Jason Schweinsberg

Prediction intervals for neural networks via nonlinear regression

Standard methods for computing prediction intervals in nonlinear regression can be effectively applied to neural networks when the number of training points is large. Simulations show, however, that these methods can generate unreliable prediction intervals on smaller datasets when the network is trained to convergence. Stopping the training algorithm prior to convergence, to avoid overfitting, reduces the effective number of parameters but can lead to prediction intervals that are too wide. We present an alternative approach to estimating prediction intervals using weight decay to fit the network and show via a simulation study that this method may be effective in overcoming some of the shortcomings of the other approaches.

Coalescent processes obtained from supercritical Galton-Watson processes

Consider a population model in which there are N individuals in each generation. One can obtain a coalescent tree by sampling n individuals from the current generation and following their ancestral lines backwards in time. It is well-known that under certain conditions on the joint distribution of the family sizes, one gets a limiting coalescent process as N-->[infinity] after a suitable rescaling. Here we consider a model in which the numbers of offspring for the individuals are independent, but in each generation only N of the offspring are chosen at random for survival. We assume further that if X is the number of offspring of an individual, then P(X[greater-or-equal, slanted]k)~Ck-a for some a>0 and C>0. We show that, depending on the value of a, the limit may be Kingmans coalescent, in which each pair of ancestral lines merges at rate one, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.

Beta-coalescents and continuous stable random trees

Coalescents with multiple collisions, also known as A-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure A is the Beta(2 - a, a) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly-Kurtz lookdown process using continuous random trees, which is of independent interest. This produces a number of results concerning the small-time behavior of Beta-coalescents. Most notably, we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics.

Small-time behavior of beta coalescents

For a finite measureon (0,1), the �-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate R 1 0 x k 2 (1 x) b k �(dx). It has recently been shown that if 1 < � < 2, the �-coalescent in whichis the Beta(2 �,�) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an �-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other �-coalescents for whichhas the same asymptotic behavior near zero as the Beta(2 �,�) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of �-coalescents.

The genealogy of branching Brownian motion with absorption

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order (logN) 3 , in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu’s continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the non-rigorous predictions by Brunet, Derrida, Muller, and Munier for a closely related model.

A waiting time problem arising from the study of multi-stage carcinogenesis

We consider the population genetics problem: How long does it take before some member of the population has m specified mutations? The case m = 2 is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for larger m are needed for colon cancer and other diseases where a sequence of mutations leads to cells with uncontrolled growth. 1. Introduction. It has long been known that cancer is the end result of several mutations that disrupt normal cell division. Armitage and Doll [1] did a statistical analysis of the age of onset of several cancers and fit power laws to estimate the number of mutations. Knudson [14] discovered that the incidence of retinoblastoma grows as a linear function of time in the group of children who have multiple cancers in both eyes, but as a slower quadratic function in children who only have one cancer. Based on this, Knudson proposed the concept of a tumor suppressor gene. Later it was confirmed that in the first group of children, one copy is already inactivated at birth, while in the second group both copies must be mutated before cancer occurs. Since that time, about 30 tumor suppressor genes have been identified. They have the property that inactivating the first copy does not cause a change, while inactivating the second increases the cells’ net reproductive rate, which is a step toward cancer. Over the last decade, a number of studies have been carried out to identify molecular pathways in the development of colorectal cancer. Among the earliest premalignant lesions are aberrant crypt foci (ACF). ACF are widely believed to be precursors to the adenomatous polyps, which in turn lead to colon carcinoma. The widespread use of colonoscopy is motivated by the fact that the early stages in this process can be seen long before cancer occurs. � RD and DS are partially supported by DMS 0202935 from the probability program

Random partitions approximating the coalescence of lineages during a selective sweep

When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep. Suppose we sample n individuals at the end of a selective sweep. If we focus on a site on the chromosome that is close to the location of the beneficial mutation, then many of the lineages will likely be descended from the individual that had the beneficial mutation, while others will be descended from a dierent individual because of recombination between the two sites. We introduce two approximations for the eect of a selective sweep. The first one is simple but not very accurate: flip n independent coins with probability p of heads and say that the lineages whose coins come up heads are those that are descended from the individual with the beneficial mutation. A second approximation, which is related to Kingman’s paintbox construction, replaces the coin flips by integer-valued random variables and leads to very accurate results.

Survival of Near-Critical Branching Brownian Motion

Consider a system of particles performing branching Brownian motion with negative drift

Self-similar fragmentations and stable subordinators

\mu= \sqrt{2 - \varepsilon}

Power laws for family sizes in a duplication model

and killed upon hitting zero. Initially there is one particle at x>0. Kesten (Stoch. Process. Appl. 7:9–47, 1978) showed that the process survives with positive probability if and only if ε>0. Here we are interested in the asymptotics as ε→0 of the survival probability Qμ(x). It is proved that if