**Redundancy Analysis (RDA)**

Redundancy analysis (RDA) is a form of constrained ordination that examines how much of the variation in one set of variables explains the variation in another set of variables. It is the multivariate analog of simple linear regression. Redundancy analysis is based on similar principles as principal components analysis and thus makes similar assumptions about the data. It is appropriate when the expected relationship between dependent and independent variables is linear (e.g. climate and allele frequency). … **Augmented Interval Markov Chains (AIMC)**

In this paper we propose augmented interval Markov chains (AIMCs): a generalisation of the familiar interval Markov chains (IMCs) where uncertain transition probabilities are in addition allowed to depend on one another. This new model preserves the flexibility afforded by IMCs for describing stochastic systems where the parameters are unclear, for example due to measurement error, but also allows us to specify transitions with probabilities known to be identical, thereby lending further expressivity. The focus of this paper is reachability in AIMCs. We study the qualitative, exact quantitative and approximate reachability problem, as well as natural subproblems thereof, and establish several upper and lower bounds for their complexity. We prove the exact reachability problem is at least as hard as the famous square-root sum problem, but, encouragingly, the approximate version lies in $\mathbf{NP}$ if the underlying graph is known, whilst the restriction of the exact problem to a constant number of uncertain edges is in $\mathbf{P}$. Finally, we show that uncertainty in the graph structure affects complexity by proving $\mathbf{NP}$-completeness for the qualitative subproblem, in contrast with an easily-obtained upper bound of $\mathbf{P}$ for the same subproblem with known graph structure. … **Variance Reduction (VR)**

In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates that can be obtained for a given number of iterations. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main ones are: Common random numbers, antithetic variates, control variates, importance sampling and stratified sampling. Under these headings are a variety of specialized techniques; for example, particle transport simulations make extensive use of ‘weight windows’ and ‘splitting/Russian roulette’ techniques, which are a form of importance sampling. …

# If you did not already know

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Nov 2017

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