gradient {rootSolve} | R Documentation |
Given a vector of variables (x), and a function (f) that estimates one function value or a set of function values (f(x)), estimates the gadient matrix, containing, on rows i and columns j
d(f(x)_i)/d(x_j)
The gradient matrix is not necessarily square
gradient(f, x, centered = FALSE, pert=1e-8, ...)
f |
function returning one function value, or a vector of function values |
x |
either one value or a vector containing the x-value(s) at which the gradient matrix should be estimated |
centered |
if TRUE, uses a centered difference approximation, else a forward difference approximation |
pert |
numerical perturbation factor; increase depending on precision of model solution |
... |
other arguments passed to function f |
the function f
that estimates the function values will be called as
f(x, ...). If x
is a vector, then the first argument passed to f
should also be a vector.
The gradient is estimated numerically, by perturbing the x-values.
The gradient matrix where the number of rows equals the length of f
and the number of columns equals the length of x
.
the elements on i-th row and j-th column contain: d((f(x))_i)/d(x_j)
gradient
can be used to calculate so-called sensitivity functions
,
that estimate the effect of parameters on output variables.
Karline Soetaert <k.soetaert@nioo.knaw.nl>
Soetaert, K. and P.M.J. Herman (2008). A practical guide to ecological modelling - using R as a simulation platform. Springer.
jacobian.full
, for generating a full and square gradient (jacobian) matrix and where the function call is more complex
hessian
, for generating the hessian matrix
############################################################ # 1. Sensitivity analysis of the logistic differential equation # dN/dt = r*(1-N/K)*N , N(t0)=N0. # analytical solution of the logistic equation: logistic <- function (x,times) { with (as.list(x), { N=K/(1+(K-N0)/N0*exp(-r*times)) return(c(N=N)) }) } # parameters for the US population from 1900 x=c(N0=76.1,r=0.02,K=500) # Sensitivity function: SF: dfi/dxj at # output intervals from 1900 to 1950 SF<-gradient(f=logistic,x,times=0:50) # sensitivity, scaled for the value of the parameter: # [dfi/(dxj/xj)]= SF*x (columnise multiplication) sSF<-(t(t(SF)*x)) matplot(sSF,xlab="time",ylab="relative sensitivity ", main = "logistic equation",pch=1:3) legend("topleft",names(x),pch=1:3,col=1:3) # mean scaled sensitivity colMeans(sSF) ############################################################ # 2. Stability of the budworm model, as a function of its # rate of increase. # # Example from the book of Soetaert and Herman(2008) # A practical guide to ecological modelling # using R as a simulation platform. Springer # code and theory are explained in this book r <- 0.05 K <- 10 bet <- 0.1 alf <- 1 # density-dependent growth and sigmoid-type mortality rate rate <- function(x,r=0.05) r*x*(1-x/K)-bet*x^2/(x^2+alf^2) # Stability of a root ~ sign of eigenvalue of Jacobian stability <- function (r) { Eq <- uniroot.all(rate,c(0,10),r=r) eig <- vector() for (i in 1:length(Eq)) eig[i] <- sign (gradient(rate,Eq[i],r=r)) return(list(Eq=Eq,Eigen=eig)) } # bifurcation diagram rseq <- seq(0.01,0.07,by=0.0001) plot(0,xlim=range(rseq),ylim=c(0,10),type="n", xlab="r",ylab="B*",main="Budworm model, bifurcation", sub="Example from book of Soetaert and Herman") for (r in rseq) { st <- stability(r) points(rep(r,length(st$Eq)),st$Eq,pch=22, col=c("darkblue","black","lightblue")[st$Eigen+2], bg =c("darkblue","black","lightblue")[st$Eigen+2]) } legend("topleft",pch=22,pt.cex=2,c("stable","unstable"), col=c("darkblue","lightblue"),pt.bg=c("darkblue","lightblue"))