu.pdf

Consistencies and rates of convergence of jump-penalized least squares estimators

We introduce a new estimator SMUCE (simultaneous multiscale change-point estimator) for the change-point problem in exponential family regression. An unknown step function is estimated by minimizing the number of change-points over the acceptance region of a multiscale test at a level \alpha. The probability of overestimating the true number of change-points K is controlled by the asymptotic null distribution of the multiscale test statistic. Further, we derive exponential bounds for the probability of underestimating K. By balancing these quantities, \alpha will be chosen such that the probability of correctly estimating K is maximized. All results are even non-asymptotic for the normal case. Based on the aforementioned bounds, we construct asymptotically honest confidence sets for the unknown step function and its change-points. At the same time, we obtain exponential bounds for estimating the change-point locations which for example yield the minimax rate O(1/n) up to a log term. Finally, SMUCE asymptotically achieves the optimal detection rate of vanishing signals. We illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions. The performance of the proposed multiscale approach is illustrated by simulations and in two cutting-edge applications from genetic engineering and photoemission spectroscopy.

Multiscale Change-Point Inference

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in $L^2([0,1))$ our results cover other metrics like Skorokhod metric on the space of c\`{a}dl\`{a}g functions and uniform metrics on $C([0,1])$. We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) H\"{o}lder continuous functions of order $0<\alpha\le1$ and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.

We present very fast algorithms for the exact computation of estimators for time series, based on complexity penalized log-likelihood or M-functions. The algorithms apply to a wide range of functionals with morphological constraints, in particular to Potts or Blake–Zisserman functionals. The latter are the discrete versions of the celebrated Mumford–Shah functionals. All such functionals contain model parameters. Our algorithms allow for optimization not only for each separate parameter, but even for all parameters simultaneously. This allows for the examination of the models in the sense of a family approach. The algorithms are accompanied by a series of illustrative examples from molecular biology.

https://www.jstor.org/stable/info/27594299

Complexity Penalized M-Estimation

Among the rich material on graphical presentation of information in "La Graphique et le Traitement Graphique de l'Information" (1977), engl. "Graphics and Graphic Information Processing" (1981), Jaques Bertin discusses the presentation of data matrices, with a particular view to seriation.
"A Tribute to J. Bertin's Graphical Data Analysis",
presented at the SoftStat Conference '97, gives an appraisal of this aspect of Bertin's work.

Bertin's approach has been implemented in the Voyager system for data analysis. This is a video of the SoftStat '97 presentation using Voyager. Video recorded in Feb. 2014.

References:

J. Bertin: La Graphique et le Traitement Graphique de l'Information. Flammerion, Paris (1977)

J. Bertin: Graphics and Graphic Information Processing. De Gruyter, Berlin(1981)

A. de Falguerolles; F. Friedrich, G. Sawitzki: A Tribute to J. Bertin's Graphical Data Analysis. In: W. Bandilla, F. Faulbaum (eds.) Advances in Statistical Software 6.
Lucius&Lucius Stuttgart 1997 ISBN 3-8282-0032-X pp. 11 - 20.

G. Sawitzki: Extensible Statistical Software: On a Voyage to Oberon. Journal of Computational and Graphical Statistics Vol. 5 No 3 (1996)

SoftStat '97: the 9th Conference on the Scientific Use of Statistical Software, March 3-6, 1997, Heidelberg.


Project home page:
 

Author's home page:

http://www.statlab.uni-heidelberg.de/projects/bertin/Bertin.pdf

A Tribute to J. Bertin's Graphical Data Analysis

We present a heuristic algorithm for the choice of the wedgelet regularization parameter for the purpose of denoising in the case where the noise variance σ 2 is not known. Numerical experiments comparing wavelet thresholding with wedgelet denoising, and with the related schemes quadtree approximation and platelet approximation , allow to assess the respective strengths of the different approaches. For small values of σ 2 , wavelets are clearly superior to wedgelets, and they are better at restoring textured regions. For large σ 2 , or for images of a predominantly geometric nature, wedgelets yield consistently better results. Moreover, the tests reveal that the heuristic algorithm is quite effective in choosing the regularization parameter.

Multiscale wedgelet denoising algorithms

We introduce coupling from the past, a recently developed method for exact sampling from a given distribution. Focus is on rigour and thorough proofs. We stay on an elementary level which requires little or no prior knowledge from probability theory. This should fill an obvious gap between innumerable intuitive and incomplete reviews, and few precise derivations on an abstract level.

/pdf/an-elementary-rigorous-introduction-to-exact-sampling-5ausdgynjt.pdf

An elementary rigorous introduction to exact sampling

The thesis deals with variational approaches to the segmentation of images with particular emphasis on geometry-based algorithms. Criteria for the choice of segmentations are given by minimization of a suitable functional ranging over a certain prescribed segmentation class. The arising minimization problem is in general intractable. Therefore various geometric restrictions on the class of segmentations are formulated and characterized. Main contribution of the thesis is the design of fast and flexible algorithms for the computation of wedgelet-type approximations, based on a particularly efficient solution of a local regression problem. Here, the angular resolution can be prescribed in an arbitrary manner. For dyadic partitions, the implementation gives fast access to the entire scale of minimizers. Consistency results for the wedgelet estimators are formulated and proved. These results describe the asymptotic behavior of the wedgelet-minimizers of discretized continuous mages, as the pixel size tends to zero. They provide heuristics for the choice of the scaling parameter and yield convergence rates that are close to theoretically optimal. Experimental results substantiate the efficiency of the algorithms, they rate the dependency of the angular resolution and give hints for the choice of the scaling parameter.

Complexity Penalized Segmentations in 2D

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