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Metropolis Hastings R Code
This page explains the basic ideas behind DRAM and provides examples and Matlab code for the computations. Translated the New York Fed's DSGE model code from MATLAB to Julia, implementing solving linear rational expectations models, gradient-descent optimization, the Kalman filter, and sampling with. Suppose you want to simulate samples from a random variable which can be described by an arbitrary PDF, i. The most basic use in applied or indeed theoretical disciplines is to repeatedly and. Below is a list of all packages provided by project gregmisc. You'd think this is hard, but in fact there are methods that work in principle for any density. A later paper by Hastings (1970) expanded on the technique ¥Gibbs sampling was invented later, and Þrst described. Familiarity with MCMC methods in general is assumed, however. Keith Hastings (1970) recognized the potential of the Metropolis algorithm to solve statistical problems. Click here to download the code. The Metropolis{Hastings algorithm C. Massively parallel CMB map-making code development status (Hamza); Using S4CMB software package for quick, realistic simulations (Clara). Matlab code taken from Professor Joo-Ho Choi. Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models: Abstract: This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. Think of recording the number of steps. EE641DigitalImageProcessingII:PurdueUniversityVISE-November14,2012 13 Coding Method/Maximum Pseudolikelihood[1, 2] 4 pt Neighborhood Code 1 Code 2 Code 3 Code 4 • Assume a 4 point neighborhood • Separate points into four groups or codes. Kockelman, Associate Professor & William J. MCMC Review ¥The ideas have been known for a long time ¥Metropolis-Hastings sampling was developed in the 1950s by physicists. Then the M-H algorithm is deﬁned by two. A more comprehensive list of Markov chain Monte Carlo (MCMC) algorithms available in a public R package, LaplacesDemon. Metropolis-Hastings R. In this website you will find R code for several worked examples that appear in our book Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. , Urbana, IL, 61801 Abstract. I am an open source contributor on a number of libraries, notably PyMC3, which is a library for probabilistic programming in Python. The code about to calculate the energy ( get_dH ) appears first. The Rayleigh distribution is used to model lifetime subject to rapid aging, because the hazard rate is linearly increasing. Illustrating the vagaries of Metropolis-Hastings Markov chain generation: different trace plots and histograms for the same model (Maxwell target with Exponential proposal). Official MapQuest website, find driving directions, maps, live traffic updates and road conditions. Like MATLAB and Java, R is an interpreted language, although like both of those languages R can be compiled or interact with compiled modules from the “big three” of compiled mathematical languages (C, C++, and FORTRAN descendents) to improve efficiency. Metropolis-Hastings algorithm on xβ; Matlab implementation of Random-Walk Metropolis; R implementation of Random-Walk Metropolis; IA2RMS is a Matlab code of the Independent Doubly Adaptive Rejection Metropolis Sampling method for drawing from the full-conditional densities within a Gibbs sampler. Massively parallel CMB map-making code development status (Hamza); Using S4CMB software package for quick, realistic simulations (Clara). Alternately, fGarch or rugarch for R. You can add to it!. Implementation of the Berry and Berry model is shown in S-PLUS code, using S+flexBayes and links to the BUGS. Click here to download the code. on Metropolis-Hastings sampling). Course Announcement. Random-walk Metropolis-Hastings¶ A practical implementation of the Metropolis-Hastings algorithm makes use of a random-walk proposal. an R function that evaluates the unnormalized probability density of the desired equilibrium distribution of the Markov chain. The Metropolis-Hastings algorithm enables us to draw a few time series realizations {θ t}, t= 0 to N, from a Markov chain with a speciﬁed stationary distribution p(θ) The algorithm works for any f(θ) ∝ p(θ), i. Major changes made in this edition from the previous edition include: more examples with discussion of computational details in chapters on Gibbs sampling and. Stochastic Volatility An experimental approach The Metropolis-Hastings Algorithm Monte Carlo Methods based on Markov Chains Running Monte Carlo via Markov Chains (2) Idea For an arbitrary starting value x(0), an ergodic chain (X(t)) is generated using a transition kernel with stationary distribution f. Details: It is a capture recapture study, with seven total draws from the population. One really interesting question from a CS 281 assignment this past semester involved comparing Metropolis-Hastings and slice sampling on a joint distribution. We're going to look at two methods for sampling a distribution: rejection sampling and Markov Chain Monte Carlo Methods (MCMC) using the Metropolis Hastings algorithm. You can only submit a maximum of two files, one PDF and one R file with code. In this paper, we developed a 1D Bayesian inversion code based Metropolis-Hastings algorithm whereby the uncertainty of the earth model parameters were quantified by examining the posterior model distribution. One of the benefits of PyMC3 is the friendly, simple API. Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models: Abstract: This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. Series and solutions. This database contains models, codes, data and publications pertaining to smart and multifunctional materials including PZT, PLZT, SMA, terfenol-D, lead titanate and lead zirconate. Simple Metropolis Hastings sampler. How can I do thinning in R program (I wrote the code not a function in R)? So, I used Metropolis-Hastings to generate MCMC samples from conditional posterior density of each parameter. I will suggest several tips, and discuss common beginner's mistakes occuring when coding from scratch a Metropolis-Hastings algorithm. This means that going from a 1 to a 5 is coded as equivalent to going from a 4 to a 5 which is obviously not ideal from a measurement standpoint. Translated the New York Fed's DSGE model code from MATLAB to Julia, implementing solving linear rational expectations models, gradient-descent optimization, the Kalman filter, and sampling with. In a previous post, I demonstrated how to use my R package MHadapive to do general MCMC to estimate Bayesian models. of steps needed to get independent sample is of order (σ max. Link to: My R and Python Video Tutorials. These extend beyond the current (Stata 14. [Reading: R cookbook Chapter 6] Lecture 8 : Split/apply/combine part 2 [ R code ] [Reading: Wickham, The Split/Apply/Combine Strategy for Data Analysis ]. seed(101) #loglikelihood logl <;- function(b,data) { ly = len. The only reason why the Metropolis works for the function is because I have added a step function to make areas outside the interval of $[0,\pi]$ to be zero. R Code 10, Blocked Sampling; R Code 8 / Metropolis Hastings Steps. The Metropolis-Hastings Sampler is the most common Markov-Chain-Monte-Carlo (MCMC) algorithm used to sample from arbitrary probability density functions (PDF). In comparison to rejection sampling where we always throw away the rejected samples, here we sometimes keep those samples as well. Wiley Examples (2nd ed) Examples for the second edition. on Metropolis-Hastings sampling). Rather as I did with my Metropolis-Hastings Mata code, I have stored the values of the random effects in the data matrix, X, so they do not get written to the results file. In this post, we will investigate the Metropolis-Hastings algorithm, which is still one of the most popular algorithms in the field of Markov chain Monte Carlo methods, even though its first appearence (see [1]) happened in 1953, more than 60 years in the past. 1 (Metropolis-Hastings algorithm) data for Examples 7. In such cases, the Metropolis-Hastings algorithm is used to produce a Markov chain say X 1,X 2,. Major changes made in this edition from the previous edition include: more examples with discussion of computational details in chapters on Gibbs sampling and. This database contains models, codes, data and publications pertaining to smart and multifunctional materials including PZT, PLZT, SMA, terfenol-D, lead titanate and lead zirconate. One continues a random search, making proposed moves if they decrease f, but only with some probability if they don't, as in Metropolis-Hastings. By the way, if you are new to this then write this code yourself, don't just read it. Rizzo ### ### Chapman & Hall / CRC ### ### ISBN 9781584885450 ### ### ### ### R code for Chapter 9 Examples. Python is the used a lot in many fields including physics and is strong in the big-data arena, more so than any of the other packages mentioned above. Design matrix for regression. Keith Hastings (1970) recognized the potential of the Metropolis algorithm to solve statistical problems. This function allows a user to construct a sample from a user-defined R function using a random walk Metropolis algorithm. smpl = mhsample(,'symmetric',sym) draws nsamples random samples from a target stationary distribution pdf using the Metropolis-Hastings algorithm. describe what is known as the Metropolis algorithm (see the section Metropolis and Metropolis-Hastings Algorithms). This article is a self‐contained introduction to the Metropolis-Hastings algorithm, the ubiquitous tool for producing dependent simulations from an arbitrary distribution. What does RWMH mean in Statistics? This page is about the meanings of the acronym/abbreviation/shorthand RWMH in the Academic & Science field in general and in the Statistics terminology in particular. Rizzo ### ### Chapman & Hall / CRC ### ### ISBN 9781584885450 ### ### ### ### R code for Chapter 9 Examples. Maybe (in my view probably) also easier to code your own samplers than R as I really prefer python syntax to R. We work every day to bring you discounts on new products across our entire store. Metropolis-Hastings and Shuffled Complex Evolution Metropolis algorithms for three case studies with increasing complexity. Metropolis-Hastings algorithm¶ There are numerous MCMC algorithms. Update from (s 1) 1. This is an R package developed and re-cently released by Buckner et al. Moudud Alam Advanced Statistical Modelling. Ask Question It is indeed a very poor idea to start learning a topic just from an on-line code with no explanation. Generalized through work done by Hastings in the 1970's. I couldn't find a simple R code for random-walk Metropolis sampling (the symmetric proposal version of Metropolis Hastings sampling) from a multivariate target distribution in arbitrary dimensions, so I wrote one. Syllabus: mcmc_syllabus. All that's required is the ability to sample the distribution at x. Metropolis-Hastings Generative Adversarial Networks Ryan Turner Uber AI Labs Jane Hung Uber AI Labs Yunus Saatci Uber AI Labs Jason Yosinski Uber AI Labs Abstract We introduce the Metropolis-Hastings generative adversarial network (MH-GAN), which combines aspects of Markov chain Monte Carlo and GANs. R Code 10, Blocked Sampling; R Code 8 / Metropolis Hastings Steps. In the code below a part of the proposal function for the item parameters is given. Number of MCMC draws to take. 85 sec, cpu time: 2. edu/~cgates/PERSI/papers/MCMCRev. DRAM is a combination of two ideas for improving the efficiency of Metropolis-Hastings type Markov chain Monte Carlo (MCMC) algorithms, Delayed Rejection and Adaptive Metropolis. The 'Metropolis' function is the main function for all Metropolis based samplers in this package. The purpose of this "answer" is to provide a clear statement of the Metropolis-Hastings algorithm and its relation to the Metropolis algorithm in hopes that this would aid the OP in modifying the code him- or herself. Metropolis-Hastings algorithm: In statistics and in statistical physics, the Metropolis-Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. Generalized linear model: Probit. Internals The internals section of the manual provides some insight into how Hakaru is implemented and offers guidance into how the system can be extended. Massively parallel CMB map-making code development status (Hamza); Using S4CMB software package for quick, realistic simulations (Clara). } \item{newtonSteps}{ Only used if \code{whichModel=2}. Hastings (1970) general-ized the Metropolis algorithm, and simulations following his scheme are said to use the Metropolis-Hastings algorithm. Metropolis-Hastings MCMC: Intro & some history An implementation of MCMC. edu/~cgates/PERSI/papers/MCMCRev. Results are shown when the number of iterations (N. Difficulty level: Not rated yet Previously, we introduced Bayesian Inference with R using the Markov Chain Monte Carlo (MCMC) techniques. From my CSE845 class at Michigan State University. Simple Example of a Metropolis-Hastings Algorithm in R (www. To do this, run the R code named "quakemetrophf15. The MH algorithm generalizes the Metropolis to use asymmetric proposal distributions and uses an $$r$$ to correct for asymmetry 52. Using simple toy examples we review their theoretical underpinnings, and in particular show why adaptive MCMC algorithms might fail when some fundamental properties are not satisfied. By the way, if you are new to this then write this code yourself, don't just read it. Rizzo ### ### Chapman & Hall / CRC ### ### ISBN 9781584885450 ### ### ### ### R code for Chapter 9 Examples. In Metropolis based unbiased rendering however, the next sample is usually constructed by applying small mutations to the current sample. You can add to it!. This is an R package developed and re-cently released by Buckner et al. Metropolis-Hastings Algorithm. Stan can run from various data analysis environments such as Stata, R, Python, and Julia and also has a command-line interface (CmdStan). Tobias The Metropolis-Hastings Algorithm. /r/programming is a reddit for discussion and news about computer programming. You might call call this strategy static adaptive Metropolis-Hastings. Stats 535 Lecture 6: More Regression with Linear Models, Math and Probability in R, Metropolis-Hastings Algorithm, k-Nearest Neighbors and Classification of Handwritten Digits Thomas Fiore May 29, 2019. com and find the best online deals on everything for your home. ) Stata now has Bayesian methods using Metropolis. However, we do not have to compute these values! (Show R-code demo_toyMCMC2. Metropolis-Hasting Example in R. MCMC: Metropolis Algorithm Proposition (Metropolis works): - The p ij 's from Metropolis Algorithm satisfy detailed balance property w. I’ll illustrate the algorithm, give some R code results (all code posted on my GitHub ), and then profile the R code to identify the bottlenecks in. Easiest to prove for random scan Metropolis which just chooses a component to update at random, and then updates according to a 1-dimensional Metropolis. Some R code helpful for some of the solutions in HW 1; Sketches of partial solutions for selected HW 1 problems; Some R code helpful for some of the solutions in HW 2. Before using this code, the following libraries must be installed in R: MASS,. Note: the code here is designed to be readable by a beginner, rather than "efficient". In the Metropolis-Hastings algorithm you have the extra part added in the second code block but in the Metropolis there isn't such a thing. After that, in standard_approach , comes the MCMC implementation. via the multivariate sampling technique described in Gamerman (1997). Gibbs Sampling is a special case of the Metropolis-Hastings algorithm which generates a Markov chain by sampling from the full set of conditional distributions. PyMCMC contains classes for Gibbs, Metropolis Hastings, independent Metropolis Hastings, random walk Metropolis Hastings, orientational bias Monte Carlo and slice samplers as well as specific modules for common models such as a module for Bayesian regression analysis. This toolbox provides tools to generate and analyse Metropolis-Hastings MCMC chains using multivariate Gaussian proposal distribution. Markov Chain Monte Carlo for Bayesian Inference - The Metropolis Algorithm By QuantStart Team In previous discussions of Bayesian Inference we introduced Bayesian Statistics and considered how to infer a binomial proportion using the concept of conjugate priors. zip" ﬁle contains R code to implement the algorithms and Bayesian model comparison methods discussed in the paper. L’Algorithme Metropolis-Hastings Projet de recherche CRSNG Vanessa Bergeron Laperrière (supervisée par Mylène Bédard) Été 2010 Département de Mathématiques et Statistique Université de Montréal 1 Introduction Ce rapport conclut un stage de recherche de premier cycle du CRSNG, effectué sous la supervision de la professeure Mylène Bédard. MCMC and likelihood-free methods The Metropolis-Hastings Algorithm (Fortran)code, and of a uniform prior on a hypercube. The generalisation of the Metropolis algorithm is the Metropolis-Hastings algorithm. are called \Random-walk Metropolis algorithm. · The R code for IRLS. 4 MCMC AND GIBBS SAMPLING The probability that the chain has state value s i at time (or step) t+1is given by the Chapman-Kolomogrov equation, which sums over the probability of being in a particular state at the current step and the transition probability from. Probability Density Estimation Univariate Density Estimation Kernel Density Estimation Bivariate and Multivariate Density Estimation Other Methods of Density Estimation Exercises R Code. ouY need to specify the type, if you can the range (e. Fellow of Civil Engineering. Efficiently sampling the conformational space of complex molecular systems is a difficult problem that frequently arises in the context of molecular dynamics (MD) si. I am making this list from the top of my mind, so feel free to propose suggestions by commenting to this post. A special case of the Metropolis–Hastings. draws from f is often infeasible. Chapter 4 Example: Dempster, Laird, and Rubin (1977) multinomial example. The estimate is based on the fast Cholesky decomposition of the sparse block band (tridiagonal) matrix. The basic version is the Metropolis algorithm (Metropolis et al, 1953), which was generalized by Hastings (1970). A function which is proportional to the distribution we wish to sample from is passed to the algorithm. Flegal University of California, Riverside, CA February 12, 2016. In the code below a part of the proposal function for the item parameters is given. This is an R package developed and re-cently released by Buckner et al. Simulations were not readily accepted by the statistical community at that time. MCMC algorithms such as Metropolis--Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. Metropolis Algorithm 1) Start from some initial parameter value c 2) Evaluate the unnormalized posterior p( c) 3) Propose a new parameter value Random draw from a "jump" distribution centered on the current parameter value 4) Evaluate the new unnormalized posterior p( ) 5) Decide whether or not to accept the new value. 3 Zones: Identifying Random Choices 5. #sample from a standard normal using MH with a random walk proposals. , the proposal is always accepted Thus, Gibbs sampling produces a Markov chain whose stationary distribution is the posterior distribution, for all the same reasons that the Metropolis-Hastings algorithm works Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 23/30. Metropolis-within-Gibbs sampler for Binomial and Poisson Metropolis-Hastings rdrr. 85 sec, cpu time: 2. (1953) • It was then generalized by Hastings in Hastings (1970) • Made into mainstream statistics and engineering via the articles Gelfand and Smith (1990) and Gelfand et al. This blog is where we post additional examples for our books about SAS and R (Amazon: SAS and R. Moudud Alam Advanced Statistical Modelling. The button below opens a separate window from your browser containing a demonstation of some of the most common chains which are used for this purpose. Comparison of sampling techniques for Bayesian parameter estimation. Metropolis-Hastings Equilibrium states to communicate. See chapters 29 and 30 in MacKay’s ITILA for a very nice introduction to Monte-Carlo algorithms. He wanders off briefly in the middle, but then goes back to the good stuff. the following is an implementation of the standard Metropolis Hastings Monte Carlo sampler. GPU functions have been developed by following the sample code available in the gputools. In this example the parameter estimates are not too bad, a little off given the small number of data points but this at least demonstrates the implementation of the Metropolis algorithm. MCMC algorithms such as the Metropolis-Hastings algorithm (Metropolis et al. DRAM is a combination of two ideas for improving the efficiency of Metropolis-Hastings type Markov chain Monte Carlo (MCMC) algorithms, Delayed Rejection and Adaptive Metropolis. For MLE, you might look at the source code for Kevin Sheppard's MFE toolbox for Matlab. However, with simulation method, we can also reach a satisfying result. Metropolis-Hastings algorithms are a class of Markov chains which are commonly used to perform large scale calculations and simulations in Physics and Statistics. Some Notes on Markov Chain Monte Carlo (MCMC) John Fox 2016-11-21 1 Introduction These notes are meant to describe, explain (in a non-technical manner), and illustrate the use of Markov Chain Monte Carlo (MCMC) methods for sampling from a distribution. This post shows how we can use Metropolis-Hastings (MH) to sample from non-conjugate conditional posteriors within each blocked Gibbs iteration – a much better alternative than the grid method. These extend beyond the current (Stata 14. Stats 535 Lecture 6: More Regression with Linear Models, Math and Probability in R, Metropolis-Hastings Algorithm, k-Nearest Neighbors and Classification of Handwritten Digits Thomas Fiore May 29, 2019. Order pizza, pasta, sandwiches & more online for carryout or delivery from Domino's. MCMC algorithms for ﬁtting Bayesian models - p. Tweedie, Exponential Convergence of Langevin Distributions and Their Discrete Approximations (1996) [4] Li, Tzu-Mao, et al. The R implementation runs about 4-7 times slower. 13 in Robert and Casella, 2004. on Metropolis-Hastings sampling). Springer, New York. This function generates simulated realisations from any of a range of spatial point processes, using the Metropolis-Hastings algorithm. MCMC method dubbed Metropolis-Hastings. Markov chain Monte Carlo for Ordinal Data Factor Analysis Model Description. "Tune" the Metropolis sampler by finding a value of alpha which results in an acceptance probability of around 0. Jones, et al recommend a batch size of $$N^{1/2}$$ (or the smallest integer closest to that) and that is the default setting for their software. From my CSE845 class at Michigan State University. Ivan Jeliazkov Department of Economics, University of California, Irvine, 3151 Social Science Plaza, Irvine, CA. com, amazon. Metropolis Light Transport is an application of the Metropolis-Hastings algorithm. "Tune" the Metropolis sampler by finding a value of alpha which results in an acceptance probability of around 0. Lecture 7: The Metropolis-Hastings Algorithm. AFAIK no other MCMC software is veri ed this well. Metropolis-Hastings Algorothm. I'm storing it here so it is not lost in history. 2 Lecture Notes - Part A Simulation - Oxford TT 2011 of view, the eﬃciency of such generation of random variables can be analysed. Metropolis-Hastings Sampling I When the full conditionals for each parameter cannot be obtained easily, another option for sampling from the posterior is the Metropolis-Hastings (M-H) algorithm. Being secret, the work of von Neumann and Ulam required a code name. One continues a random search, making proposed moves if they decrease f, but only with some probability if they don't, as in Metropolis-Hastings. The Rayleigh distribution is used to model lifetime subject to rapid aging, because the hazard rate is linearly increasing. uk Department of Statistics, University of Leeds, UK Adaptive rejection sampling (ARS) is a method for efficiently sampling from any univariate probability density function which is log-concave. In this post, I give an educational example of the Bayesian equivalent of a linear regression, sampled by an MCMC with Metropolis-Hastings steps, based on an earlier…. Some R code helpful for some of the solutions in HW 1; Sketches of partial solutions for selected HW 1 problems; Some R code helpful for some of the solutions in HW 2. The optimal Bayes multitarget tracking problem is formulated in the random finite set framework and a particle marginal Metropolis-Hastings (PMMH) technique which is a combination of the Metropolis-Hastings (MH) algorithm and sequential Monte Carlo methods is applied to compute the multi-target posterior distribution. This function uses the MetropolisHastings algorithm to draw a sample from a correlated bivariate normal target density using a random walk candidate and an independent candidate density respectively where we are drawing both parameters in a single draw. Aperiodicity A Markov chain taking only ﬁnite number of values is aperiodic if greatest common divisor of return times to any particular state, say, is 1. 8) MATLAB code for Metropolis-Hastings with burn-in and lag (COMPSCI 3016: Computational Cognitive Science by Dan Navarro & Amy Perfors, University of Adelaide) 9) Blog on metropolis-hastings 10) Gibbs sampling (wikipedia). Some of the codes are my own and the rest are either derived or taken from the R codes are taken from various resources such as matrix examples in R tutorial by Prof. Just because it has a computer in it doesn't make it programming. (1990) which presented the Gibbs sampler as used in Geman and Geman (1984). Now, for the weirdness. py python wrapper mcmc. strategies are compared using conventional means for assessing Metropolis–Hastings eﬃciency, as wellaswithanovelmetric called the conditional acceptance rate for assessing theconsequencesof usinganestimated,andnotexact,likelihood. If updating a single scalar, it is recommended that r be around 40%. Implementation of the Berry and Berry model is shown in S-PLUS code, using S+flexBayes and links to the BUGS. IMPROVING ON THE INDEPENDENT METROPOLIS-HASTINGS ALGORITHM Yves F. The optimal Bayes multitarget tracking problem is formulated in the random finite set framework and a particle marginal Metropolis-Hastings (PMMH) technique which is a combination of the Metropolis-Hastings (MH) algorithm and sequential Monte Carlo methods is applied to compute the multi-target posterior distribution. See chapters 29 and 30 in MacKay’s ITILA for a very nice introduction to Monte-Carlo algorithms. 2 Gibbs Sampling Gibbs sampling is a special case of Metropolis-Hastings updating, and MCM-Cped uses Gibbs sampling to sample genotypes and parents. As usual, I'll be providing a mix of intuitive explanations, theory and some examples with code. After that, in standard_approach , comes the MCMC implementation. Problem 7: Metropolis-Hastings Algorithm. ##### ### R code for Lec 8 Examples ### ##### ### Example 1 (Metropolis-Hastings sampler) f - function(x, sigma) { if (any(x 0)) return (0) stopifnot(sigma > 0. Order pizza, pasta, sandwiches & more online for carryout or delivery from Domino's. For a given target distribution …, the proposal q is admissible if supp …(x) ‰ [xsupp q(¢jx): Metropolis-Hastings Algorithm is universal. Metropolis Algorithm vs. : R package Statistical inference for partially observed models, includes bootstrap filter, iterated filtering and particle Marginal Metropolis-Hastings. The second case. • Metropolis-Hastings – allows use of nonsymmetric trial functions, t – Markov Chain Monte Carlo in Practice, W. Kockelman, Associate Professor & William J. R code to run an **MCMC** chain using a **Metropolis-Hastings** algorithm with a Gaussian proposal distribution. Glenn Meyers Introduction to Bayesian MCMC Models. The source code for this procedure is available at Murali Haran’s web site. I am trying to write a program to estimate AR(1) coefficients using metropolis-hastings algorithm. We assume that this target distribution has a density. of Physics University of Illinois Urbana-Champaign 1110 W. R Code 10, Blocked Sampling; R Code 8 / Metropolis Hastings Steps. Metropolis Hastings sampler for several parameters are not closed form, so I need to use Metropolis Hastings sampler within Gibbs sampler. Chapter 4 Example: Dempster, Laird, and Rubin (1977) multinomial example. In addition, R is designed to interface well with other technologies, the Metropolis Hastings algorithm – that use Markov chains and appear. Markov Chain Monte Carlo for Bayesian Inference - The Metropolis Algorithm By QuantStart Team In previous discussions of Bayesian Inference we introduced Bayesian Statistics and considered how to infer a binomial proportion using the concept of conjugate priors. Other arguments arbitrary. com, See also here. View source: R/bivnormMH. de Zea Bermudez, K. Calculated the distribution's mean and variance-covariance matrix. Practical Data Analysis with JAGS using R Department of Biostatistics Institute of Public Health, University of Copenhagen Tuesday 1st January, 2013 Computer practicals. (Another is Gibbs sampling. Another resource is the StatSoftEquivs wiki. This is an R package developed and re-cently released by Buckner et al. metropolis_hastings. Description. Metropolis-Hastings Algorothm. For the moment, we only consider the Metropolis-Hastings algorithm, which is the simplest type of MCMC. Try out the R code for the Metropolis-Hastings independence sampler. A statistical approach in nature is a possible solution to this problem as it can provide extensive information about unknown parameters. JAGS Code 1: My first few models; R Code 1 : Bayes Rule; R Code 2, Beta Binomial; R Code 3, Normal + R Code 4: My first chain; R Code 5: Hierarchical; R Code 6, Mixtures; R Code 7, Race; R Code 8, Metropolis Hastings; R Code 9: Probit Model; Readings; R Code 10, Blocked Sampling. A function which is proportional to the distribution we wish to sample from is passed to the algorithm. Yosinski We introduce the Metropolis-Hastings generative adversarial network (MH-GAN), which combines aspects of Markov chain Monte Carlo and GANs. In the Metropolis-Hastings example above, the Markov Chain was allowed to move in both di-. You'd think this is hard, but in fact there are methods that work in principle for any density. I'll illustrate the algorithm, give some R code results (all code posted on my GitHub ), and then profile the R code to identify the bottlenecks in. jp, barnesandnoble. ouY need to specify the type, if you can the range (e. Guidelines. Bayesian Statistics and R. A nice feature of the book is the use of real data, called by the author ‘Case Studies’, for examples. Again we can use V(x) = ˇ(x) 1=2. Use the Metropolis-Hastings-within-Gibbs algorithm to obtain estimates of the posterior means and 95% equal-tail credible intervals of $$\theta$$ and $$b$$ for the earthquake data described above. algorithm and the random walk Metropolis algorithms. http://statweb. However, we do not have to compute these values! (Show R-code demo_toyMCMC2. via the multivariate sampling technique described in Gamerman (1997). Civil engineering Dept. Metropolis-Hastings ratio of 1 { i. The Metropolis algorithm is a special case of the general Metropolis-Hastings algorithm (Hoff, 2009). R code The "SMLN codes. Based on two well-known sampling techniques: the Modified Metropolis–Hastings algorithm and the Metropolis–Hastings algorithm with delayed rejection , the new algorithm is designed specially for sampling from high dimensional conditional distributions. The next block is arpameters. Section 2 takes up the original MCMC method, the Metropolis-Hastings algorithm, outlining. I The M-H algorithm also produces a Markov chain whose values approximate a sample from the posterior distribution. This function allows a user to construct a sample from a user-defined R function using a random walk Metropolis algorithm. A homemade Metropolis algorithm animation using R and the animation package. Recall that a random walk is a Markov chain that evolves according to:. A function which is proportional to the distribution we wish to sample from is passed to the algorithm. At the end of this session, participants will have code fragments that can be readily used or easily adopted for their own scientific work. David Draper and the R codes accompanying the ISLR book. We employ literate programming 1 to build up the code using a top-down approach, see Section 1. More importantly, the self-explanatory nature of the codes will enable modification of the inputs to the codes and variation on many directions will be available for further exploration. Statistics for Data Scientists: Monte Carlo and MCMC Simulations James M. Ivan Jeliazkov Department of Economics, University of California, Irvine, 3151 Social Science Plaza, Irvine, CA. metropolis_hastings. · The R code for IRLS. Again we can use V(x) = ˇ(x) 1=2. One can select an arbitrary proposal distribution that is admissible. Prerequisites Experience with programming in a high level language. (1990) which presented the Gibbs sampler as used in Geman and Geman (1984). The Metropolis-Hastings algorithm is the most general MCMC method used in practice and includes. ##### ### Statistical Computing with R ### ### Maria L. Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models: Abstract: This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. For more complex probability distributions, you might need more advanced methods for generating samples than the methods described in Common Pseudorandom Number Generation Methods. The algorithm can be used to generate sequences of samples from the joint distribution of multiple variables, and it is the foundation of MCMC. A simple Metropolis-Hastings independence sampler Let's look at simulating from a gamma distribution with arbitrary shape and scale parameters, using a Metropolis-Hastings independence sampling algorithm with normal proposal distribution with the same mean and variance as the desired gamma. Prerequisites Experience with programming in a high level language. The purpose of this study was to examine the performance of the Metropolis–Hastings Robbins–Monro (MH-RM) algorithm in the estimation of multilevel multidimensional item response theory (ML-MIRT) m. This is performed in the ARMS package in R. In order to evaluate the integral We can rewrite it as where. The Metropolis-Hastings acceptance rates are then printed to the screen during model tting. smallest, value and its location). Metropolis-within-Gibbs sampler for Binomial and Poisson Metropolis-Hastings rdrr. via the multivariate sampling technique described in Gamerman (1997). is itself another rich and very broadly useful class of MCMC methods, and the MH framework extends it enormously. One really interesting question from a CS 281 assignment this past semester involved comparing Metropolis-Hastings and slice sampling on a joint distribution. edu/~cgates/PERSI/papers/MCMCRev. Let's look at this in generic/pseudo R code for additional clarity (in practice we can take the difference in the log values for step 3):. In this post, we will investigate the Metropolis-Hastings algorithm, which is still one of the most popular algorithms in the field of Markov chain Monte Carlo methods, even though its first appearence (see [1]) happened in 1953, more than 60 years in the past. Set the working directory by typing into the R prompt the command: > setwd(“C:\Example”) 8. 1 Metropolis-Hastings Transitions 5. At the end I am going to give you a link to the Rust playground, so you can test the code yourself! First I start with some traits, that define probability functions: type Float=f64; /*Standard trait for distributions. MCMC: Metropolis Algorithm Proposition (Metropolis works): - The p ij 's from Metropolis Algorithm satisfy detailed balance property w. This is an R package developed and re-cently released by Buckner et al. Practical Data Analysis with JAGS using R Department of Biostatistics Institute of Public Health, University of Copenhagen Tuesday 1st January, 2013 Computer practicals. Different functions are sampled by the Metropolis-Hastings algorithm. 25) % page8: Metropolis(-Hastings) algorithm % true (target) pdf is p(x) where we know it but can¡¯t. r” with the data named “quake_data. Description. This is the algorithm that I always teach first, because it is so simple that it can fit inside a single (old school 140 character) tweet: y=sum(rpois(20,2)) n=1e4 p=rep(1,n) for(i in 2:n){r=p[i-1]. The first set of exercises gave insights on the Bayesian paradigm, while the second set focused on well-known sampling techniques that can be used to generate a sample from the posterior distribution. Monte Carlo Methods with R: Basic R Programming [16] Probability distributions in R R , or the web, has about all probability distributions Preﬁxes: p, d,q, r Distribution Core Parameters Default Values Beta beta shape1, shape2 Binomial binom size, prob Cauchy cauchy location, scale 0, 1 Chi-square chisq df Exponential exp 1/mean 1 F f df1, df2. Some R code helpful for some of the solutions in HW 1; Sketches of partial solutions for selected HW 1 problems; Some R code helpful for some of the solutions in HW 2.


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# Use the form of the definition of the integral given in the theorem to evaluate the integral. b=4 a=0 (6-x^2)dx

thilina-g | Certified Educator

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The definition of integral is (limit definition) is,

`int_a^bf(x)dx = lim_(n->oo)sum_(n=1)^nf(x_1^*)Deltax`

our function is, `int_0^4(6-x^2)dx`

the range is 0 to 4,

therefore, `Deltax = (4-0)/n = 4/n`

therefore,  `x_i^* = (4i)/n`

`sum_(n=1)^nf(x_1^*)Deltax = sum_(n=1)^nf((4i)/n)*(4/n)`

`= sum_(n=1)^n(6 - ((4i)/n)^2)*(4/n)`

`= sum_(n=1)^n(6 - (16i^2)/n^2)*(4/n)`

`= sum_(n=1)^n24/n - (64i^2)/n^3`

`= sum_(n=1)^n24/n - sum_(n=1)^n(64i^2)/n^3`

`= 24/nsum_(n=1)^n1 - 64/n^3sum_(n=1)^ni^2`

we know, `sum_(n=1)^ni^2 = (n(n+1)(2n+1))/6`

we get,

`=24/n * n - 64/n^3 * (n(n+1)(2n+1))/6`

`=24 - (32(n+1)(2n+1))/(3n^2)`

`=(72n^2 - 32(2n^2+3n+1))/(3n^2)`

`=(72n^2 - 64n^2-96n-32)/(3n^2)`

`=(8n^2-96n-32)/(3n^2)`

`int_0^4(6-x^2)dx = lim_(n->oo)sum_(n=1)^nf(x_1^*)Deltax`

`int_0^4(6-x^2)dx = lim_(n->oo)(8n^2-96n-32)/(3n^2) =8/3 `

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Pion mass

04/09/2008

We have seen in preceding posts how good was the computation of ground states of quarkonia obtaining the interquark potential from the gluon propagator and solving the Dirac equation (see here). Here we try a more ambitious aim: We compute the pion mass from the interquark potential in the limit of very light quarks but assuming them to be not relativistic that is a drastic assumption. So, the interquark potential is given by (see here)

$V(r)=-\frac{\alpha_s}{r}\sum_{n=0}^\infty B_n e^{-m_n r}$

but in case of light quarks we can take just the first term, that is

$V(r)\approx -\frac{\alpha_s}{r}B_0e^{-m_0r}$

and so our problem reduces to the one of solving the Schroedinger equation with the Yukawa potential. This is a well-known problem. To get the ground state we have used this paper by A.E.S. Green. Then, our final formula is

$m_\pi=2m_q-\frac{m_0^2}{m_q}\frac{\alpha_s}{2}B_0\left(\frac{\alpha_s}{2}B_0-As^2\right)$

being $m_q=350$ MeV the constituent mass quark, $\alpha_s=1.47$ the strong coupling constant, $m_0=1.19814\sqrt{\sigma}$ with $\sigma=(440MeV)^2$ the string tension, $B_0=1.144231$, $A=1.9875$ and $s=0.03951$ two constants of the energy level computation from Green’s paper. So, finally we get the satisfactory value $m_\pi\approx 140$ MeV in good agreement with experimental value taking into account of how rough was our computation.

We cannot claim this as a full success but rather as a simple exercise showing how knowing the proper gluon propagator can give a serious hint on computation of all the relevant quantities in QCD and this has been the main aim of such analysis.

The interquark potential

27/08/2008

In our initial post about quarkonia we have derived the interquark potential from the gluon propagator. In this post we want to deepen this matter being this central to all hadronic bound states. The gluon propagator is given by

$G(p^2)=\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}$

being

$B_n=(2n+1)\frac{\pi^2}{K^2(i)}\frac{(-1)^{n+1}e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}$

and

$m_n = (2n+1)\frac{\pi}{2K(i)}\sqrt{\sigma}$

with $\sqrt{\sigma}=0.44$ GeV being $\sigma$ the string tension that is just an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally. We have obtained this propagator in a series of papers starting from a massless scalar theory. The most relevant of them is here. It is immediate to recognize that this propagator is just an infinite superposition of Yukawa propagators. But the expectations from effective theories are quite different (see Brambilla’s CERN yellow report here). Indeed, a largely used interquark potential is given by

$V(r)=-\frac{a}{r}+\sigma r +b$

but this potential is just phenomenological and not derived from QCD. Rather, as pointed out by Gocharov (see here) this potential is absolutely not a solution of QCD. We note that it would be if the linear term is just neglected as happens at very small distances where

$V_C(r)\approx -\frac{a}{r}+b$.

We can derive this potential from the gluon propagator imposing $p_0=0$ and Fourier transforming in space obtaining

$V(r)=-\frac{\alpha_s}{r}\sum_{n=0}^\infty B_n e^{-m_n r}$

and we can Taylor expand the exponential in r obtaining

$V(r)\approx -\frac{\alpha_s}{r}+Ar+b$

but we see immediately that $A=0$ and so no linear term exists in the potential for heavy quarkonia! This means that we can formulate a relativistic theory of heavy quarkonia by solving the Dirac equation for the corresponding Coulombic-like potential whose solution is well-known and adding the b constant. We will discuss such a spectrum in future posts.

For lighter quarks the situation is more involved as we have to take into account the full potential and in this case no solution is known and one has to use numerical computation. But solving Dirac equation on a computer should be surely easier than treating full QCD.


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# Journal of Integrative Bioinformatics

Editor-in-Chief: Schreiber, Falk / Hofestädt, Ralf

Managing Editor: Sommer, Björn

Ed. by Baumbach, Jan / Chen, Ming / Orlov, Yuriy / Allmer, Jens

Editorial Board: Giorgetti, Alejandro / Harrison, Andrew / Kochetov, Aleksey / Krüger, Jens / Ma, Qi / Matsuno, Hiroshi / Mitra, Chanchal K. / Pauling, Josch K. / Rawlings, Chris / Fdez-Riverola, Florentino / Romano, Paolo / Röttger, Richard / Shoshi, Alban / Soares, Siomar de Castro / Taubert, Jan / Tauch, Andreas / Yousef, Malik / Weise, Stephan / Hassani-Pak, Keywan

CiteScore 2017: 0.77

SCImago Journal Rank (SJR) 2017: 0.336

Open Access
Online
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1613-4516
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Volume 14, Issue 1

# Towards a Consistent, Quantitative Evaluation of MicroRNA Evolution

Ali M. Yazbeck
• Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for Bioinformatics, University Leipzig, Härtelstraße 16-18, D-04107 Leipzig, Germany
• Lebanese University, Doctoral School for Science and Technology, Rafic Hariri University Campus, Hadath, Lebanon
• Other articles by this author:
/ Kifah R. Tout
• Lebanese University, Doctoral School for Science and Technology, Rafic Hariri University Campus, Hadath, Lebanon
• Other articles by this author:
• Corresponding author
• Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for Bioinformatics, University Leipzig, Härtelstraße 16-18, D-04107 Leipzig, Germany
• German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Competence Center for Scalable Data Services and Solutions, and Leipzig Research Center for Civilization Diseases, University Leipzig, Leipzig, Germany
• Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany
• Fraunhofer Institute for Cell Therapy and Immunology, Perlickstraße 1, D-04103 Leipzig, Germany
• Department of Theoretical Chemistry of the University of Vienna, Währingerstraße 17, A-1090 Vienna, Austria
• Center for RNA in Technology and Health, Univ. Copenhagen, Grønnegårdsvej 3, Frederiksberg C, Denmark
• Santa Fe Institute, 1399 Hyde Park Road, Santa Fe NM 87501, USA
• Helmholtz Centre for Environmental Research – UFZ, Young Investigators Group Bioinformatics and Transcriptomics, Permoserstraße 15, D-04318 Leipzig, Germany
• Email
• Other articles by this author:
/ Jana Hertel
• Helmholtz Centre for Environmental Research – UFZ, Young Investigators Group Bioinformatics and Transcriptomics, Permoserstraße 15, D-04318 Leipzig, Germany
• Other articles by this author:
Published Online: 2017-06-05 | DOI: https://doi.org/10.1515/jib-2016-0013

## Abstract

The miRBase currently reports more than 25,000 microRNAs in several hundred genomes that belong to more than 1000 families of homologous sequences. Quantitative investigations of miRNA gene evolution requires the construction of data sets that are consistent in their coverage and include those genomes that are of interest in a given study. Given the size and structure of data, this can be achieved only with the help of a fully automatic pipeline that improves the available seed alignments, extends the set of available sequences by homology search, and reliably identifies true positive homology search results. Here we describe the current progress towards such a system, emphasizing the task of improving and completing the initial seed alignment.

## 1 Introduction

MicroRNAs (miRNAs) are an important class of abundant, endogenous small non-coding RNAs that are produced by almost all animals and plants as well as many unicellular eukaryotes. They function in post-transcriptional gene silencing making use of the evolutionarily ancient RNA interference pathways, through which double-stranded RNA can inactivate cognate sequences [1], [2]. The overwhelming majority of miRNAs is produced from mRNA-like primary precursor transcripts, the pri-miRNAs, through a common processing pathway. The hairpin-shaped pre-miRNAs are cut from the pri-miRNA in the nucleus. Pre-miRNAs are subject to stringent structural constraints that are different between major clades. The next processing steps differ substantially between animals and plants and lead to miRNA/miRNA* duplex structures [3]. These are separated to produce mature miRNAs that are then incorporated in the Argonaut complex.

The innovation of miRNA-like endogenous RNAs clearly has occurred multiple times. Despite their functional analogy, there is no evidence for any homology between the plant miRNAs, animal miRNAs, and the miRNAs reported for several unicellular lineages [4], [5], [6]. Albeit, miRNAs and their hairpin precursors are most highly conserved genetic elements [7] within the single kingdoms. This makes it possible to accurately pinpoint the evolutionary origin of individual miRNAs [8]. Since miRNAs readily form paralogs by segmental or chromosomal duplications [9], [10] they often appear as members of families of homologous sequences, which form the basis of microRNA nomenclature used by miRBase [11]. The phylogenetic distribution of miRNA families appears to show that entire families are rarely lost. Hence they have been advertised as excellent, nearly homoplasy-free phylogenetic markers [12], [13], [14], [15], although there are some well described exceptions to this rule [16].

With the rapid increase of sequencing data covering an even denser distribution of animal taxa there has also been an increasing interest in quantifying innovation, turn-over, and loss of miRNAs and miRNA families [8], [15], [17], [18]. Despite best efforts, such quantitative questions are difficult to answer directly from the data provided e.g. by the miRBase. Recent critical evaluations of the miRBase inventory concluded that less than a third of the miRBase entries are robustly supported as miRNA genes [19], [20] and indicated that – without further curation – the database does not provide information that is sufficiently consistent for machine learning applications [21]. The reasons are manifold and, at least in part, a consequence of the non-trivial biology of small RNA genes. In fact, the term “microRNA” is far from being used in a well-defined and unambiguously manner in the literature. A broad range of biogenesis pathways for small RNAs have been discovered, reviewed e.g. in [22], [23]. These overlap with the canonical scheme of miRNA processing to various degrees. For instance, should mir-451 [24], which is not processed by Dicer, or mirtrons, whose precursor hairpins are produced by the splicing machinery rather than with the help of Drosha [25], be counted as bona fide microRNAs? This issue is aggravated by small RNAs produced from structured RNAs such as tRNAs or snoRNAs. Here the consensus secondary structure often does not match the minimum free energy fold deposited in the database and/or deviates substantially from the criteria usually expected for a miRNA precursor [26], [27], [28].

Additional issues plague a quantitative evolutionary analysis. First, there is a strong ascertainment bias favoring the miRNAs from heavily researched model organisms (human, mouse, fruit fly, C. elegans). Second, automatic homology searches are largely restricted to the genomes curated by ensembl. Third, some homologous groups of miRNAs have been assigned to different or no miRBase families as a consequence of sufficient sequence divergence in the mature sequences (e.g. mir-100 and the “tunicate specific” mir-1473 [10]). Pre-miRNAs, furthermore, are annotated with quite different lengths within the same family. This concerns both different paralogs from the same family and orthologs from different species. An unfortunate consequence of this inconsistency is that the distinction between orthologous and paralogous sequences is blurred. Families with uncertain conserved structures tend to result in gap-rich, unreliable alignments. These give rise to poor covariance models and thus less sensitive homology searches leading eventually to underestimating the family’s phylogenetic distribution. In response to all these difficulties, all large scale studies of miRNA evolution start with homology-based searches and/or the integration of alternative miRNA annotations, e.g. from MirGeneDB [20], to complete the data set, as well as extensive, usually manual, curation of the data; see e.g. [8], [29], [30], [31], [32], [18]. This approach, however, has reached its limits with more than a hundred genomes and thousands of groups of miRNAs.

In this contribution we give a preliminary report on progress towards a completely automated pipeline for the analysis of microRNA evolution. We focus here on the key step of processing the data from the miRBase into a consistent collection of miRNA alignments. These in turn are then extended by homology search across a large set of additional genomes, using the ideas of the initial data cleaning now as filter criteria for deciding on the inclusion of initial homology search candidates. Using the current set of metazoan miRNAs as an example we demonstrate that such an approach is feasible.

## 2 Methods

A quantitative analysis of miRNA evolution requires data sets that are comparable between miRNA families in two respects: (i) within each species, all paralogs should be included, and (ii) across species, homologs should be detected with comparable sensitivity and specificity to avoid strong correlations of coverage and completeness of the included sequence data e.g. with the evolutionary age of the miRNA families. The data included in any repository resource unavoidably is subject to ascertainment biases deriving from the research history of individual genes and gene families and necessarily lags behind the available genomic data that can be included in a particular analysis. Therefore, it is indispensable to prepare a consistent data set by means of homology search. We propose, based on previous experience [8], [18], to break down this task into well-separated steps that can be automatized independently and then combined to a single work-flow:

• The completion and correction of (seed) alignments requires the identification of incomplete and problematic sequences and their correction. We will focus on this step below.

• The homology search itself is conducted with covariance models using infernal and produced new candidates in all genomes, including those from which the seed alignments are taken. This has the potential to identify new paralogs that have not yet been included in the data source.

• The decision to include a homology search candidate into the data set is taken according to the same criteria that are applied to the initial data curation step.

As a consequence, the entire data processing pipeline can be run until no further candidates are detected and thus a self-consistent data set is obtained. To exploit all the available information, alignments, consensus structures, and homology searches are conducted at the level of miRNA precursors. The evaluation of the homology search candidates also explicitly uses the mature miRNA products.

Our first task is to construct consistent, high quality alignments of each miRBase family. Given a multiple sequence alignment 𝒜 we compute the average column-wise Shannon entropy

$S(A)=−1m∑i=1n(∑αP(α;i)lnP(α;i))$(1)

as a measure of alignment quality that is independent of the method and scoring system that has been used for the original construction of 𝒜. Here, m is the number of alignment columns, P(α;i) is the frequency of the symbol α (which may also be IUPAC nucleotide code or a gap character ’’), and n is the number of sequences in the alignment. A reduction of S(𝒜) upon modification of 𝒜 signifies an improvement of the alignment. At present we consider only column-wise entropies since these provide a good first order approximation to quantifying sequence conservation and thus also alignment quality. Given that dinucleotide distributions do matter also for miRNAs [33], one could conceivably extend the quality measure to incorporate mutual information measures of adjacent and/or paired columns.

The most frequently observed errors within the miRNA alignments are too short or too long sequences, and sequences that do not fit at all into the alignments. All these issues produce bad (high) entropy values, since there are more regions containing gaps or unmatched bases, and thus are readily identified by our pipeline. Our basic approach to improve the alignment is to correct for these errors. This is reasonable because the ends of the precursor sequences submitted to miRBase are in almost all cases not determined experimentally but are based on the individual contributor’s perception of the highly conserved region, the region with conserved base pairs, or simply the extent of the infernal hit included in an ensembl genome annotation. We therefore truncate database entries that are too long and complete sequences that are too short by retrieving the ostensibly missing flanking regions from the genomic DNA sequences. Finally, we remove sequences that are most likely false positives from a previous homology search step. These cases are recognizable as those that introduce a lot of heterogeneity into the alignment. This filtering step can lead to the complete removal of short input sequences that were extended in the previous step.

Upon modification of an input alignment by extension, cropping, or outright removal of one or more sequences, the sequences were realigned and the entropy S(𝒜′) was computed for 𝒜′. We tested two re-alignment strategies: (i) sequence-based clustalw alignments (version 2.0.12) [34] and (ii) structure based mlocarna alignments [35], which in turn depend on secondary structure predictions using the Vienna RNA package [36]. Since the sequence-based approach usually resulted in larger entropy reductions in test sample of miRBase families, we opted for clustalw.

To determine plausible 5′ and 3′ ends for a precursor we use the mature miR and miR* sequences as anchors. The latter are also provided by the Rfam database. To obtain a consistent query set we define our precursors to extent 10nt beyond the ends of the mature sequences and prune longer database entries. This step is made difficult by the fact that for many families mature sequences are annotated in some or all sequences on only one of the two arms of the precursor hairpin. For each miRNA family, the precursors that have both mature sequences are pruned to at most 10nt upstream of the start of the 5′ mature sequence and 10nt downstream of the end of the 3′ mature sequence, respectively.

In some cases precursor sequences in the initial data basis are reverse complements of each other. In a few well-described cases this is biologically correct. Famously, iab-4 and iab-8 are produced from almost exactly the same genomic location but opposite DNA strands in arthropods [37], [38], and a similar situation was reported for mir-3120 and miR-214 [39]. Such cases may also appear as artifacts, however, when a mature miR* is erroneously identified with a miR produced from the opposite strand e.g. due to a chemical modification such as adenosine-inosine (A-I) editing [40], [41] in the mature product. To catch such cases we also test whether the replacement of a sequence with its reverse complement would lead to an improvement in the alignment quality.

At this stage, the improved alignment still contains some entries that are too short. It suffices to run cmsearch with the covariance model for the improved alignment against the genome for which only a truncated precursor was available. The result of this search is then used to replace the original entry.

The sequences belonging to a family are finally re-aligned with clustalw and their consensus secondary structure is computed using RNAalifold [36]. The resulting structure-annotated alignment is then converted to Stockholm format and passed to cmbuild, a component of the infernal suite (release 1.1.1) [42], [43], to construct revised covariance models. We used cmscan (another component of the infernal suite) to test the precursors of the same family with only one annotated mature sequence (if available) for membership.

Covariance models are used for homology search in more than 300 animal genomes in an ongoing study. The candidate sequences are added to query alignments using cmalign and then subjected again to data cleaning procedure. These steps can be repeated if necessary. The final data are converted to a presence/absence table that, for each species, reports the total number of paralogs of each miRNA family. The ePoPe tool [18] implements a Dollo parsimony approach that determines the most likely point of origin (as the lineage leading up to the last common ancestor of all observed paralogs) as well as the most parsimonious scenario of duplications and losses in each family.

## 3 Application and Results

We focus here on the data cleaning steps for metazoan microRNAs as an illustrative example. The same pipeline, however, could just as well be applied to plant miRNA data. Our starting point is miRBase release 21. It comprises 21263 pre-miRNAs in animal species. A total of 14712 of these are assigned to a family and thus, come in at least two copies in this kingdom. The number of metazoan miRNA families is 1415. MiRNAs may have one or two annotated mature sequence, Figure 1. We selected the subgroup of miRNA families that have at least two pre-miRNAs with an annotated 5′ and 3′ mature sequence. These sum up to 367 families only. Albeit, these families comprise ∼72 % of all metazoan pre-miRNAs. Of these we obtained improved alignments for 359 families. Only in 8 cases, the cleaning procedure did not lead to an improvement.

Figure 1:

Distribution of mature microRNA sequence entries for miRBase (v. 21) families. The majority of families reports a miR and miR* for all (red) or at least some (blue) of the family members. Families in which only a single mature product is reported for every member are shown in yellow.

Figure 2 compares the entropies S(𝒜) and S(𝒜′) before and after processing of the alignments. We find that the entropy systematically increases roughly proportional to the per column entropy in the input. Not surprisingly, noisy, high-entropy alignments are thus improved more than alignments that already show very little variation upon input. The average improvement is about 33 % and does not vary dramatically with number of sequences per alignment. This constant rate of improvement suggests that the input alignments with higher entropy not only contain more diverse sequences but also substantial levels of alignment problems. Only the large alignments (group 4 in Figure 2), which often correspond to well-studied miRNA families, have been improved by only about 20 % on average. It is likely that this outlier can be explained by a more extensive manual curation of the corresponding miRBase alignments.

Figure 2:

Comparison of S(𝒜) for original and processed alignments. Each data point represents one miRNA family. Data are stratified into six groups of miRBase families depending on the number of members in the initial alignment (1: 2-10 pre-miRNAs, 139 families; 2: 11-20 pre-miRNAs, 92 families; 3: 21-40 pre-miRNAs, 57 families; 4: 41-100 pre-miRNAs, 52 families; 5: 101-200 pre-miRNAs, 15 families; 6: >200 pre-miRNAs, 4 families, indicating that the improvements in alignment quality depends much more strongly on the entropy of the input alignment than on the number of sequences in the miRNA family.

To demonstrate the value of consistent data for quantitative analyses we select the miRNA family mir-3 (MIPF0000140) from the miRBase database. It comprises at total of 30 entries of the two paralogous groups traditionally named mir-3 and mir-309. While mir-3 can only be observed in Drosophila species, mir-309 is also found in Nematocera. A naïve analysis of the miRBase data thus would results in mir-3 being identified as the result of a drosophilid-specific duplication of an ancestral mir-309 gene. Processing the MIPF0000140 alignment into a revised query eventually results in homologs in most of the 35 sequenced genomes of Hexapoda, 20 of which were not present in the miRBase. In particular, the pipeline uncovered unambiguous mir-309 homologs in Apis mellifera and Nasonia vitripennis, pushing the origin of this family back from the Diptera to the stem of the Endopterygota. In the bees and wasps, however, miRs are observed only on the 3′ side of the hairpin. As a second example we briefly discuss the mir-723 family, which in miRbase is annotated in the three teleost species, namely zebrafish, atlantic halibut and atlantic salmon. Homology search with the automatically improved alignment uncovered homologs not only in additional teleosts but also in the gar Lepisosteus oculatus. This pushed the origin of the miRNA family back to the stem of the Actinopterygii. Furthermore, the 3R genome duplication, which pre-dates the radiation of the teleosts, is clearly visible in the data, see Figure 3.

Figure 3:

Distribution of miRNA homologs of miRNA families mir-3 (left) and mir-723 (right). Numbers at the nodes denote potential observations of the number of paralogs. The black triangle assigns the LCA to the phylogenetic tree. ’3R’ denotes the third round of whole genome duplication events that is assumed to have happened during the evolution of vertebrates.

## 4 Discussion

The current version 21 of miRBase contains 28,645 entries in 223 species [44]. Nevertheless the data set is far from providing a complete catalog of miRNAs for any given species. A quantitative analysis of miRNA evolution therefore requires substantial computational efforts to complete the data by homology search. This in turn requires extensive efforts to curate the miRBase data, facilitate homology search and to make the data comparable between families. In the present report we have focused on the strategies to obtain virtually complete sets of sequences and plausible alignments for each of the microRNA families. With more than 1000 families, a number that is rapidly increasing further, it is clear that a fully automatic pipeline is indispensable – and it is indeed under development. In this preliminary report we concentrated on the basic strategy and demonstrated in a pilot application to animal microRNAs that a fully automatized work-flow is feasible. The detailed evaluation of several example families showed, furthermore, that the last common ancestor of a miRBase family is frequently older than what would be inferred from the raw miRBase data themselves. A quantitative comparison will be described elsewhere.

The data cleaning procedure developed so far does not solve all problems and certainly will be subject to further improvements. Solutions to several issues remain under construction at this point. We therefore advocate to abandon the goal of computationally completing the data base itself. We suspect that discrepancies between manual and computational curation cannot be entirely avoided. The aim for data consistency in quantitative analyses and the aim of representing the published knowledge hence will remain at odds, at least on a – as one would hope – small subset of the data. We therefore aim instead at the construction of a pipeline that can simply be rerun when updates of the miRBase or the relevant sequenced genomes become available. This also allows us to seek to improve pipeline components gradually that address specific biases and technical limitations.

Very small initial seed alignments, and in particular miRNAs that are not incorporated into a miRBase family, for instance, remain a problem. The reason is that covariance models cannot be trained from a single sequence or a few very similar sequences, and the prediction of secondary structures is much less reliable from a single sequence than for an alignment of sequences with moderate mutual differences. A simple blastn-based homology search in closely related species, however, usually provides an easy remedy.

There is no guarantee, however, that the results of pre-processing the data by extending small families retrieves a bona fide miRNA family. In fact miRBase also occasionally includes fragments from misclassified ncRNAs other than miRNAs. For instance, mir-1940, which is a fragment of the snoRNA SCARNA4 [44] and a survey of many more such cases can be found in [27]. Upon identification, these erroneous miRBase families are usually quickly retired [44]. A check for miRNA properties with a tool such as RNAmicro [45] is nevertheless strongly advised. Such a filter, on the other hand, will tend to reject also special cases such as mir-451. The mature mir-451 sequence is not produced from the stems in the canonical way but originates from the loop [24], hence it is (correctly) rejected by tools that are designed to recognize canonical miRNAs. The “fuzzy” limits of the notion of microRNAs not only affects machine learning efforts [21] but also has a potential impact for any quantification of microRNA evolution since there is no reason to assume that related classes of small RNAs should show the same patterns in evolution. Since miRBase will likely continue to hold a diverse collection of “miRNA-like” genes, we suggest that subclassifications that e.g. identify mirtrons, semi-mirtrons, etc., will be required. Whether annotation on the side of the database will provide this information or whether it will also need to be retrieved and/or computed by post-processing pipelines is still an open question.

We have focused here on properly treating miRNA precursors for which both miR and miR* sequences are known. In cases where a mature product is reported for only one of the two arms of the precursor, a different strategy is required. The most natural approach is to start from the predicted secondary structure for the precursor, obtained e.g. with RNAalifold. The predicted structure and the known position of the miR implies the position of the hypothetical miR* by taking into account that processing produces 2nt overhangs. After this extra processing step the alignment improvement can proceed as outlined in the previous section.

Conceivably, further improvements to the miRNA alignments could be achieved by explicitly using secondary structure information already in the alignment process. The small size of the precursors certainly would allow the use of one of the many variants of simultaneous alignment of folding algorithms [46]. However, the very definition of families as groups of miRNAs that are unambiguous homologs implies a sufficiently high level of sequence similarity to obtain decent sequence-based alignments. In addition, the positioning of the mature miRs yields a powerful anchor for the alignment that is consistent with the structure due the mechanism of Dicer processing. We suspect, therefore, that there is not much to be gained for the data curation pipeline itself. The presence of an appropriate consensus secondary structure relative to the positioning of mature products and the proper matching of patterns of sequence conservation extracted from the sequence conservation with the consensus structure becomes an important issue for the discrimination of bona fide microRNAs from their more distant small RNA relatives.

It is important to keep in mind that the exact size of the pre-miRNA stem-loop has not been determined experimentally for most miRNAs. In the absence of direct information, we have opted for a simple heuristic to delimit the alignments, giving precedence to consistency. There are some exceptional cases, however, where the 3′ and 5′ ends of the pre-miRNA are known precisely. This is in particular the case for mirtrons. These are introns that constitute pre-miRNAs [25], [47]. Our pragmatic “10nt from the miR and miR*” rule will in general not respect this information. Ideally, the relative entropy criterion should be able to determine the boundaries of the precursor. This is bound to fail, however, in cases such as mirtrons, where the surrounding coding exons may exhibit an even larger level of sequence conservation than much of the pre-miRNA sequence. Further research into the patterns of sequence conservation and turnover around the boundaries of pre-miRNAs will be required to arrive at improved criteria. The pipeline outline here can, however, be easily adapted to such a task since it is a trivial matter to retrieve and align the precursor sequence with more extensive flanking regions. It is not difficult, however, to use pattern-based tools such as MaxEntScan [48] to identify conserved splice junctions close to the estimated ends of the precursor hairpin.

Albeit a fully automatized pipeline to process miRBase alignments into a complete and consistent set of miRNA sequences has proved to be feasible, much work still lies ahead. In addition to the incorporation and improvement of components that address more difficult special cases and exceptions, it also remains a non-trivial task to devise informative benchmarking strategies to test and optimize methods for the quantitative analysis of microRNA evolution.

## Acknowledgement

A.Y. was funded by a doctoral stipend of the National Council for Scientific Research of Lebanon (CNRS-L).

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Revised: 2017-02-12

Accepted: 2017-02-16

Published Online: 2017-06-05

Conflict of interest statement: Authors state no conflict of interest. All authors have read the journal’s Publication ethics and publication malpractice statement available at the journal’s website and hereby confirm that they comply with all its parts applicable to the present scientific work.

Citation Information: Journal of Integrative Bioinformatics, Volume 14, Issue 1, 20160013, ISSN (Online) 1613-4516,

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