sensFun: Local Sensitivity Analysis

Description

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

$\frac{\partial y_{i}}{\partial Θ_{j}} \cdot \frac{Δ Θ_{j}}{Δ y_{i}}$

and where $y_{i}$ is an output variable (at a certain time instance), $Θ_{j}$ is a parameter, and $Δ y_{i}$ is the scaling of variable $y_{i}$ , $Δ Θ_{j}$ is the scaling of parameter $Θ_{j}$ .

Usage

sensFun(func, parms, sensvar = NULL, senspar = names(parms),
        varscale = NULL, parscale = NULL, tiny = 1e-8, map = 1, ...)
# S3 method for sensFun
summary(object, vars = FALSE, ...)
# S3 method for sensFun
pairs(x, which = NULL, ...)
# S3 method for sensFun
plot(x, which = NULL, legpos="topleft", ask = NULL, ...)
# S3 method for summary.sensFun
plot(x, which = 1:nrow(x), ...)

Value

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

The data.frame returned by sensFun has methods for the generic functions summary, plot,

pairs -- see note.

Arguments

func: an R-function that has as first argument parms and that returns a matrix or data.frame with the values of the output variables (columns) at certain output intervals (rows), and -- optionally -- a mapping variable (by default the first column).
parms: parameters passed to func; should be either a vector, or a list with named elements. If NULL, then the first element of parInput is taken.
sensvar: the output variables for which the sensitivity needs to be estimated. Either NULL, the default, which selects all variables, or a vector with variable names (which should be present in the matrix returned by func), or a vector with indices to variables as present in the output matrix (note that the column of this matrix with the mapping variable should not be selected).
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 parms.
varscale: the scaling (weighing) factor for sensitivity variables, NULL indicates that the variable value is used.
parscale: the scaling (weighing) factor for sensitivity parameters, NULL indicates that the parameter value is used.
tiny: the perturbation, or numerical difference, factor, see details.
map: the column number with the (independent) mapping variable in the output matrix returned by func. For dynamic models solved by integration, this will be the (first) column with time. For 1-D spatial output, this column will be some distance variable. Set to NULL if there is no mapping variable. Mapping variables should not be selected for estimating sensitivity functions; they are used for plotting.
...: additional arguments passed to func or to the methods.
object: an object of class sensFun.
x: an object of class sensFun.
vars: if FALSE: summaries per parameter are returned; if TRUE, summaries per parameter and per variable are returned.
which: the name or the index to the variables that should be plotted. Default = all variables.
legpos: position of the legend; set to NULL to avoid plotting a legend.
ask: logical; if TRUE, the user is asked before each plot, if NULL the user is only asked if more than one page of plots is necessary and the current graphics device is set interactive, see par(ask = ...) and dev.interactive.

Author

Karline Soetaert <karline.soetaert@nioz.nl>

Details

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 $Θ_{j}$ is perturbed as $max (t i n y, Θ_{j} \cdot (1 + t i n y))$

References

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. tools:::Rd_expr_doi("10.1029/2000WR900350")

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. tools:::Rd_expr_doi("10.18637/jss.v033.i03")

Examples

Run this code

## =======================================================================
## 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)])

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