Sensitivity analysis and calibration

About this vignette

The present document shows how to conduct a sensitivity analyses and calibration exercises on the simulation models included in package medfate. The document is written assuming that the user is familiarized with the basic water balance model (i.e. function spwb). The aim of the exercises presented here are:

To determine which spwb() model parameters are more influential in determining stand transpiration and plant drought stress.
To determine which model parameters are more influential to determine model fit to soil water content dynamics.
To reduce the uncertainty in parameters determining fine root distribution, given an observed data set of soil water content dynamics.

As an example data set we will use here the same data sets provided to illustrate simulation functions in medfate. We begin by loading the package and the example forest data:

library(medfate)

## Package 'medfate' [ver. 4.8.0]

data(exampleforest)
exampleforest

## $treeData
##            Species   N   DBH Height Z50  Z95
## 1 Pinus halepensis 168 37.55    800 100  600
## 2     Quercus ilex 384 14.60    660 300 1000
## 
## $shrubData
##             Species Cover Height Z50  Z95
## 1 Quercus coccifera  3.75     80 200 1000
## 
## $herbCover
## [1] 10
## 
## $herbHeight
## [1] 20
## 
## $seedBank
## [1] Species Percent
## <0 rows> (or 0-length row.names)
## 
## attr(,"class")
## [1] "forest" "list"

We also load the species parameter table and the example weather dataset:

data(SpParamsMED)
data(examplemeteo)

Preparing model inputs

We will focus on three species/cohorts of the example data set:

PH_coh = paste0("T1_", SpParamsMED$SpIndex[SpParamsMED$Name=="Pinus halepensis"])
QI_coh = paste0("T2_", SpParamsMED$SpIndex[SpParamsMED$Name=="Quercus ilex"])
QC_coh = paste0("S1_", SpParamsMED$SpIndex[SpParamsMED$Name=="Quercus coccifera"])

The data set consists of a forest with two tree species (Pinus halepensis/T1_148 and Quercus ilex/T2_168) and one shrub species (Quercus coccifera/S1_165 or Kermes oak).

We first define a soil with four layers (default values of texture, bulk density and rock content) and the species input parameters for simulation function spwb():

examplesoil <- defaultSoilParams(4)
x1 <- spwbInput(exampleforest,examplesoil, SpParamsMED, control = defaultControl())

Although it is not necessary, we make an initial call to the model (spwb()) with the default parameter settings:

S1<-spwb(x1, examplemeteo, latitude = 41.82592, elevation = 100)

## Package 'meteoland' [ver. 2.2.2]

## Initial plant water content (mm): 4.73001
## Initial soil water content (mm): 290.875
## Initial snowpack content (mm): 0
## Performing daily simulations
## 
##  [Year 2001]:............
## 
## Final plant water content (mm): 4.7285
## Final soil water content (mm): 274.723
## Final snowpack content (mm): 0
## Change in plant water content (mm): -0.00151775
## Plant water balance result (mm): -0.00151775
## Change in soil water content (mm): -16.1521
## Soil water balance result (mm): -16.1521
## Change in snowpack water content (mm): 0
## Snowpack water balance result (mm): -7.10543e-15
## Water balance components:
##   Precipitation (mm) 513 Rain (mm) 462 Snow (mm) 51
##   Interception (mm) 92 Net rainfall (mm) 370
##   Infiltration (mm) 400 Infiltration excess (mm) 21 Saturation excess (mm) 0 Capillarity rise (mm) 0
##   Soil evaporation (mm) 26  Herbaceous transpiration (mm) 14 Woody plant transpiration (mm) 249
##   Plant extraction from soil (mm) 249  Plant water balance (mm) -0 Hydraulic redistribution (mm) 5
##   Runoff (mm) 21 Deep drainage (mm) 128

Function spwb() will be implicitly called multiple times in the sensitivity analyses and calibration analyses that we will illustrate below.

Sensitivity analysis

Introduction and input factors

Model sensitivity analyses are used to investigate how variation in the output of a numerical model can be attributed to variations of its input factors. Input factors are elements that can be changed before model execution and may affect its output. They can be model parameters, initial values of state variables, boundary conditions or the input forcing data (Pianosi et al. 2016).

According to Saltelli et al. (2016), there are three main purposes of sensitivity analyses:

Ranking aims at generating the ranking of the input factors according to their relative contribution to the output variability.
Screening aims at identifying the input factors, if any, which have a negligible influence on the output variability.
Mapping aims at determining the region of the input variability space that produces significant output values.

Here we will mostly interested in ranking parameters according to different objectives. We will take as input factors three plant traits (leaf area index, fine root distribution and the water potential corresponding to a reduction in plant conductance) in the three plant cohorts (species), and two soil factors (the rock fragment content of soil layer 1 and 2). In total, eleven model parameters will be studied. The following shows the initial values for plant trait parameters:

x1$above$LAI_live

## [1] 0.84874773 0.70557382 0.03062604

x1$below$Z50

## [1] 100 300 200

x1$paramsTransp$Psi_Extract

## [1] -0.9218219 -1.9726871 -2.1210726

x1$soil$rfc[1:2]

## [1] 25 45

In the following code we define a vector of parameter names (using naming rules of function modifyInputParams()) as well as the input variability space, defined by the minimum and maximum parameter values:

#Parameter names of interest
parNames = c(paste0(PH_coh,"/LAI_live"), paste0(QI_coh,"/LAI_live"), paste0(QC_coh,"/LAI_live"),
             paste0(PH_coh,"/Z50"), paste0(QI_coh,"/Z50"), paste0(QC_coh,"/Z50"),
             paste0(PH_coh,"/Psi_Extract"), paste0(QI_coh,"/Psi_Extract"), paste0(QC_coh,"/Psi_Extract"),
             "rfc@1", "rfc@2")
parNames

##  [1] "T1_148/LAI_live"    "T2_168/LAI_live"    "S1_165/LAI_live"   
##  [4] "T1_148/Z50"         "T2_168/Z50"         "S1_165/Z50"        
##  [7] "T1_148/Psi_Extract" "T2_168/Psi_Extract" "S1_165/Psi_Extract"
## [10] "rfc@1"              "rfc@2"

#Parameter minimum and maximum values
parMin = c(0.1,0.1,0.1,
           100,100,50,
           -5,-5,-5,
           25,25)
parMax = c(2,2,2,
           500,500,300,
           -1,-1,-1,
           75,75)

Model output functions

In sensitivity analyses, model output is summarized into a single variable whose variation is to be analyzed. Pianosi et al. (2016) distinguish two types of model output functions:

objective functions (also called loss or cost functions), which are measures of model performance calculated by comparison of modelled and observed variables.
prediction functions, which are scalar values that are provided to the model-user for their practical use, and that can be computed even in the absence of observations.

Here we will use examples of both kinds. First, we define a function that, given a simulation result, calculates total transpiration (mm) over the simulated period (one year):

sf_transp<-function(x) {sum(x$WaterBalance$Transpiration, na.rm=TRUE)}
sf_transp(S1)

## [1] 249.0844

Another prediction function can focus on plant drought stress. We define a function that, given a simulation result, calculates the average drought stress of plants (measured using the water stress index) over the simulated period:

sf_stress<-function(x) {
  lai <- x$spwbInput$above$LAI_live
  lai_p <- lai/sum(lai)
  stress <- droughtStress(x, index="WSI", draw = F)
  mean(sweep(stress,2, lai_p, "*"), na.rm=T)
}
sf_stress(S1)

## [1] 2.301471

Sensitivity analysis requires model output functions whose parameters are the input factors to be studied. $\begin{equation} y = g(\mathbf{x}) = g(x_1, x_2, \dots, x_n) \end{equation}$ where $y$ is the output, $g$ is the output function and $\mathbf{x} = \{x_1, x_2, \dots, x_n\}$ is the vector of parameter input factors. Functions of_transp and of_stress take simulation results as input, not values of input factors. Instead, we need to define functions that take soil and plant trait values as input, run the soil plant water balance model and return the desired prediction or performance statistic. These functions can be generated using the function factory optimization_function(). The following code defines one of such functions focusing on total transpiration:

of_transp<-optimization_function(parNames = parNames,
                                 x = x1,
                                 meteo = examplemeteo, 
                                 latitude = 41.82592, elevation = 100,
                                 summary_function = sf_transp)

Note that we provided all the data needed for simulations as input to optimization_function(), as well as the names of the parameters to study and the function sf_transp. The resulting object of_transp is a function itself, which we can call with parameter values (or sets of parameter values) as input:

of_transp(parMin)

## [1] 47.63267

of_transp(parMax)

## [1] 349.1011

It is important to understand the steps that are done when we call of_transp():

The function of_transp() calls spwb() using all the parameters specified in its construction (i.e. in the call to the function factory), except for the input factors indicated in parNames, which are specified as input at the time of calling of_transp().
The result of soil plant water balance is then passed to function sf_transp() and the output of this last function is returned as output of of_transp().

We can build a similar model output function, in this case focusing on plant stress (note that the only difference in the call to the factory is in the specification of sf_stress as summary function, instead of sf_transp).

of_stress<-optimization_function(parNames = parNames,
                                 x = x1, 
                                 meteo = examplemeteo, 
                                 latitude = 41.82592, elevation = 100,
                                 summary_function = sf_stress)
of_stress(parMin)

## [1] 0.9115247

of_stress(parMax)

## [1] 109.9439

As mentioned above, another kind of output function can be the evaluation of model performance. Here we will assume that performance in terms of predictability of soil water content is desired; and use a data set of ‘observed’ values (actually simulated values with gaussian error) as reference:

data(exampleobs)
head(exampleobs)

##        dates       SWC       ETR   E_T1_148   E_T2_168 FMC_T1_148 FMC_T2_168
## 1 2001-01-01 0.3007733 2.2436218 0.09187857 0.14142950   125.9071   93.07915
## 2 2001-01-02 0.3091627 2.3236565 0.26480973 0.19095008   125.9137   93.07863
## 3 2001-01-03 0.2996498 0.7409083 0.15345643 0.17546363   125.8760   93.10512
## 4 2001-01-04 0.3042764 1.7173522 0.23470647 0.04643454   125.8643   93.07022
## 5 2001-01-05 0.3054886 2.0002562 0.37687792 0.10623552   125.8493   93.08487
## 6 2001-01-06 0.3062005 2.0722706 0.16342360 0.05550329   125.9367   93.07343
##     BAI_T1_148 BAI_T2_168    DI_T1_148 DI_T2_168
## 1 6.222625e-06          0 9.948881e-08         0
## 2 3.091274e-10          0 1.071090e-11         0
## 3 1.298482e-13          0 0.000000e+00         0
## 4 2.886195e-11          0 5.552753e-13         0
## 5 1.287020e-03          0 1.367289e-05         0
## 6 1.471202e-03          0 1.000411e-05         0

where soil water content dynamics is in column SWC. The model fit to observed data can be measured using the Nash-Sutcliffe coefficient, which we calculate for the initial run using function evaluation_metric():

evaluation_metric(S1, measuredData = exampleobs, type = "SWC", 
                  metric = "NSE")

## [1] 0.9372955

A call to evaluation_metric() provides the coefficient given a model simulation result, but is not a model output function as we defined above. Analogously to the measures of total transpiration and average plant stress, we can use a function factory to define a model output function that takes input factors as inputs, runs the model and performs the evaluation:

of_eval<-optimization_evaluation_function(parNames = parNames,
                x = x1,
                meteo = examplemeteo, latitude = 41.82592, elevation = 100,
                measuredData = exampleobs, type = "SWC", 
                metric = "NSE")

Function of_eval() stores internally both the data needed for conducting simulations and the data needed for evaluating simulation results, so that we only need to provide values for the input factors:

of_eval(parMin)

## [1] 0.5982837

of_eval(parMax)

## [1] -11.76721

Global sensitivity analyses

Sensitivity analysis is either referred to as local or global, depending on variation of input factors is studied with respect to some initial parameter set (local) or the whole space of input factors is taken into account (global). Here we will conduct global sensitivity analyses using package sensitivity (Ioss et al. 2020):

library(sensitivity)

## Registered S3 method overwritten by 'sensitivity':
##   method    from 
##   print.src dplyr

## 
## Attaching package: 'sensitivity'

## The following object is masked from 'package:medfate':
## 
##     extract

This package provides a suite of approaches to global sensitivity analysis. Among them, we will follow the Elementary Effect Test implemented in function morris(). We call this function to analyze sensitivity of total transpiration simulated by spwb() to input factors (500 runs are done, so be patient):

sa_transp <- morris(of_transp, parNames, r = 50, 
             design = list(type = "oat", levels = 10, grid.jump = 3), 
             binf = parMin, bsup = parMax, scale=TRUE, verbose=FALSE)

Apart from indicating the sampling design to sample the input factor space, the call to morris() includes the response model function (in our case of_transp), the parameter names and parameter value boundaries (i.e. parMin and parMax).

print(sa_transp)

## 
## Call:
## morris(model = of_transp, factors = parNames, r = 50, design = list(type = "oat",     levels = 10, grid.jump = 3), binf = parMin, bsup = parMax,     scale = TRUE, verbose = FALSE)
## 
## Model runs: 600 
##                             mu     mu.star      sigma
## T1_148/LAI_live    153.6591445 162.6752000 84.4307909
## T2_168/LAI_live     95.3652583 107.8969236 62.7946309
## S1_165/LAI_live    146.7424176 152.7440550 72.7860722
## T1_148/Z50          -3.1140109   4.9123925 10.9257630
## T2_168/Z50          -1.0121568   2.3895855  8.0897898
## S1_165/Z50           0.1440795   0.4817739  0.9125476
## T1_148/Psi_Extract  -3.1010337   6.1051877 10.9368259
## T2_168/Psi_Extract  -2.6449397   3.5882482  5.8285725
## S1_165/Psi_Extract  -0.4657024   3.3910221  9.9270164
## rfc@1              -10.0489723  10.0489723 17.5544947
## rfc@2              -16.0083538  16.0112776 33.9905052

mu.star values inform about the mean of elementary effects of each $i$ factor and can be used to rank all the input factors, whereas sigma inform about the degree of interaction of the $i$ -th factor with others. According to the result of this sensitivity analysis, leaf area index (LAI_live) parameters are the most relevant to determine total transpiration, much more than fine root distribution (Z50), the rock fragment content in the soil and the water potentials corresponding to whole-plant conductance reduction (i.e. Psi_Extract).

plot(sa_transp, xlim=c(0,150))

We can run the same sensitivity analysis but focusing on the input factors relevant for predicted plant drought stress (i.e. using of_stress as model output function):

sa_stress <- morris(of_stress, parNames, r = 50, 
             design = list(type = "oat", levels = 10, grid.jump = 3), 
             binf = parMin, bsup = parMax, scale=TRUE, verbose=FALSE)

print(sa_stress)

## 
## Call:
## morris(model = of_stress, factors = parNames, r = 50, design = list(type = "oat",     levels = 10, grid.jump = 3), binf = parMin, bsup = parMax,     scale = TRUE, verbose = FALSE)
## 
## Model runs: 600 
##                            mu    mu.star     sigma
## T1_148/LAI_live    39.3701732 40.6280153 43.012050
## T2_168/LAI_live    24.2422700 24.2422700 28.709054
## S1_165/LAI_live    34.2099445 34.2099445 37.462036
## T1_148/Z50          1.7496071  3.5329408  8.654708
## T2_168/Z50          0.2240808  1.5457806  3.052235
## S1_165/Z50         -0.2410193  0.7326172  1.020798
## T1_148/Psi_Extract -0.9947753  4.5214926 14.289468
## T2_168/Psi_Extract -3.1169767  3.1169767  9.000830
## S1_165/Psi_Extract -1.0580752  1.0580752  1.693242
## rfc@1               8.2724188  8.2724188 11.264721
## rfc@2               3.2186280 11.1825579 24.324838

Again, LAI values parameters are the most relevant, but closely followed by the water potentials corresponding to whole-plant conductance reduction (i.e. Psi_Extract), which appear as more relevant than parameters of fine root distribution (Z50) and rock fragment content (rfc).

plot(sa_stress, xlim=c(0,300))

Finally, we can study the contribution of input factors to model performance in terms of soil water content dynamics (i.e. using of_eval as model output function):

sa_eval <- morris(of_eval, parNames, r = 50, 
             design = list(type = "oat", levels = 10, grid.jump = 3), 
             binf = parMin, bsup = parMax, scale=TRUE, verbose=FALSE)

print(sa_eval)

## 
## Call:
## morris(model = of_eval, factors = parNames, r = 50, design = list(type = "oat",     levels = 10, grid.jump = 3), binf = parMin, bsup = parMax,     scale = TRUE, verbose = FALSE)
## 
## Model runs: 600 
##                             mu    mu.star     sigma
## T1_148/LAI_live    -12.8813849 12.8813849 6.7216712
## T2_168/LAI_live     -7.1523876  7.2482813 4.3856065
## S1_165/LAI_live    -10.3023170 10.3023170 5.9723072
## T1_148/Z50           2.9462338  2.9655605 2.8234231
## T2_168/Z50           0.7120910  0.7132248 0.4806898
## S1_165/Z50           0.4317170  0.4401409 0.2578594
## T1_148/Psi_Extract   0.2343639  0.3865185 0.7713386
## T2_168/Psi_Extract   0.1737744  0.1996813 0.3235109
## S1_165/Psi_Extract   0.1981883  0.2465299 0.5361140
## rfc@1               -2.0935668  2.2512164 2.6786947
## rfc@2               -1.3197193  1.6729610 1.9113855

Contrary to the previous cases, the contribution of LAI parameters is similar to that of parameters of fine root distribution (Z50), which appear as more relevant than the water potentials corresponding to whole-plant conductance reduction (i.e. Psi_Extract).

plot(sa_eval, xlim=c(0,15))

Calibration

By model calibration we mean here the process of finding suitable parameter values (or suitable parameter distributions) given a set of observations. Hence, the idea is to optimize the correspondence between model predictions and observations by changing model parameter values.

Defining parameter space and objective function

To simplify our analysis and avoid problems of parameter identifiability, we focus here on the calibration of parameter Z50 of fine root distribution. Below we redefine vectors parNames, parMin, and parMax; and we specify a vector of initial values.

#Parameter names of interest
parNames = c(paste0(PH_coh,"/Z50"), paste0(QI_coh,"/Z50"), paste0(QC_coh,"/Z50"))
#Parameter minimum and maximum values
parMin = c(50,50,50)
parMax = c(500,500,300)
parIni = x1$below$Z50

In order to run calibration analyses we need to define an objective function. Many evaluation metrics could be used but it is common practice to use likelihood functions . We can use the function factory optimization_evaluation_function and the ‘observed’ data to this aim, but in this case we specify a log-likelihood with Gaussian error as the evaluation metric for of_eval().

of_eval<-optimization_evaluation_function(parNames = parNames,
                x = x1,
                meteo = examplemeteo, latitude = 41.82592, elevation = 100,
                measuredData = exampleobs, type = "SWC", 
                metric = "loglikelihood")

Calibration by gradient search

Model calibration can be performed using a broad range of approaches. Many of them - including simulated annealing, genetic algorithms, gradient methods, etc. - focus on the maximization or minimization of the objective function. To illustrate this common approach, we will use function optim from package stats, which provides several optimization methods. In particular we will use “L-BFGS-B”, which is the “BFGS” quasi-Newton method published by Broyden, Fletcher, Goldfarb and Shanno, modified by the inclusion of minimum and maximum boundaries. By default, function optim performs the minimization of the objective function (here of_eval), but we can specify a negative value for control parameter fnscale to turn the process into a maximization of maximum likelihood:

opt_cal = optim(parIni, of_eval, method = "L-BFGS-B",
                control = list(fnscale = -1), verbose = FALSE)

The calibration result is the following:

print(opt_cal)

## $par
## [1] 305.8826 110.5760 187.2690
## 
## $value
## [1] 909.4165
## 
## $counts
## function gradient 
##       25       25 
## 
## $convergence
## [1] 0
## 
## $message
## [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"

Note that the optimized parameters are relatively close to those of Z50 in the original x1.

cbind( x1$below[,"Z50", drop = FALSE], opt_cal$par)

##        Z50 opt_cal$par
## T1_148 100    305.8826
## T2_168 300    110.5760
## S1_165 200    187.2690

This occurs because these default values were used to generate the ‘observed’ data in exampleobs, which contains a small amount of non-systematic error.

Bayesian calibration

As an example of a more sophisticated model calibration, we will conduct a Bayesian calibration analysis using package BayesianTools (Hartig et al. 2019):

library(BayesianTools)

## Warning in check_dep_version(): ABI version mismatch: 
## lme4 was built with Matrix ABI version 1
## Current Matrix ABI version is 2
## Please re-install lme4 from source or restore original 'Matrix' package

In a Bayesian analysis one evaluates how the uncertainty in model parameters is changed (hopefully reduced) after observing some data, because observed values do not have the same likelihood under all regions of the parameter space. For a Bayesian analysis we need to specify a (log)likelihood function and the prior distribution (i.e. the initial uncertainty) of the input factors. The central object in the BayesianTools package is the BayesianSetup. This class, created by calls to createBayesianSetup(), contains the information about the model to be fit (likelihood), and the priors for model parameters. In absence of previous data, we specify a uniform distribution between the minimum and maximum values, which in the BayesianTools package can be done using function createUniformPrior():

prior <- createUniformPrior(parMin, parMax, parIni)
mcmc_setup <- createBayesianSetup(likelihood = of_eval, 
                                  prior = prior, 
                                  names = parNames)

Function createBayesianSetup() automatically creates the posterior and various convenience functions for the Markov Chain Monte Carlo (MCMC) samplers. The runMCMC() function is the main wrapper for all other implemented MCMC functions. Here we call it with nine chains of 1000 iterations each.

mcmc_out <- runMCMC(
  bayesianSetup = mcmc_setup, 
  sampler = "DEzs",
  settings = list(iterations = 1000, nrChains = 9))

By default runMCMC() uses parallel computation, but the calibration process is nevertheless rather slow.

A summary function is provided to inspect convergence results and correlation between parameters:

summary(mcmc_out)

## Parameter values  241.786576775968 295.749078259277 177.667831744866

## Problem encountered in the calculation of the likelihood with parameter 241.786576775968295.749078259277177.667831744866
##  Error message wasError in eval(expr, envir): Index out of bounds: [index='Z100'].
## 
##  set result of the parameter evaluation to -Inf ParameterValues

## # # # # # # # # # # # # # # # # # # # # # # # # # 
## ## MCMC chain summary ## 
## # # # # # # # # # # # # # # # # # # # # # # # # # 
##  
## # MCMC sampler:  DEzs 
## # Nr. Chains:  27 
## # Iterations per chain:  334 
## # Rejection rate:  0.751 
## # Effective sample size:  673 
## # Runtime:  2212.486  sec. 
##  
## # Parameters
##              psf     MAP    2.5%  median   97.5%
## T1_148/Z50 1.037 306.754 101.883 245.343 345.003
## T2_168/Z50 1.034 109.695  59.929 306.980 490.651
## S1_165/Z50 1.036 179.784  57.204 182.498 294.762
## 
## ## DIC:  -Inf 
## ## Convergence 
##  Gelman Rubin multivariate psrf:  1.077 
##  
## ## Correlations 
##            T1_148/Z50 T2_168/Z50 S1_165/Z50
## T1_148/Z50      1.000     -0.801     -0.128
## T2_168/Z50     -0.801      1.000      0.066
## S1_165/Z50     -0.128      0.066      1.000

According to the Gelman-Rubin diagnostic, the convergence can be accepted because the multivariate potential scale reduction factor was ≤ 1.1. We can plot the Markov Chains and the posterior density distribution of parameters that they generate using:

plot(mcmc_out)

We can also plot the marginal prior and posterior density distributions for each parameter. In this case, we see a similar Z50 distribution for the two trees, which is more informative than the prior distribution. In contrast, the posterior distribution of Z50 for the kermes oak remains as uncertain as the prior one. This happens because the LAI value of kermes oak is low, so that it has small influence on soil water dynamics regardless of its root distribution.

marginalPlot(mcmc_out, prior = T)

Plots can also be produced to display the correlation between parameter values.

correlationPlot(mcmc_out)

Here it can be observed the large correlation between Z50 of the two tree cohorts. Since their LAI values are similar, a similar effect on soil water depletion can be obtained to some extent by exchanging their fine root distribution.

Posterior model prediction distributions can be obtained if we take samples from the Markov chains and use them to perform simulations (here we use sample size of 100 but a larger value is preferred).

s = getSample(mcmc_out, numSamples = 100)
head(s)

##      T1_148/Z50 T2_168/Z50 S1_165/Z50
## [1,]   217.9150  478.40434  258.52315
## [2,]   260.5909  213.55127  184.20214
## [3,]   243.8444  244.94568  233.38867
## [4,]   166.7885   73.53628  230.78928
## [5,]   313.5510  258.36418  216.23578
## [6,]   276.2950  276.41855   71.93304

To this aim, medfate includes function multiple_runs() that allows running a simulation model with a matrix of parameter values. For example, the following code runs spwb() with all combinations of fine root distribution specified in s.

MS = multiple_runs(s, x = x1, meteo = examplemeteo,
                   latitude = 41.82592, elevation = 100, verbose = FALSE)

Function multiple_runs() determines the model to be called inspecting the class of x (here x1 is a spwbInput). Once we have conducted the simulations we can inspect the posterior distribution of several prediction variables, for example total transpiration:

plot(density(unlist(lapply(MS, sf_transp))), main = "Posterior transpiration", 
     xlab = "Total transpiration (mm)")

or average plant drought stress:

plot(density(unlist(lapply(MS, sf_stress))), 
     xlab = "Average plant stress", main="Posterior stress")

Finally, we can use object prior to generate another sample under the prior parameter distribution, perform simulations:

s_prior = prior$sampler(100)
colnames(s_prior)<- parNames
MS_prior = multiple_runs(s_prior, x = x1, meteo = examplemeteo,
                         latitude = 41.82592, elevation = 100, verbose = FALSE)

and compare the prior prediction uncertainty with the posterior prediction uncertainty for the same output variables:

plot(density(unlist(lapply(MS_prior, sf_transp))), main = "Transpiration", 
     xlab = "Total transpiration (mm)",
     xlim = c(100,200), ylim = c(0,6))
lines(density(unlist(lapply(MS, sf_transp))), col = "red")
legend("topleft", legend = c("Prior", "Posterior"), 
       col = c("black", "red"), lty=1, bty="n")

plot(density(unlist(lapply(MS_prior, sf_stress))), main = "Plant stress", 
     xlab = "Average plant stress",
     xlim = c(0,30), ylim = c(0,2))
lines(density(unlist(lapply(MS, sf_stress))), col = "red")
legend("topleft", legend = c("Prior", "Posterior"), col = c("black", "red"), lty=1, bty="n")

References

Pianosi, F., Beven, K., Freer, J., Hall, J.W., Rougier, J., Stephenson, D.B., Wagener, T., 2016. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environ. Model. Softw. 79, 214–232. https://doi.org/10.1016/j.envsoft.2016.02.008
Bertrand Iooss, Sebastien Da Veiga, Alexandre Janon, Gilles Pujol, with contributions from Baptiste Broto, Khalid Boumhaout, Thibault Delage, Reda El Amri, Jana Fruth, Laurent Gilquin, Joseph Guillaume, Loic Le Gratiet, Paul Lemaitre, Amandine Marrel, Anouar Meynaoui, Barry L. Nelson, Filippo Monari, Roelof Oomen, Oldrich Rakovec, Bernardo Ramos, Olivier Roustant, Eunhye Song, Jeremy Staum, Roman Sueur, Taieb Touati and Frank Weber (2020). sensitivity: Global Sensitivity Analysis of Model Outputs. R package version 1.23.1. https://CRAN.R-project.org/package=sensitivity
Florian Hartig, Francesco Minunno and Stefan Paul (2019). BayesianTools: General-Purpose MCMC and SMC Samplers and Tools for Bayesian Statistics. R package version 0.1.7. https://CRAN.R-project.org/package=BayesianTools
Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis. The Primer. Wiley.

Miquel De Caceres

2024-11-05