error_function_c3_aci: Generate an error function for C3 A-Ci curve fitting

Description

Creates a function that returns an error value (the negative of the natural logarithm of the likelihood) representing the amount of agreement between modeled and measured An values. When this function is minimized, the likelihood is maximized.

Internally, this function uses apply_gm to calculate Cc, and then uses link{calculate_c3_assimilation} to calculate assimilation rate values that are compared to the measured ones.

Usage

error_function_c3_aci(
    replicate_exdf,
    fit_options = list(),
    sd_A = 1,
    Wj_coef_C = 4.0,
    Wj_coef_Gamma_star = 8.0,
    a_column_name = 'A',
    ci_column_name = 'Ci',
    gamma_star_norm_column_name = 'Gamma_star_norm',
    gmc_norm_column_name = 'gmc_norm',
    j_norm_column_name = 'J_norm',
    kc_norm_column_name = 'Kc_norm',
    ko_norm_column_name = 'Ko_norm',
    oxygen_column_name = 'Oxygen',
    rl_norm_column_name = 'RL_norm',
    total_pressure_column_name = 'total_pressure',
    tp_norm_column_name = 'Tp_norm',
    vcmax_norm_column_name = 'Vcmax_norm',
    cj_crossover_min = NA,
    cj_crossover_max = NA,
    hard_constraints = 0,
    debug_mode = FALSE,
    ...
  )

Value

A function with one input argument guess, which should be a numeric vector representing values of the parameters to be fitted (which are specified by the fit_options input argument.) Each element of guess is the value of one parameter (arranged in alphabetical order.) For example, with the default settings, guess should contain values of alpha_old,

J_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25 (in that order).

Arguments

replicate_exdf: An exdf object representing one CO2 response curve.
fit_options: A list of named elements representing fit options to use for each parameter. Values supplied here override the default values (see details below). Each element must be 'fit', 'column', or a numeric value. A value of 'fit' means that the parameter will be fit; a value of 'column' means that the value of the parameter will be taken from a column in replicate_exdf of the same name; and a numeric value means that the parameter will be set to that value. For example, fit_options = list(alpha_g = 0, Vcmax_at_25 = 'fit', Tp_at_25 = 'column') means that alpha_g will be set to 0, Vcmax_at_25 will be fit, and Tp_at_25 will be set to the values in the Tp_at_25 column of replicate_exdf.
sd_A: The standard deviation of the measured values of the net CO2 assimilation rate, expressed in units of micromol m^(-2) s^(-1). If sd_A is not a number, then there must be a column in replicate_exdf called sd_A with appropriate units. A numeric value supplied here will overwrite the values in the sd_A column of replicate_exdf if it exists.
Wj_coef_C: A coefficient in the equation for RuBP-regeneration-limited carboxylation, whose value depends on assumptions about the NADPH and ATP requirements of RuBP regeneration; see calculate_c3_assimilation for more information.
Wj_coef_Gamma_star: A coefficient in the equation for RuBP-regeneration-limited carboxylation, whose value depends on assumptions about the NADPH and ATP requirements of RuBP regeneration; see calculate_c3_assimilation for more information.
a_column_name: The name of the column in replicate_exdf that contains the net assimilation in micromol m^(-2) s^(-1).
ci_column_name: The name of the column in replicate_exdf that contains the intercellular CO2 concentration in micromol mol^(-1).
gamma_star_norm_column_name: The name of the column in replicate_exdf that contains the normalized Gamma_star values (with units of normalized to Gamma_star at 25 degrees C).
gmc_norm_column_name: The name of the column in replicate_exdf that contains the normalized mesophyll conductance values (with units of normalized to gmc at 25 degrees C).
j_norm_column_name: The name of the column in replicate_exdf that contains the normalized J values (with units of normalized to J at 25 degrees C).
kc_norm_column_name: The name of the column in replicate_exdf that contains the normalized Kc values (with units of normalized to Kc at 25 degrees C).
ko_norm_column_name: The name of the column in replicate_exdf that contains the normalized Ko values (with units of normalized to Ko at 25 degrees C).
oxygen_column_name: The name of the column in replicate_exdf that contains the concentration of O2 in the ambient air, expressed as a percentage (commonly 21% or 2%); the units must be percent.
rl_norm_column_name: The name of the column in replicate_exdf that contains the normalized RL values (with units of normalized to RL at 25 degrees C).
total_pressure_column_name: The name of the column in replicate_exdf that contains the total pressure in bar.
tp_norm_column_name: The name of the column in replicate_exdf that contains the normalized Tp values (with units of normalized to Tp at 25 degrees C).
vcmax_norm_column_name: The name of the column in replicate_exdf that contains the normalized Vcmax values (with units of normalized to Vcmax at 25 degrees C).
cj_crossover_min: The minimum value of Cc (in ppm) where Aj is allowed to become the overall rate-limiting factor. If cj_crossover_min is set to NA, this restriction will not be applied.
cj_crossover_max: The maximim value of Cc (in ppm) where Wj is allowed to be smaller than Wc. If cj_crossover_max is set to NA, this restriction will not be applied.
hard_constraints: To be passed to calculate_c3_assimilation; see that function for more details.
debug_mode: A logical (TRUE or FALSE) variable indicating whether to operate in debug mode. In debug mode, information about the guess is printed each time the error function is called; this can be helpful when troubleshooting issues with an optimizer.
...: Additional arguments to be passed to calculate_c3_assimilation.

Details

When fitting A-Ci curves using a maximum likelihood approach, it is necessary to define a function that calculates the likelihood of a given set of alpha_g, alpha_old, alpha_s, alpha_t, Gamma_star_at_25, gmc_at_25, J_at_25, Kc_at_25, Ko_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25 values by comparing a model prediction to a measured curve. This function will be passed to an optimization algorithm which will determine the values that produce the largest likelihood.

The error_function_c3_aci returns such a function, which is based on a particular A-Ci curve and a set of fitting options. It is possible to just fit a subset of the available fitting parameters; by default, the fitting parameters are alpha_old, J_at_25, RL_at_25, Tp_at_25, and Vcmax_at_25. This behavior can be changed via the fit_options argument.

For practical reasons, the function actually returns values of -ln(L), where L is the likelihood. The logarithm of L is simpler to calculate than L itself, and the minus sign converts the problem from a maximization to a minimization, which is important because most optimizers are designed to minimize a value.

Sometimes an optimizer will choose biologically unreasonable parameter values that nevertheless produce good fits to the supplied assimilation values. A common problem is that the fit result may not indicate Ac-limited assimilation at low CO2 values, which should be the case for any A-Ci curves measured at saturating light. In this case, the optional cj_crossover_min and cj_crossover_max can be used to constrain the range of Cc values (in ppm) where Aj is allowed to be the overall rate limiting factor. If the crossover from Rubisco-limited to RuBP-regeneration limited assimilation occurs outside these bounds (when they are supplied), a heavy penalty will be added to the error function, preventing the optimizer from choosing those parameter values.

A penalty is also added for any parameter combination where An is not a number, or where calculate_c3_assimilation produces an error.

Examples

Run this code

# Read an example Licor file included in the PhotoGEA package
licor_file <- read_gasex_file(
  PhotoGEA_example_file_path('c3_aci_1.xlsx')
)

# Define a new column that uniquely identifies each curve
licor_file[, 'species_plot'] <-
  paste(licor_file[, 'species'], '-', licor_file[, 'plot'] )

# Organize the data
licor_file <- organize_response_curve_data(
    licor_file,
    'species_plot',
    c(9, 10, 16),
    'CO2_r_sp'
)

# Calculate the total pressure in the Licor chamber
licor_file <- calculate_total_pressure(licor_file)

# Calculate temperature-dependent values of C3 photosynthetic parameters
licor_file <- calculate_temperature_response(licor_file, c3_temperature_param_bernacchi)

# Define an error function for one curve from the set
error_fcn <- error_function_c3_aci(
  licor_file[licor_file[, 'species_plot'] == 'tobacco - 1', , TRUE]
)

# Evaluate the error for:
#  alpha_old = 0
#  J_at_25 = 236
#  RL_at_25 = 4e-8
#  Tp_at_25 = 22.7
#  Vcmax_at_25 = 147
error_fcn(c(0, 236, 4e-8, 22.7, 147))

# Make a plot of likelihood vs. Vcmax when other parameters are fixed to the
# values above.
vcmax_error_fcn <- function(Vcmax) {error_fcn(c(0, 236, 4e-8, 22.7, Vcmax))}
vcmax_seq <- seq(135, 152, length.out = 41)

lattice::xyplot(
  exp(-sapply(vcmax_seq, vcmax_error_fcn)) ~ vcmax_seq,
  type = 'b',
  xlab = 'Vcmax_at_25 (micromol / m^2 / s)',
  ylab = 'Negative log likelihood (dimensionless)'
)

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