Phylogenetic effective sample size

In this paper I address the question - how large is a phylogenetic sample I propose a definition of a phylogenetic effective sample size for Brownian motion and Ornstein-Uhlenbeck processes - the regression effective sample size. I discuss how mutual information can be used to define an effective sample size in the non-normal process case and compare these two definitions to an already present concept of effective sample size (the mean effective sample size). Through a simulation study I find that the AICc is robust if one corrects for the number of species or effective number of species. Lastly I discuss how the concept of the phylogenetic effective sample size can be useful for biodiversity quantification, identification of interesting clades and deciding on the importance of phylogenetic correlations

The phylogenetic effective sample size and jumps

The phylogenetic effective sample size is a parameter that has as its goal the quantification of the amount of independent signal in a phylogenetically correlated sample. It was studied for Brownian motion and Ornstein-Uhlenbeck models of trait evolution. Here, we study this composite parameter when the trait is allowed to jump at speciation points of the phylogeny. Our numerical study indicates that there is a non-trivial limit as the effect of jumps grows. The limit depends on the value of the drift parameter of the Ornstein-Uhlenbeck process

PCA consistency in high dimension, low sample size context

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a $\rho$-mixing condition and a broad range of sphericity measures of the covariance matrix.Comment: Published in at http://dx.doi.org/10.1214/09-AOS709 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rethinking the Effective Sample Size

The effective sample size (ESS) is widely used in sample-based simulation methods for assessing the quality of a Monte Carlo approximation of a given distribution and of related integrals. In this paper, we revisit and complete the approximation of the ESS in the specific context of importance sampling (IS). The derivation of this approximation, that we will denote as $\widehat{\text{ESS}}$, is only partially available in Kong [1992]. This approximation has been widely used in the last 25 years due to its simplicity as a practical rule of thumb in a wide variety of importance sampling methods. However, we show that the multiple assumptions and approximations in the derivation of $\widehat{\text{ESS}}$, makes it difficult to be considered even as a reasonable approximation of the ESS. We extend the discussion of the ESS in the multiple importance sampling (MIS) setting, and we display numerical examples. This paper does not cover the use of ESS for MCMC algorithms

On bootstrap sample size in extreme value theory

It has been known for a long time that for bootstrapping theprobability distribution of the maximum of a sample consistently,the bootstrap sample size needs to be of smaller order than theoriginal sample size. See Jun Shao and Dongsheng Tu (1995), Ex.3.9,p. 123. We show that the same is true if we use the bootstrapfor estimating an intermediate quantile.Bootstrap;Regular variation