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Elasticity in Leslie Matrix

 Dear R-users, I would like to calculate elasticities and sensitivities of each parameters involved in the following transition matrix: A <- matrix(c( sigma*s0*f1,  sigma*s0*f2,                s,                 v ), nrow=2, byrow=TRUE,dimnames=list(stage,stage)) The command "eigen.analysis" avaliable in package "popbio" provides sensibility matrix and elasticity matrix (same dimension than A). I would like to know if there is a way to calculate separetely the elasticity of sigma, s0, f1, f2, s and v ? Thanks and regards, privalan
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Re: Elasticity in Leslie Matrix

 > I would like to calculate elasticities and sensitivities of each parameters > involved in the following transition matrix: > > A <- matrix(c( > sigma*s0*f1,  sigma*s0*f2, >                s,                 v > ), nrow=2, byrow=TRUE,dimnames=list(stage,stage)) > > The command "eigen.analysis" avaliable in package "popbio" provides > sensibility matrix and elasticity matrix (same dimension than A). I would > like to know if there is a way to calculate separetely the elasticity of > sigma, s0, f1, f2, s and v ? > If you want the eigenvalue second derivatives, check the secder and fullsecder functions in the demogR package (also elassens for partial derivatives of the eigenvalue elasticities).  A description of that package is also found in the recent special issue of Ecology in R in Journal of Statistical Software.   Chris ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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Re: Elasticity in Leslie Matrix

 In reply to this post by privalan On Mon, 2007-10-22 at 06:24 -0700, privalan wrote: > Dear R-users, > > I would like to calculate elasticities and sensitivities of each parameters > involved in the following transition matrix: > > A <- matrix(c( > sigma*s0*f1,  sigma*s0*f2, >                s,                 v > ), nrow=2, byrow=TRUE,dimnames=list(stage,stage)) > > > The command "eigen.analysis" avaliable in package "popbio" provides > sensibility matrix and elasticity matrix (same dimension than A). I would > like to know if there is a way to calculate separetely the elasticity of > sigma, s0, f1, f2, s and v ? You should first calculate the matrix element sensitivities, then calculate the derivatives of the elements of your matrix with respect to the parameter of interest. Then for each element in the matrix, the sensitivity of the parameter is just the product of the derivative and the appropriate element sensitivity, by the chain rule. Then add up all the sensitivities, multiply by your parameter and divide by lambda. ie plot(1:10, 1:10, type="n") text(5,6, expression(e[s]  == over(s, lambda)~sum(~over(paste(partialdiff,~lambda),paste(partialdiff,~a[ij])) %.% over(paste(partialdiff, ~a[ij]), paste(partialdiff, ~s)), ij)), cex=2) Be aware that these elasticities will in general not sum to unity, and cannot be interpreted as "contributions" to lambda. See Caswell (2001), page 232. Cheers, Simon. > > Thanks and regards, > > privalan -- Simon Blomberg, BSc (Hons), PhD, MAppStat. Lecturer and Consultant Statistician Faculty of Biological and Chemical Sciences The University of Queensland St. Lucia Queensland 4072 Australia Room 320 Goddard Building (8) T: +61 7 3365 2506 email: S.Blomberg1_at_uq.edu.au Policies: 1.  I will NOT analyse your data for you. 2.  Your deadline is your problem. The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data. - John Tukey. ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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Re: Elasticity in Leslie Matrix

 After a short exchange with the original questioner, I wrote the following function to calculate the elasticities of lower level variables in population transition matrices (Leslie matrices etc.) Perhaps it will be of use to others. There is no error-checking, so use with care. Users should consult Caswell (2001) for reference. Cheers, Simon. # example values to construct leslie matrix vl <- list(f1=1, f2=4, s0=.6, s=.4, v=.9, sigma=.5) # Expressions for each matrix element F1 <- expression(sigma*s0*f1) F2 <- expression(sigma*s0*f2) S <- expression(s) V <- expression(v) el <- c(F1, F2, S, V) elas.var <- function (elements, varlist) {   # elements should be a vector of expressions corresponding to the elements   # of the leslie matrix, in terms of the variables in varlist   require(demogR)   res <- vector("list", length(varlist))   deriv.funcs <- sapply(elements, deriv, namevec=names(varlist), function.arg=TRUE)   devs <- lapply(deriv.funcs, function (x) do.call(x, varlist))   leslie.mat <-  matrix(as.numeric(devs), nrow=sqrt(length(elements)), byrow=TRUE)   eig <- eigen.analysis(leslie.mat)   for (i in 1:length(varlist)) {     derivs <- matrix( as.numeric(lapply(devs, function (x) x@gradient[i])), nrow=sqrt(length(elements)), byrow=TRUE)     res[[i]] <- varlist[[i]]/eig\$lambda1*sum(derivs*eig\$sensitivities)     names(res)[i] <- names(varlist)[i]   }   res } # example output > elas.var(el, vl) \$f1  0.06671376 \$f2  0.2346064 \$s0  0.3013201 \$s  0.2346064 \$v  0.4640735 \$sigma  0.3013201 Simon Blomberg, BSc (Hons), PhD, MAppStat. Lecturer and Consultant Statistician Faculty of Biological and Chemical Sciences The University of Queensland St. Lucia Queensland 4072 Australia T: +61 7 3365 2506 email: S.Blomberg1_at_uq.edu.au Policies: 1.  I will NOT analyse your data for you. 2.  Your deadline is your problem. The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data. - John Tukey.         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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Re: Elasticity in Leslie Matrix

 Sorry, I didn't think carefully about your question on my first reply, so please ignore it.   Chapter 9 in Morris and Doak (2002) cover vital rate elasticities and sensitivities (see Box 9.1 for Matlab code http://www.sinauer.com/PVA/vitalsens.m).   Simon, thanks for posting your code, I will add a function called vitalsens to the next release of the popbio package and output both sensitivities and elasiticities. Here's the example for the emperor goose vital rates in chapter 9. vl<-list(  Ss0=0.1357,  Ss1=0.8926,  Sf2=0.6388,  Sf3= 0.8943) el<-expression( 0,  0,  Sf2*Ss1,Sf3*Ss1, Ss0,0,  0,      0, 0,  Ss1,0,      0, 0,  0,  Ss1,    Ss1) ## and the matching elasticities in table 9.1 elas.var(el, vl) \$Ss0  0.0797 \$Ss1  0.92 \$Sf2  0.00569 \$Sf3  0.074 Also, the expression el should be formatted so this command returns the projection matrix. matrix(sapply(el, eval, vl), nrow=4, byrow=TRUE) Thanks, Chris Simon Blomberg-4 wrote After a short exchange with the original questioner, I wrote the following function to calculate the elasticities of lower level variables in population transition matrices (Leslie matrices etc.) Perhaps it will be of use to others. There is no error-checking, so use with care. Users should consult Caswell (2001) for reference. Cheers, Simon. # example values to construct leslie matrix vl <- list(f1=1, f2=4, s0=.6, s=.4, v=.9, sigma=.5) # Expressions for each matrix element F1 <- expression(sigma*s0*f1) F2 <- expression(sigma*s0*f2) S <- expression(s) V <- expression(v) el <- c(F1, F2, S, V) elas.var <- function (elements, varlist) {   # elements should be a vector of expressions corresponding to the elements   # of the leslie matrix, in terms of the variables in varlist   require(demogR)   res <- vector("list", length(varlist))   deriv.funcs <- sapply(elements, deriv, namevec=names(varlist), function.arg=TRUE)   devs <- lapply(deriv.funcs, function (x) do.call(x, varlist))   leslie.mat <-  matrix(as.numeric(devs), nrow=sqrt(length(elements)), byrow=TRUE)   eig <- eigen.analysis(leslie.mat)   for (i in 1:length(varlist)) {     derivs <- matrix( as.numeric(lapply(devs, function (x) x@gradient[i])), nrow=sqrt(length(elements)), byrow=TRUE)     res[[i]] <- varlist[[i]]/eig\$lambda1*sum(derivs*eig\$sensitivities)     names(res)[i] <- names(varlist)[i]   }   res } # example output > elas.var(el, vl) \$f1  0.06671376 \$f2  0.2346064 \$s0  0.3013201 \$s  0.2346064 \$v  0.4640735 \$sigma  0.3013201 Simon Blomberg, BSc (Hons), PhD, MAppStat. Lecturer and Consultant Statistician Faculty of Biological and Chemical Sciences The University of Queensland St. Lucia Queensland 4072 Australia T: +61 7 3365 2506 email: S.Blomberg1_at_uq.edu.au Policies: 1.  I will NOT analyse your data for you. 2.  Your deadline is your problem. The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data. - John Tukey.         [[alternative HTML version deleted]] ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.