In most situations, growth predictions are surrounded by different sources of uncertainty and variability. For this reason, discrete growth predictions (i.e. growth curves) can be, in some cases, misleading. Consequently, biogrowth includes several functions to calculate growth predictions accounting for parameter uncertainty. Namely, it can account for the uncertainty of parameter estimates of primary growth models defined manually using
predict_growth_uncertainty(). Also, it can include the uncertainty of a model fitted using a Monte Carlo algorithm with the
predict_growth_uncertainty() allows the definition of the distribution of the parameters of the primary growth model. Then, it includes this uncertainty in the model predictions through Monte Carlo simulations. It has 6 arguments:
model_namedefines the primary growth model,
timesdefines the time points where to make the calculations,
n_simsdefines the number of Monte Carlo simulations,
parsdefines the distribution of the model parameters,
corr_matrixcorrelation matrix between the model parameters. By default, this argument is set to an identity matrix (i.e. no correlation between parameters).
checkstates whether to do validity checks of the model parameters (
The calculations are done by taking a sample of size
n_sims of the model parameters according to a multivariate normal distribution. For simulation, the population growth is predicted and the quantiles of the predicted population size is used as an estimate of the credible interval.
For this example, we will use the modified Gompertz model
pars argument defines the distribution of the model parameters. It must be a tibble with 4 columns (
scale) and as many rows as model parameters. Then, for the modified Gompertz model, we will need 4 rows. The column
par defines the parameter that is defined on each row. It must be a parameter identifier according to
primary_model_data(). This function considers that each model parameter follows a marginal normal distribution with the mean defined in the
mean column and the standard deviation defined in
sd. This distribution can be defined in log-scale (by setting the value in
scale to “log”), square-root scale (“sqrt”) or in the original scale (“original”). Note that, in order to omit the variability/uncertainty of any model parameter, one just has to set its corresponding standard error to zero.
For the time points, we will take 100 points uniformly distributed between 0 and 15:
For the example, we will set the number of simulations to 1000. Nevertheless, it is advisable to repeat the calculations for various number of simulations to ensure convergence.
Once the arguments have been defined, we can call the
Before doing any calculations,
predict_growth_uncertainty() makes several consistency checks of the model parameters (this can be turned off by passing
check=FALSE). It returns an instance of
GrowthUncertainty with the results of the simulation. It implements several S3 methods to facilitate the interpretation of the model simulations. The
print(unc_growth) #> Growth prediction based on primary models with uncertainty #> #> Primary model: modGompertz #> #> Mean values of the model parameters: #>  0.0 2.0 0.5 6.0 #> #> Variance-covariance matrix: #> [,1] [,2] [,3] [,4] #> [1,] 0.04 0.00 0.00 0.00 #> [2,] 0.00 0.09 0.00 0.00 #> [3,] 0.00 0.00 0.01 0.00 #> [4,] 0.00 0.00 0.00 0.25
It also implements an S3 method for plot that can be used to visualize the credible intervals
In this plot, the solid line represents the mean of the simulations. Then, the two shaded areas represent, respectively, the space between the 10th and 90th, and the 5th and 95th quantiles.
The plot method includes additional arguments to edit the aesthetics of the plot.
GrowthUncertainty is a subclass of
list, making it easy to access several results of the simulation. Namely, it includes the following items:
sample: sample of model parameters used for the simulations
simulations: results of the individual simulations
quantiles: quantiles of the population size predicted in the simulations
model: model used for the simulations
mus: expected values of the model parameters used for the simulations.
sigma: variance-covariance matrix used for the simulations
By default, the function considers that there is no correlation between the model parameters. This can be varied by defining a correlation matrix. Note that the rows and columns of this matrix are defined in the the same order as in
pars, and the correlation is defined in the scale of
pars. For instance, we can define a correlation of -0.7 between the square root of \(\mu\) and the logarithm of \(\lambda\):
Then, we can include it in the call to the function
An alternative approach to account for uncertainty in model predictions is to use the distribution of the model parameters estimated from experimental data. In biogrowth, this can be done using the
predictMCMC() S3 method of
GlobalGrowthFit. Note that this function is only available for models fitted using an Adaptive Monte Carlo algorithm (i.e.,
algorithm="MCMC"). This function takes 5 arguments:
modelthe instance of
GlobalGrowthFitto use for the calculations
timesa numeric vector stating the time points where the solutions are calculated
env_conditionsa tibble (or data.frame) describing the values of the environmental conditions
niterthe number of Monte Carlo simulations
newparsa named list defining values for some model parameters
First, we need a model fitted using
fit_growth(). In this example, we will use a model fitted using
approach="global". Nonetheless, the calculations of the predictions are exactly the same for a model fitted using
approach="single". Note that, in this case, the model is fitted using an Adaptive Monte Carlo algorithm by setting
algorithm="MCMC". This uses
FME::modMCMC() instead of
FME::modFit() for model fitting. We encourage the reader to read in detail the documentation of this function and references therein, as the interpretation of this fitting algorithm has several differences with respect to regression.
The following code chunk fits the growth model using a global approach. For a detailed description of the
fit_growth() function, the reader is referred to the vignette dedicated to model fitting.
## We will use the data included in the package data("multiple_counts") data("multiple_conditions") ## We need to assign a model equation for each environmental factor sec_models <- list(temperature = "CPM", pH = "CPM") ## Any model parameter (of the primary or secondary models) can be fixed known_pars <- list(Nmax = 1e8, N0 = 1e0, Q0 = 1e-3, temperature_n = 2, temperature_xmin = 20, temperature_xmax = 35, pH_n = 2, pH_xmin = 5.5, pH_xmax = 7.5, pH_xopt = 6.5) ## The rest, need initial guesses my_start <- list(mu_opt = .8, temperature_xopt = 30) set.seed(12421) global_MCMC <- fit_growth(multiple_counts, sec_models, my_start, known_pars, environment = "dynamic", algorithm = "MCMC", approach = "global", env_conditions = multiple_conditions, niter = 100, lower = c(.2, 29), # lower limits of the model parameters upper = c(.8, 34) # upper limits of the model parameters ) #> number of accepted runs: 13 out of 100 (13%)
Note that the number of iterations used in the model is too low for convergence of the fitting algorithm. This can be visualized by inspecting the trace plot of the Markov chain:
Therefore, this model should only be considered as an illustration of the functions included in the package, not as a “serious” model. For additional details on how to evaluate the convergence of this fitting algorithm, the reader is referred to the documentation of FME and references therein.
Once the model has been fitted, we can define the settings for the simulation. The
predictMCMC() method requires the definition of the variation of the environmental conditions during the simulation. This is defined as a tibble (or data.frame) with the same conventions as for
fit_growth(). Note that this argument must defined the same environmental factors as the ones considered in the original model. In this example, we will describe a constant environmental profile where
pH=7. Note that this is just an example, and dynamic profiles can also be calculated.
Then, we need to define the time points where the solution is calculated. We will use a uniformly distributed vector of length 50 between 0 and 40
The last argument we need to define is the number of Monte Carlo simulations to use for the calculations. The calculations are performed by taking a sample of size
niter from the Markov chain of the model parameters. Then, for each parameter vector, the functions calculates the corresponding growth curve at each time point defined in
times. For this example we will use 50 Monte Carlo simulations. This value is extremely low, and should only be used as an illustration of the implementation of the function.
Once every argument has been defined, we can call the
The function returns an instance of
MCMCgrowth with the results of the simulation. It includes several S3 methods to facilitate the interpretation of the results. The print method provides an overview of the simulation settings.
print(uncertain_prediction) #> Growth prediction under dynamic conditions with parameter uncertainty #> #> Environmental factors included: time, temperature, pH #> #> Simulations based on the following model: #> #> Growth model fitted to data following a global approach conditions using MCMC #> #> Number of experiments: 2 #> #> Environmental factors included: temperature, pH #> #> Secondary model for temperature: CPM #> Secondary model for pH: CPM #> #> Parameter estimates: #> mu_opt temperature_xopt #> 0.5095069 29.9829208 #> #> Fixed parameters: #> Nmax N0 Q0 temperature_n #> 1.0e+08 1.0e+00 1.0e-03 2.0e+00 #> temperature_xmin temperature_xmax pH_n pH_xmin #> 2.0e+01 3.5e+01 2.0e+00 5.5e+00 #> pH_xmax pH_xopt #> 7.5e+00 6.5e+00 #> Parameter mu defined in log-10 scale #> Population size defined in log-10 scale
plot() method illustrates the distribution of the population size during the experiment.
The solid line illustrates the median of the simulations for each time point. Then, the ribbons show the space between the 10th and 90th, and 5th and 95th percentiles of the simulations for each time point.
MCMCgrowth is defined as a subclass of
list, so it is easy to access several aspects of the simulations. Namely, it has 5 entries:
samplea tibble with the sample of model parameters used for the simulations.
simulationsa tibble with the results of every individual simulation used.
quantilesa tibble providing the calculated quantiles (5th, 10th, 50th, 90th, 95th) of the population size for each time point.
modelthe instance of
FitDynamicGrowthMCMCused for the predictions.
env_conditionsa tibble with the environmental conditions of the simulations.
As already mentioned, the
newpars argument can be used to define new values of the model parameters. As an illustration, we can fix
sample entry of the outcome shows that the value of
mu_opt is fixed to 0.5 for these simulations, whereas
temperature_xopt varies according to the fitted model.
The biogrowth package includes the
time_to_size() function to estimate the elapsed time required to reach a given population size for growth models. By default, this function calculates a discrete value for the elapsed time. Nevertheless, this can be modified passing
type=distribution. In this case, the function takes two arguments:
modelan instance of
sizethe target population size (in log units)
The distribution of the time to reach the given is estimated by linear interpolation for each of the growth curves included in
model. Therefore, its precision is strongly dependent on the number of simulations (i.e. the number of growth curves) and the density of the time points around the solution. For that reason, considering the low number of iterations and time points included in the simulations, the results presented here should only be considered as an illustration of the functions.
The function returns an instance of
TimeDistribution with the results of the calculation. It includes several S3 methods to facilitate the interpretation of the results. The
summary method provides a table with several statistical indexes of the distribution of the time.
plot method shows an histogram of the distribution. In this plot, the median of the simulations is shown as a dashed, red line. The 10th and 90th percentiles are shown as grey, dashed lines.
When interpreting the results of the calculation is important to consider how
NAs are considered by the function. It is possible that, for some simulations, the target population size is not included in the growth curve. In that case, that particular calculation would results in a time of
NA. This value is then omitted when making the calculations (medians, quantiles, histograms…). This can be relevant for population size in the edge of the simulations (e.g. close to
logNmax) and can introduce a bias in the results.