.. _pyomo.doe:
Pyomo.DoE
=========
**Pyomo.DoE** (Pyomo Design of Experiments) is a Python library for model-based design
of experiments using science-based models.
Pyomo.DoE was originally created by **Jialu Wang** and **Alexander W. Dowling** at the
University of Notre Dame as part of the `Carbon Capture Simulation for Industry Impact
(CCSI2) `_
project, funded through the U.S. Department Of Energy Office of Fossil Energy with
assistance from **John Siirola**, **Bethany Nicholson**, **Miranda Mundt**, and **Hailey Lynch**.
Significant improvements and extensions were contributed by **Dan Laky**, and
**Shuvashish Mondal** with funding from the
`Process Optimization & Modeling for Minerals Sustainability (PrOMMiS) `_ project
and the `University of Notre Dame `_.
If you use Pyomo.DoE, please cite:
[Wang and Dowling, 2022] Wang, Jialu, and Alexander W. Dowling.
"Pyomo.DOE: An open‐source package for model‐based design of experiments in Python."
AIChE Journal 68.12 (2022): e17813. `https://doi.org/10.1002/aic.17813`
Methodology Overview
---------------------
Model-based Design of Experiments (MBDoE) is a technique to maximize the information
gain from experiments by directly using science-based models with physically meaningful
parameters. It is one key component in the model calibration and uncertainty
quantification workflow shown below:
.. figure:: pyomo_workflow_new.png
:align: center
:scale: 90 %
The parameter estimation, uncertainty analysis, and MBDoE are
combined into an iterative framework to select, refine, and calibrate science-based
mathematical models with quantified uncertainty. Currently, Pyomo.DoE focuses on
increasing parameter precision.
Pyomo.DoE provides the exploratory analysis and MBDoE capabilities to the
Pyomo ecosystem. The user provides one Pyomo model, a set of parameter nominal values,
the allowable design spaces for design variables, and the assumed observation error model.
During exploratory analysis, Pyomo.DoE checks if the model parameters can be
inferred from the postulated measurements or preliminary data.
MBDoE then recommends optimized experimental conditions for collecting more data.
Parameter estimation packages such as :ref:`Parmest ` can perform
parameter estimation using the available data to infer values for parameters,
and facilitate an uncertainty analysis to approximate the parameter covariance matrix.
If the parameter uncertainties are sufficiently small, the workflow terminates
and returns the final model with quantified parametric uncertainty.
If not, MBDoE recommends optimized experimental conditions to generate new data
that will maximize information gain and eventually reduce parameter uncertainty.
Below is an overview of the type of optimization models Pyomo.DoE can accommodate:
* Pyomo.DoE is suitable for optimization models of **continuous** variables
* Pyomo.DoE can handle **equality constraints** defining state variables
* Pyomo.DoE supports (Partial) Differential-Algebraic Equations (PDAE) models via :ref:`Pyomo.DAE `
* Pyomo.DoE also supports models with only algebraic equations
The general form of a DAE problem that can be passed into Pyomo.DoE is shown below:
.. math::
:nowrap:
\[\begin{array}{l}
\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}}, \boldsymbol{\theta}) \\
\mathbf{g}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\theta})=\mathbf{0} \\
\mathbf{y} =\mathbf{h}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\theta}) \\
\mathbf{f}^{\mathbf{0}}\left(\dot{\mathbf{x}}\left(t_{0}\right), \mathbf{x}\left(t_{0}\right), \mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)=\mathbf{0} \\
\mathbf{g}^{\mathbf{0}}\left( \mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)=\mathbf{0}\\
\mathbf{y}^{\mathbf{0}}\left(t_{0}\right)=\mathbf{h}\left(\mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\theta}\right)
\end{array}\]
where:
* :math:`\boldsymbol{\theta} \in \mathbb{R}^{N_p}` are unknown model parameters.
* :math:`\mathbf{x} \subseteq \mathcal{X}` are dynamic state variables which characterize trajectory of the system, :math:`\mathcal{X} \in \mathbb{R}^{N_x \times N_t}`.
* :math:`\mathbf{z} \subseteq \mathcal{Z}` are algebraic state variables, :math:`\mathcal{Z} \in \mathbb{R}^{N_z \times N_t}`.
* :math:`\mathbf{u} \subseteq \mathcal{U}` are time-varying decision variables, :math:`\mathcal{U} \in \mathbb{R}^{N_u \times N_t}`.
* :math:`\overline{\mathbf{w}} \in \mathbb{R}^{N_w}` are time-invariant decision variables.
* :math:`\mathbf{y} \subseteq \mathcal{Y}` are measurement response variables, :math:`\mathcal{Y} \in \mathbb{R}^{N_r \times N_t}`.
* :math:`\mathbf{f}(\cdot)` are differential equations.
* :math:`\mathbf{g}(\cdot)` are algebraic equations.
* :math:`\mathbf{h}(\cdot)` are measurement functions.
* :math:`\mathbf{t} \in \mathbb{R}^{N_t \times 1}` is a union of all time sets.
.. note::
Parameters and design variables should be defined as Pyomo ``Var`` components
when building the model using the ``Experiment`` class so that users can use both
``Parmest`` and ``Pyomo.DoE`` seamlessly.
Based on the above notation, the form of the MBDoE problem addressed in Pyomo.DoE is shown below:
.. math::
:nowrap:
\begin{equation}
\begin{aligned}
\underset{\boldsymbol{\varphi}}{\max} \quad & \Psi (\mathbf{M}(\boldsymbol{\hat{\theta}}, \boldsymbol{\varphi})) \\
\text{s.t.} \quad & \mathbf{M}(\boldsymbol{\hat{\theta}}, \boldsymbol{\varphi}) = \sum_r^{N_r} \sum_{r'}^{N_r} \tilde{\sigma}_{(r,r')}\mathbf{Q}_r^\mathbf{T} \mathbf{Q}_{r'} + \mathbf{M}_0 \\
& \dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}}, \boldsymbol{\hat{\theta}}) \\
& \mathbf{g}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{y}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\hat{\theta}})=\mathbf{0} \\
& \mathbf{y} =\mathbf{h}(\mathbf{x}(t), \mathbf{z}(t), \mathbf{u}(t), \overline{\mathbf{w}},\boldsymbol{\hat{\theta}}) \\
& \mathbf{f}^{\mathbf{0}}\left(\dot{\mathbf{x}}\left(t_{0}\right), \mathbf{x}\left(t_{0}\right), \mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\hat{\theta}})\right)=\mathbf{0} \\
& \mathbf{g}^{\mathbf{0}}\left( \mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{y}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\hat{\theta}}\right)=\mathbf{0}\\
&\mathbf{y}^{\mathbf{0}}\left(t_{0}\right)=\mathbf{h}\left(\mathbf{x}\left(t_{0}\right),\mathbf{z}(t_0), \mathbf{u}\left(t_{0}\right), \overline{\mathbf{w}}, \boldsymbol{\hat{\theta}}\right)
\end{aligned}
\end{equation}
where:
* :math:`\boldsymbol{\varphi}` are design variables, which are manipulated to maximize the information content of experiments. It should consist of one or more of :math:`\mathbf{u}(t), \mathbf{y}^{\mathbf{0}}({t_0}),\overline{\mathbf{w}}`. With a proper model formulation, the timepoints for control or measurements :math:`\mathbf{t}` can also be degrees of freedom.
* :math:`\mathbf{M}` is the Fisher information matrix (FIM), approximated as the inverse of the covariance matrix of parameter estimates :math:`\boldsymbol{\hat{\theta}}`. A large FIM indicates more information contained in the experiment for parameter estimation.
* :math:`\mathbf{Q}` is the dynamic sensitivity matrix, containing the partial derivatives of :math:`\mathbf{y}` with respect to :math:`\boldsymbol{\theta}`.
* :math:`\Psi` is the scalar design criteria to measure the information content in the FIM.
* :math:`\mathbf{M}_0` is the sum of all the FIMs from previous experiments.
Pyomo.DoE provides five design criteria :math:`\Psi` to measure the information in the FIM.
The covariance matrix of parameter estimates is approximated as the inverse of the FIM,
i.e., :math:`\mathbf{V} \approx \mathbf{M}^{-1}`.
We can use the FIM or the covariance matrix to define the design criteria.
.. list-table:: Pyomo.DoE design criteria
:header-rows: 1
:class: tight-table
* - Design criterion
- Computation
- Geometrical meaning
* - A-optimality
- :math:`\text{trace}(\mathbf{V}) = \text{trace}(\mathbf{M}^{-1})`
- Minimizing this is equivalent to minimizing the enclosing box of the confidence ellipse
* - Pseudo A-optimality
- :math:`\text{trace}(\mathbf{M})`
- Maximizing this is equivalent to maximizing the dimensions of the enclosing box of the Fisher Information Matrix
* - D-optimality
- :math:`\det(\mathbf{M}) = 1/\det(\mathbf{V})`
- Maximizing this is equivalent to minimizing confidence-ellipsoid volume
* - E-optimality
- :math:`\lambda_{\min}(\mathbf{M}) = 1/\lambda_{\max}(\mathbf{V})`
- Maximizing this is equivalent to minimizing the longest axis of the confidence ellipse
* - Modified E-optimality
- :math:`\text{cond}(\mathbf{M}) = \text{cond}(\mathbf{V})`
- Minimizing this is equivalent to minimizing the ratio of the longest axis to the shortest axis of the confidence ellipse
.. note::
A confidence ellipse is a geometric representation of the uncertainty in parameter
estimates. It is derived from the covariance matrix :math:`\mathbf{V}`.
In order to solve problems of the above, Pyomo.DoE implements the 2-stage stochastic program. Please see Wang and Dowling (2022) for details.
Pyomo.DoE Required Inputs
-------------------------
To use Pyomo.DoE, a user must implement a subclass of the :ref:`Parmest ` ``Experiment`` class.
The subclass must have a ``get_labeled_model`` method which returns a Pyomo `ConcreteModel`
containing four Pyomo ``Suffix`` components identifying the parts of the model used in
MBDoE analysis. This is in line with the convention used in the parameter estimation tool,
:ref:`Parmest `. The four Pyomo ``Suffix`` components are:
* ``experiment_inputs`` - The experimental design decisions
* ``experiment_outputs`` - The values measured during the experiment
* ``measurement_error`` - The error associated with individual values measured during the experiment. It is passed as a standard deviation or square root of the diagonal elements of the observation error covariance matrix. Pyomo.DoE currently assumes that the observation errors are Gaussain and independent both in time and across measurements.
* ``unknown_parameters`` - Those parameters in the model that are estimated using the measured values during the experiment
An example of the subclassed ``Experiment`` object that builds and labels the model is shown in the next few sections.
Pyomo.DoE Usage Example
-----------------------
We illustrate the use of Pyomo.DoE using a reaction kinetics example (Wang and Dowling, 2022).
.. math::
:nowrap:
\begin{equation}
A \xrightarrow{k_1} B \xrightarrow{k_2} C
\end{equation}
The Arrhenius equations model the temperature dependence of the reaction rate coefficients :math:`k_1` and :math:`k_2`. Assuming a first-order reaction mechanism gives the reaction rate model shown below. Further, we assume only species A is fed to the reactor.
.. math::
:nowrap:
\begin{equation}
\begin{aligned}
k_1 & = A_1 e^{-\frac{E_1}{RT}} \\
k_2 & = A_2 e^{-\frac{E_2}{RT}} \\
\frac{d{C_A}}{dt} & = -k_1{C_A} \\
\frac{d{C_B}}{dt} & = k_1{C_A} - k_2{C_B} \\
C_{A0}& = C_A + C_B + C_C \\
C_B(t_0) & = 0 \\
C_C(t_0) & = 0 \\
\end{aligned}
\end{equation}
:math:`C_A(t), C_B(t), C_C(t)` are the time-varying concentrations of the species A, B, C, respectively.
:math:`k_1, k_2` are the rate constants for the two chemical reactions using an Arrhenius equation with activation energies :math:`E_1, E_2` and pre-exponential factors :math:`A_1, A_2`.
The goal of MBDoE is to optimize the experiment design variables :math:`\boldsymbol{\varphi} = (C_{A0}, T(t))`, where :math:`C_{A0},T(t)` are the initial concentration of species A and the time-varying reactor temperature, to maximize the precision of unknown model parameters :math:`\boldsymbol{\theta} = (A_1, E_1, A_2, E_2)` by measuring :math:`\mathbf{y}(t)=(C_A(t), C_B(t), C_C(t))`.
The observation errors are assumed to be independent both in time and across measurements with a constant standard deviation of 1 M for each species.
Step 0: Import Pyomo and the Pyomo.DoE module and create an ``Experiment`` class
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. note::
This example uses the data file ``result.json``, located in the Pyomo repository at:
``pyomo/contrib/doe/examples/result.json``, which contains the nominal parameter
values, and measurements for the reaction kinetics experiment.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_experiment.py
:start-after: # === Required imports ===
:end-before: # ========================
Subclass the :ref:`Parmest ` ``Experiment`` class to define the reaction
kinetics experiment and build the Pyomo ConcreteModel.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_experiment.py
:start-after: ========================
:end-before: End constructor definition
Step 1: Define the Pyomo process model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The process model for the reaction kinetics problem is shown below. Here, we build
the model without any data or discretization.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_experiment.py
:start-after: Create flexible model without data
:end-before: End equation definition
Step 2: Finalize the Pyomo process model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Here, we add data to the model and finalize the discretization using a new method to
the class. This step is required before the model can be labeled.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_experiment.py
:start-after: End equation definition
:end-before: End model finalization
Step 3: Label the information needed for DoE analysis
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
We label the four important groups as Pyomo Suffix components as mentioned before by
adding a ``label_experiment`` method. This method is required by Pyomo.DoE to identify
the design variables (experimental inputs), measurements, measurement errors, and
unknown parameters in the model.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_experiment.py
:start-after: End model finalization
:end-before: End model labeling
Step 4: Implement the ``get_labeled_model`` method
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This method utilizes the previous 3 steps and is used by `Pyomo.DoE` to build the model
to perform optimal experimental design.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_experiment.py
:start-after: End constructor definition
:end-before: Create flexible model without data
Step 5: Exploratory analysis (Enumeration)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
After creating the subclass of the ``Experiment`` class, exploratory analysis is
suggested to enumerate the design space to check if the problem is identifiable,
i.e., ensure that D-, E-optimality metrics are not small numbers near zero, and
Modified E-optimality is not a big number.
Additionally, it helps to initialize the model for the optimal experimental design step.
Pyomo.DoE can perform exploratory sensitivity analysis with the ``compute_FIM_full_factorial`` method.
The ``compute_FIM_full_factorial`` method generates a grid over the design space as specified by the user.
Each grid point represents an MBDoE problem solved using the ``compute_FIM`` method.
In this way, sensitivity of the FIM over the design space can be evaluated.
Pyomo.DoE supports plotting the results from the ``compute_FIM_full_factorial`` method
with the ``draw_factorial_figure`` method.
The following code defines the ``run_reactor_doe`` function. This function encapsulates
the workflow for both sensitivity analysis (Step 5) and optimal design (Step 6).
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_example.py
:language: python
:start-after: # === Required imports ===
:end-before: if __name__ == "__main__":
After defining the function, we will call it to perform the exploratory analysis and
the optimal experimental design.
.. literalinclude:: /../../pyomo/contrib/doe/examples/reactor_example.py
:language: python
:start-after: if __name__ == "__main__":
:dedent: 4
A design exploration for the initial concentration and temperature as experimental
design variables with 9 values for each, produces the the five figures for
five optimality criteria using the ``compute_FIM_full_factorial`` and
``draw_factorial_figure`` methods as shown below:
|plot1| |plot2|
|plot3| |plot4|
|plot5|
.. |plot1| image:: example_reactor_compute_FIM_D_opt.png
:width: 48 %
.. |plot2| image:: example_reactor_compute_FIM_A_opt.png
:width: 48 %
.. |plot3| image:: example_reactor_compute_FIM_pseudo_A_opt.png
:width: 48 %
.. |plot4| image:: example_reactor_compute_FIM_E_opt.png
:width: 48 %
.. |plot5| image:: example_reactor_compute_FIM_ME_opt.png
:width: 48 %
The heatmaps show the values of the objective functions, a.k.a. the
experimental information content, in the design space. Horizontal
and vertical axes are the two experimental design variables, while
the color of each grid shows the experimental information content.
For example, the D-optimality (upper left subplot) heatmap figure shows that the
most informative region is around :math:`C_{A0}=5.0` M, :math:`T=500.0` K with
a :math:`\log_{10}` determinant of FIM being around 19,
while the least informative region is around :math:`C_{A0}=1.0` M, :math:`T=300.0` K,
with a :math:`\log_{10}` determinant of FIM being around -5. For D-, Pseudo A-, and
E-optimality we want to maximize the objective function, while for A- and Modified
E-optimality we want to minimize the objective function.
In this sensitivity analysis plot (heatmap), we only varied the initial
concentration and the initial temperature, while the temperature at other time
points is fixed at 300 K.
.. math::
:nowrap:
\[
T(t) = \begin{cases}
T_0, & t \le 0.125 \\
300\ \text{K}, & t > 0.125
\end{cases}
\]
If :math:`T_0 = 300\ \text{K}`, the reaction is conducted under strictly isothermal
conditions. Because the temperature is constant, the sensitivities of the species
concentrations with respect to the Arrhenius parameters (:math:`A_i` and :math:`E_i`)
become linearly dependent. This high correlation means the effects of the
pre-exponential factor and the activation energy cannot be uniquely distinguished
from the measurements. Consequently, the Fisher Information Matrix (FIM) becomes
ill-conditioned, resulting in a near-zero determinant and a very large condition number.
To break this correlation and make the parameters identifiable, introducing a time-
varying temperature profile (for example, a temperature step or a ramp) is required.
As shown in the heatmap, when the initial temperature :math:`T_0` differs from the
subsequent 300 K baseline, such a temperature change breaks the linear dependence,
yielding a well-conditioned FIM and identifiable parameters.
Step 6: Performing an optimal experimental design
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In Step 5, we defined the ``run_reactor_doe`` function. This function constructs
the DoE object and performs the exploratory sensitivity analysis. The way the function
is defined, it also proceeds immediately to the optimal experimental design step
(applying ``run_doe`` on the ``DesignOfExperiments`` object).
We can initialize the model with the result we obtained from the exploratory
analysis (optimal point from the heatmaps) to help the optimal design step to speed
up convergence. However, implementation of this initialization is not shown here.
After applying ``run_doe`` on the ``DesignOfExperiments`` object,
the optimal design is an initial concentration of 5.0 mol/L and
an initial temperature of 494 K with all other temperatures being 300 K.
The corresponding :math:`\log_{10}` determinant of the FIM is 19.32.