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
predictMCMC() method.
Growth predictions with uncertainty using
predict_growth_uncertainty()
The function 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 8 arguments:
model_name defines the primary growth model,times defines the time points where to make the
calculations,n_sims defines the number of Monte Carlo
simulations,pars defines the distribution of the model
parameters,corr_matrix correlation matrix between the model
parameters. By default, this argument is set to an identity matrix
(i.e. no correlation between parameters).check states whether to do validity checks of the model
parameters (TRUE by default).logbase_mu defines the log-base used for the definition
of the growth rate (as in predict_growth()).logbase_logN defines the log-base used for the
definition of the log-microbial concentration (as in
predict_growth()).
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
my_model <- "modGompertz"
The pars argument defines the distribution of the model
parameters. It must be a tibble with 4 columns (par,
mean, sd and 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.
pars <- tribble(
~par, ~mean, ~sd, ~scale,
"logN0", 0, .2, "original",
"mu", 2, .3, "sqrt",
"lambda", .5, .1, "log",
"C", 6, .5, "original"
)
For the time points, we will take 100 points uniformly distributed
between 0 and 15:
my_times <- seq(0, 15, length = 100)
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
predict_growth_uncertainty() function.
unc_growth <- predict_growth_uncertainty(my_model, my_times, n_sims, pars)
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 method provides an overview of
the simulation setting.
print(unc_growth)
#> Growth prediction based on primary models with uncertainty
#>
#> Primary model: modGompertz
#>
#> Mean values of the model parameters:
#> [1] 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.
plot(unc_growth, ribbon80_fill = "purple", ribbon90_fill = "pink", alpha80 = .8)
Note that 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
simulationssimulations: results of the individual simulationsquantiles: quantiles of the population size predicted
in the simulationsmodel: model used for the simulationsmus: 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 \(\lambda\):
my_cor <- matrix(c(1, 0, 0, 0,
0, 1, -0.7, 0,
0, -0.7, 1, 0,
0, 0, 0, 1),
nrow = 4)
Then, we can include it in the call to the function
unc_growth2 <- predict_growth_uncertainty(my_model, my_times, n_sims, pars, my_cor)
plot(unc_growth2)
Accounting for the uncertainty of a fitted model using
predictMCMC()
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 GrowthFit
or 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:
model the instance of GrowthFit or
GlobalGrowthFit to use for the calculationstimes a numeric vector stating the time points where
the solutions are calculatedenv_conditions a tibble (or data.frame) describing the
values of the environmental conditionsniter the number of Monte Carlo simulationsnewpars a 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:
plot(global_MCMC$fit_results)
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 temperature=30 and
pH=7. Note that this is just an example, and dynamic
profiles can also be calculated.
my_conditions <- tibble(time = c(0, 40),
temperature = c(30, 30),
pH = c(7, 7)
)
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
my_times <- seq(0, 40, length = 50)
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
predictMCMC() method
set.seed(124)
uncertain_prediction <- predictMCMC(global_MCMC,
my_times,
my_conditions,
niter = niter
)
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
The plot() method illustrates the distribution of the
population size during the experiment.
plot(uncertain_prediction)
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:
sample a tibble with the sample of model parameters
used for the simulations.simulations a tibble with the results of every
individual simulation used.quantiles a tibble providing the calculated quantiles
(5th, 10th, 50th, 90th, 95th) of the population size for each time
point.model the instance of FitDynamicGrowthMCMC
used for the predictions.env_conditions a 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 mu_opt=0.5.
uncertain_prediction2 <- predictMCMC(global_MCMC,
my_times,
my_conditions,
niter = 5,
newpars = list(mu_opt = 0.5)
)
Inspecting the 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.
uncertain_prediction2$sample
#> # A tibble: 5 × 3
#> mu_opt temperature_xopt iter
#> <dbl> <dbl> <int>
#> 1 0.5 29.3 1
#> 2 0.5 29.5 2
#> 3 0.5 29.5 3
#> 4 0.5 29.0 4
#> 5 0.5 29.3 5
Accounting for uncertainty when calculating the time to reach a
population size
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:
model an instance of GrowthUncertainty or
MCMCgrowthsize the 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.
unc_distrib <- time_to_size(type = "distribution", unc_growth, 3)
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 print
method provides an overall view of the results
print(unc_distrib)
#> Distribution of the time required to reach a target population size
#>
#> # A tibble: 1 × 5
#> m_time sd_time med_time q10 q90
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 3.99 0.815 3.94 3.03 5
The summary method provides a table with several
statistical indexes of the distribution of the time.
summary(unc_distrib)
#> # A tibble: 1 × 5
#> m_time sd_time med_time q10 q90
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 3.99 0.815 3.94 3.03 5
The 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.
plot(unc_distrib)
#> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
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 logN0 or logNmax) and can
introduce a bias in the results.