Part-products of SS-restricted integer compositions

If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the random variable ${\bf B}(\vec{\lambda})=\prod_i \lambda_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.Comment: 18 page

Preprocessing algorithms for the digital histology of colorectal cancer

Pre-processing techniques were developed for cell identification algorithms. These algorithms which locate and classify cells in digital microscopy images are important in digital pathology. The pre-processing methods included image sampling and colour normalisation for standard Haemotoxilyn and Eosin (H&E) images and co-localisation algorithms for multiplexed images. Data studied in the thesis came from patients with colorectal cancer. Patient histology images came from `The Cancer Genome Atlas' (TCGA), a repository with contributions from many different institutional sites. The multiplexed images were created by TIS, the Toponome Imaging System. Experiments with image sampling were applied to TCGA diagnostic images. The effect of sample size and sampling policy were evaluated. TCGA images were also used in experiments with colour normalisation algorithms. For TIS multiplexed images, probabilistic graphical models were developed as well as clustering applications. NW-BHC, an extension to Bayesian Hierarchical Clustering, was developed and, for TIS antibodies, applied to TCGA expression data. Using image sampling with a sample size of 100 tiles gave accurate prediction results while being seven to nine times faster than processing the entire image. The two most accurate colour normalisation methods were that of Macenko and a `Nave' algorithm. Accuracy varied by TCGA site, indicating that researchers should use several independent data sets when evaluating colour normalisation algorithms. Probabilistic graphical models, applied to multiplexed images, calculated links between pairs of antibodies. The application of clustering to cell nuclei resulted in two main groups, one associated with epithelial cells and the second associated with the stromal environment. For TCGA expression data and for several clustering metrics, NW-BHC improved on the standard EM algorithm

Part-products of random integer compositions

A composition of n is a sequence of positive integers, called parts, that sum to n. Given a set S of positive integers, we consider compositions chosen randomly from a uniform distribution on the set of all compositions of n with parts in S. Three progressively more di cult choices of S are considered: unrestricted compositions, where S = Z+; 1-free compositions, where S = Z+nf1g; and S-restricted compositions, where S is an arbitrary co nite subset of Z+. For each choice of S, we regard the product of the parts as a random variable. We begin by deriving formulas for the moments of both the part-product and its logarithm and then proceed to the more challenging problem of proving that the part-product is asymptotically lognormal. In the case of unrestricted compositions, the calculations are relatively easy to complete using classical methods. However, those methods break down for the remaining two choices of S. We therefore introduce and formalize two new techniques for studying random compositions, the \embedding" technique and the \blocking" technique, which lead to proofs of the asymptotic lognormality of the product of parts for 1-free and S-restricted compositions respectively.Ph.D., Mathematics -- Drexel University, 201