| 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, ...)
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 |
... |
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"))