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In this vignette, we will discuss two important points which are related to each other: the parameters physical units and how to choose parameter priors. Those two are related since the biological meaning of the priors necessarily depends on the units used for parameters.
Some parameters of the network models are unitless, but those which aren’t have units which depend on the units used for the data.
This is an overview of the different types of parameters that are used in a network model:
| Parameter name | Description | Unit | Domain |
|---|---|---|---|
upsilon_<src>_to_<dst> |
Uptake rate from src to dst |
Depends on the data | \(\left[0;+\infty\right[\) |
lambda_<src> |
Loss rate from src |
Depends on the data | \(\left[0;+\infty\right[\) |
portion.act_<src> |
Active portion of src |
Unitless | \(\left[0;1\right]\) |
eta |
Default: coefficient of variation (tracer proportion) | Unitless | \(\left[0;+\infty\right[\) |
zeta |
Default: coefficient of variation (biomass) | Unitless | \(\left[0;+\infty\right[\) |
Uptake rates (upsilon_<src>_to_<dst>) are
expressed in proportion of compartment material transferred per
unit of time. For example: 0.05 \(\textrm{day}^{-1}\) means that 5% of
src is transferred to dst per day.
The actual time unit corresponds to the time unit used in the observation data. For example, if the observation data give sampling times as hours since the beginning of the experiment, then the uptake rates will be in \(\textrm{hour}^{-1}\).
Loss rates (lambda_<src>) are also expressed in
proportion of compartment material transferred (lost) per unit
of time. For example: 0.1 \(\textrm{week}^{-1}\) means that 10% of
src is lost and exits the network system per week.
Again, the actual time unit corresponds to the time unit used in the observation data. It is the same as for uptake rates.
An active portion is the proportion of a compartment initial standing stock that is being involved in flows in the network, and must be comprised between 0 and 1. For example, an active portion of 0.25 means that only 25% of the initial compartment standing stock is used when calculating uptakes by other compartments and losses, while 75% of the initial compartment standing stock behaves as a refractory portion which is not involved in the network.
Active portions are useful to model compartments for which only some of the content is involved in exchanges at the time scale of the experiment (e.g. algal and bacterial layers growing on dead leaves in a stream). They can explain why consumers feeding selectively on this active portion might exhibit higher isotopic enrichments than the apparent enrichment of the consumed compartment.
By default, \(\eta\) and \(\zeta\) are the coefficients of variation for the distribution of observed isotopic proportions and compartment sizes, respectively, around their predicted mean values. As coefficients of variation, they are the ratio between the standard deviation and the mean of the distributions they describe, and are thus unitless.
For \(\eta\), several
parameterizations are actually implemented in isotracer and
can be set with set_prop_family():
"gamma_cv": observed proportions are assumed to
follow a gamma distribution with mean
the predicted proportions and coefficient of variation
\(\eta\) (this is the
default).
"normal_cv": observed proportions are assumed to
follow a normal distribution with mean
the predicted proportions and coefficient of variation
\(\eta\).
"normal_sd": observed proportions are assumed to
follow a normal distribution with mean
the predicted proportions and the standard deviation
\(\eta\) (note that in this case \(\eta\) has the same physical unit as the
provided compartment sizes).
"beta_phi": observed proportions are assumed to
follow a beta distribution with mean
the predicted proportions a precision parameter \(\phi = \eta\).
Once we understand what the physical units of each parameter are, we can make sensible choices for priors.
For uptake and loss rates, normal priors centered at 0 are reasonable choices since they allow for any positive value but gives more weight to smaller values and can be scaled by their standard deviation. For example, if we consider a stream ecosystem for which the time data was given in days, allowing for 25% of a compartment biomass to be renewed every day is already pretty generous, so a normal prior with standard deviation 1 is a slightly informative but still very permissive prior.
For a normal prior centered at 0 (and given that priors are truncated
at 0 for rates in isotracer), about 68% of the probability
mass is located between 0 and one standard deviation, and about 95% of
the probability mass is between 0 and two standard deviations.
For example, for a prior normal_p(0, 5):
For active portions, beta priors can be used if the user has any preliminary knowledge about expected active portions, but since the interval is bounded on \(\left[0;1\right]\) a uniform prior is proper and acceptable.
For coefficients of variation, normal priors are
again a reasonable choice. For example, a normal prior
normal_p(0, 2) with standard deviation 2 is not very
informative but gives more weight to coefficients of variation below
200%.
For now, a few different types of priors can be specified by the user when building a network model. Let’s illustrate this with our usual example of an aquarium with NH4, algae and Daphnia:
library(isotracer)
mod <- new_networkModel() %>% set_topo("NH4 -> algae -> daphnia -> NH4")What are the parameters of the model?
params(mod)## # A tibble: 8 × 2
## in_model value
## <chr> <dbl>
## 1 eta NA
## 2 lambda_algae NA
## 3 lambda_daphnia NA
## 4 lambda_NH4 NA
## 5 upsilon_algae_to_daphnia NA
## 6 upsilon_daphnia_to_NH4 NA
## 7 upsilon_NH4_to_algae NA
## 8 zeta NAThe priors available in isotracer are:
constant_p(value): This is a special prior, where the
parameter value is actually fixed to a constant value given by the
user.uniform_p(min, max): A uniform prior.hcauchy_p(scale): The half-Cauchy prior we mentioned
earlier, with hcauchy_p(0.1) being the default prior.scaled_beta_p(alpha, beta, scale): A scaled Beta prior.
This prior is basically a regular Beta prior with parameters \(\alpha\) and \(\beta\), which is then stretched so that it
is defined on \(\left[0;\textrm{scale}\right]\). A
scaled_beta(alpha, beta, 1) is thus a regular Beta prior
defined on \(\left[0;1\right]\).
Another way to see it is that if a random variable \(X\) follows a scaled Beta distribution with
parameters (alpha, beta, scale) then \(X\)/scale follows a Beta
distribution with parameters (alpha, beta).exponential_p(lambda): An exponential prior with rate =
lambda.gamma_p(alpha, beta): A gamma prior with shape =
alpha and rate = beta.Priors of a network model are set using the set_priors()
function:
mod <- mod %>% set_priors(uniform_p(0, 10), "zeta")
priors(mod)## # A tibble: 8 × 2
## in_model prior
## <chr> <list>
## 1 eta <NULL>
## 2 lambda_algae <NULL>
## 3 lambda_daphnia <NULL>
## 4 lambda_NH4 <NULL>
## 5 upsilon_algae_to_daphnia <NULL>
## 6 upsilon_daphnia_to_NH4 <NULL>
## 7 upsilon_NH4_to_algae <NULL>
## 8 zeta <uniform(min=0,max=10)>Note that by default set_priors() assigns priors to all
parameters that match the name provided:
mod <- mod %>% set_priors(scaled_beta_p(4, 8, 5), "lambda")
priors(mod)## # A tibble: 8 × 2
## in_model prior
## <chr> <list>
## 1 eta <NULL>
## 2 lambda_algae <scaled_beta(alpha=4,beta=8,scale=5)>
## 3 lambda_daphnia <scaled_beta(alpha=4,beta=8,scale=5)>
## 4 lambda_NH4 <scaled_beta(alpha=4,beta=8,scale=5)>
## 5 upsilon_algae_to_daphnia <NULL>
## 6 upsilon_daphnia_to_NH4 <NULL>
## 7 upsilon_NH4_to_algae <NULL>
## 8 zeta <uniform(min=0,max=10)>If you want to avoid using partial matching, you can use the argument
use_regexp = FALSE:
mod <- mod %>% set_priors(constant_p(0.2), "eta", use_regexp = FALSE)
priors(mod)## # A tibble: 8 × 2
## in_model prior
## <chr> <list>
## 1 eta <constant(value=0.2)>
## 2 lambda_algae <scaled_beta(alpha=4,beta=8,scale=5)>
## 3 lambda_daphnia <scaled_beta(alpha=4,beta=8,scale=5)>
## 4 lambda_NH4 <scaled_beta(alpha=4,beta=8,scale=5)>
## 5 upsilon_algae_to_daphnia <NULL>
## 6 upsilon_daphnia_to_NH4 <NULL>
## 7 upsilon_NH4_to_algae <NULL>
## 8 zeta <uniform(min=0,max=10)>Notice in the last example how eta was affected while
zeta was not.
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