We consider regularized least-squares (RLS) with a Gaussian kernel. We prove that if we let the Gaussian bandwidth σ → ∞ while letting the regularization parameter λ → 0, the RLS solution tends to a polynomial whose order is controlled by the rielative rates of decay of 1 σ2 and λ: if λ = σ−(2k+1), then, as σ → ∞, the RLS solution tends to the kth order polynomial with minimal empirical error. We illustrate the result with an example.

Recent sequencing of the human and other mammalian genomes has brought about the necessity to align them, to identify and characterize their commonalities and differences. Programs that align whole genomes generally use a seed-and-extend technique, starting from exact or near-exact matches and selecting a reliable subset of these, called anchors, and then filling in the remaining portions between the anchors using a combination of local and global alignment algorithms, but their choices for the parameters so far have been primarily heuristic. We present a statistical framework and practical methods for selecting a set of matches that is both sensitive and specific and can constitute a reliable set of anchors for a one-to-one mapping of two genomes from which a whole-genome alignment can be built. Starting from exact matches, we introduce a novel per-base repeat annotation, the $Z$-score, from which noise and repeat filtering conditions are explored. Dynamic programming-based chaining algorithms are also evaluated as context-based filters. We apply the methods described here to the comparison of two progressive assemblies of the human genome, NCBI build 28 and build 34 http://genome.ucsc.edu), and show that a significant portion of the two genomes can be found in selected exact matches, with very limited amount of sequence duplication.

With the consensus human genome sequenced and many other sequencing projects at varying stages of completion, greater attention is being paid to the genetic differences among individuals and the abilities of those differences to predict phenotypes. A significant obstacle to such work is the difficulty and expense of determining haplotypes--sets of variants genetically linked because of their proximity on the genome--for large numbers of individuals for use in association studies. This paper presents some algorithmic considerations in a new approach for haplotype determination: inferring haplotypes from localised polymorphism data gathered from short genome 'fragments.' Formalised models of the biological system under consideration are examined, given a variety of assumptions about the goal of the problem and the character of optimal solutions. Some theoretical results and algorithms for handling haplotype assembly given the different models are then sketched. The primary conclusion is that some important simplified variants of the problem yield tractable problems while more general variants tend to be intractable in the worst case.