# Simulation of Pumas Models

## The `simobs`

Function

Simulation of a `PumasModel`

are performed via the `simobs`

function. The function is given by the values:

`Pumas.simobs`

— Function```
simobs(
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 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 a`PumasModel`

or a`PumasEMModel`

.`population`

may either be a`Population`

of`Subject`

s or a single`Subject`

.`param`

should be either a single parameter set, in the form of a`NamedTuple`

, 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 in`population`

. Such random effects are specified by`NamedTuple`

s for`PumasModels`

(e.g.`(; tvCL=1., tvKa=2.)`

) and by`Tuples`

for`PumasEMModel`

s (e.g.`(1., 2.)`

). If`population`

is a single`Subject`

(without being enclosed in a vector) then a single such random effect specifier should be passed. If, however,`population`

is a`Population`

of multiple`Subject`

s then`randeffs`

should be a vector containing one such specifier for each`Subject`

. The functions`init_randeffs`

,`zero_randeffs`

, and`sample_randeffs`

are sometimes convenient for generating`randeffs`

:

```
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 each`Subject`

.`ensemblealg`

is a keyword argument that allows you to choose between different modes of parallelization. Options include`EnsembleSerial()`

,`EnsembleThreads()`

and`EnsembleDistributed()`

.`diffeq_options`

is a keyword argument that takes a`NamedTuple`

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.

`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 multi-variate normal distribution for the parameter values. The optimal parameter values are used 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.

## Handling Simulated Returns

When running

`sim = simobs(m, data, param)`

`sim`

is a `SimulatedPopulation`

which a collection of `SimulatedObservation`

s, and when indexed like `sim[i]`

it returns the `SimulatedObservation`

of the `i`

th simulation subject. The fields of `SimulatedObservation`

are:

`subject`

: the`Subject`

used to generate the observation`time`

: the times associated with the observations`observations`

: the resulting observations of the simulation`retcode`

: the status of the simulation process, success or failure.

If the `@model`

DSL is used, then `observed`

is a `NamedTuple`

where the names give the associated values. From the function-based interface, `observed`

is the chosen return type of the `observed`

function in the model specification.

## Visualizing Simulated Returns

These objects have automatic plotting and dataframe visualization. To plot a simulation return, simply call `sim_plot`

on the output. For example, the following will run a simulation and plot the observed variables:

```
sim = simobs(m, data, param)
using PumasUtilities
sim_plot(m, obs, observations = [:dv])
```

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 of the specified attributes can be used in this `plot`

command. For more advanced use of the underlying plotting ecosystem, please see the CairoMakie 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:

```
sim = simobs(m, data, param)
df = DataFrame(sim)
```

## Different usage patterns of `simobs`

Below, we showcase the different usage patterns of `simobs`

. Consider the following model, population and parameters that we will use to simulate from.

```
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), 1:3)
```

NOTE: In all the examples below, the default simulation time is till 24 hours.

### Subject level simulation

In this pattern, you can use `simobs`

to do a single subject simulation using the syntax below.

`simobs(model::PumasModel, subject::Subject, param::NamedTuple)`

Particularly, for this example, it translates to

`sims2 = simobs(onecomp, pop[1], ext_param)`

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 using the syntax below.

`simobs(model::PumasModel, subject::Subject, param::NamedTuple, randeffs::Union{Nothing, NamedTuple})`

Here individual subject random effects can be passed in 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],)`

Particularly, for this example, it translates to

`sims3 = simobs(onecomp, pop[1], ext_param, ext_randeffs)`

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 using the syntax below.

`simobs(model::PumasModel, subject::Subject, vparam::AbstractArray{<:NamedTuple,1})`

Here, a subject 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[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)

```
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)]
```

Particularly, for this example, it translates to

`sims4 = simobs(onecomp, pop[1], ext_param_array)`

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

```
julia> 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)
3×4 DataFrame
│ 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`

```
julia> using Tables
julia> myparam_tuple = Tables.rowtable(myparams)
3-element Array{NamedTuple{(:tvcl, :tvvc, :Ω, :σ),Tuple{Int64,Int64,Diagonal{Float64,Array{Float64,1}},Float64}},1}:
(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

`simobs(model::PumasModel, subject::Subject, vparam::AbstractArray{<:NamedTuple,1}, vrandeffs::Union{Nothing, AbstractArray{<:NamedTuple,1}})`

```
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)`

### Population level simulation

### Population level simulation with passing in parameters

In this pattern, you can use `simobs`

to do a population simulation using the syntax below.

`simobs(model::PumasModel, population::AbstractArray{<:Subject,1}, param::NamedTuple)`

Particularly, for this example, it translates to

`sims7 = simobs(onecomp, pop, ext_param)`

where, `onecomp`

is the model, `pop`

represents the `Population`

, `pop`

. 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 using the syntax below.

`simobs(model::PumasModel, population::AbstractArray{<:Subject,1}, param::NamedTuple, vrandeffs::Union{Nothing, AbstractArray{<:NamedTuple,1}})`

Here individual subject random effects can be passed in 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),)]
```

Particularly, for this example, it translates to

`sims8 = simobs(onecomp, pop, ext_param, ext_randeffs)`

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

In this pattern, you can use `simobs`

to do a population simulation using the syntax below.

`simobs(model::PumasModel, population::AbstractArray{<:Subject,1}, vparam::AbstractArray{<:NamedTuple,1})`

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)]
```

Particularly, for this example, it translates to

`sims9 = simobs(onecomp, pop, ext_param_array)`

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

`simobs(model::PumasModel, population::AbstractArray{T,1} where T<:Subject, vparam::AbstractArray{T,1} where T, vrandeffs::Union{Nothing, AbstractArray{T,1} where T}, args...; rng, kwargs...)`

`ext_randeffs_array = [(η = rand(2),) for i in 1:length(ext_param_array)*length(pop)]`

`sims10 = simobs(onecomp, pop, ext_param_array, ext_randeffs_array)`

### Simulation using `FittedPumasModel`

The usage pattern is represented below

`simobs(fpm::Pumas.FittedPumasModel)`

Any model that has been *fitted* results in a `FittedPumasModel`

. `simobs`

can simulate from the object where it utilizes the final parameter estimates of the `fit`

and the `design`

of the data that was *fitted*.

### Simulation using a `FittedPumasModelInference`

The usage pattern is represented below

`simobs(fpm::Pumas.FittedPumasModelInference)`

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*. The output this produces is then a *simulation with uncertainty* that you can read about more here.