Author: Josefa Arán

# Parameter calibration and cost functions

#### Josefa Arán

The `rsofun` package allows to calibrate parameters of the `pmodel` and `biomee` models via the `calib_sofun()` function. The implementation of the calibration is fairly flexible and can be adapted to a specific use-case via a tailor-made cost function (used as metrics for the optimization routines in `calib_sofun()`). The package provides a set of standard cost functions named `cost_*`, which can be used for a variety of calibrations (different sets of model parameters, using various target variables, etc.). Alternatively, it’s possible to write a more specific new cost function to be used together with `calib_sofun()`.

In this vignette, we go over some examples on how to use the `rsofun` cost functions for parameter calibration with `calib_sofun()` and how to write your own custom one from scratch.

### Calibration to GPP using RMSE and GenSA optimizer

A simple approach to parameter calibration is to find the parameter values that lead to the best prediction performance, in terms of the RMSE (root mean squared error). The function `cost_rmse_pmodel()` runs the P-model internally to calculate the RMSE between predicted target values (in this case GPP) and the corresponding observations.

The implementations of `cost_rmse_pmodel()` and `calib_sofun()` allows flexibility in various ways. We can simultaneously calibrate a subset of model parameters and also replicate the different calibration setups in Stocker et al., 2020 GMD, simply by providing the appropriate inputs as `settings`, `par_fixed` and `targets` to `calib_sofun()`. Since the P-model is run internally to make predictions, we must always specify the values of the model parameters that aren’t calibrated (via argument `par_fixed`). For example, following the `ORG` setup, only parameter `kphio` is calibrated with below code:

    # Define calibration settings and parameter ranges from previous work
    settings_rmse <- list(
      method = 'GenSA',                   # minimizes the RMSE
      metric = cost_rmse_pmodel,          # our cost function returning the RMSE
      control = list(                     # control parameters for optimizer GenSA
        maxit = 100),
      par = list(                         # bounds for the parameter space
        kphio = list(lower=0.02, upper=0.2, init=0.05)
      )
    )

    # Calibrate the model and optimize the free parameters using
    # demo datasets
    pars_calib_rmse <- calib_sofun(
      # calib_sofun arguments:
      drivers = p_model_drivers,
      obs = p_model_validation,
      settings = settings_rmse,
      # extra arguments passed to the cost function:
      par_fixed = list(         # fix all other parameters
        kphio_par_a        = 0.0,        # set to zero to disable temperature-dependence
                                         # of kphio, setup ORG
        kphio_par_b        = 1.0,
        soilm_thetastar    = 0.6 * 240,  # to recover paper setup with soil moisture stress
        soilm_betao        = 0.0,
        beta_unitcostratio = 146.0,
        rd_to_vcmax        = 0.014,      # value from Atkin et al. 2015 for C3 herbaceous
        tau_acclim         = 30.0,
        kc_jmax            = 0.41
      ),
      targets = "gpp"           # define target variable GPP
    )

    pars_calib_rmse
    #> $par
    #>      kphio
    #> 0.03580938
    #>
    #> $mod
    #> $mod$value
    #> [1] 1.20014
    #>
    #> $mod$par
    #>      kphio
    #> 0.03580938
    #>
    #> $mod$trace.mat
    #>      nb.steps temperature function.value current.minimum
    #> [1,]        1        5230       4.241255         1.20014
    #>
    #> $mod$counts
    #> [1] 286

The output of `calib_sofun()` is a list containing the calibrated parameter values (element `par`) and the raw optimization output from the optimizer (element `mod`; here from `GenSA` or, as we see next, from `BayesianTools::runMCMC`).

Note that the standard cost functions allow to calibrate to several targets (fluxes and leaf traits predicted by the P-model) simultaneously and to parallelize the simulations.

### Calibration to GPP using a simple likelihood function and BayesianTools

Let’s calibrate the parameters involved in the temperature dependency of the quantum yield efficiency, `kphio`, `kphio_par_a` and `kphio_par_b`. Taking a Bayesian calibration approach, we need to define a likelihood as cost function (we’ll use `cost_likelihood_pmodel()`). We assume that the target variable (`'gpp'`) follows a normal distribution centered at the observations and with its unknown standard deviation (`'err_gpp'`), that we need to add to the calibratable parameters. We also assume a uniform prior distribution for all calibratable parameters. By maximizing the log-likelihood, the MAP (maximum a posteriori) estimators for all 4 parameters are computed. With the functions `cost_likelihood_pmodel()` and `calib_sofun()`, we can easily perform this type of calibration:

    # Define calibration settings
    settings_likelihood <- list(
      method = 'BayesianTools',
      metric = cost_likelihood_pmodel,            # our cost function
      control = list(                             # optimization control settings for
        sampler = 'DEzs',                           # BayesianTools::runMCMC
        settings = list(
          burnin = 1500,
          iterations = 3000
        )),
      par = list(
        kphio = list(lower = 0, upper = 0.2, init = 0.05),
        kphio_par_a = list(lower = -0.5, upper = 0.5, init = -0.1),
        kphio_par_b = list(lower = 10, upper = 40, init =25),
        err_gpp = list(lower = 0.1, upper = 4, init = 0.8)
      )
    )

    # Calibrate the model and optimize the free parameters using
    # demo datasets
    pars_calib_likelihood <- calib_sofun(
      # calib_sofun arguments:
      drivers = p_model_drivers,
      obs = p_model_validation,
      settings = settings_likelihood,
      # extra arguments passed ot the cost function:
      par_fixed = list(         # fix all other parameters
        soilm_thetastar    = 0.6 * 240,  # to recover paper setup with soil moisture stress
        soilm_betao        = 0.0,
        beta_unitcostratio = 146.0,
        rd_to_vcmax        = 0.014,      # value from Atkin et al. 2015 for C3 herbaceous
        tau_acclim         = 30.0,
        kc_jmax            = 0.41
      ),
      targets = "gpp"
    )

    pars_calib_likelihood
    #> $par
    #>        kphio  kphio_par_a  kphio_par_b      err_gpp
    #>  0.036462810 -0.001859453 14.614170205  1.209760614
    #>
    #> $mod
    #> [1] "mcmcSampler - you can use the following methods to summarize, plot or reduce this class:"
    #> [1] plot    print   summary
    #> see '?methods' for accessing help and source code

Furthermore, there are equivalent cost functions available for the BiomeE model. Check out the reference pages for more details on how to use `cost_likelihood_biomee()` and `cost_rmse_biomee()`.

### Calibration to GPP and Vcmax25 using the joint log-likelihood and BayesianTools

You may be interested in calibrating the model to different target variables simultaneously, like flux and leaf trait measurements. Here we present an example, where we use `cost_likelihood_pmodel()` to compute the joint likelihood of all the targets specified (that is, by summing the log-likelihoods of GPP and Vcmax25) and ultimately calibrate the `kc_jmax` parameter. It would be possible to follow this workflow for several-target calibration also with RMSE as the optimization metric, using `cost_rmse_pmodel()` and `GenSA` optimization.

    # Define calibration settings for two targets
    settings_joint_likelihood <- list(
      method = "BayesianTools",
      metric = cost_likelihood_pmodel,
      control = list(
        sampler = "DEzs",
        settings = list(
          burnin = 1500,             # kept artificially low
          iterations = 3000
        )),
      par = list(kc_jmax = list(lower = 0.2, upper = 0.8, init = 0.41),  # uniform priors
                 err_gpp = list(lower = 0.001, upper = 4, init = 1),
                 err_vcmax25 = list(lower = 0.000001, upper = 0.0001, init = 0.00001))
    )

    # Run the calibration on the concatenated data
    par_calib_join <- calib_sofun(
      drivers = rbind(p_model_drivers,
                      p_model_drivers_vcmax25),
      obs = rbind(p_model_validation,
                  p_model_validation_vcmax25),

      settings = settings_joint_likelihood,
      # arguments for the cost function
      par_fixed = list(         # fix parameter value from previous calibration
        kphio              = 0.041,
        kphio_par_a        = 0.0,
        kphio_par_b        = 16,
        soilm_thetastar    = 0.6 * 240,  # to recover paper setup with soil moisture stress
        soilm_betao        = 0.0,
        beta_unitcostratio = 146.0,
        rd_to_vcmax        = 0.014,      # value from Atkin et al. 2015 for C3 herbaceous
        tau_acclim         = 30.0
      ),
      targets = c('gpp', 'vcmax25')
    )

    par_calib_join
    #> $par
    #>      kc_jmax      err_gpp  err_vcmax25
    #> 4.824008e-01 1.189628e+00 3.289346e-05
    #>
    #> $mod
    #> [1] "mcmcSampler - you can use the following methods to summarize, plot or reduce this class:"
    #> [1] plot    print   summary
    #> see '?methods' for accessing help and source code

Note that GPP predictions are directly compared to GPP observations on that day, but Vcmax25 predicted by the P-model (being a leaf trait) is averaged over the growing season and compared to a single Vcmax25 observation taken per site.

The cost functions provided in the package tell apart fluxes and leaf traits by the presence of a `"date"` column in the nested validation data frames `p_model_validation` and `p_model_validation_vcmax25`.

### Write your custom cost function

If the RMSE or log-likelihood (for one or several targets) cost functions that we provide do not fit your use case, you can easily write a custom one. In this section, we drive you through the main ideas with an example. To run the calibration, you can still use `calib_sofun()` in combination with your custom cost function.

The routine `calib_sofun()` requires `drivers`, `obs` and `settings` as mandatory arguments. These provide data.frames with driver and observational data, as well as settings for the calibration. The optional argument `optim_out` defines if the raw optimization output should be returned. All other (optional) arguments to `calib_sofun()` are passed through to the cost function (e.g. `par_fixed` in above example). They can be used freely inside of your custom cost function, e.g. to control the simulation setup or the processing. On top of these optional arguments, it is also possible to extend the `drivers` and `obs` data.frames with additional columns that can be used freely for fine-grained control within your custom cost function.

All cost functions must take at least three arguments:

- `par`: A named vector of calibratable model parameters. In each iteration of the optimization, a new set of values of `par` is used to run the model and compute the cost.

- `obs`: A data frame of observations, against which to compare the simulation results.

- `drivers`: A data frame of driver data, used to run the simulations.

- Additional optional arguments can be used, An example would be model parameter values that should be fixed across simulations, etc.

Below we’ll walk you through the definition of a custom cost function. In this example, we’ll calibrate the soil moisture stress parameters and use the mean absolute error (MAE) as custom cost function.

Since we are calibrating the parameters based on model outputs, the cost function will eventually need to run the P-model and compare its output to observed validation data.

To get started we suggest to write a dummy cost function and use it together with `calib_sofun()` as shown below. Note that one way of developing the cost function would be to use a `browser()` statement during the development. It allows you to explore the variables that you have access to from within the cost function.

    # Define the custom cost function to be used
    cost_mae <- function(par, obs, drivers, my_own_message){
      # Your code
      browser() # can facilitate the development, remove afterwards
    }

    # Define calibration settings and parameter ranges
    settings_mae <- list(
      method = 'GenSA',
      metric = cost_mae, # directly uses the custom cost function
      control = list(
        maxit = 100
      ),
      par = list(
        soilm_thetastar = list(lower=0.0, upper=3000, init=0.6*240),
        soilm_betao = list(lower=0, upper=1, init=0.2)
      )
    )

    # Calibrate the model and optimize the free parameters
    pars_calib_mae <- calib_sofun(
      drivers = p_model_drivers,
      obs = p_model_validation,
      settings = settings_mae,
      # optional arguments if needed in the cost function
      my_own_message = "Hi from inside the cost_mae function."
    )

During the optimization procedure, the cost function receives as argument a suggestion of the parameters `par`. This might be just a subset of all needed parameters (defined via `settings$par`). Thus to call `runread_pmodel_f()` within the cost function, a full set of model parameters is needed. Here, we’ll hardcode the parameters that aren’t being calibrated inside the cost function. (Note, that in above examples they were passed as an additional argument `par_fixed`).

    cost_mae <- function(par, obs, drivers, my_own_message){

      # Set values for the list of calibrated and non-calibrated model parameters
      params_modl <- list(
        kphio              = 0.09423773,
        kphio_par_a        = 0.0,
        kphio_par_b        = 25,
        soilm_thetastar    = par[["soilm_thetastar"]],
        soilm_betao        = par[["soilm_betao"]],
        beta_unitcostratio = 146.0,
        rd_to_vcmax        = 0.014,
        tau_acclim         = 30.0,
        kc_jmax            = 0.41
      )

      # Run the model
      df <- runread_pmodel_f(
        drivers,
        par = params_modl,
        makecheck = TRUE,
        parallel = FALSE
      )

      # Your code to compute the cost
      print(my_own_message) # useless, but showcases how to use additional arguments
      browser() # can facilitate the development, remove afterwards
    }

The following chunk defines the final function. We clean the observations and model output and align the data according to site and date, to compute the mean absolute error (MAE) on GPP. Finally, the function should return a scalar value, in this case the MAE, which we want to minimize. Keep in mind that the GenSA optimization will minimize the cost, but with the BayesianTools method the cost (i.e. the likelihood) is always maximized.

    cost_mae <- function(par, obs, drivers){

      # Set values for the list of calibrated and non-calibrated model parameters
      params_modl <- list(
        kphio              = 0.09423773,
        kphio_par_a        = 0.0,
        kphio_par_b        = 25,
        soilm_thetastar    = par[["soilm_thetastar"]],
        soilm_betao        = par[["soilm_betao"]],
        beta_unitcostratio = 146.0,
        rd_to_vcmax        = 0.014,
        tau_acclim         = 30.0,
        kc_jmax            = 0.41
      ) # Set values for the list of calibrated and non-calibrated model parameters

      # Run the model
      df <- runread_pmodel_f(
        drivers = drivers,
        par = params_modl,
        makecheck = FALSE,
        parallel = FALSE
      )

      # Clean model output to compute cost
      df <- df %>%
        dplyr::select(sitename, data) %>%
        tidyr::unnest(data)

      # Clean validation data to compute cost
      obs <- obs %>%
        dplyr::select(sitename, data) %>%
        tidyr::unnest(data) %>%
        dplyr::rename('gpp_obs' = 'gpp') # rename for later

      # Left join model output with observations by site and date
      df <- dplyr::left_join(df, obs, by = c('sitename', 'date'))

      # Compute mean absolute error
      cost <- mean(abs(df$gpp - df$gpp_obs), na.rm = TRUE)

      # Return the computed cost
      return(cost)
      # browser() # can facilitate the development, remove afterwards
    }

As a last step, let’s verify that the calibration procedure runs using this cost function.

    # Define calibration settings and parameter ranges
    settings_mae <- list(
      method = 'GenSA',
      metric = cost_mae, # our cost function
      control = list(
        maxit = 100),
      par = list(
        soilm_thetastar = list(lower=0.0, upper=3000, init=0.6*240),
        soilm_betao = list(lower=0, upper=1, init=0.2)
      )
    )

    # Calibrate the model and optimize the free parameters
    pars_calib_mae <- calib_sofun(
      drivers = p_model_drivers,
      obs = p_model_validation,
      settings = settings_mae
    )

    pars_calib_mae
    #> $par
    #> soilm_thetastar     soilm_betao
    #>    2942.7566560       0.2649316
    #>
    #> $mod
    #> $mod$value
    #> [1] 1.072192
    #>
    #> $mod$par
    #> soilm_thetastar     soilm_betao
    #>    2942.7566560       0.2649316
    #>
    #> $mod$trace.mat
    #>      nb.steps temperature function.value current.minimum
    #> [1,]        1  5230.00000       3.918748        1.078549
    #> [2,]       37    30.00603       1.072921        1.072192
    #>
    #> $mod$counts
    #> [1] 742