I just wanted to clarify one thing. Agus wrote: " Under the case of phylo dependence, all off-diagonal values are expected to be non zero "
That is incorrect for species on opposite sides of the root of the phylogenetic tree, as they have zero shared phylogenetic history in the context of the set of species under analysis. So, your expected variance-covariance matrix for the trait (or for residuals from a regression model) will have some blocks of zeros on the off-diagonals. Cheers, Ted Theodore Garland, Jr., Distinguished Professor Department of Evolution, Ecology, and Organismal Biology (EEOB) University of California, Riverside Riverside, CA 92521 Office Phone: (951) 827-3524 Facsimile: (951) 827-4286 (not confidential) Email: tgarl...@ucr.edu http://www.biology.ucr.edu/people/faculty/Garland.html http://scholar.google.com/citations?hl=en&user=iSSbrhwAAAAJ Director, UCR Institute for the Development of <http://idea.ucr.edu/>Educational Applications <http://idea.ucr.edu/> Editor in Chief, *Physiological and Biochemical Zoology <http://www.press.uchicago.edu/ucp/journals/journal/pbz.html>* Fail Lab: Episode One *https://www.youtube.com/watch?v=c0msBWyTzU0 <https://www.youtube.com/watch?v=c0msBWyTzU0>* On Mon, Aug 27, 2018 at 7:28 AM, Agus Camacho <agus.cama...@gmail.com> wrote: > Many thanks to all for your very helpful references, commands and examples. > > For the sake of future readers, a summary of the off-list discussions and > further reading. > > Andrew Hipp provided a very didactic way to look at the vcv matrix as a > triangular distance matrix > > library(geiger) > tr <- sim.bdtree(n = 100) > C = as.dist(vcv(tr)) > C > > Now, it made sense that for n species where a trait X has been measured, > Cov[xi,xj] is an nxn matrix with species coordinates, which for any pair of > tip values Xi and Xj, represents the shared distances of these tips from > the root, multiplied for a rate of evolution, or matrix of rates. The > results of such multiplication constitute the *expected *covariances among > values at the tips. > > Under the case of phylo independence, all off-diagonal values of Cov[xi,xj] > are expected to become zero (because of shared history=0, and is a > multiplicative term in the covariance formula). At the diagonal, each trait > covariance with itself consists in the squared distance from the tip to the > root of the tree multiplied by the rate of evolution (or rates, if they are > not homogeneous). That can be represented by a unity matrix multiplied by > the squared term "sigma", which represents the rate of evolution, or a > matrix of rates of evolution along the tree. > > Under the case of phylo dependence, all off-diagonal values are expected to > be non zero (because of shared history>0 and is a multiplicative term in > the covariance formula). The rest of Cov [xi,xj] is similar to a case of > phylo independence. > > Thus, during a PGLS model fit procedure, R calculates Cov [xi,xj] for a > given model of evolution over the residuals of the OLS regression. These > residuals should contain phylogenetic dependence coming from shared > evolution of trait values in X and Y. That is because the OLS regression > does not account for such dependence. The residuals of an OLS do not need > to be one unity matrix, but the dependence in residuals should look like it > to assume phylo independence in trait values. Finally, the PGLS procedure > modifies the predictive and predicted variables accordingly, to avoid a > case of phylo dependence under the specified model of evolution, and > finally performs an OLS. > > Many thanks again to the refs pointed by Brian, and Liam's 2010 paper on > phylo signal. Matterial from Andrew, Julen and Diogo Provete were also > super didactic. > > Hope that help others too, > Cheers > > Dr. Agustín Camacho Guerrero. Universidade de São Paulo. > http://www.agustincamacho.com > Laboratório de Comportamento e Fisiologia Evolutiva, Departamento de > Fisiologia, > Instituto de Biociências, USP.Rua do Matão, trav. 14, nº 321, Cidade > Universitária, > São Paulo - SP, CEP: 05508-090, Brasil. > > > > El dom., 26 ago. 2018 a las 21:55, Julien Clavel (< > julien.cla...@hotmail.fr>) > escribió: > > > Hi Agus, > > > > I just posted some courses I did a while ago to understand the variances > > and covariances of a BM process on trees and time-series (R markdown): > > https://github.com/JClavel/Examples <https://github.com/JClavel/Teaching > > > > > > This is also based on the previous mentioned references illustrated with > > some simulations. > > Hope it may helps... > > > > Best, > > > > Julien > > > > ------------------------------ > > *De :* R-sig-phylo <r-sig-phylo-boun...@r-project.org> de la part de > > Andrew Hipp <ah...@mortonarb.org> > > *Envoyé :* dimanche 26 août 2018 05:36 > > *À :* bome...@utk.edu > > *Cc :* mailman, r-sig-phylo; Agus Camacho > > *Objet :* Re: [R-sig-phylo] understanding variance-covariance matrix > > > > I'll second Brian's self-citation. O'Meara et al. 2006 is I think one of > > the best introductions to the phylogenetic covariance matrix, and I often > > direct students to it. > > > > Brian's point about the relationship between observed and expected > > covariance is illustrated here in a brief note I wrote up for students > this > > spring: > > > > > > https://github.com/andrew-hipp/PCM-2018/blob/master/R- > tutorials/2018-PCM-covarianceMatrixRuminations.ipynb > > > > It might be helpful, or it might not. I hope so! > > > > Take care, > > Andrew > > > > On Sat, Aug 25, 2018 at 1:33 PM, Brian O'Meara <omeara.br...@gmail.com> > > wrote: > > > > > Hi, Agus. The variance-covariance matrix comes from the tree and the > > > evolutionary model, not the data. Each entry between taxa A and B in > the > > > VCV is how much covariance I should expect between data for taxa A and > B > > > simulated up that tree using that model. I don't want to be *that guy*, > > but > > > O'Meara et al. (2006) > > > https://onlinelibrary.wiley.com/doi/10.1111/j.0014-3820.2006.tb01171.x > > has > > > a fairly accessible explanation of this (largely b/c I was just > learning > > > about VCVs when working on that paper). Hansen and Martins (1996) > > > https://onlinelibrary.wiley.com/doi/10.1111/j.1558-5646.1996.tb03914.x > > > have > > > a much more detailed description of how you get these covariance > matrices > > > from microevolutionary processes. > > > > > > Typically, ape::vcv() is how you get a variance covariance for a > > phylogeny, > > > assuming Brownian motion and no measurement error. It just basically > > takes > > > the history two taxa share to create the covariance (or variance, if > the > > > two taxa are the same taxon). A different approach, which seems to be > > what > > > you're doing, would be to simulate up a tree many times, and then for > > each > > > pair of taxa (including the pair of a taxon with itself, the diagonal > of > > > the VCV), calculate the covariance. These approaches should get the > same > > > results, though the shared history on the tree approach is faster. > > > > > > Best, > > > Brian > > > > > > > > > ____________________________________________________________ > ___________ > > > Brian O'Meara, http://www.brianomeara.info, especially Calendar > > > <http://brianomeara.info/calendars/omeara/>, CV > > > <http://brianomeara.info/cv/>, and Feedback > > > <http://brianomeara.info/teaching/feedback/> > > > > > > Associate Professor, Dept. of Ecology & Evolutionary Biology, UT > > Knoxville > > > Associate Head, Dept. of Ecology & Evolutionary Biology, UT Knoxville > > > > > > > > > > > > On Sat, Aug 25, 2018 at 1:16 PM Agus Camacho <agus.cama...@gmail.com> > > > wrote: > > > > > > > Dear list users, > > > > > > > > I am trying to make an easy R demonstration to teach the > > > > variance-covariance matrix to students. However, After consulting the > > > > internet and books, I found myself facing three difficulties to > > > understand > > > > the math and code behind this important matrix. As this list is > > answered > > > by > > > > several authors of books of phylocomp methods, thought this might > make > > an > > > > useful general discussion. > > > > > > > > Here we go, > > > > > > > > 1) I dont know how to generate a phyloVCV matrix in R (Liams kindly > > > > described some options here > > > > < > > > > http://blog.phytools.org/2013/12/three-different-ways-to- > > > calculate-among.html > > > > > > > > > but I cannot tell for sure what is X made of. It would seem a > dataframe > > > of > > > > some variables measured across species. But then, I get errors when I > > > > write: > > > > > > > > tree <- pbtree(n = 10, scale = 1) > > > > tree$tip.label <- sprintf("sp%s",seq(1:n)) > > > > x <- fastBM(tree) > > > > y <- fastBM(tree) > > > > X=data.frame(x,y) > > > > rownames(X)=tree$tip.label > > > > ## Revell (2009) > > > > A<-matrix(1,nrow(X),1)%*%apply(X,2,fastAnc,tree=tree)[1,] > > > > V1<-t(X-A)%*%solve(vcv(tree))%*%(X-A)/(nrow(X)-1) > > > > ## Butler et al. (2000) > > > > Z<-solve(t(chol(vcv(tree))))%*%(X-A) > > > > V2<-t(Z)%*%Z/(nrow(X)-1) > > > > > > > > ## pics > > > > Y<-apply(X,2,pic,phy=tree) > > > > V3<-t(Y)%*%Y/nrow(Y) > > > > > > > > 2) The phyloVCV matrix has n x n coordinates defined by the n > species, > > > and > > > > it represents covariances among observations made across the n > species, > > > > right?. Still, I do no know whether these covariances are calculated > > over > > > > a) X vs Y values for each pair of species coordinates in the matrix, > > > across > > > > the n species, or b) directly over the vector of n residuals of Y, > > after > > > > correlating Y vs X, across all pairs of species coordinates. I think > it > > > may > > > > be a) because, by definition, variance cannot be calculated for a > > single > > > > value. I am not sure though, since it seems the whole point of PGLS > is > > to > > > > control phylosignal within the residuals of a regression procedure, > > prior > > > > to actually making it. > > > > > > > > 3) If I create two perfeclty correlated variables with independent > > > > observations and calculate a covariance or correlation matrix for > > them, I > > > > do not get a diagonal matrix, with zeros at the off diagonals (ex. > here > > > > < > > > > > > https://www.dropbox.com/s/y8g3tkzk509pz58/vcvexamplewithrandomvariables. > > > xlsx?dl=0 > > > > >), > > > > why expect then a diagonal matrix for the case of independence among > > the > > > > observations? > > > > > > > > Thanks in advance and sorry if I missed anything obvious here! > > > > Agus > > > > Dr. Agustín Camacho Guerrero. Universidade de São Paulo. > > > > http://www.agustincamacho.com > > > > Laboratório de Comportamento e Fisiologia Evolutiva, Departamento de > > > > Fisiologia, > > > > Instituto de Biociências, USP.Rua do Matão, trav. 14, nº 321, Cidade > > > > Universitária, > > > > São Paulo - SP, CEP: 05508-090, Brasil. > > > > > > > > [[alternative HTML version deleted]] > > > > > > > > _______________________________________________ > > > > R-sig-phylo mailing list - R-sig-phylo@r-project.org > > > > https://stat.ethz.ch/mailman/listinfo/r-sig-phylo > > > > Searchable archive at > > > > http://www.mail-archive.com/r-sig-phylo@r-project.org/ > > > > > > > > > > [[alternative HTML version deleted]] > > > > > > _______________________________________________ > > > R-sig-phylo mailing list - R-sig-phylo@r-project.org > > > https://stat.ethz.ch/mailman/listinfo/r-sig-phylo > > > Searchable archive at http://www.mail-archive.com/r- > > > sig-ph...@r-project.org/ > > > > > > > > > > > -- > > Andrew Hipp, PhD > > Senior Scientist in Plant Systematics and Herbarium Curator, The Morton > > Arboretum > > Lecturer, Committee on Evolutionary Biology, University of Chicago > > > > The Morton Arboretum > > 4100 Illinois Route 53 / Lisle IL 60532-1293 / USA > > +1 630 725 2094 > > > > Lab: http://systematics.mortonarb.org/lab > > Hebarium data: http://vplants.org > > U of Chicago, CEB: http://evbio.uchicago.edu/ > > Phenology of the East Woods: http://systematics.mortonarb.org/phenology > > > > [[alternative HTML version deleted]] > > > > _______________________________________________ > > R-sig-phylo mailing list - R-sig-phylo@r-project.org > > https://stat.ethz.ch/mailman/listinfo/r-sig-phylo > > Searchable archive at > > http://www.mail-archive.com/r-sig-phylo@r-project.org/ > > > > [[alternative HTML version deleted]] > > _______________________________________________ > R-sig-phylo mailing list - R-sig-phylo@r-project.org > https://stat.ethz.ch/mailman/listinfo/r-sig-phylo > Searchable archive at http://www.mail-archive.com/r- > sig-ph...@r-project.org/ > [[alternative HTML version deleted]] _______________________________________________ R-sig-phylo mailing list - R-sig-phylo@r-project.org https://stat.ethz.ch/mailman/listinfo/r-sig-phylo Searchable archive at http://www.mail-archive.com/r-sig-phylo@r-project.org/
