StructuralIdentifiability.jl

https://docs.sciml.ai/StructuralIdentifiability/stable
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StructuralIdentifiability.jl is a Julia toolbox for assessing local and global structural...
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### Install StructuralIdentifiability.jl

Source: https://docs.sciml.ai/StructuralIdentifiability/stable

Use the Julia package manager to add the StructuralIdentifiability package to your environment.

```julia
using Pkg
Pkg.add("StructuralIdentifiability")
```

--------------------------------

### Example: find_identifiable_functions with @ODEmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Complete example showing how to define an ODE model with @ODEmodel and call find_identifiable_functions. The output prints two identifiable functions: a12 + a01 + a21 and a12*a01.

```julia
using StructuralIdentifiability

ode = @ODEmodel(
    x0'(t) = -(a01 + a21) * x0(t) + a12 * x1(t),
    x1'(t) = a21 * x0(t) - a12 * x1(t),
    y(t) = x0(t)
)

find_identifiable_functions(ode)

# prints
3-element Vector{AbstractAlgebra.Generic.FracFieldElem{Nemo.QQMPolyRingElem}}:
 a12 + a01 + a21
 a12*a01
```

--------------------------------

### find_submodels method signature and example

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Calculates and returns all valid submodels of the given ODE system as a list of ODE objects. The example defines a simple ODE model and calls find_submodels, showing the resulting submodel.

```julia
find_submodels(ode)
```

```julia
>ode = @ODEmodel(x1'(t) = x1(t)^2, 
                 x2'(t) = x1(t) * x2(t), 
                 y1(t) = x1(t), 
                 y2(t) = x2(t))
>find_submodels(ode)
    ODE{QQMPolyRingElem}[
        
        x1'(t) = a(t)*x2(t)^2 + x1(t)
        y1(t) = x1(t)
    ]
```

--------------------------------

### Create a linear compartment model

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/input/input

Define a linear compartment model using a graph adjacency list. The example shows a bidirected cycle with four nodes, specifying outputs, inputs, and leaks.

```julia
julia> ode = linear_compartment_model([[2, 4], [1, 3], [2, 4], [1, 3]], outputs = [1], inputs = [2], leaks = [2, 3])
x1' = -x1*a_2_1 - x1*a_4_1 + x2*a_1_2 + x4*a_1_4
x3' = x2*a_3_2 - x3*a_2_3 - x3*a_4_3 - x3*a_0_3 + x4*a_3_4
x2' = x1*a_2_1 - x2*a_1_2 - x2*a_3_2 - x2*a_0_2 + x3*a_2_3 + u2
x4' = x1*a_4_1 + x3*a_4_3 - x4*a_1_4 - x4*a_3_4
y1 = x1
```

--------------------------------

### find_ioequations(ode, [var_change_policy=:default])

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/ioequations/ioequations

Finds the input-output equations of an ODE system. Returns a dictionary from leaders to the corresponding input-output equations.

```APIDOC
## find_ioequations(ode, [var_change_policy=:default])

### Description
Finds the input-output equations of an ODE system.

### Method
Function

### Signature
`find_ioequations(ode, [var_change_policy=:default])`

### Parameters
- `ode` - The ODE system.
- `var_change_policy` - Whether to perform automatic variable change, can be one of `:default`, `:yes`, `:no`.
- `loglevel` - Logging level (default: Logging.Info).

### Output
A dictionary from "leaders" to the corresponding input-output equations; if an extra projection is needed, it will be the value corresponding to `rand_proj_var`.
```

--------------------------------

### Define and assess a discrete-time system using ModelingToolkitBase

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/discrete_time

Defines a system using ModelingToolkitBase and assesses its local identifiability, noting that shifts must be negative in this approach.

```julia
using ModelingToolkitBase
using StructuralIdentifiability

@independent_variables t
@parameters α β
@variables S(t) I(t) R(t)
k = ShiftIndex(t)

eqs = [
    S(k) ~ S(k - 1) - β * S(k - 1) * I(k - 1),
    I(k) ~ I(k - 1) + β * S(k - 1) * I(k - 1) - α * I(k - 1),
    R(k) ~ R(k - 1) + α * I(k - 1),
]

@named sys = System(eqs, t)

assess_local_identifiability(sys, measured_quantities = [I])
```

--------------------------------

### check_primality(polys::Dict{QQMPolyRingElem, QQMPolyRingElem}, extra_relations::Array{QQMPolyRingElem, 1})

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/primality

Checks if the ideal generated by the given polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals. Additional polynomials can be provided via extra_relations without affecting the saturation.

```APIDOC
## check_primality(polys::Dict{QQMPolyRingElem, QQMPolyRingElem}, extra_relations::Array{QQMPolyRingElem, 1})

### Description
Checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals. The `extra_relations` allows adding more polynomials to the generators (not affecting the saturation).

### Method
Function call

### Signature
```julia
check_primality(polys::Dict{QQMPolyRingElem, QQMPolyRingElem}, extra_relations::Array{QQMPolyRingElem, 1})
```

### Parameters
- `polys` (Dict{QQMPolyRingElem, QQMPolyRingElem}) - Required - Dictionary of polynomials generating the ideal.
- `extra_relations` (Array{QQMPolyRingElem, 1}) - Required - Additional polynomials to add to the generators without affecting saturation.

### Returns
- Boolean indicating whether the ideal is prime over rationals.
```

--------------------------------

### StructuralIdentifiability.io_switch!(pbr)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/ioequations/ioequations

In a single-output pb-representation, makes the leading variable to be the first of the inputs.

```APIDOC
## StructuralIdentifiability.io_switch!(pbr)

### Description
In a single-output pb-representation `pbr` makes the leading variable to be the first of the inputs.

### Method
Function

### Signature
`io_switch(pbr)`

### Parameters
- `pbr` - A projection-based representation.
```

--------------------------------

### check_primality(polys::Dict{QQMPolyRingElem, QQMPolyRingElem})

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/primality

Checks if the ideal generated by the given polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.

```APIDOC
## check_primality(polys::Dict{QQMPolyRingElem, QQMPolyRingElem})

### Description
Checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.

### Method
Function call

### Signature
```julia
check_primality(polys::Dict{QQMPolyRingElem, QQMPolyRingElem})
```

### Parameters
- `polys` (Dict{QQMPolyRingElem, QQMPolyRingElem}) - Required - Dictionary of polynomials generating the ideal.

### Returns
- Boolean indicating whether the ideal is prime over rationals.
```

--------------------------------

### Define model with ModelingToolkit System

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/creating_ode

Defines the same mutualism model using ModelingToolkit's System. The first block declares variables, parameters, and the differential operator; the second block builds the equations, measured quantities, and the System; the third block shows the printed System summary.

```julia
using StructuralIdentifiability, ModelingToolkitBase

@independent_variables t
@parameters r1, r2, c1, c2, beta1, beta2, chi1, chi2
@variables x1(t), x2(t), y(t), u(t)

D = Differential(t)
```

```julia
eqs = [
    D(x1) ~ r1 * x1 * (1 - c1 * x1) + beta1 * x1 * x2 / (chi1 + x2) + u,
    D(x2) ~ r2 * x2 * (1 - c2 * x2) + beta2 * x1 * x2 / (chi2 + x1),
]

measured_quantities = [y ~ x1]

ode_mtk = System(eqs, t, name = :mutualist)
```

```text
Model mutualist:
Equations (2):
  2 standard: see equations(mutualist)
Unknowns (3): see unknowns(mutualist)
  x1(t)
  x2(t)
  u(t)
Parameters (8): see parameters(mutualist)
  beta1
  chi1
  c1
  r1
  ⋮
```

--------------------------------

### Macro @ODEmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/input/input

Macro for creating an ODE model from a list of differential and output equations. It also injects all variables into the global scope.

```APIDOC
## Macro @ODEmodel

### Description
Macro for creating an ODE from a list of equations. It also injects all variables into the global scope.

### Syntax
```julia
@ODEmodel(equations...)
```

### Parameters
- `equations` (list of equations) - Required - State derivative equations (e.g., `x'(t) = ...`) and output equations (e.g., `y(t) = ...`). The equations define state variables, output variables, input variables, and parameters.

### Example
```julia
using StructuralIdentifiability

ode = @ODEmodel(
    x1'(t) = a * x1(t) + u(t),
    x2'(t) = b * x2(t) + c*x1(t)*x2(t),
    y(t) = x1(t)
)
```

Here,
- `x1`, `x2` are state variables
- `y` is an output variable
- `u` is an input variable
- `a`, `b`, `c` are time-independent parameters

```

--------------------------------

### assess_identifiability with known_ic = [x2, x3]

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiability

Call assess_identifiability with the experimental known_ic keyword argument to assume initial conditions for x2 and x3 are known. The output shows that alpha and gama become locally identifiable.

```julia
assess_identifiability(ode, known_ic = [x2, x3])
```

```text
OrderedCollections.OrderedDict{Nemo.QQMPolyRingElem, Symbol} with 11 entries:
  x1(0) => :globally
  x2(0) => :globally
  x3(0) => :globally
  x4(0) => :globally
  alpha => :locally
  b     => :globally
  beta  => :locally
  c     => :globally
  delta => :locally
  gama  => :locally
  sigma => :globally
```

--------------------------------

### Create an ODE system with @ODEmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/input/input

Use the @ODEmodel macro to define an ODE system from a list of equations. This macro automatically injects state, output, input, and parameter variables into the global scope.

```julia
using StructuralIdentifiability

ode = @ODEmodel(
    x1'(t) = a * x1(t) + u(t),
    x2'(t) = b * x2(t) + c*x1(t)*x2(t),
    y(t) = x1(t)
)
```

--------------------------------

### View the reparametrized ODE

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/reparametrization

Display the original ODE rewritten in terms of the new variables.

```julia
reparam[:new_ode]
```

```text
X1'(t) = -X1(t)*a2 + X4(t)*a1
X2'(t) = (-X2(t)*X3(t))//a3
X3'(t) = -X3(t)*a2 + X4(t)*a1
X4'(t) = (X2(t)*X3(t) - X4(t)*a1*a3)//a3
y1(t) = X1(t)

```

--------------------------------

### Find identifiable functions with known initial condition

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiable_functions

Calls `find_identifiable_functions` with `known_ic = [x1]` to assume the initial condition `x1(0)` is known. The output includes identifiable functions of parameters and initial conditions, such as `x3(0)`, `x2(0)`, and `x1(0)`, along with the parameter combinations.

```julia
find_identifiable_functions(LLW1987, known_ic = [x1])
```

```text
8-element Vector{AbstractAlgebra.Generic.FracFieldElem{Nemo.QQMPolyRingElem}}:
 x3(0)
 x2(0)
 x1(0)
 p1 + p3
 p2*p4
 p1*p3
 p2*x2(0) + p4*x1(0)
 p1*p4*x1(0) + p2*p3*x2(0)
```

--------------------------------

### monomial_compress(io_equation, ode)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/wronskian

Compresses an input-output equation for rank computation, producing a representation with coefficients involving only parameters and terms involving only inputs and outputs, with minimal length.

```APIDOC
## monomial_compress(io_equation, ode)

### Description
Compresses an input-output equation for rank computation, producing a representation with coefficients involving only parameters and terms involving only inputs and outputs, with minimal length.

### Method
Function call

### Endpoint
`StructuralIdentifiability.monomial_compress(io_equation, ode)`

### Parameters
- **io_equation** (input-output equation) - Required - The input-output equation to compress
- **ode** (ODE model) - Required - The corresponding ODE model

### Response
#### Success Response
- **coeffs** - Coefficients involving only parameters
- **terms** - Terms involving only inputs and outputs
- **sum** - Sum of `coeffs[i] * terms[i]` equals the input-output equation
- **length** - The representation length is the smallest possible

#### Response Example
```
(coeffs, terms)
```
```

--------------------------------

### Macro @DDSmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/input/input

Macro for creating a discrete dynamical system (DDS) from a list of equations. It also injects all variables into the global scope.

```APIDOC
## Macro @DDSmodel

### Description
Macro for creating a DDS (discrete dynamical system) from a list of equations. It also injects all variables into the global scope.

### Syntax
```julia
@DDSmodel(equations...)
```

### Parameters
- `equations` (list) - Required - List of difference equations (e.g., `x(t+1) = ...`) and output equations (e.g., `y(t) = ...`). The equations define state variables, output variables, input variables, and parameters.

### Example
```julia
using StructuralIdentifiability

dds = @DDSmodel(
    x1(t + 1) = a * x1(t) + u(t),
    x2(t + 1) = b * x2(t) + c*x1(t)*x2(t),
    y(t) = x1(t)
)
```

Here,
  - `x1`, `x2` are state variables
  - `y` is an output variable
  - `u` is an input variable
  - `a`, `b`, `c` are time-independent parameters
```

--------------------------------

### reduce_ode_mod_p method signature

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Reduces an ODE over QQ modulo a prime p. Throws an exception if p divides any denominator in the ODE.

```julia
reduce_ode_mod_p(ode, p)
```

--------------------------------

### find_identifiable_functions(ode::ODE; options...)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Finds all functions of parameters/states that are identifiable in the given ODE system. Supports options for including states, simplification level, known initial conditions, probability threshold, seed, and log level.

```APIDOC
## find_identifiable_functions(ode::ODE; options...)

### Description
Finds all functions of parameters/states that are identifiable in the given ODE system. The function returns a vector of identifiable functions.

### Method
Function call

### Endpoint
find_identifiable_functions(ode::ODE; options...)

### Parameters
#### Input Arguments
- **ode** (ODE) - Required - The ODE model
- **with_states** (Bool) - Optional - When true, also reports the identifiable functions in the ODE states. Default is false.
- **simplify** (Symbol) - Optional - The extent to which the output functions are simplified. Possible options: :standard (default), :weak, :strong, :absent.
- **known_ic** (list) - Optional - A list of functions whose initial conditions are assumed to be known; then the returned identifiable functions will be functions of parameters and initial conditions, not states (experimental)
- **prob_threshold** (Float) - Optional - Probability of correctness (default: 0.99)
- **seed** (Int) - Optional - The rng seed (default: 42)
- **loglevel** (LogLevel) - Optional - Minimal level of log messages to display (default: Logging.Info)

### Request Example
```julia
using StructuralIdentifiability

ode = @ODEmodel(
    x0'(t) = -(a01 + a21) * x0(t) + a12 * x1(t),
    x1'(t) = a21 * x0(t) - a12 * x1(t),
    y(t) = x0(t)
)

find_identifiable_functions(ode)
```

### Response
#### Success Response
Returns a vector of identifiable functions. Example output:
```julia
3-element Vector{AbstractAlgebra.Generic.FracFieldElem{Nemo.QQMPolyRingElem}}:
 a12 + a01 + a21
 a12*a01
```

```

--------------------------------

### power_series_solution method signature

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Computes a power series solution of the ODE with given parameter values, initial conditions, and input power series, up to the specified precision. Returns a dictionary mapping each variable to its coordinate in the solution.

```julia
power_series_solution(ode, param_values, initial_conditions, input_values, prec)
```

--------------------------------

### Create a discrete dynamical system with @DDSmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/input/input

Use the @DDSmodel macro to define a discrete-time dynamical system. Similar to @ODEmodel, it injects all variables into the global scope.

```julia
using StructuralIdentifiability

dds = @DDSmodel(
    x1(t + 1) = a * x1(t) + u(t),
    x2(t + 1) = b * x2(t) + c*x1(t)*x2(t),
    y(t) = x1(t)
)
```

--------------------------------

### set_parameter_values method signature

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Substitutes the given parameter values into the ODE, producing a new ODE with those parameters replaced by numbers. The parameter values dictionary maps parameter names to values.

```julia
set_parameter_values(ode, param_values)
```

--------------------------------

### wronskian(io_equations, ode)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/wronskian

Computes the Wronskians of a set of input-output equations evaluated at a point modulo prime.

```APIDOC
## wronskian(io_equations, ode)

### Description
Computes the Wronskians of a set of input-output equations evaluated at a point modulo prime.

### Method
Function call

### Endpoint
`StructuralIdentifiability.wronskian(io_equations, ode)`

### Parameters
- **io_equations** (set of io-equations) - Required - A set of io-equations in the form of the `Dict` as returned by `find_ioequations`
- **ode** (ODE object) - Required - The `ODE` object

### Response
#### Success Response
- **list of Wronskians** - The Wronskians evaluated at a point modulo prime

#### Response Example
```
[wronskian1, wronskian2, ...]
```
```

--------------------------------

### assess_local_identifiability for DDS

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Signature for checking local identifiability/observability of functions in a discrete dynamical system (DDS). Accepts a list of known_ic quantities whose initial conditions are assumed known and generic. Result is correct with probability at least prob_threshold.

```julia
assess_local_identifiability(dds::DDS{P}; funcs_to_check::Array{<: Any, 1}, known_ic, prob_threshold::Float64=0.99, loglevel=Logging.Info) where P <: MPolyRingElem{Nemo.QQFieldElem}
```

--------------------------------

### power_series_solution

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Computes a power series solution of an ODE with given parameters, initial conditions, and input values up to a specified precision.

```APIDOC
## power_series_solution(ode, param_values, initial_conditions, input_values, prec)

### Description
Computes a power series solution of the given ODE with the specified precision.

### Method
power_series_solution

### Parameters
- **ode** (ODE) - Required - The ODE to solve.
- **param_values** (Dict) - Required - Parameter values, mapping each parameter to a value.
- **initial_conditions** (Dict) - Required - Initial conditions, mapping each state variable to a value.
- **input_values** (Dict) - Required - Power series for the inputs, mapping each variable to a list of coefficients.
- **prec** (Integer) - Required - The precision of the solution.

### Output
Returns a dictionary mapping each variable to the corresponding coordinate of the power series solution.
```

--------------------------------

### Display the LLW1987 model equations

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiable_functions

Shows the expanded form of the LLW1987 model equations as they would be printed, with terms expanded and reordered. This is the output of displaying the model object.

```text
x1'(t) = -x1(t)*p1 + u(t)*p2
x2'(t) = -x2(t)*p3 + u(t)*p4
x3'(t) = x1(t)*u(t)*p4 + x2(t)*u(t)*p2 - x3(t)*p1 - x3(t)*p3
y1(t) = x3(t)
```

--------------------------------

### Define a discrete-time SIR model with @DDSmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/discrete_time

Uses the @DDSmodel macro to define a discrete-time SIR model using shift notation.

```julia
using StructuralIdentifiability

dds = @DDSmodel(
    S(t + 1) = S(t) - β * S(t) * I(t),
    I(t + 1) = I(t) + β * S(t) * I(t) - α * I(t),
    R(t + 1) = R(t) + α * I(t),
    y(t) = I(t)
)
```

--------------------------------

### assess_local_identifiability for ODE

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Signature for checking local identifiability/observability of specific functions in an ODE model. The type parameter can be :SE (single-experiment) or :ME (multi-experiment); for :ME, states are not allowed in funcs_to_check. Result is correct with probability at least prob_threshold.

```julia
assess_local_identifiability(ode::ODE{P}; funcs_to_check::Array{<: Any, 1}, prob_threshold::Float64=0.99, type=:SE, loglevel=Logging.Info) where P <: MPolyRingElem{Nemo.QQFieldElem}
```

--------------------------------

### Check Identifiability of Specific Functions

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiability

Uses the funcs_to_check parameter to investigate the identifiability of specific parameter combinations in the Goodwin oscillator model.

```julia
assess_identifiability(ode, funcs_to_check = [beta + delta, beta * delta])
```

```text
OrderedCollections.OrderedDict{Any, Symbol} with 2 entries:
  beta + delta => :globally
  beta*delta   => :globally
```

--------------------------------

### assess_local_identifiability(ode::ODE{P}; funcs_to_check, prob_threshold, type, loglevel)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Checks local identifiability/observability of specified functions in an ODE model. The result is correct with probability at least prob_threshold. Supports single-experiment (:SE) or multi-experiment (:ME) identifiability.

```APIDOC
## assess_local_identifiability(ode::ODE{P}; funcs_to_check, prob_threshold, type, loglevel)

### Description
Checks the local identifiability/observability of the functions in funcs_to_check. The result is correct with probability at least prob_threshold. Call this function if you have a specific collection of parameters of which you would like to check local identifiability.

### Method
Function call

### Endpoint
assess_local_identifiability(ode::ODE{P}; funcs_to_check::Array{<: Any, 1}, prob_threshold::Float64=0.99, type=:SE, loglevel=Logging.Info) where P <: MPolyRingElem{Nemo.QQFieldElem}

### Parameters
#### Input Arguments
- **ode** (ODE) - Required - The ODE model
- **funcs_to_check** (Array) - Required - List of functions to check local identifiability
- **prob_threshold** (Float) - Optional - Probability of correctness (default: 0.99)
- **type** (Symbol) - Optional - Can be :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). If :ME, states are not allowed to appear in funcs_to_check.
- **loglevel** (LogLevel) - Optional - Minimal level of log messages to display (default: Logging.Info)

### Response
#### Success Response
Returns the local identifiability/observability of the functions in funcs_to_check.

```

--------------------------------

### assess_local_identifiability(dds::DDS{P}; funcs_to_check, known_ic, prob_threshold, loglevel)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Checks local identifiability/observability of specified functions in a discrete dynamical system (DDS). The result is correct with probability at least prob_threshold. A list of quantities can be provided as known_ic for which initial conditions are assumed known and generic.

```APIDOC
## assess_local_identifiability(dds::DDS{P}; funcs_to_check, known_ic, prob_threshold, loglevel)

### Description
Checks the local identifiability/observability of the functions in funcs_to_check. The result is correct with probability at least prob_threshold. A list of quantities can be provided as known_ic for which the initial conditions can be assumed to be known and generic.

### Method
Function call

### Endpoint
assess_local_identifiability(dds::DDS{P}; funcs_to_check::Array{<: Any, 1}, known_ic, prob_threshold::Float64=0.99, loglevel=Logging.Info) where P <: MPolyRingElem{Nemo.QQFieldElem}

### Parameters
#### Input Arguments
- **dds** (DDS) - Required - The discrete dynamical system
- **funcs_to_check** (Array) - Required - List of functions to check local identifiability
- **known_ic** (list) - Optional - A list of quantities for which the initial conditions can be assumed to be known and generic
- **prob_threshold** (Float) - Optional - Probability of correctness (default: 0.99)
- **loglevel** (LogLevel) - Optional - Minimal level of log messages to display (default: Logging.Info)

### Response
#### Success Response
Returns the local identifiability/observability of the functions in funcs_to_check.

```

--------------------------------

### Find identifiable functions of parameters and states

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiable_functions

Calls `find_identifiable_functions` with `with_states = true` to also include state variables in the search. The output includes functions of states and parameters, such as `x3(t)`, `x1(t)*x2(t)`, and combinations involving `x1(t)` and `x2(t)`.

```julia
find_identifiable_functions(LLW1987, with_states = true)
```

```text
7-element Vector{AbstractAlgebra.Generic.FracFieldElem{Nemo.QQMPolyRingElem}}:
 x3(t)
 p1 + p3
 p2*p4
 p1*p3
 x1(t)*x2(t)
 x1(t)*p4 + x2(t)*p2
 x1(t)*p1*p4 + x2(t)*p2*p3
```

--------------------------------

### reduce_ode_mod_p

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Reduces an ODE defined over the rationals modulo a prime number, throwing an exception if the prime divides any denominator.

```APIDOC
## reduce_ode_mod_p(ode, p)

### Description
Reduces an ODE over QQ modulo a prime number p.

### Method
reduce_ode_mod_p

### Parameters
- **ode** (ODE) - Required - The ODE over QQ to reduce.
- **p** (Prime) - Required - The prime number to reduce modulo.

### Output
Returns the reduction of the ODE modulo p. Throws an exception if p divides one of the denominators.
```

--------------------------------

### set_parameter_values

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Substitutes given numeric values for specified parameters in an ODE, returning a new ODE with those parameters replaced.

```APIDOC
## set_parameter_values(ode, param_values)

### Description
Sets parameter values in an ODE by substituting given numeric values for the specified parameters.

### Method
set_parameter_values

### Parameters
- **ode** (ODE) - Required - The ODE in which to set parameter values.
- **param_values** (Dict) - Required - A dictionary mapping parameters to their numeric values.

### Output
Returns a new ODE with the specified parameters replaced by their given values.
```

--------------------------------

### find_identifiable_functions signature

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Signature for finding all identifiable functions of parameters/states in an ODE system. Options include with_states, simplify (with :standard, :weak, :strong, :absent), known_ic, prob_threshold (default 0.99), seed (default 42), and loglevel.

```julia
find_identifiable_functions(ode::ODE; options...)
```

--------------------------------

### Reparametrize the model

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/reparametrization

Perform global reparametrization and inspect the new variables.

```julia
reparam = reparametrize_global(ode)
@assert isempty(reparam[:implicit_relations]) # checking that the result is an ODE on the whole space, not on a manifold
reparam[:new_vars]
```

```text
Dict{Nemo.QQMPolyRingElem, AbstractAlgebra.Generic.FracFieldElem{Nemo.QQMPolyRingElem}} with 8 entries:
  y1(t) => y(t)
  a3    => (N*a)//b
  X1(t) => I(t)
  a2    => d
  a1    => g
  X2(t) => S(t)*a
  X3(t) => U(t)*a + I(t)*a
  X4(t) => E(t)*a
```

--------------------------------

### Assess local identifiability of a Lotka-Volterra model

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiability

Defines an ODE model using the @ODEmodel macro and assesses the local identifiability of its parameters and states.

```julia
using StructuralIdentifiability

ode = @ODEmodel(
    x1'(t) = a * x1(t) - b * x1(t) * x2(t) + u(t),
    x2'(t) = -c * x2(t) + d * x1(t) * x2(t),
    y(t) = x1(t)
)

assess_local_identifiability(ode)
```

```text
OrderedCollections.OrderedDict{Any, Bool} with 6 entries:
  x1(t) => 1
  x2(t) => 0
  a     => 1
  b     => 0
  c     => 1
  d     => 1
```

--------------------------------

### find_submodels

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/ode

Finds and returns all valid submodels of a given ODE system, each represented as an ODE object.

```APIDOC
## find_submodels(ode)

### Description
Calculates and returns all valid submodels of a given system of ODEs.

### Method
find_submodels

### Parameters
- **ode** (ODE) - Required - The ODE system to be studied.

### Output
Returns a list of submodels, each represented as an ODE object.

### Example
```julia
>ode = @ODEmodel(x1'(t) = x1(t)^2, 
                 x2'(t) = x1(t) * x2(t), 
                 y1(t) = x1(t), 
                 y2(t) = x2(t))
>find_submodels(ode)
    ODE{QQMPolyRingElem}[
        
        x1'(t) = a(t)*x2(t)^2 + x1(t)
        y1(t) = x1(t)
    ]
```
```

--------------------------------

### Assess model identifiability

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/reparametrization

Check the identifiability of the defined SEUIR model states and parameters.

```julia
assess_identifiability(ode)
```

```text
OrderedCollections.OrderedDict{Any, Symbol} with 9 entries:
  S(t) => :nonidentifiable
  E(t) => :nonidentifiable
  U(t) => :nonidentifiable
  I(t) => :globally
  N    => :nonidentifiable
  a    => :nonidentifiable
  b    => :nonidentifiable
  d    => :globally
  g    => :globally
```

--------------------------------

### Define ODE model with @ODEmodel macro

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/creating_ode

Defines a mutualism ODE model using the @ODEmodel macro from StructuralIdentifiability.jl. The second block shows the expanded form of the equations as printed by the macro.

```julia
using StructuralIdentifiability

ode = @ODEmodel(
    x1'(t) =
        r1 * x1(t) * (1 - c1 * x1(t)) + beta1 * x1(t) * x2(t) / (chi1 + x2(t)) + u(t),
    x2'(t) = r2 * x2(t) * (1 - c2 * x2(t)) + beta2 * x1(t) * x2(t) / (chi2 + x1(t)),
    y(t) = x1(t)
)
```

```text
x1'(t) = (-x1(t)^2*x2(t)*c1*r1 - x1(t)^2*c1*chi1*r1 + x1(t)*x2(t)*beta1 + x1(t)*x2(t)*r1 + x1(t)*chi1*r1 + x2(t)*u(t) + u(t)*chi1)//(x2(t) + chi1)
x2'(t) = (-x1(t)*x2(t)^2*c2*r2 + x1(t)*x2(t)*beta2 + x1(t)*x2(t)*r2 - x2(t)^2*c2*chi2*r2 + x2(t)*chi2*r2)//(x1(t) + chi2)
y(t) = x1(t)
```

--------------------------------

### assess_identifiability(ode; funcs_to_check, prob_threshold, loglevel)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/identifiability/identifiability

Assesses identifiability of a given ODE model. Returns a dictionary mapping functions to their identifiability properties (:nonidentifiable, :locally, :globally).

```APIDOC
## assess_identifiability(ode; funcs_to_check, prob_threshold, loglevel)

### Description
Assesses identifiability of a given ODE model. The result is guaranteed to be correct with probability at least prob_threshold. Returns an ordered dictionary from the functions to check to their identifiability properties (one of :nonidentifiable, :locally, :globally).

### Method
Function call

### Endpoint
assess_identifiability(ode; funcs_to_check = [], prob_threshold=0.99, loglevel=Logging.Info)

### Parameters
#### Input Arguments
- **ode** (ODE) - Required - The ODE model
- **funcs_to_check** (list) - Optional - List of functions to check identifiability for; if empty, all parameters and states are taken
- **known_ic** (list) - Optional - A list of functions whose initial conditions are assumed to be known; then the returned identifiable functions will be functions of parameters and initial conditions, not states (experimental)
- **prob_threshold** (Float) - Optional - Probability of correctness (default: 0.99)
- **loglevel** (LogLevel) - Optional - Minimal level of log messages to display (default: Logging.Info)

### Response
#### Success Response
Returns an ordered dictionary from the functions to check to their identifiability properties (one of :nonidentifiable, :locally, :globally).

```

--------------------------------

### massive_eval(polys, eval_dict)

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/utils/wronskian

Evaluates a list of polynomials at a point, assuming multiplications are relatively expensive. Precomputes all monomials and caches values of lower-degree monomials to compute higher-degree ones.

```APIDOC
## massive_eval(polys, eval_dict)

### Description
Evaluates a list of polynomials at a point, assuming multiplications are relatively expensive. Precomputes all monomials and caches values of lower-degree monomials to compute higher-degree ones.

### Method
Function call

### Endpoint
`StructuralIdentifiability.massive_eval(polys, eval_dict)`

### Parameters
- **polys** (list of polynomials) - Required - A list of polynomials to evaluate
- **eval_dict** (dictionary) - Required - Dictionary from variables to values; missing values are treated as zeroes

### Response
#### Success Response
- **list of values** - The values of the polynomials at the given point

#### Response Example
```
[value1, value2, ...]
```
```

--------------------------------

### Assess identifiability of ModelingToolkit model

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/creating_ode

Assesses identifiability for the ModelingToolkit-based model, passing the measured quantities. The output shows the identifiability results for states and parameters.

```julia
assess_identifiability(ode_mtk, measured_quantities = measured_quantities)
```

```text
OrderedCollections.OrderedDict{SymbolicUtils.BasicSymbolicImpl.var"typeof(BasicSymbolicImpl)"{SymbolicUtils.SymReal}, Symbol} with 10 entries:
  x1(t) => :globally
  x2(t) => :nonidentifiable
  beta1 => :globally
  chi1  => :nonidentifiable
  c1    => :globally
  r1    => :globally
  beta2 => :globally
  chi2  => :globally
  c2    => :nonidentifiable
  r2    => :globally
```

--------------------------------

### Define the LLW1987 ODE model with @ODEmodel

Source: https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/identifiable_functions

Defines the LLW1987 model as a system of ODEs with three state variables and one output. This model is used throughout the page to demonstrate identifiability analysis.

```julia
LLW1987 = @ODEmodel(
    x1'(t) = -p1 * x1(t) + p2 * u(t),
    x2'(t) = -p3 * x2(t) + p4 * u(t),
    x3'(t) = -(p1 + p3) * x3(t) + (p4 * x1(t) + p2 * x2(t)) * u(t),
    y1(t) = x3(t)
)
```
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