Local Sensitivity Analysis
Given a model consisting of differential equations, estimates the local effect of certain parameters on selected sensitivity variables by calculating a matrix of so-called sensitivity functions. In this matrix the (i,j)-th element contains
dy_i/dpar_j*parscale_j/varscale_i
and where y_i is an output variable (at a certain time instance), par_j is a parameter, and varscale_i is the scaling of variable y_i, parscale_j is the scaling of parameter par_j.
sensFun(func, parms, sensvar = NULL, senspar = names(parms),
varscale = NULL, parscale = NULL, tiny = 1e-8, map = 1, ...)
## S3 method for class 'sensFun'
summary(object, vars = FALSE, ...)
## S3 method for class 'sensFun'
pairs(x, which = NULL, ...)
## S3 method for class 'sensFun'
plot(x, which = NULL, legpos="topleft", ask = NULL, ...)
## S3 method for class 'summary.sensFun'
plot(x, which = 1:nrow(x), ...)func |
an R-function that has as first argument |
parms |
parameters passed to |
sensvar |
the output variables for which the sensitivity needs
to be estimated. Either |
senspar |
the parameters whose sensitivity needs to be
estimated, the default=all parameters. Either a vector with
parameter names, or a vector with indices to positions
of parameters in |
varscale |
the scaling (weighing) factor for sensitivity
variables, |
parscale |
the scaling (weighing) factor for sensitivity
parameters, |
tiny |
the perturbation, or numerical difference, factor, see details. |
map |
the column number with the (independent) mapping variable
in the output matrix returned by |
... |
additional arguments passed to |
object |
an object of class |
x |
an object of class |
vars |
if FALSE: summaries per parameter are returned; if
|
which |
the name or the index to the variables that should be plotted. Default = all variables. |
legpos |
position of the legend; set to |
ask |
logical; if |
There are essentially two ways in which to use function sensFun.
When func returns a matrix or data frame with output
values, sensFun can be used for sensitivity analysis,
estimating the impact of parameters on output variables.
When func returns an instance of class modCost
(as returned by a call to function modCost), then
sensFun can be used for parameter
identifiability. In this case the results from sensFun are
used as input to function collin. See the help file for
collin.
For each sensitivity parameter, the number of sensitivity functions
estimated is: length(sensvar) * length(mapping variable), i.e. one for
each element returned by func (except the mapping variable).
The sensitivity functions are estimated numerically. This means that each parameter value par_j is perturbed as max(tiny,par_j)*(1+tiny)
a data.frame of class sensFun containing the sensitivity
functions this is one row for each sensitivity variable at each
independent (time or position) value and the following columns:
x, the value of the independent (mapping) variable, usually
time (solver= "ode.."), or distance (solver= "steady.1D")
var, the name of the observed variable,
..., a number of columns, one for each sensitivity parameter
Sensitivity functions are generated by perturbing one by one the parameters with a very small amount, and quantifying the differences in the output.
It is important that the output is generated with high precision, else it
is possible, that the sensitivity functions are just noise.
For instance, when used with a dynamic model (using solver from deSolve)
set the tolerances atol and rtol to a lower value, to see if
the sensitivity results make sense.
The following methods are provided:
summary. Produces summary statistics of the sensitivity
functions, a data.frame with: one row for each parameter
and the following columns:
L1: the L1-norm sum(abs(Sij))/n,
L2: the L2-norm sqrt(sum(Sij^2)/n),
Mean: the mean of the sensitivity functions,
Min: the minimal value of the sensitivity functions,
Max: the maximal value of the sensitivity functions.
var the summary of the variables sensitivity functions, a
data.frame with the same columns as model
and one row for each parameter + variable combination.
This is only outputted if the variable names are effectively known
plot plots the sensitivity functions for each parameter; each parameter has its own color.
By default, the sensitivity functions for all variables are plotted
in one figure, unless which gives a selection of variables;
in that case, each variable will be plotted in a separate figure, and
the figures aligned in a rectangular grid, unless par mfrow is
passed as an argument.
pairs produces a pairs plot of the sensitivity results; per parameter.
By default, the sensitivity functions for all variables are plotted
in one figure, unless which gives a selection of variables.
Overrides the default gap = 0, upper.panel = NA, and
diag.panel.
Karline Soetaert <karline.soetaert@nioz.nl>
Soetaert, K. and Herman, P. M. J., 2009. A Practical Guide to Ecological Modelling – Using R as a Simulation Platform. Springer, 390 pp.
Brun, R., Reichert, P. and Kunsch, H.R., 2001. Practical Identificability Analysis of Large Environmental Simulation Models. Water Resour. Res. 37(4): 1015–1030.
Soetaert, K. and Petzoldt, T., 2010. Inverse Modelling, Sensitivity and Monte Carlo Analysis in R Using Package FME. Journal of Statistical Software 33(3) 1–28. http://www.jstatsoft.org/v33/i03
## =======================================================================
## Bacterial growth model as in Soetaert and Herman, 2009
## =======================================================================
pars <- list(gmax = 0.5, eff = 0.5,
ks = 0.5, rB = 0.01, dB = 0.01)
solveBact <- function(pars) {
derivs <- function(t, state, pars) { # returns rate of change
with (as.list(c(state, pars)), {
dBact <- gmax * eff * Sub/(Sub + ks) * Bact - dB * Bact - rB * Bact
dSub <- -gmax * Sub/(Sub + ks) * Bact + dB * Bact
return(list(c(dBact, dSub)))
})
}
state <- c(Bact = 0.1, Sub = 100)
tout <- seq(0, 50, by = 0.5)
## ode solves the model by integration ...
return(as.data.frame(ode(y = state, times = tout, func = derivs,
parms = pars)))
}
out <- solveBact(pars)
plot(out$time, out$Bact, ylim = range(c(out$Bact, out$Sub)),
xlab = "time, hour", ylab = "molC/m3", type = "l", lwd = 2)
lines(out$time, out$Sub, lty = 2, lwd = 2)
lines(out$time, out$Sub + out$Bact)
legend("topright", c("Bacteria", "Glucose", "TOC"),
lty = c(1, 2, 1), lwd = c(2, 2, 1))
## sensitivity functions
SnsBact <- sensFun(func = solveBact, parms = pars,
sensvar = "Bact", varscale = 1)
head(SnsBact)
plot(SnsBact)
plot(SnsBact, type = "b", pch = 15:19, col = 2:6,
main = "Sensitivity all vars")
summary(SnsBact)
plot(summary(SnsBact))
SF <- sensFun(func = solveBact, parms = pars,
sensvar = c("Bact", "Sub"), varscale = 1)
head(SF)
tail(SF)
summary(SF, var = TRUE)
plot(SF)
plot(SF, which = c("Sub","Bact"))
pm <- par(mfrow = c(1,3))
plot(SF, which = c("Sub", "Bact"), mfrow = NULL)
plot(SF, mfrow = NULL)
par(mfrow = pm)
## Bivariate sensitivity
pairs(SF) # same color
pairs(SF, which = "Bact", col = "green", pch = 15)
pairs(SF, which = c("Bact", "Sub"), col = c("green", "blue"))
mtext(outer = TRUE, side = 3, line = -2,
"Sensitivity functions", cex = 1.5)
## pairwise correlation
cor(SnsBact[,-(1:2)])Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.