# Sensitivity Analysis

#### Fernando Miguez

#### 2024-05-06

# Introduction

Sensitivity analysis of crop models such as APSIM is a valuable task which allows us to develop an intuition of the model response to changing a variety of inputs. These inputs can generally be included in the following categories:

- weather
- soil profiles
- crop/genetics
- management

The weather component can depend on different sources of data or manipulations to simulate different conditions and stresses. The soil profiles can be from different locations or specifically manipulated to understand the impact of particular soil properties. Crop properties or cultivar specific parameters can also be investigated as a source of model behavior. Finally, examples of management usually include planting date, fertilizer application, tillage or residue management and irrigation. All of these components can be sources of uncertainty and can be evaluated in a sensitivity analysis.

When multiple parameters are evaluated performing model runs for all combinations can become prohibitive. For this reason, there are some combinations which can be more efficient and provide nearly the same amount of information. The suggestion here is to build complexity in the sensitivity analysis task gradually. For example, first choose one parameter and select 5-10 reasonable values, run the model and evaluate the model output. This is a one parameter at a time approach and it can be of great value in developing an intuition for the model behavior.

For a general introduction to sensitivity analysis see Saltelli et al. 2008. Global sensitivity analysis. The Primer.

Here they define sensitivity analysis as:

The study of how uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input

A preliminary step in sensitivity analysis often involves uncertainty analysis and, as an example, it can involve computing the output from a model based on various samples from the relevant inputs. Usually, this involves identifying parameters, choosing appropriate distributions which we sample from and creating candidate values which are then used in the model.

## Uncertainty Analysis using for loops

The steps needed to perform a uncertainty analysis can involve simple use of for loops.

1. Identify a parameter of interest
2. Sample (or select) a new value for the parameter of interest
3. Edit the APSIM file with the new value
4. Run the model
5. Collect the results and repeat as needed

A simple example

    ## Create a temporary directory and copy the Wheat.apsimx file
    tmp.dir <- tempdir()
    extd.dir <- system.file("extdata", package = "apsimx")
    file.copy(file.path(extd.dir, "Wheat.apsimx"), tmp.dir)
    ## Identify a parameter of interest
    ## In this case we want to know the impact of varying the fertilizer amount
    pp <- inspect_apsimx("Wheat.apsimx", src.dir = tmp.dir,
                          node = "Manager", parm = list("SowingFertiliser", 1))
    ## For simplicity, we create a vector of fertilizer amounts (instead of sampling)
    ferts <- seq(5, 200, length.out = 7)
    col.res <- NULL
    for(i in seq_along(ferts)){

      edit_apsimx("Wheat.apsimx", src.dir = tmp.dir,
                  node = "Other",
                  parm.path = pp, parm = "Amount",
                  value = ferts[i])

      sim <- apsimx("Wheat.apsimx", src.dir = tmp.dir)
      col.res <- rbind(col.res, data.frame(fertilizer.amount = ferts[i],
                                           wheat.aboveground.wt = mean(sim$Wheat.AboveGround.Wt, na.rm = TRUE)))
    }

The code here is shown for illustration so that the basic strategy can be replicated. The results collected in object col.res can be further processed to better understand the impact of the chosen parameter (input) on the output.

There is a function in the package which simplifies this task. The one below is for APSIM Next Gen and there is a similar one for APSIM Classic.

## sens_apsimx

- file = .apsimx file

- src.dir = source directory where the APSIM file is located.

- parm.paths = character vector with the full path to the parameters to be evaluated. This can be obtained by using inspect_apsimx or inspect_apsim_json. The length of the parameter vector should be equal to the number of parameters being evaluated (sometimes less is more).

- convert = whether to convert strings to numeric

- replacement = logical indicating whether each parameter is in the replacement component or not.

- grid = data frame with combinations of parameters to be evaluated.

- soil.profiles = soil profiles, which can be passed as a list

- summary = optional summary function to apply if the APSIM output returns multiple rows. The mean is used by default.

- root = indicate the root simulation in case it is not the default one.

- verbose = printing progress and other details

- cores = number of cores to use. The package ‘future’ is currently used.

- save = whether to save intermediate (and final) results.

- … = additional arguments (none used at the moment).

## Example using sens_apsimx

The example below shows the following steps.

1. Copy the example ‘Wheat’ file to at temporary directory
2. Identify parameters of interest (using inspect_apsimx)
3. Create a simple grid for evaluating the model

Here the grid is created using expand.grid and it has 9 rows. This means that this will require 9 simulation runs of the APSIM model. This takes a couple of minutes to run. The resulting object is of class sens_apsim and it can be investigated using the summary function (for now).

    tmp.dir <- tempdir()
    extd.dir <- system.file("extdata", package = "apsimx")
    file.copy(file.path(extd.dir, "Wheat.apsimx"), tmp.dir)
    ## Identify a parameter of interest
    ## In this case we want to know the impact of varying the fertilizer amount
    ## and the plant population
    pp1 <- inspect_apsimx("Wheat.apsimx", src.dir = tmp.dir,
                          node = "Manager", parm = list("SowingFertiliser", 1))
    pp1 <- paste0(pp1, ".Amount")

    pp2 <- inspect_apsimx("Wheat.apsimx", src.dir = tmp.dir,
                          node = "Manager", parm = list("SowingRule1", 9))
    pp2 <- paste0(pp2, ".Population")

    ## The names in the grid should (partially) match the parameter path names
    grd <- expand.grid(Fertiliser = c(50, 100, 150), Population = c(100, 200, 300))

    ## This takes 2-3 minutes
    sns <- sens_apsimx("Wheat.apsimx", src.dir = tmp.dir,
                        parm.paths = c(pp1, pp2),
                        grid = grd)

## Creating grids

There are many ways of creating grids. The example above using function expand.gird returned all possible combinations of the parameter values. This might not be desirable or feasible and other methods are available. For this task, the package sensitivity is especially useful.

### First option: parameterSets

    library(sensitivity)

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

    ## Simple example: see documentation for other options
    X.grid <- parameterSets(par.ranges = list(Fertiliser = c(1, 300),
                                              Population = c(1, 300)),
                            samples = c(3,3), method = "grid")
    X.grid

    ##       Fertiliser Population
    ##  [1,]        1.0        1.0
    ##  [2,]      150.5        1.0
    ##  [3,]      300.0        1.0
    ##  [4,]        1.0      150.5
    ##  [5,]      150.5      150.5
    ##  [6,]      300.0      150.5
    ##  [7,]        1.0      300.0
    ##  [8,]      150.5      300.0
    ##  [9,]      300.0      300.0

## Second option: Morris

A second option is to use the function morris.

    X.mrrs <- morris(factors = c("Fertiliser", "Population"),
                       r = 3, design = list(type = "oat", levels = 3, grid.jump = 1),
                       binf = c(0, 5), bsup = c(200, 300))
    X.mrrs$X

    ##       Fertiliser Population
    ##  [1,]        100      152.5
    ##  [2,]        100        5.0
    ##  [3,]        200        5.0
    ##  [4,]        100      152.5
    ##  [5,]        200      152.5
    ##  [6,]        200      300.0
    ##  [7,]        100      300.0
    ##  [8,]        100      152.5
    ##  [9,]        200      152.5

The sensitivity package allows for decoupling the simulation runs and the analysis of the results. For example:

    ## This takes 2-3 minutes
    sns2 <- sens_apsimx("Wheat.apsimx", src.dir = tmp.dir,
                         parm.paths = c(pp1, pp2),
                         grid = X.mrrs$X)
    ## These are the sensitivity results for AboveGround.Wt only
    sns.res.ag <- tell(X.mrrs, sns2$grid.sims$Wheat.AboveGround.Wt)
    sns.res.ag
    ## Call: morris(factors = c("Fertiliser", "Population"),
    ## r = 3, design = list(type = "oat", levels = 3, grid.jump = 1),
    ## binf = c(0, 5), bsup = c(200, 300))
    ##
    ## Model runs: 9
    ##                 mu  mu.star    sigma
    ## Fertiliser 916.6674 916.6674 662.8798
    ## Population 448.4530 448.4530 640.6807
    plot(sns.res.ag)

I’m not running the code in this vignette to save time, but hopefully this is sufficient information to illustrate performing sensitivity analysis using package apsimx and sensitivity.