Simulation of Pumas Models
Simulation of a PumasModel
is performed via the simobs
function. The function is given by the values:
Pumas.simobs
— Functionsimobs(
model::AbstractPumasModel,
population::Union{Subject, Population},
param,
randeffs=sample_randeffs(model, param, population);
obstimes=nothing,
ensemblealg=EnsembleSerial(),
diffeq_options=NamedTuple(),
rng=Random.default_rng(),
simulate_error=Val(true),
)
Simulate random observations from model
for population
with parameters param
at obstimes
(by default, use the times of the existing observations for each subject). If no randeffs
is provided, then random ones are generated according to the distribution in the model.
Arguments
model
may either be aPumasModel
or aPumasEMModel
.population
may either be aPopulation
ofSubject
s or a singleSubject
.param
should be either a single parameter set, in the form of aNamedTuple
, or a vector of such parameter sets. If a vector then each of the parameter sets in that vector will be applied in turn. Example:(; tvCL=1., tvKa=2., tvV=1.)
randeffs
is an optional argument that, if used, should specify the random effects for each subject inpopulation
. Such random effects are specified byNamedTuple
s forPumasModels
(e.g.(; tvCL=1., tvKa=2.)
) and byTuples
forPumasEMModel
s (e.g.(1., 2.)
). Ifpopulation
is a singleSubject
(without being enclosed in a vector) then a single such random effect specifier should be passed. If, however,population
is aPopulation
of multipleSubject
s thenrandeffs
should be a vector containing one such specifier for eachSubject
. The functionsinit_randeffs
,zero_randeffs
, andsample_randeffs
are sometimes convenient for generatingrandeffs
:
randeffs = zero_randeffs(model, param, population)
solve(model, population, param, randeffs)
If no randeffs
is provided, then random ones are generated according to the distribution in the model.
obstimes
is a keyword argument where you can pass a vector of times at which to simulate observations. The default,nothing
, ensures the use of the existing observation times for eachSubject
.ensemblealg
is a keyword argument that allows you to choose between different modes of parallelization. Options includeEnsembleSerial()
,EnsembleThreads()
andEnsembleDistributed()
.diffeq_options
is a keyword argument that takes aNamedTuple
of options to pass on to the differential equation solver.rng
is a keyword argument where you can specify which random number generator to use.simulate_error
is a keyword argument that allows you to choose whether a sample (Val(true)
ortrue
) or the mean (Val(false)
orfalse
) of the predictive distribution's error model will be returned for each subject.
simobs(model::PumasModel, population::Population; samples::Int = 10, simulate_error = true, rng::AbstractRNG = default_rng())
Simulates model predictions for population
using the prior distributions for the population and subject-specific parameters. samples
is the number of simulated predictions returned per subject. If simulate_error
is false
, the mean of the predictive distribution's error model will be returned, otherwise a sample from the predictive distribution's error model will be returned for each subject. rng
is the pseudo-random number generator used in the sampling.
simobs(model::PumasModel, subject::Subject; samples::Int = 10, simulate_error = true, rng::AbstractRNG = default_rng())
Simulates model predictions for subject
using the prior distributions for the population and subject-specific parameters. samples
is the number of simulated predictions returned. If simulate_error
is false
, the mean of the predictive distribution's error model will be returned, otherwise a sample from the predictive distribution's error model will be returned for each subject. rng
is the pseudo-random number generator used in the sampling.
simobs(b::BayesMCMCResults; subject = nothing, samples::Int = 10, simulate_error = false, rng::AbstractRNG = default_rng())
Simulates model predictions using the posterior samples for the population and subject-specific parameters. The predictions are either for the subject with index subject
or the whole population if no index is input. samples
is the number of simulated predictions returned per subject. If simulate_error
is false
, the mean of the predictive distribution's error model will be returned, otherwise a sample from the predictive distribution's error model will be returned for each subject. rng
is the pseudo-random number generator used in the sampling.
simobs(pmi::FittedPumasModelInference, population::Population = pmi.fpm.data, randeffs::Union{Nothing, AbstractVector{<:NamedTuple}} = nothing; samples::Int, rng = default_rng(), kwargs...,)
Simulates observations from the fitted model using a truncated multivariate normal distribution for the parameter values when a covariance estimate is available and using non-parametric sampling for simulated inference and bootstrapping. When a covariance estimate is available, the optimal parameter values are used as the mean and the variance-covariance estimate is used as the covariance matrix. Rejection sampling is used to avoid parameter values outside the parameter domains. Each sample uses a different parameter value. samples
is the number of samples to sample. population
is the population of subjects which defaults to the population associated the fitted model. randeffs
can be set to a vector of named tuples, one for each sample. If randeffs
is not specified (the default behaviour), it will be sampled from its distribution.
simobs(fpm::FittedPumasModel, [population::Population,] vcov::AbstractMatrix, randeffs::Union{Nothing, AbstractVector{<:NamedTuple}} = nothing; samples::Int, rng = default_rng(), kwargs...)
Simulates observations from the fitted model using a truncated multi-variate normal distribution for the parameter values. The optimal parameter values are used for the mean and the user supplied variance-covariance (vcov
) is used as the covariance matrix. Rejection sampling is used to avoid parameter values outside the parameter domains. Each sample uses a different parameter value. samples
is the number of samples to sample. population
is the population of subjects which defaults to the population associated the fitted model so it's optional to pass. randeffs
can be set to a vector of named tuples, one for each sample. If randeffs
is not specified (the default behaviour), it will be sampled from its distribution.
For the examples showcase here, consider the following, model, population and parameters that we will use to simulate from:
using Pumas
using PumasUtilities
onecomp = @model begin
@param begin
tvcl ∈ RealDomain(; lower = 0, init = 4)
tvvc ∈ RealDomain(; lower = 0, init = 70)
Ω ∈ PDiagDomain(init = [0.04, 0.04])
σ ∈ RealDomain(; lower = 0.00001, init = 0.1)
end
@random begin
η ~ MvNormal(Ω)
end
@pre begin
CL = tvcl * exp(η[1])
Vc = tvvc * exp(η[2])
end
@dynamics Central1
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, cp * σ)
end
end
model_params = init_params(onecomp)
ext_param = (; tvcl = 3, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1)
regimen = DosageRegimen(100)
pop = map(i -> Subject(; id = i, events = regimen, observations = (; dv = nothing)), 1:3)
sims = simobs(onecomp, pop, ext_param; obstimes = 1:24)
Simulated population (Vector{<:Subject})
Simulated subjects: 3
Simulated variables: dv
Simulations from simobs
are multithreaded by default. To change the mode of parallelization it is possible to set the ensemblealg
keyword to alternative settings such as EnsembleSerial()
or EnsembleDistributed()
.
Visualizing Simulated Returns
The object returned by simobs
have automatic plotting and DataFrame
conversion. To plot a simulation return, simply call sim_plot
on the output:
sim_plot(sims)
This generates a plot for each variable(s) specified in observations
keyword. For example, sim_plot(m, obs; observations=[:dv1, :dv2])
would only plot the values dv1
and dv2
. In addition, all the specified attributes can be used in this plot
command. For more advanced use of the underlying plotting ecosystem, please see the Makie layout tutorial. Note that if the simulated return is a SimulatedPopulation
, then the plots overlay the results of the various subjects.
To generate the DataFrame
associated with the observed outputs, simply call DataFrame
on the simulated return. For example, the following builds the tabular output from the returned object:
df = DataFrame(sims)
first(df, 5)
Row | id | time | dv | evid | amt | cmt | rate | duration | ss | ii | route | Central | CL | Vc | η_1 | η_2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
String | Float64 | Float64? | Int64 | Float64? | Symbol? | Float64? | Float64? | Int8? | Float64? | NCA.Route? | Float64? | Float64? | Float64? | Float64 | Float64 | |
1 | 1 | 0.0 | missing | 1 | 100.0 | Central | 0.0 | 0.0 | 0 | 0.0 | NullRoute | missing | 3.31311 | 78.1868 | 0.0992754 | 0.264757 |
2 | 1 | 1.0 | 1.2759 | 0 | missing | missing | missing | missing | missing | missing | missing | 95.8511 | 3.31311 | 78.1868 | 0.0992754 | 0.264757 |
3 | 1 | 2.0 | 0.9999 | 0 | missing | missing | missing | missing | missing | missing | missing | 91.8743 | 3.31311 | 78.1868 | 0.0992754 | 0.264757 |
4 | 1 | 3.0 | 1.42134 | 0 | missing | missing | missing | missing | missing | missing | missing | 88.0625 | 3.31311 | 78.1868 | 0.0992754 | 0.264757 |
5 | 1 | 4.0 | 1.14577 | 0 | missing | missing | missing | missing | missing | missing | missing | 84.4089 | 3.31311 | 78.1868 | 0.0992754 | 0.264757 |
Different usage patterns of simobs
Below, we showcase the different usage patterns of simobs
.
Subject level simulation
In this pattern, you can use simobs
to do a single subject simulation by passing a Subject
instead of a Population
:
sims2 = simobs(onecomp, pop[1], ext_param; obstimes = 1:24)
SimulatedObservations
Simulated variables: dv
Time: 1:24
where, onecomp
is the model, pop[1]
represents one subject in the Population
, pop
. The parameters of the model are passed in as the NamedTuple
defined earlier.
Subject level simulation with passing random effects
In this pattern, you can use simobs
to do a single subject simulation by passing an additional positional argument of individual subject random effects represented as vector. The length of the random effect vector should be the same as the number of parameters for each subject:
ext_randeffs = (; η = [0.7, -0.44])
sims3 = simobs(onecomp, pop[1], ext_param, ext_randeffs; obstimes = 1:24)
SimulatedObservations
Simulated variables: dv
Time: 1:24
where, onecomp
is the model, pop[1]
represents one subject in the Population
, pop
. The parameters of the model are passed in as the NamedTuple
defined earlier. ext_randeffs
are the specific values of the η
's that are passed in, which means that simobs
uses these values as opposed to those sampled from the distributions specified in the @random
block.
Subject level simulation with passing in an array of parameters
In this pattern, you can use simobs
to do a single subject simulation by passing a positional argument, an array of parameters that generates a unique simulation solution for each element of the parameter array. E.g. let's simulate pop[1]
with three different combinations of tvcl
and tvvc
. In the example below, we change the value of tvcl
to represent lo
, med
, hi
of 3
, 6
, 9
, but we don't change the other values (Note that are more efficient programmatic ways to generate such arrays) using the syntax below:
ext_param_array = [
(tvcl = 3, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1),
(tvcl = 6, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1),
(tvcl = 9, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1),
]
sims4 = simobs(onecomp, pop[1], ext_param_array; obstimes = 1:24)
Simulated population (Vector{<:Subject})
Simulated subjects: 3
Simulated variables: dv
where, onecomp
is the model, pop[1]
represents one subject in the Population
, pop
. The parameters of the model are passed in as the vector of NamedTuple
s ext_param_array
.
A DataFrame of parameter values can be converted to a NamedTuple
easily using Tables.rowtable
syntax. See the example below.
myparams = DataFrame(;
tvcl = [1, 2, 3],
tvvc = [20, 30, 40],
Ω = [Diagonal([0.04, 0.04]), Diagonal([0.04, 0.04]), Diagonal([0.04, 0.04])],
σ = 0.1,
)
Row | tvcl | tvvc | Ω | σ |
---|---|---|---|---|
Int64 | Int64 | Diagonal… | Float64 | |
1 | 1 | 20 | [0.04 0.0; 0.0 0.04] | 0.1 |
2 | 2 | 30 | [0.04 0.0; 0.0 0.04] | 0.1 |
3 | 3 | 40 | [0.04 0.0; 0.0 0.04] | 0.1 |
This can be converted to a NamedTuple
myparam_tuple = Tables.rowtable(myparams)
3-element Vector{@NamedTuple{tvcl::Int64, tvvc::Int64, Ω::Diagonal{Float64, Vector{Float64}}, σ::Float64}}:
(tvcl = 1, tvvc = 20, Ω = [0.04 0.0; 0.0 0.04], σ = 0.1)
(tvcl = 2, tvvc = 30, Ω = [0.04 0.0; 0.0 0.04], σ = 0.1)
(tvcl = 3, tvvc = 40, Ω = [0.04 0.0; 0.0 0.04], σ = 0.1)
myparam_tuple
is of the same form as the ext_param_array
.
Subject level simulation with passing in an array of parameters and random effects
The two usage patterns above can be combined such that one can pass in an array of parameters and corresponding random effects with the usage pattern below:
ext_randeffs = [(; η = [0.7, -0.44]), (; η = [0.6, -0.9]), (; η = [-0.7, 0.6])]
sims5 = simobs(onecomp, pop[1], ext_param_array, ext_randeffs; obstimes = 1:24)
Simulated population (Vector{<:Subject})
Simulated subjects: 3
Simulated variables: dv
Population level simulations
In this pattern, you can use simobs
to do a whole population simulation by passing a Population
instead of a Subject
:
sims6 = simobs(onecomp, pop, ext_param; obstimes = 1:24)
Simulated population (Vector{<:Subject})
Simulated subjects: 3
Simulated variables: dv
where, onecomp
is the model, pop
is the Population
. The parameters of the model are passed in as the NamedTuple
defined earlier.
Population level simulation passing random effects
In this pattern, you can use simobs
to do a population simulation by passing an additional positional argument of individual subject random effects represented as vector. The length of the random effect vector should be the same as the number of parameters for each subject:
You can use this method to pass in the empirical_bayes
of a FittedPumasModel
.
ext_randeffs = [(; η = rand(2)), (; η = rand(2)), (; η = rand(2))]
sims8 = simobs(onecomp, pop, ext_param, ext_randeffs; obstimes = 1:24)
Simulated population (Vector{<:Subject})
Simulated subjects: 3
Simulated variables: dv
where, onecomp
is the model, pop
represents the Population
, pop
. The parameters of the model are explicitly passed in as the NamedTuple
defined earlier. ext_randeffs
are the specific values of the η
's that are passed in, which means that simobs
uses these values as opposed to those sampled from the distributions specified in the @random
block.
Population level simulation with passing in an array of parameters
The two usage patterns above can be combined such that one can pass in an array of parameters and corresponding random effects.
Here, a population can be simulated with an array of parameters that generates a unique simulation solution for each element of the parameter array. E.g. let's simulate pop
with three different combinations of tvcl
and tvvc
. In the example below, we change the value of tvcl
to represent lo
, med
, hi
of 3, 6, 9, but we don't change the other values (Note that are more efficient programmatic ways to generate such arrays):
ext_param_array = [
(; tvcl = 3, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1),
(; tvcl = 6, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1),
(; tvcl = 9, tvvc = 60, Ω = Diagonal([0.04, 0.04]), σ = 0.1),
]
sims9 = simobs(onecomp, pop, ext_param_array; obstimes = 1:24)
3-element Vector{Vector{Pumas.SimulatedObservations{PumasModel{(tvcl = 1, tvvc = 1, Ω = 2, σ = 1), 2, (:Central,), (:CL, :Vc), ParamSet{@NamedTuple{tvcl::RealDomain{Int64, TransformVariables.Infinity{true}, Int64}, tvvc::RealDomain{Int64, TransformVariables.Infinity{true}, Int64}, Ω::PDiagDomain{PDMats.PDiagMat{Float64, Vector{Float64}}}, σ::RealDomain{Float64, TransformVariables.Infinity{true}, Float64}}}, Main.var"#1#6", Main.var"#2#7", Nothing, Main.var"#3#8", Central1, Pumas.DerivedObj{(:dv,), Main.var"#4#9"}, Main.var"#5#10", Nothing, Pumas.PumasModelOptions}, Subject{@NamedTuple{dv::Vector{Missing}}, Pumas.ConstantCovar{@NamedTuple{}}, DosageRegimen{Float64, Symbol, Float64, Float64, Float64, Float64}, Nothing}, UnitRange{Int64}, @NamedTuple{tvcl::Int64, tvvc::Int64, Ω::Diagonal{Float64, Vector{Float64}}, σ::Float64}, @NamedTuple{η::Vector{Float64}}, @NamedTuple{}, @NamedTuple{callback::Nothing, continuity::Symbol, saveat::UnitRange{Int64}, alg::CompositeAlgorithm{1, Tuple{OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, true, LinearSolve.GenericLUFactorization{LinearAlgebra.RowMaximum}, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!)}}, OrdinaryDiffEq.AutoSwitch{OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, true, LinearSolve.GenericLUFactorization{LinearAlgebra.RowMaximum}, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!)}, Rational{Int64}, Int64}}}, Pumas.PKPDAnalyticalSolution{StaticArraysCore.SVector{1, Float64}, 2, Vector{StaticArraysCore.SVector{1, Float64}}, Vector{Float64}, Vector{StaticArraysCore.SVector{1, Float64}}, Vector{StaticArraysCore.SVector{1, Float64}}, Returns{@NamedTuple{CL::Float64, Vc::Float64}}, Pumas.AnalyticalPKPDProblem{StaticArraysCore.SVector{1, Float64}, Float64, false, Central1, Vector{Pumas.Event{Float64, Float64, Float64}}, Vector{Float64}, Returns{@NamedTuple{CL::Float64, Vc::Float64}}}}, @NamedTuple{}, @NamedTuple{CL::Vector{Float64}, Vc::Vector{Float64}}, @NamedTuple{dv::Vector{Float64}}, @NamedTuple{}}}}:
Simulated population (Vector{<:Subject}), n = 3, variables: dv
Simulated population (Vector{<:Subject}), n = 3, variables: dv
Simulated population (Vector{<:Subject}), n = 3, variables: dv
where, onecomp
is the model, pop
represents the Population
, pop
. The parameters of the model are passed in as a vector of the NamedTuple
s ext_param_array
as defined earlier represents the array of parameter values we want to simulate the model at.
Population level simulation with passing in an array of parameters and random effects
The two usage patterns above can be combined such that one can pass in an array of parameters and corresponding random effects with the usage pattern below:
ext_randeffs_array = [(; η = rand(2)) for i = 1:length(ext_param_array)*length(pop)]
sims10 = simobs(onecomp, pop, ext_param_array, ext_randeffs_array; obstimes = 1:24)
3-element Vector{Vector{Pumas.SimulatedObservations{PumasModel{(tvcl = 1, tvvc = 1, Ω = 2, σ = 1), 2, (:Central,), (:CL, :Vc), ParamSet{@NamedTuple{tvcl::RealDomain{Int64, TransformVariables.Infinity{true}, Int64}, tvvc::RealDomain{Int64, TransformVariables.Infinity{true}, Int64}, Ω::PDiagDomain{PDMats.PDiagMat{Float64, Vector{Float64}}}, σ::RealDomain{Float64, TransformVariables.Infinity{true}, Float64}}}, Main.var"#1#6", Main.var"#2#7", Nothing, Main.var"#3#8", Central1, Pumas.DerivedObj{(:dv,), Main.var"#4#9"}, Main.var"#5#10", Nothing, Pumas.PumasModelOptions}, Subject{@NamedTuple{dv::Vector{Missing}}, Pumas.ConstantCovar{@NamedTuple{}}, DosageRegimen{Float64, Symbol, Float64, Float64, Float64, Float64}, Nothing}, UnitRange{Int64}, @NamedTuple{tvcl::Int64, tvvc::Int64, Ω::Diagonal{Float64, Vector{Float64}}, σ::Float64}, @NamedTuple{η::Vector{Float64}}, @NamedTuple{}, @NamedTuple{callback::Nothing, continuity::Symbol, saveat::UnitRange{Int64}, alg::CompositeAlgorithm{1, Tuple{OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, true, LinearSolve.GenericLUFactorization{LinearAlgebra.RowMaximum}, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!)}}, OrdinaryDiffEq.AutoSwitch{OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, Rosenbrock23{1, true, LinearSolve.GenericLUFactorization{LinearAlgebra.RowMaximum}, typeof(OrdinaryDiffEq.DEFAULT_PRECS), Val{:forward}, true, nothing, typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!)}, Rational{Int64}, Int64}}}, Pumas.PKPDAnalyticalSolution{StaticArraysCore.SVector{1, Float64}, 2, Vector{StaticArraysCore.SVector{1, Float64}}, Vector{Float64}, Vector{StaticArraysCore.SVector{1, Float64}}, Vector{StaticArraysCore.SVector{1, Float64}}, Returns{@NamedTuple{CL::Float64, Vc::Float64}}, Pumas.AnalyticalPKPDProblem{StaticArraysCore.SVector{1, Float64}, Float64, false, Central1, Vector{Pumas.Event{Float64, Float64, Float64}}, Vector{Float64}, Returns{@NamedTuple{CL::Float64, Vc::Float64}}}}, @NamedTuple{}, @NamedTuple{CL::Vector{Float64}, Vc::Vector{Float64}}, @NamedTuple{dv::Vector{Float64}}, @NamedTuple{}}}}:
Simulated population (Vector{<:Subject}), n = 3, variables: dv
Simulated population (Vector{<:Subject}), n = 3, variables: dv
Simulated population (Vector{<:Subject}), n = 3, variables: dv
Simulation using a FittedPumasModelInference
The usage pattern is represented below:
Pumas.simobs
— Methodsimobs(pmi::FittedPumasModelInference, population::Population = pmi.fpm.data, randeffs::Union{Nothing, AbstractVector{<:NamedTuple}} = nothing; samples::Int, rng = default_rng(), kwargs...,)
Simulates observations from the fitted model using a truncated multivariate normal distribution for the parameter values when a covariance estimate is available and using non-parametric sampling for simulated inference and bootstrapping. When a covariance estimate is available, the optimal parameter values are used as the mean and the variance-covariance estimate is used as the covariance matrix. Rejection sampling is used to avoid parameter values outside the parameter domains. Each sample uses a different parameter value. samples
is the number of samples to sample. population
is the population of subjects which defaults to the population associated the fitted model. randeffs
can be set to a vector of named tuples, one for each sample. If randeffs
is not specified (the default behaviour), it will be sampled from its distribution.
Any model that has been fitted results in a FittedPumasModelInference
. simobs
can simulate from the inferred object where it utilizes the variance-covariance-matrix of the infer
and the design
of the data that was fitted.
Post-processing simulations
The output of a simobs
call stores the simulated observations but also all the intermediate values computed such as the parameters used, individual coefficients, dose control parameters, covariates, differential equation solution, etc. There are a number of post-processing operations you can do on the simulation output to compute various queries and summary statistics based on the simulations.
The postprocess
function is a powerful tool that allows you to make various queries using the simulation results.
Pumas.postprocess
— Functionpostprocess(obs::SimulatedObservations)
postprocess(func::Function, obs::SimulatedObservations)
postprocess(obs::SimulatedPopulation; stat = identity)
postprocess(func::Function, obs::SimulatedPopulation; stat = identity)
Post-processes the output of simobs
. The output of a simobs
call stores the simulated observations but also all the intermediate values computed such as the parameters used, individual coefficients, dose control parameters, covariates, differential equation solution, etc. There are a number of post-processing operations you can do on the simulation output to compute various queries and summary statistics based on the simulations.
The postprocess
function is a powerful tool that allows you to make various queries using the simulation results. There are multiple ways to use the postprocess
function. The first way to use the postprocess
function is to extract all the information stored in the simulation result in the form of a vector of named tuples. Each named tuple has all the intermediate values evaluated when simulating 1 run. Let sims
be the output of any simobs
operation.
generated = Pumas.postprocess(sims)
generated
is a vector of simulation runs. Hence, generated[i]
is the result from the i
th simulation run. Time-dependent variables are stored as a vector with 1 element for each time point where there is an observation. Alternatively, if obstimes
was set instead when calling simobs
, the time-dependent variables will be evaluated at the time points in obstimes
instead.
The second way to use postprocess
is by passing in a post-processing function. The post-processing function can be used to:
- Transform the simulated quantities, or
- Compare the simulated quantities to the observations.
We use the do
syntax here which is short for passing in a function as the first argument to postprocess
. The do
syntax to pass in a post-processing function is:
Pumas.postprocess(sims) do gen, obs
...
end
where gen
is the named tuple of all generated quantities from 1 simulation run and obs
is the named tuple of observations. For instance to query the ratio of simulated observations dv
that are higher than the observed quantity dv
at the respective time points, you can use:
Pumas.postprocess(sims) do gen, obs
sum(gen.dv .> obs.dv) / length(gen.dv)
end
gen.dv
is the simulated vector of dv
whose length is the same as the number of observation time points. obs.dv
is the observation vector dv
. gen.dv .> obs.dv
returns a vector of true
/false
, with one element for each time point. The sum of this vector gives the number of time points where the simulation was higher than the observation. Dividing by the number of time points gives the ratio. When using postprocess
in this way, the output is always a vector of the query results. In the query function body, you can choose to use only gen
or only obs
but the header must always have both gen
and obs
:
Pumas.postprocess(sims) do gen, obs
...
end
The third way to use the postprocess
function is to compute summary statistics of the simulated quantities or of functions of the simulated quantities. Summary statistics can be computed by passing a statistic function as the stat
keyword argument. For example in order to estimate the probability that a simulated value is higher than an observation, you can use:
Pumas.postprocess(sims, stat = mean) do gen, obs
gen.dv .> obs.dv
end
This function will do 2 things:
- Concatenate the query results (e.g.
gen.dv .> obs.dv
) from all the simulation runs into a single vector. - Compute the mean value of the combined vector.
Alternatively, you can use the mean
function to do the same thing without using the keyword argument. The following will call the postprocess
function under the hood.
mean(sims) do gen, obs
gen.dv .> obs.dv
end
The result of this operation will be a scalar equal to the mean value of the concatenated vector of queries.
In order to get the probability that the simulated quantity is higher than the observation for each time point, you can call the mean
function externally as such:
generated = Pumas.postprocess(sims) do gen, obs
gen.dv .> obs.dv
end
mean(generated)
This returns a vector of probabilities of the same length as the number of time points without concatenating all the queries together.
To compute a summary statistic of all the generated quantity, you can also call:
Pumas.postprocess(sims, stat = mean)
without specifying a post-processing function or for short:
mean(sims)
Besides mean
, you can also use any of the following summary statistic functions in the same way:
std
for element-wise standard deviationvar
for element-wise variancecor
for correlation between multiple quantitiescov
for covariance between multiple quantities
These functions can be passed in as the stat
keyword argument to postprocess
or they can be used in the short form, e.g.:
generated = Pumas.postprocess(sims, stat = std) do gen, obs
...
end
std(sims) do gen, obs
...
end
std(sims)
generated = Pumas.postprocess(sims, stat = var) do gen, obs
...
end
var(sims) do gen, obs
...
end
var(sims)
The cor
and cov
statistics are unique in that they require a post-processing function that outputs a vector. For example, to estimate the correlation between the CL
and Vc
parameters in a 1-compartment model, you can use any of the following:
Pumas.postprocess(s, stat = cor) do gen, obs
[gen.CL[1], gen.Vc[1]]
end
cor(s) do gen, obs
[gen.CL[1], gen.Vc[1]]
end
Note that gen.CL
is a vector of simulated CL
values for all the time points. But since the value is constant across time, we can use the first element gen.CL[1]
only. cov
can be used instead of cor
to compute the covariance matrix. The output of this operation is either a correlation or a covariance matrix.