coda Manual
Contents
1. varnames 41 subset indices of the subset of the series to plot The default is constructed from the start and thin arguments Value An object of class trellis The relevant update method can be used to update components of the object and the pr int method usually called by default will plot it on an appropriate plotting device Author s Deepayan Sarkar Deepayan Sarkar R project org See Also Lattice for a brief introduction to lattice displays and links to further documentation Examples data line Not run xyplot line xyplot line 1 start 10 densityplot line start 10 qqmath line start 10 levelplot line 2 acfplot line outer TRUE End Not run varnames Named dimensions of MCMC objects Description varnames returns the variable names and chanames returns the chain names or NULL if these are not set If allow null FALSE then NULL values will be replaced with canonical names Usage varnames x allow null TRUE chanames x allow null TRUE varnames x lt value chanames x lt value 42 window mcmc Arguments x an meme or momc list object allow null Logical argument that determines whether the function may return NULL value A character vector or NULL Value A character vector or NULL See Also meme Meme List window mcmc Time windows for mcmc objects Description window mcmc is a m2. uk a e e a aaa Smeg a buds ores 23 memcUperade x iu ane eoe xe OE e maus ege Sb do aede ave S 24 MEP aoc oe he ee SAE A AEE o deck e Y Pow Ld 24 multiinen 44 cR e ae Rhe be ole Re eee HS 3 Fue do fos 229 HENAN pM rar re a e a Ss 25 Plot MME 0 RR E CRUS AU e a Ae dede os e e eod 26 Yaftery diag wns ewe b Roue te Bree Ro Re we ER Ups pene a RE c OR Reds 27 read and check tdi EGG vC C OY A 4545 2 OS OSRAM S eR 40 28 read cOda ovd oem a Sede EROR A RE Rex E 29 read coda interactive ees 30 read openbugs us dor dee Snc d eq ide IS A A 31 rejecBonRdte s s doo e vox ee a LU E Oe de Re Ue e CY ev y DR e ene d 32 spectrum ici aa hans a a E poe dogs 32 SpectruMU dados den dl eS dd S Us eA eS 34 SUMMALY MCMC ec a e Voy e a 35 i MACC 35 time meme 20 5 4 Robs rr xm A SY WOES SUE NUES 36 traceplot e s Sakata s e E X ew he E gero a p E X XD e Yes Le e ee ae 37 trelhsplots s sos ce cd e RORORBOR a ed ee bero REOR REGE E a 37 MEIDEHICS MEC TS is dad ee see dee eo aa doo las Ge eee hoe 41 WINdOW MCMG ec al e ee ee ee al wie ae Ee eN 42 Index 43 Cramer The Cramer von Mises Distribution Description Distribution function of the Cramer von Mises distribution Usage pcramer q eps Arguments q vector of quantiles eps accuracy required Value pcramer gives the distribution function HPDinterval 3 References Anderson TW and Darling DA Asymptotic theory of certain goodness of fit criteria based on stochastic processes Ann Ma
3. S3 method for class mcmc list acfplot x data outer FALSE groups outer prepanel panel type if groups b else h aspect xy start 1 thin 1 lag max NULL ylab Autocorrelation xlab Lag main attr x title 40 Arguments x data outer groups aspect trellisplots subset an momc or memc list object ignored present for consistency with generic for the mcmc list methods a logical flag to control whether multiple runs of a series are displayed in the same panel they are if FALSE not if TRUE If specified in the mcmc methods this argument is ignored with a warning for the memc 1ist methods a logical flag to control whether the underlying lattice call will be supplied a groups arguments indicating which run a data point originated from The panel function is responsible for handling such an argument and will usually differentiate runs within a panel by using different graphical parameters When out er FALSE the default of groups is TRUE if the corresponding default panel function is able to make use of such information When outer FALSE groups TRUE will be ignored with a warning controls the physical aspect ratio of the panel See xyplot for details The default for these methods is chosen carefully check what the default plot looks like before changing this parameter default scales type thin start plot points layout
4. TRUE mcemc subset Extract or replace parts of MCMC objects Description These are methods for subsetting mcmc objects You can select iterations using the first dimension and variables using the second dimension Selecting iterations will return a vector or matrix not an meme object If you want to do row subsetting of an memc object and preserve its dimensions use the window function Subsetting applied to an memc 1ist object will simultaneously affect all the parallel chains in the object Usage S3 method for class mcmc x i j drop missing i S3 method for class mcmc list x i j drop TRUE Arguments x An memc object di Row to extract j Column to extract drop if TRUE the redundant dimensions are dropped See Also window mcmc 24 mcpar mcmcUpgrade Upgrade mcmc objects in obsolete format Description In previous releases of CODA an mcmc object could be a single or multiple chains A new class mcmc list has now been introduced to deal with multiple chains and memc objects can only have data from a single chain Objects stored in the old format are now obsolete and must be upgraded Usage mcmcUpgrade x Arguments x an obsolete memc object Author s Martyn Plummer See Also mcmc mcpar Mcpar attribute of MCMC objects Description The mcpar attribute of an MCMC object gives the start iteration the end iteration and the thinning interval of the cha
5. fracl fraction to use from beginning of chain frac2 fraction to use from end of chain nbins Number of segments pvalue p value used to plot confidence limits for the null hypothesis auto layout If TRUE then set up own layout for plots otherwise use existing one ask If TRUE then prompt user before displaying each page of plots Default is dev interactive inR and interactive in S PLUS Graphical parameters Note The graphical implementation of Geweke s diagnostic was suggested by Steve Brooks See Also geweke diag heidel diag 19 heidel diag Heidelberger and Welch s convergence diagnostic Description heidel diag is a run length control diagnostic based on a criterion of relative accuracy for the estimate of the mean The default setting corresponds to a relative accuracy of two significant digits heidel diag also implements a convergence diagnostic and removes up to half the chain in order to ensure that the means are estimated from a chain that has converged Usage heidel diag x eps 0 1 pvalue 0 05 Arguments x eps Target value for ratio of halfwidth to sample mean pvalue significance level to use Details The convergence test uses the Cramer von Mises statistic to test the null hypothesis that the sampled values come from a stationary distribution The test is successively applied firstly to the whole chain then after discarding the first 10 20 of the chain until either the n
6. this parameter provides a reasonable default value of the scales parameter for the method It is unlikely that a user will wish to change this parameter Pass a value for scales see xyplot instead which will override values specified here a character vector that determines if lines points etc are drawn on the panel The default values for the methods are carefully chosen See panel xyplot for possible values an optional thinning interval that is applied before the plot is drawn an optional value for the starting point within the series Values before the start ing point are considered part of the burn in of the series and dropped character argument giving the style in which points are added to the plot See panel densityplot for details a method specific default for the layout argument to the lattice functions xlab ylab main cuts at col regions lag max Used to provide default axis annotations and plot labels defines number and location of values where colors change color palette used maximum lag for which autocorrelation is computed By default the value cho sen by acf is used prepanel panel suitable prepanel and panel functions for ac plot The prepanel function omits the lag 0 auto correlation which is always 1 from the range calculations other arguments passed to the lattice function Documentation of the corre sponding generics in the lattice package should be consulted for possible arguments
7. coda options function which groups the options into sections Available options are bandwidth Bandwidth function used when smoothing samples to produce density estimates De faults to Silverman s Rule of thumb combine corr Logical option that determines whether to combine multiple chains when calculating cross correlations combine plots Logical option that determines whether to combine multiple chains when plotting combine plots Logical option that determines whether to combine multiple chains when calculat ing summary statistics coda options 9 data saved For internal use only densplot Logical option that determines whether to plot a density plot when plot methods are called for mcmc objects digits Number of significant digits to use when printing fracl For Geweke diagnostic fraction to use from start of chain Defaults to 0 1 frac2 For Geweke diagnostic fraction to use from end of chain Default to 0 5 gr bin For Geweke Brooks plot number of iterations to use per bin gr max For Geweke Brooks plot maximum number of bins to use This option overrides gr bin halfwidth For Heidelberger and Welch diagnostic the target value for the ratio of half width to sample mean lowess Logical option that controls whether to plot a smooth line through a trace plot when plotting MCMC output q For Raftery and Lewis diagnostic the target quantile to be estimated r For Raftery and Lewis diagnostic the required precisio
8. main attr x title xlab plot points rug tg subset for class memei data main attr x title start 1 thin 1 RIT xlab ylab cuts 10 at col regions topo colors 100 subset for class mcmc outer aspect xy default scales list y list relation free prepanel prepanel qqmathline start 1 thin 1 main attr x title ylab tg subset for class mcmc list outer FALSE groups outer aspect xy default scales list y list relation free prepanel prepanel qqmathline start 1 thin 1 main attr x title trellisplots 39 ylab E subset S3 method for class mcmc xyplot x data outer layout c 1 ncol x default scales list y list relation free type l start 1 thin 1 ylab xlab Iteration number main attr x title tg subset S3 method for class mcmc list xyplot x data outer FALSE groups outer aspect xy layout c 1 ncol x I 111 default scales list y list relation free type l start 1 thin 1 main attr x title ylab tg subset acfplot x data ss S3 method for class mcmc acfplot x data outer prepanel panel type h aspect xy start 1 thin 1 lag max NULL ylab Autocorrelation xlab Lag main attr x title d i subset
9. of equality of means between the first and last parts of the chain A separate statistic is calculated for each variable in each chain References Geweke J Evaluating the accuracy of sampling based approaches to calculating posterior mo ments In Bayesian Statistics 4 ed JM Bernado JO Berger AP Dawid and AFM Smith Clarendon Press Oxford UK See Also geweke plot 18 geweke plot geweke plot Geweke Brooks plot Description If geweke diag indicates that the first and last part of a sample from a Markov chain are not drawn from the same distribution it may be useful to discard the first few iterations to see if the rest of the chain has converged This plot shows what happens to Geweke s Z score when succes sively larger numbers of iterations are discarded from the beginning of the chain To preserve the asymptotic conditions required for Geweke s diagnostic the plot never discards more than half the chain The first half of the Markov chain is divided into nbins 1 segments then Geweke s Z score is repeatedly calculated The first Z score is calculated with all iterations in the chain the second after discarding the first segment the third after discarding the first two segments and so on The last Z score is calculated using only the samples in the second half of the chain Usage geweke plot x fracl 0 1 frac2 0 5 nbins 20 pvalue 0 05 auto layout TRUE ask Arguments x an mcmc object
10. than max length If this is set to NULL no aggregation occurs Details The raw periodogram is calculated for the series x and a generalized linear model with family Gamma and log link is fitted to the periodogram The linear predictor is a polynomial in terms of the frequency The degree of the polynomial is determined by the parameter order Value A list with the following values spec The predicted value of the spectral density at frequency zero Theory Heidelberger and Welch 1991 observed that the usual non parametric estimator of the spectral den sity obtained by smoothing the periodogram is not appropriate for frequency zero They proposed an alternative parametric method which consisted of fitting a linear model to the log periodogram of the batched time series Some technical problems with model fitting in their original proposal can be overcome by using a generalized linear model Batching of the data originally proposed in order to save space has the side effect of flattening the spectral density and making a polynomial fit more reasonable Fitting a polynomial of degree zero is equivalent to using the batched means method Note The definition of the spectral density used here differs from that used by spec pgram We con sider the frequency range to be between 0 and 0 5 not between 0 and frequency x 2 The model fitting may fail on chains with very high autocorrelation References Heidelberger P an
11. See Also mcmc memc list plot mecmc Summary plots of mcmc objects Description plot mcmc summarizes an memc or memc list object with a trace of the sampled output and a density estimate for each variable in the chain Usage S3 method for class mcmc plot x trace TRUE density TRUE smooth TRUE bwf auto layout TRUE ask dev interactive Arguments x an object of class memc or momc list trace Plot trace of each variable density Plot density estimate of each variable smooth Draw a smooth line through trace plots bwf Bandwidth function for density plots auto layout Automatically generate output format ask Prompt user before each page of plots Further arguments Author s Martyn Plummer raftery diag 27 See Also densplot traceplot raftery diag Raftery and Lewis s diagnostic Description raftery diag is a run length control diagnostic based on a criterion of accuracy of estimation of the quantile q It is intended for use on a short pilot run of a Markov chain The number of iterations required to estimate the quantile q to within an accuracy of r with probability p is calculated Separate calculations are performed for each variable within each chain If the number of iterations in data is too small an error message is printed indicating the minimum length of pilot run The minimum length is the required sample size for a chain with no correlation between consec
12. The coda Package February 16 2008 Version 0 13 1 Date 2007 12 11 Title Output analysis and diagnostics for MCMC Author Martyn Plummer Nicky Best Kate Cowles Karen Vines Maintainer Martyn Plummer plummerQG iarc fr Depends R gt 2 5 0 lattice Description Output analysis and diagnostics for Markov Chain Monte Carlo simulations License GPL gt 2 H topics documented Cramer 2h ett ek bint Re ee eee di edet i e b er es HPDintetvall ense e Ge Gok Be m OR Rene Ue RR A EUR SONS ROS ee d ASIS MCG uos morcm ee Roe AU ow Re ee epo D WU NOx n E dox e qe A APP autocorEdiag se c egon e a e E Eh Ro EROS Eee RR doe RR Roe E oe d AULOCOMT P O PC b tchSE nce caes A eRe eu kw POR y woe ee E bugs2 agS 2 4e RR ER x RUE BES OS ded BOOS dom ewe xq T GOdAa OplOfS voice oh oe doe bd eee Ge ee Pee S EHE Soe ue codamenu x 2 sp RR e bebe SSNS SHEE mb e get EE b estu ais CIOSSCOM pl Ra A POS Bee d CIOSSCOFEDlOL 4 42 4 4 a A qu eA e ARX E A hus CUMUPIOL cios TIT TT densplot 2 xo Ro Roos e Eo RO eee Ro PAR E Moos nd T pelmandiag APER CL Selman Plots see 5 4 C r gpeweke diag 4 2 2b wa P or wm e or Y oo gpeweke plot i e ones A does UE S eee eA a dee eG 4 held IAS hacg ans mem ee eR AA EUR SR ee Neg 2 Cramer lne BPs ce rl Sy ee ede a BO Sale A A ae A Eee aS 20 uuu EMPTIS 20 Incmc convert 6 6 Rs a ee eee eh Eom Mox VUE Hoe x dU qo PUE S 21 MCMC St e rr Rw ER A S ee Ee ed 22 mcmcsubset3
13. art end or thin are incompatible with the data they are ignored Usage read coda output file index file start end thin quiet FALSE read jags file jags out start end thin quiet FALSE Arguments output file The name of the file containing the monitored output index file The name of the file containing the index showing which rows of the output file correspond to which variables file For JAGS output the name of the output file The extension out may be omitted There must be a corresponding ind file with the same file stem start First iteration of chain end Last iteration of chain thin Thinning interval for chain quiet Logical flag If true a progress summary will be printed 30 read coda interactive Value An object of class mcmc containing a representation of the data in the file Author s Karen Vines Martyn Plummer References Spiegelhalter DJ Thomas A Best NG and Gilks WR 1995 BUGS Bayesian inference Using Gibbs Sampling Version 0 50 MRC Biostatistics Unit Cambridge See Also mcmc read coda interactive read openbugs read coda interactive Read CODA output files interactively Description read coda interactive reads Markov Chain Monte Carlo output in the format produced by the classic BUGS program No arguments are required Instead the user is prompted for the required information Usage read coda interactive Value An object of class memc 1is
14. bridge See Also read coda 32 spectrumO rejectionRate Rejection Rate for Metropolis Hastings chains Description rejectionRate calculates the fraction of time that a Metropolis Hastings type chain rejected a proposed move The rejection rate is calculates separately for each variable in the memc obj argument irregardless of whether the variables were drawn separately or in a block In the latter case the values returned should be the same Usage rejectionRate x Arguments x An mcmc or memc list object Details For the purposes of this function a rejection has occurred if the value of the time series is the same at two successive time points This test is done naively using and may produce problems due to rounding error Value A vector containing the rejection rates one for each variable Author s Russell Almond spectrum0 Estimate spectral density at zero Description The spectral density at frequency zero is estimated by fitting a glm to the low frequency end of the periodogram spect rum0 x length x estimates the variance of mean x Usage spectrum0 x max freq 0 5 order 1 max length 200 spectrumO 33 Arguments x A time series max freq The glm is fitted on the frequency range 0 max freq order Order of the polynomial to fit to the periodogram max length The data x is aggregated if necessary by taking batch means so that the length of the series is less
15. ce the individual chains have not had time to range all over the stationary distribution and the second method will overestimate the variance since the starting points were chosen to be overdispersed The convergence diagnostic is based on the assumption that the target distribution is normal A Bayesian credible interval can be constructed using a t distribution with mean jt Sample mean of all chains combined and variance P B V2o mn and degrees of freedom estimated by the method of moments d 2 0 Var V gelman plot 15 Use of the t distribution accounts for the fact that the mean and variance of the posterior distribution are estimated The convergence diagnostic itself is Values substantially above 1 indicate lack of convergence If the chains have not converged Bayesian credible intervals based on the t distribution are too wide and have the potential to shrink by this factor if the MCMC run is continued Note The multivariate a version of Gelman and Rubin s diagnostic was proposed by Brooks and Gelman 1997 References Gelman A and Rubin DB 1992 Inference from iterative simulation using multiple sequences Statistical Science 7 457 511 Brooks SP and Gelman A 1997 General methods for monitoring convergence of iterative simu lations Journal of Computational and Graphical Statistics 7 434 455 See Also gelman plot gelman plot Gelman Rubin Brooks plot Description Thi
16. d Welch P D A spectral method for confidence interval generation and run length control in simulations Communications of the ACM Vol 24 pp233 245 1981 See Also spectrum spectrum0 ar glm 34 spectrum0 ar spectrum0 ar Estimate spectral density at zero Description The spectral density at frequency zero is estimated by fitting an autoregressive model spect rum0 x length x estimates the variance of mean x Usage spectrum0 ar x Arguments x A time series Details The ar function to fit an autoregressive model to the time series x For multivariate time series separate models are fitted for each column The value of the spectral density at zero is then given by a well known formula Value A list with the following values spec The predicted value of the spectral density at frequency zero order The order of the fitted model Note The definition of the spectral density used here differs from that used by spec pgram We con sider the frequency range to be between 0 and 0 5 not between 0 and frequency x 2 See Also Spectrum spectrum0 glm summary mcmc 35 summary mcmc Summary statistics for Markov Chain Monte Carlo chains Description summary mcmc produces two sets of summary statistics for each variable Mean standard deviation naive standard error of the mean ignoring autocorrelation of the chain and time series standard error based on an estimate of the spectral de
17. ethod for memc objects which is normally called by the generic function window In addition to the generic parameters st art and end the additional parameter thin may be used to thin out the Markov chain Setting thin k selects every kth iteration starting with the first Note that the value of thin is absolute not relative The value supplied given to the parameter thin must be a multiple of thin x Values of start end and thin which are inconsistent with x are ignored but a warning message is issued Usage HH S3 method for class momc window x start end thin Arguments x an mcmc object start the first iteration of interest end the last iteration of interest thin the required interval between successive samples futher arguments for future methods See Also window thin Index Topic array crosscorr 9 mcmc convert 20 Topic datasets line 19 Topic distribution Cramer l Topic file bugs2jags 6 read coda 28 read coda interactive 29 read openbugs 30 Topic hplot autocorr plot 5 crosscorr plot 10 cumuplot 10 densplot 11 gelman plot 14 geweke plot 17 plot mecmc 25 traceplot 36 trellisplots 36 Topic htest gelman diag 12 geweke diag 16 heidel diag 18 HPDinterval 2 raftery diag 26 Topic manip varnames 40 Topic multivariate crosscorr 9 Topic ts as ts mcmc 3 autocorr 3 autocorr diag 4 batchSE 5 ffectiveSize 12 meme 19 43 mcmc list 21 mc
18. function and u is the qth quantile of U The process Z is derived from the Markov chain data by marginalization and truncation but is not itself a Markov chain However Z may behave as a Markov chain if it is sufficiently thinned out raftery diag calculates the smallest value of thinning interval k which makes the thinned chain Z7 behave as a Markov chain The required sample size is calculated from this thinned sequence Since some data is thrown away the sample size estimates are conservative The criterion for the number of burn in iterations m to be discarded is that the conditional dis tribution of ZE given Zo should be within converge eps of the equilibrium distribution of the chain Z7 Note raftery diag is based on the FORTRAN program gibbsit written by Steven Lewis and avail able from the Statlib archive References Raftery A E and Lewis S M 1992 One long run with diagnostics Implementation strategies for Markov chain Monte Carlo Statistical Science 7 493 497 Raftery A E and Lewis S M 1995 The number of iterations convergence diagnostics and generic Metropolis algorithms n Practical Markov Chain Monte Carlo W R Gilks D J Spiegel halter and S Richardson eds London U K Chapman and Hall read and check Read data interactively and check that it satisfies conditions Description Input is read interactively and checked against conditions specified by the arguments wha
19. ical flag indicating whether variables in x should be transformed to im prove the normality of the distribution If set to TRUE a log transform or logit transform as appropriate will be applied autoburnin a logical flag indicating whether only the second half of the series should be used in the computation If set to TRUE default and start x is less than end x 2 then start of series will be adjusted so that only second half of series is used Theory Gelman and Rubin 1992 propose a general approach to monitoring convergence of MCMC output in which m gt 1 parallel chains are run with starting values that are overdispersed relative to the posterior distribution Convergence is diagnosed when the chains have forgotten their initial val ues and the output from all chains is indistinguishable The gelman diag diagnostic is applied to a single variable from the chain It is based a comparison of within chain and between chain variances and is similar to a classical analysis of variance There are two ways to estimate the variance of the stationary distribution the mean of the empirical variance within each chain W and the empirical variance from all chains combined which can be expressed as A n 1W B WP n n where n is the number of iterations and B n is the empirical between chain variance If the chains have converged then both estimates are unbiased Otherwise the first method will underestimate the variance sin
20. in It resembles the tsp attribute of time series t s objects Usage mcpar x Arguments x An mcmcm or memc list object See Also ts meme meme list multi menu 25 multi menu Choose multiple options from a menu Description multi menu presents the user with a menu of choices labelled from 1 to the number of choices The user may choose one or more options by entering a comma separated list A range of values n n may also be specified using the operator Mixed expressions such as 1 3 5 6 are permitted If allow zero is set to TRUE one can select 0 to exit without choosing an item Usage El multi menu choices title header allow zero TRUI Arguments choices Character vector of labels for choices title Title printed before menu header Character vector of length 2 giving column titles allow zero Permit 0 as an acceptable response Value Numeric vector giving the numbers of the options selected or O if no selection is made Author s Martyn Plummer See Also menu nchain Dimensions of MCMC objects Description These functions give the dimensions of an MCMC object niter x returns the number of iterations nvar x returns the number of variables nchain x returns the number of parallel chains 26 plot mcmc Usage niter x nvar x nchain x Arguments x An mcmc or memc list object Value A numeric vector of length 1
21. in simulations Comm ACM 24 233 245 1981 Heidelberger P and Welch PD Simulation run length control in the presence of an initial transient Opns Res 31 1109 44 1983 Schruben LW Detecting initialization bias in simulation experiments Opns Res 30 569 590 1982 line Simple linear regression example Description Sample MCMC output from a simple linear regression model given in the BUGS manual Usage data line Format An meme object Source Spiegelhalter D J Thomas A Best N G and Gilks W R 1995 BUGS Bayesian inference using Gibbs Sampling Version 0 5 MRC Biostatistics Unit Cambridge mcmc Markov Chain Monte Carlo Objects Description The function memc is used to create a Markov Chain Monte Carlo object The data are taken to be a vector or a matrix with one column per variable An memc object may be summarized by the summary function and visualized with the plot function MCNC objects resemble time series t s objects and have methods for the generic functions t ime start end frequency and window Usage mcmc data NA start 1 nd numeric 0 thin 1 as mcomc x is mcmc x mcmc convert 21 Arguments data a vector or matrix of MCMC output start the iteration number of the first observation end the iteration number of the last observation thin the thinning interval between consecutive observations x An object that may be coerced to an mcmc object No
22. mc subset 22 mcmcUpgrade 23 mcpar 23 nchain 24 rejectionRate 31 spectrumo 31 spectrum0 ar 33 thin 34 time mcmc 35 window mcmc 41 Topic univar HPDinterval 2 summary mcmc 34 Topic utilities as ts mcmc 3 coda options 7 codamenu 9 multi menu 24 read and check 27 Coda Options coda options 7 22 mcmc memc subset 22 acf 4 5 39 acfplot trellisplots 36 acfplot mcmc list trellisplots 36 as array 21 as array mcmc list meme convert 20 as matrix 2 as matrix mcmc mcmc convert 20 as momc 21 as mocmc memc 19 as mcmc list memc list 21 as mcmc mcmc list memc convert 20 as ts 3 as ts mcmc 3 autocorr 3 5 9 autocorr diag 4 autocorr diag mcmc list autocorr diag 4 autocorr plot 4 5 5 batchSE 5 bugs2 Jags 6 chanames varnames 40 chanames lt varnames 40 coda options 7 codamenu 9 30 Cramer 1 crosscorr 9 10 crosscorr plot 9 10 cumuplot 10 density J1 densityplot mcmc trellisplots 36 densplot 11 26 36 display coda options coda options 7 dput 7 dump 7 ffectiveSize 6 12 end mcmc time mcmc 35 frequency 35 frequency mcmc time mcmc 35 gelman diag 12 15 gelman plot 14 14 geweke diag 16 17 geweke plot 16 17 glm 32 33 heidel diag 18 HPDinterval 2 HPDinterval mcmc list HPDinterval 2 image 10 is mcmc memc 19 is mcmc list mcmc list 21 Lattice 40 levelplot mcmc trellisplo
23. n s For Raftery and Lewis diagnostic the probability of obtaining an estimate in the interval q r qtr quantiles Vector of quantiles to print when calculating summary statistics for MCMC output trace Logical option that determines whether to plot a trace of the sampled output when plotting MCMC output user layout Logical option that determines whether current value of par mfrow should be used for plots TRUE or whether the optimal layout should be calculated FALSE Usage coda options display coda options stats FALSE plots FALSE diags FALSE Coda Options Coda Options Default Arguments stats logical flag show summary statistic options plots logical flag show plotting options diags logical flag show plotting options list of options See Also options 10 CI OSSCOIT codamenu Main menu driver for the coda package Description codamenu presents a simple menu based interface to the functions in the coda package It is designed for users who know nothing about the R S language Usage codamenu Author s Kate Cowles Nicky Best Karen Vines Martyn Plummer crosscorr Cross correlations for MCMC output Description crosscorr calculates cross correlations between variables in Markov Chain Monte Carlo output If x is an mcmc list then all chains in x are combined before calculating the correlation Usage crosscorr x Arguments x an memc or memc list objec
24. nsity at O Quantiles of the sample distribution using the quant iles argument Usage S3 method for class momc summary object quantiles c 0 025 0 25 0 5 0 75 0 975 Arguments object an object of class memc or memc list quantiles a vector of quantiles to evaluate for each variable a list of further arguments Author s Martyn Plummer See Also meme meme List thin Thinning interval Description thin returns the interval between successive values of a time series thin x is equivalent to 1 frequency x This is a generic function Methods have been implemented for mcmc objects Usage EHIN X aex 36 Arguments x a regular time series a list of arguments Author s Martyn Plummer See Also time time mcmc time mcmc Time attributes for mcmc objects Description These are methods for mcmc objects for the generic time series functions Usage S3 method for class mcmc time x S3 method for class mcmc start x S3 method for class momc end X zaa S3 method for class mcmc thin x Arguments x an memc or meme list object extra arguments for future methods See Also time start frequency thin traceplot 37 traceplot Trace plot of MCMC output Description Displays a plot of iterations vs sampled values for each variable in the chain with a separate plot per variable Usage traceplo
25. o autocorrelation Estimation of the effective sample size requires estimating the spectral density at frequency zero This is done by the function spectrum0 ar For a memc list object the effective sizes are summed across chains To get the size for each chain individually use 1apply x e fectiveSize Value A vector giving the effective sample size for each column of x See Also spectrum0 ar gelman diag Gelman and Rubin s convergence diagnostic Description The potential scale reduction factor is calculated for each variable in x together with upper and lower confidence limits Approximate convergence is diagnosed when the upper limit is close to 1 For multivariate chains a multivariate value is calculated that bounds above the potential scale reduction factor for any linear combination of the possibly transformed variables The confidence limits are based on the assumption that the stationary distribution of the variable under examination is normal Hence the transform parameter may be used to improve the normal approximation 14 gelman diag Usage gelman diag x confidence 0 95 transform FALSE autoburnin TRUE Arguments x Anmcmc list object with more than one chain and with starting values that are overdispersed with respect to the posterior distribution confidence the coverage probability of the confidence interval for the potential scale reduc tion factor transform a log
26. or happens to be close to 1 by chance By calculating the shrink factor at several points in time gelman plot shows if the shrink factor has really converged or whether it is still fluctuating References Brooks S P and Gelman A 1998 General Methods for Monitoring Convergence of Iterative Simulations Journal of Computational and Graphical Statistics 7 p434 455 See Also gelman diag geweke diag 17 geweke diag Geweke s convergence diagnostic Description Geweke 1992 proposed a convergence diagnostic for Markov chains based on a test for equality of the means of the first and last part of a Markov chain by default the first 10 and the last 50 If the samples are drawn from the stationary distribution of the chain the two means are equal and Geweke s statistic has an asymptotically standard normal distribution The test statistic is a standard Z score the difference between the two sample means divided by its estimated standard error The standard error is estimated from the spectral density at zero and so takes into account any autocorrelation The Z score is calculated under the assumption that the two parts of the chain are asymptotically independent which requires that the sum of rac1 and rac2 be strictly less than 1 Usage geweke diag x fracl 0 1 frac2 0 5 Arguments x an mcmc object fracl fraction to use from beginning of chain frac2 fraction to use from end of chain Value Z scores for a test
27. orr diag Autocorrelation function for Markov chains Description autocorr diag calculates the autocorrelation function for the Markov chain memc obj at the lags given by lags The lag values are taken to be relative to the thinning interval if relat ive TRUI Unlike autocorr if memc obj has many parmeters it only computes the autocorrelations with itself and not the cross correlations In cases where autocorr would return a matrix this function returns the diagonal of the matrix Hence it is more useful for chains with many parameters but may not be as helpful at spotting parameters E If memc obj is of class memc 11 st then the returned vector is the average autocorrelation across all chains Usage autocorr diag mcmc obj Arguments mcmc obj an object of class memc or memc list optional arguments to be passed to autocorr Value A vector containing the autocorrelations 6 batchSE Author s Russell Almond See Also autocorr ac autocorr plot autocorr plot Plot autocorrelations for Markov Chains Description Plots the autocorrelation function for each variable in each chain in x Usage autocorr plot x lag max auto layout TRUE ask Arguments x A Markov Chain lag max Maximum value at which to calculate acf auto layout If TRUE then set up own layout for plots otherwise use existing one ask If TRUE then prompt user before displaying each page of plots Default i
28. reated by the dput function and writes it in dump format used by JAGS NB WinBUGS stores its arrays in row order This is different from R and JAGS which both store arrays in column order This difference is taken into account by bugs2 jags which will automati cally reorder the data in arrays without changing the dimension Not yet available in S PLUS Usage bugs2jags infile outfile 8 coda options Arguments infile name of the input file outfile name of the output file Note If the input file is saved from WinBUGS it must be saved in plain text format The default format for files saved from WinBUGS is a binary compound document format with extension odc that cannot be read by bugs2jags Author s Martyn Plummer References Spiegelhalter DJ Thomas A Best NG and Lunn D 2003 WinBUGS version 1 4 user manual MRC Biostatistics Unit Cambridge UK See Also dput dump coda options Options settings for the codamenu driver Description coda options is a utility function that queries and sets options for the codamenu function These settings affect the behaviour of the functions in the coda library only when they are called via the codamenu interface The coda options function behaves just like the opt ions function in the base library with the additional feature that coda options default TRUE will reset all options to the default values Options can be pretty printed using the display
29. s 4 autocorr Examples data line HPDinterval line as ts momc Coerce mcmc object to time series Description as ts momc will coerce an memc object to a time series Usage as ts momc x Arguments x an mcmc object unused arguments for compatibility with generic as ts Author s Martyn Plummer See Also eats autocorr Autocorrelation function for Markov chains Description autocorr calculates the autocorrelation function for the Markov chain memc obj at the lags given by lags The lag values are taken to be relative to the thinning interval if relat ive TRUE High autocorrelations within chains indicate slow mixing and usually slow convergence It may be useful to thin out a chain with high autocorrelations before calculating summary statistics a thinned chain may contain most of the information but take up less space in memory Re running the MCMC sampler with a different parameterization may help to reduce autocorrelation Usage E autocorr x lags c 0 1 5 10 50 relative TRUI autocorr diag 5 Arguments x an mcmc object lags a vector of lags at which to calculate the autocorrelation relative a logical flag TRUE if lags are relative to the thinning interval of the chain or FALSE if they are absolute difference in iteration numbers Value A vector or array containing the autocorrelations Author s Martyn Plummer See Also acr autocorr plot autoc
30. s dev interactive inR and interactive in S PLUS graphical parameters See Also autocorr batchSl Batch Standard Error E Description Effective standard deviation of population to produce the correct standard errors Usage batchSE x batchSize 100 bugs2jags 7 Arguments x An mcmc or memc list object batchSize Number of observations to include in each batch Details Because of the autocorrelation the usual method of taking var x n overstates the precision of the estimate This method works around the problem by looking at the means of batches of the parameter If the batch size is large enough the batch means should be approximately uncorrelated and the normal formula for computing the standard error should work The batch standard error procedure is usually thought to be not as accurate as the time series meth ods used in summary and effectiveSize It is included here for completeness Value A vector giving the standard error for each column of x Author s Russell Almond References Roberts GO 1996 Markov chain concepts related to sampling algorithms in Gilks WR Richard son S and Spiegelhalter DJ Markov Chain Monte Carlo in Practice Chapman and Hall 45 58 See Also spectrum0 ar effectiveSize summary mcmc bugs2jags Convert WinBUGS data file to JAGS data file Description bugs2jags converts a WinBUGS data in the format called S Plus i e the format c
31. s plot shows the evolution of Gelman and Rubin s shrink factor as the number of iterations increases Usage gelman plot x bin width 10 max bins 50 confidence 0 95 transform FALSE auto layout TRUE ask col lty xlab ylab type 16 gelman plot Arguments x an mcmc object bin width Number of observations per segment excluding the first segment which always has at least 50 iterations max bins Maximum number of bins excluding the last one confidence Coverage probability of confidence interval transform Automatic variable transformation see gelman diag auto layout If TRUE then set up own layout for plots otherwise use existing one ask Prompt user before displaying each page of plots Defaultis dev interactive in R and interactive in S PLUS col graphical parameter see par lty graphical parameter see par xlab graphical parameter see par ylab graphical parameter see par type graphical parameter see par further graphical parameters Details The Markov chain is divided into bins according to the arguments bin width and max bins Then the Gelman Rubin shrink factor is repeatedly calculated The first shrink factor is calculated with observations 1 50 the second with observations 1 50 n where n is the bin width the third contains samples 1 50 2n and so on Theory A potential problem with gelman diag is that it may mis diagnose convergence if the shrink fact
32. t Value A matrix or 3 d array containing the correlations See Also ecrosscorr plot autocort crosscorr plot 11 crosscorr plot Plot image of correlation matrix Description crosscorr plot provides an image of the correlation matrix for x If x is an mcmc list object then all chains are combined The range 1 1 is divided into a number of equal length categories given by the length of co1 and assigned the corresponding color By default topographic colours are used as this makes it easier to distinguish positive and negative correlations Usage crosscorr plot x col topo colors 10 Arguments x an meme or momc list object col color palette to use graphical parameters See Also crosscorr image topo colors cumuplot Cumulative quantile plot Description Plots the evolution of the sample quantiles as a function of the number of iterations Usage cumuplot x probs c 0 025 0 5 0 975 ylab lty c 2 1 lwd c 1 2 type 1 ask auto layout TRUE col 1 12 Arguments x probs ylab lty auto layout ask Author s densplot an memc object vector of desired quantiles lwd type col graphical parameters If TRUE then set up own layout for plots otherwise use existing one If TRUE then prompt user before displaying each page of plots Default is dev interactive inR and interactive in S PLUS further graphical parameters Arni Magn
33. t lower upper and answer in If the input does not satisfy all the conditions an appropriate error message is produced and the user is prompted to provide input This process is repeated until a valid input value is entered Usage read and check message what numeric lower upper answer in default Arguments message message displayed before prompting for user input what the type of what gives the type of data to be read lower lower limit of input for numeric input only read coda 29 upper upper limit of input for numeric input only answer in the input must correspond to one of the elements of the vector answer in if supplied default value assumed if user enters a blank line Value The value of the valid input When the de ault argument is specified a blank line is accepted as valid input and in this case read and check returns the value of default Note Since the function does not return a value until it receives valid input it extensively checks the conditions for consistency before prompting the user for input Inconsistent conditions will cause an error Author s Martyn Plummer read coda Read output files in CODA format Description read coda reads Markov Chain Monte Carlo output in the CODA format produced by Open BUGS and JAGS By default all of the data in the file is read but the arguments start end and thin may be used to read a subset of the data If the arguments given to st
34. t x smooth TRUE col type ylab Arguments x An memc or memc list object smooth draw smooth line through trace plot col graphical parameter see par type graphical parameter see par ylab graphical parameter see par further graphical parameters Note You can call this function directly but it is more usually called by the plot mcmc function See Also densplot plot mcmc trellisplots Trellis plots for mcmc objects Description These methods use the Trellis framework as implemented in the 1attice package to produce space conserving diagnostic plots from mcmc and memc list objects The xyplot meth ods produce trace plots The densityplot methods and qqmath methods produce empirical density and probability plots The 1evelplot method depicts the correlation of the series The acfplot methods plot the auto correlation in the series Not yet available in S PLUS 38 Usage S3 method for class momc densityplot x data HH S3 method densityplot x data HH S3 method levelplot x HH S3 method qqmath x data HH S3 method qqmath x data trellisplots outer aspect xy default scales list relation free start 1 thin 1 main attr x title xlab plot points rug A subset for class momc list outer FALSE groups outer aspect xy default scales list relation free start 1 thin 1
35. t containing a representation of the data in one or more BUGS output files Note This function is normally called by the codamenu function but can also be used on a stand alone basis Author s Nicky Best Martyn Plummer References Spiegelhalter DJ Thomas A Best NG and Gilks WR 1995 BUGS Bayesian inference Using Gibbs Sampling Version 0 50 MRC Biostatistics Unit Cambridge read openbugs 31 See Also mcmc memc 1ist read coda codamenu read openbugs Read CODA output files produced by OpenBUGS Description read openbugs reads Markov Chain Monte Carlo output in the CODA format produced by OpenBUGS This is a convenience wrapper around the function read coda which allows you to read all the data output by OpenBUGS by specifying only the file stem Usage read openbugs stem start end thin quiet FALSE Arguments stem Character string giving the stem for the output files OpenBUGS produces files with names lt stem gt CODAindex txt lt stem gt CODAchain1 txt stem CODAchain2 txt start First iteration of chain end Last iteration of chain thin Thinning interval for chain quiet Logical flag If true a progress summary will be printed Value An object of class memc list containing output from all chains Author s Martyn Plummer References Spiegelhalter DJ Thomas A Best NG and Lunn D 2004 WinBUGS User Manual Version 2 0 June 2004 MRC Biostatistics Unit Cam
36. te The format of the memc class has changed between coda version 0 3 and 0 4 Older memc objects will now cause is mcmc to fail with an appropriate warning message Obsolete memc objects can be upgraded with the mcmcUpgrade function Author s Martyn Plummer See Also mcmc list mcmcUpgrade thin window mcmc summary mcmc plot mcmc meme convert Conversions of MCMC objects Description These are methods for the generic functions as matrix as array and as memc as matrix strips the MCMC attributes from an mcmc object and returns a matrix If iters TRUE then a column is added with the iteration number For mcmc 1list objects the rows of multiple chains are concatenated and if chains TRUE a column is added with the chain number mcmc list objects can be coerced to 3 dimensional arrays with the as array function An mcmc list object with a single chain can be coerced to an memc object with as memc If the argument has multiple chains this causes an error Usage HH S3 method for class momc as matrix x iters FALSE S3 method for class mcmc list as matrix x iters FALSE chains FALSE S3 method for class mcmc list as array x drop 22 memc list Arguments x An mcmc or memc list object iters logical flag add column for iteration number chains logical flag add column for chain number if mcmc list drop logical flag if TRUE
37. th Statist 23 192 212 1952 Csorgo S and Faraway JJ The exact and asymptotic distributions of the Cramer von Mises statistic J Roy Stat Soc B 58 221 234 1996 HPDinterval Highest Posterior Density intervals Description Create Highest Posterior Density HPD intervals for the parameters in an MCMC sample Usage HPDinterval obj prob 0 95 S3 method for class momc HPDinterval obj prob 0 95 S3 method for class mcmc list HPDinterval obj prob 0 95 Arguments obj The object containing the MCMC sample usually of class mcme or memc list prob A numeric scalar in the interval 0 1 giving the target probability content of the intervals The nominal probability content of the intervals is the multiple of 1 nrow obj nearest to prob Optional additional arguments for methods None are used at present Details For each parameter the interval is constructed from the empirical cdf of the sample as the shortest interval for which the difference in the ecdf values of the endpoints is the nominal probability Assuming that the distribution is not severely multimodal this is the HPD interval Value For an mcmc object a matrix with columns lower and upper and rows corresponding to the parameters The attribute Probability is the nominal probability content of the intervals A list of such matrices is returned for an mcmc list object Author s Douglas Bate
38. the result is coerced to the lowest possible dimension optional arguments to the various methods See Also as matrix as array as mcmc mcmc list Replicated Markov Chain Monte Carlo Objects Description The function mcmc list is used to represent parallel runs of the same chain with different starting values and random seeds The list must be balanced each chain in the list must have the same iterations and the same variables Diagnostic functions which act on memc objects may also be applied to memc list objects In general the chains will be combined if this makes sense otherwise the diagnostic function will be applied separately to each chain in the list Since all the chains in the list have the same iterations a single time dimension can be ascribed to the list Hence there are time series methods time window start end frequency and thin for memc list objects Anmomc list can be indexed as if it were a single memc object using the operator see exam ples below The operator selects a single mcmc object from the list Usage mcmc list as mcmc list x is mcmc list x Arguments a list of mcmc objects x an object that may be coerced to mcmc list Author s Martyn Plummer mcmc subset 23 See Also meme Examples data line xl lt line 1 Select first chain x2 line 1 drop FALSE Select first var from all chains varnames x2 varnames line 1
39. ts 36 INDEX line 19 meme 19 22 23 25 29 30 34 41 mcmc convert 20 mcmc list 20 21 23 25 30 34 41 mcmc subset 22 mcmcUpgrade 20 23 mcpar 23 menu 24 multi menu 24 nchain 24 niter nchain 24 nvar nchain 24 options 8 panel densityplot 39 panel xyplot 39 pcramer Cramer plot mcmc 11 20 25 36 plot mcmc list mcmc list 21 print 40 print mcmc memc 19 qqmath mcmc trellisplots 36 raftery diag 26 read and check 27 read coda 28 30 read coda interactive 29 29 read jags read coda 28 read openbugs 29 30 rejectionRate 31 rejectionRate mcmc list rejectionRate 31 spectrum 32 33 spectrumo 31 33 spectrum0 ar 6 12 32 33 start 35 start mcmc time mcmc 35 summary mcmc 6 20 34 thin 20 34 35 41 thin memc time mcmo 35 time 35 time momc 35 topo colors 10 traceplot 26 36 INDEX trellisplots 36 ts 23 update 40 varnames 40 varnames varnames 40 window 4l window mcmc 20 22 41 xyplot 39 xyplot momc trellisplots 36 45
40. ull hypothesis is accepted or 50 of the chain has been discarded The latter outcome constitutes failure of the stationarity test and indicates that a longer MCMC run is needed If the stationarity test is passed the number of iterations to keep and the number to discard are reported The half width test calculates a 95 confidence interval for the mean using the portion of the chain which passed the stationarity test Half the width of this interval is compared with the estimate of the mean If the ratio between the half width and the mean is lower than eps the halfwidth test is passed Otherwise the length of the sample is deemed not long enough to estimate the mean with sufficient accuracy Theory The heidel diag diagnostic is based on the work of Heidelberger and Welch 1983 who com bined their earlier work on simulation run length control Heidelberger and Welch 1981 with the work of Schruben 1982 on detecting initial transients using Brownian bridge theory Note If the half width test fails then the run should be extended In order to avoid problems caused by sequential testing the test should not be repeated too frequently Heidelberger and Welch 1981 suggest increasing the run length by a factor I gt 1 5 each time so that estimate has the same reasonably large proportion of new data 20 mcmc References Heidelberger P and Welch PD A spectral method for confidence interval generation and run length control
41. usson lt arnima u washington edu gt densplot Probability density function estimate from MCMC output Description Displays a plot of the density estimate for each variable in x calculated by the densit y function Usage densplot x Arguments x show obs bwf main ylim Note show obs TRUE bwf main ylim An mcmc or memc list object Show observations along the x axis Function for calculating the bandwidth If omitted the bandwidth is calculate by 1 06 times the minimum of the standard deviation and the interquartile range divided by 1 34 times the sample size to the negative one fifth power Title See par Limits on y axis See par Further graphical parameters You can call this function directly but it is more usually called by the plot mcmc function If a variable is bounded below at O or bounded in the interval 0 1 then the data are reflected at the boundary before being passed to the density function This allows correct estimation of a non zero density at the boundary See Also density plot mcmc effectiveSize 13 ffectiveSiz Effective sample size for estimating the mean Description Sample size adjusted for autocorrelation Usage ffectiveSize x Arguments x An meme or memc list object Details For a time series x of length N the standard error of the mean is var x n where n is the effective sample size n Nonly when there is n
42. utive samples Positive autocorrelation will increase the required sample size above this minimum value An estimate I the dependence factor of the extent to which autocorrela tion inflates the required sample size is also provided Values of I larger than 5 indicate strong autocorrelation which may be due to a poor choice of starting value high posterior correlations or stickiness of the MCMC algorithm The number of burn in iterations to be discarded at the beginning of the chain is also calculated Usage raftery diag data q 0 025 r 0 005 s 0 95 converge eps 0 001 Arguments data an memc object q the quantile to be estimated r the desired margin of error of the estimate S the probability of obtaining an estimate in the interval q r q 1 converge eps Precision required for estimate of time to convergence Value A list with class raftery diag A print method is available for objects of this class the contents of the list are tspar The time series parameters of data params A vector containing the parameters r s and q Niters The number of iterations in data resmatrix A 3 d array containing the results M the length of burn in N the required sample size N mn the minimum sample size based on zero autocorrelation and I M N Nmin the dependence factor 28 read and check Theory The estimated sample size for variable U is based on the process Z d U lt u where d is the indicator
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