"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

The Fractal Geometry of Nature

may 7th, 1986, Professor A. F. M. Smith in the Chair] SUMMARY A continuous two-dimensional region is partitioned into a fine rectangular array of sites or "pixels", each pixel having a particular "colour" belonging to a prescribed finite set. The true colouring of the region is unknown but, associated with each pixel, there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model. The aim is to reconstruct the true scene, with the additional knowledge that pixels close together tend to have the same or similar colours. In this paper, it is assumed that the local characteristics of the true scene can be represented by a nondegenerate Markov random field. Such information can be combined with the records by Bayes' theorem and the true scene can be estimated according to standard criteria. However, the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field. Thus, a simple, iterative method of reconstruction is proposed, which does not depend on these large-scale characteristics. The method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field. Some complications, including parameter estimation, are discussed. Potential applications are mentioned briefly.

https://stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagDirtyPicsJRSSB1986.pdf

On the statistical analysis of dirty pictures

The authors define ordered subset processing for standard algorithms (such as expectation maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialize application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al. (1981) for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT, the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained. >

Accelerated image reconstruction using ordered subsets of projection data

Feedforward neural networks trained by error backpropagation are examples of nonparametric regression estimators. We present a tutorial on nonparametric inference and its relation to neural networks, and we use the statistical viewpoint to highlight strengths and weaknesses of neural models. We illustrate the main points with some recognition experiments involving artificial data as well as handwritten numerals. In way of conclusion, we suggest that current-generation feedforward neural networks are largely inadequate for difficult problems in machine perception and machine learning, regardless of parallel-versus-serial hardware or other implementation issues. Furthermore, we suggest that the fundamental challenges in neural modeling are about representation rather than learning per se. This last point is supported by additional experiments with handwritten numerals.

Neural networks and the bias/variance dilemma

The Support Vector (SV) method was recently proposed for estimating regressions, constructing multidimensional splines, and solving linear operator equations [Vapnik, 1995]. In this presentation we report results of applying the SV method to these problems.

/pdf/support-vector-method-for-function-approximation-regression-8g30ea5q8x.pdf

Support Vector Method for Function Approximation, Regression Estimation and Signal Processing

Positron emission tomography (PET)—still in its research stages—is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner. Much as in X-ray transmission tomography and other modes of computerized tomography, the quality of the reconstructed image in PET is very sensitive to the mathematical algorithm to be used for reconstruction. In this article, we tailor a mathematical model to the physics of positron emissions, and we use the model to describe the basic image reconstruction problem of PET as a standard problem in statistical estimation from incomplete data. We describe various estimation procedures, such as the maximum likelihood (ML) method (using the EM algorithm), the method of moments, and the least squares method. A computer simulation of a PET experiment is then used to demonstrate the ML and the least squares reconstructions. The main purposes of this article are to report on what we believe is an...

A Statistical Model for Positron Emission Tomography

https://repository.upenn.edu/cgi/viewcontent.cgi?article=1463&context=statistics_papers

Risk vs. profit potential: A model for corporate strategy

Arcs of lengthsl
 n, 0<l
 n+1<=l
 n<1,n=1,2,…, are thrown independently and uniformly on a circumferenceC of unit length. The union of the arcs coversC with probability one if and only if
 
$$\sum\limits_{n = 1}^\infty {n^{ - 2} \exp \left( {l_1 + ... + 1l_n } \right) = \infty } $$

 .

Covering the circle with random ARCS

Let ${\eta}=\left({\eta_j}\right)$ be a sequence of independent Gaussian, means 0, Varriance 1, random variable in all of this paper. We shall then study the distribution of 

$$q\left( {\eta}\right)\quad {\rm or} \quad q\left({\eta- a}\right)$$

where $q: {\mathbb{R}}^{\infty}\rightarrow {\mathbb{\bar{R}}}_+=\left[0,\infty\right]$ is a seminorm and a $\in \mathbb{R^{\infty}}.$ In particular we shall study the behavior of $P\left(q{\left({\eta}\right)}\,{\leq}\, {t}\right)\quad {\rm as} \quad {t}\,{\rightarrow \, 0}.$

/pdf/on-the-lower-tail-of-gaussian-seminorms-41wokn4l9x.pdf

On the Lower Tail of Gaussian Seminorms

gree than the retained sequences (“exons”) We have applied the information theoretic notion of entropy to characterize DNA sequences We consider a genetic sequence signal that is too small for asymptotic entropy estimates to be accurate, and for which similar approaches have previously failed We prove that the match length entropy estimator has a relatively fast converge rate and demonstrate experimentally that by using this entropy estimator, we can indeed extract a meaningful signal from segments of DNA Further, we derive a method for detecting certain signals within DNA known as splice junctions with significantly better performance than previously known methods

Larry A. Shepp

Papers

A Statistical Model for Positron Emission Tomography

Risk vs. profit potential: A model for corporate strategy

Covering the circle with random ARCS

On the Lower Tail of Gaussian Seminorms

On the entropy of DNA: algorithms and measurements based on memory and rapid convergence