diff --git a/doc/OnlineDocs/explanation/analysis/doe/abstraction.rst b/doc/OnlineDocs/explanation/analysis/doe/abstraction.rst new file mode 100644 index 00000000000..4aabe3a9f3a --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/doe/abstraction.rst @@ -0,0 +1,8 @@ +.. _abstractexp: + +Experiment Abstraction +====================== + +.. note:: + + Detailed descriptions and example code for experiment abstraction in Pyomo.DoE will be added in a future update. diff --git a/doc/OnlineDocs/explanation/analysis/doe/doe.rst b/doc/OnlineDocs/explanation/analysis/doe/doe.rst index 77fc8b1a08a..7cefb99e418 100644 --- a/doc/OnlineDocs/explanation/analysis/doe/doe.rst +++ b/doc/OnlineDocs/explanation/analysis/doe/doe.rst @@ -21,363 +21,15 @@ If you use Pyomo.DoE, please cite: "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. +Index of Pyomo.DoE documentation +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +.. toctree:: + :maxdepth: 2 + + overview.rst + guide.rst + abstraction.rst + objective.rst + multexp.rst + uncertainty.rst \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/doe/guide.rst b/doc/OnlineDocs/explanation/analysis/doe/guide.rst new file mode 100644 index 00000000000..cb14c08d6e0 --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/doe/guide.rst @@ -0,0 +1,230 @@ +.. _startguide: + +Quick Start Guide +================= + +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. + +This guide illustrates 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. diff --git a/doc/OnlineDocs/explanation/analysis/doe/multexp.rst b/doc/OnlineDocs/explanation/analysis/doe/multexp.rst new file mode 100644 index 00000000000..0c943b7e010 --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/doe/multexp.rst @@ -0,0 +1,8 @@ +.. _multexperiments: + +Multiple Experiments +==================== + +.. note:: + + Detailed descriptions and example code for simultaneous design of multiple experiments will be added in a future update. \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/doe/objective.rst b/doc/OnlineDocs/explanation/analysis/doe/objective.rst new file mode 100644 index 00000000000..28bd39972e9 --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/doe/objective.rst @@ -0,0 +1,8 @@ +.. _objectives: + +Objective Options +================= + +.. note:: + + Detailed descriptions and example code for the objective options in Pyomo.DoE will be added in a future update. \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/doe/overview.rst b/doc/OnlineDocs/explanation/analysis/doe/overview.rst new file mode 100644 index 00000000000..2c25f5d02fc --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/doe/overview.rst @@ -0,0 +1,131 @@ +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. \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/doe/uncertainty.rst b/doc/OnlineDocs/explanation/analysis/doe/uncertainty.rst new file mode 100644 index 00000000000..50b31dbe040 --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/doe/uncertainty.rst @@ -0,0 +1,10 @@ +.. _paramuncertainty: + +Parameter Uncertainty +===================== + +.. note:: + + Detailed descriptions and example code for experiment design under parameter uncertainty will be added in a + future update. + diff --git a/doc/OnlineDocs/explanation/analysis/parmest/covariance.rst b/doc/OnlineDocs/explanation/analysis/parmest/covariance.rst index b7142e0e103..971c411c358 100644 --- a/doc/OnlineDocs/explanation/analysis/parmest/covariance.rst +++ b/doc/OnlineDocs/explanation/analysis/parmest/covariance.rst @@ -1,14 +1,20 @@ .. _covariancesection: +Uncertainty Quantification +========================== +The goal of parameter estimation (see :ref:`driversection` Section) is to estimate unknown model parameters +from experimental data. Uncertainty quantification is required to ensure that the estimates of the parameters are +close to their true values. This parameter uncertainty can be computed using four methods in parmest: +covariance matrix, likelihood ratio test, bootstrapping, and leave-N-out. + Covariance Matrix Estimation -============================ +---------------------------- -The goal of parameter estimation (see :ref:`driversection` Section) is to estimate unknown model parameters -from experimental data. When the model parameters are estimated from the data, their accuracy is measured by -computing the covariance matrix. The diagonal of this covariance matrix contains the variance of the -estimated parameters which is used to calculate their uncertainty. Assuming Gaussian independent and identically -distributed measurement errors, the covariance matrix of the estimated parameters can be computed using the -following methods which have been implemented in parmest. +The uncertainty in estimated model parameters can be quantified by computing the covariance matrix. +The diagonal of this covariance matrix contains the variance of the estimated parameters which is used to +calculate their uncertainty. Assuming Gaussian independent and identically distributed measurement errors, +the covariance matrix of the estimated parameters can be computed using the following methods which have been +implemented in parmest. 1. Reduced Hessian Method @@ -138,4 +144,90 @@ e.g., >>> pest = parmest.Estimator(exp_list, obj_function="SSE") >>> obj_val, theta_val = pest.theta_est() >>> cov_method = "reduced_hessian" - >>> cov = pest.cov_est(method=cov_method) \ No newline at end of file + >>> cov = pest.cov_est(method=cov_method) + +Bootstrapping +------------- + +Bootstrapping is a non-intrusive method that uses resampling to approximate the variance of the parameter estimates. +By repeatedly fitting the model to resampled datasets, the parameter uncertainty (e.g., variance or covariance) +can be computed without relying on strong distributional assumptions. This method is summarized as follows: + +1. Step 0: Define the input data + + Given input data: :math:`[y_1, \dots, y_n]` + +2. Step 1: Generate :math:`B` artificial datasets through sampling + + Sample with replacement from the original data to create :math:`B` bootstrap datasets: + + .. math:: + \left\{y_1^{(1)}, \dots, y_n^{(1)} \right\}, \dots, \left\{y_1^{(B)}, \dots, y_n^{(B)} \right\} + +3. Step 2: Compute the estimator of the parameters + + Fit the model to each bootstrap dataset to obtain parameter estimates: + + .. math:: + \left\{ \hat{\theta}^{(1)}, \dots, \hat{\theta}^{(B)} \right\} + +4. Step 3: Compute the approximate variance of the parameter estimates + + The variability across bootstrap estimates approximates the estimator variance: + + .. math:: + \text{Var}(\hat{\theta}) = + \frac{1}{B} \sum_{j=1}^{B} \left(\hat{\theta}^{(j)}\right)^2 + - + \left( + \frac{1}{B} \sum_{j=1}^{B} \hat{\theta}^{(j)} + \right)^2 + +.. note:: + + The example code for this method will soon be provided. + +Likelihood Ratio Test +--------------------- + +The likelihood ratio test is a non-intrusive method that compares how well two parameter sets explain the +observed data: an unconstrained set and a constrained set defined by the null hypothesis. It is commonly used +to assess whether restricting parameters significantly degrades model fit. This method is summarized as follows: + +1. Step 1: State the hypothesis + + Null hypothesis: :math:`\theta \in \Theta_0` vs. Alternative hypothesis: :math:`\theta \notin \Theta_0` + +2. Step 2: Define the test statistic + + The test statistic is the ratio of the maximum likelihood under the full parameter space to that under the + constrained space: + + .. math:: + \lambda_n = + \frac{\sup_{\theta \in \Theta} L(\theta; y_1, \dots, y_n)}{ + \sup_{\theta \in \Theta_0} L(\theta; y_1, \dots, y_n)} + +3. Step 3: Define the decision rule + + Reject the null hypothesis if: + + .. math:: + \lambda_n > c_{\alpha} + + Equivalently, using maximum likelihood estimates: + + .. math:: + \lambda_n = + \frac{L(\hat{\theta}; y_1, \dots, y_n)}{L(\hat{\theta}_0; y_1, \dots, y_n)} > c_{\alpha} + +.. note:: + + The example code for this method will soon be provided. + +Leave-N-Out +----------- + +.. note:: + + Detailed descriptions and example code for this method will be added in a future update. \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/parmest/estimability.rst b/doc/OnlineDocs/explanation/analysis/parmest/estimability.rst new file mode 100644 index 00000000000..4cf40728689 --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/parmest/estimability.rst @@ -0,0 +1,55 @@ +.. _estimabilitysection: + +Estimability Analysis +===================== + +After estimating the model parameters with their associated uncertainty, as demonstrated in the +:ref:`driversection` and :ref:`covariancesection` Section, estimability analysis is required to identify +parameters that cannot be reliably estimated from the available data due to limitations in the mathematical +model structure. If such parameters are identified, the model may need to be reformulated, replaced with an +alternative structure, or augmented with additional prior information. In parmest, estimability analysis can +be performed using eigen-decomposition of the parameter covariance matrix, profile likelihood methods, or +multi-start initialization routines. + +Eigen-decomposition +------------------- + +The estimability of model parameters can be analyzed through eigen-decomposition of the covariance matrix +obtained from parameter estimation. This covariance matrix quantifies parameter uncertainty and captures both +parameter variances and correlations. Eigen-decomposition of this matrix identifies principal directions +in parameter space along which uncertainty is largest or smallest. These directions provide insight into +parameter identifiability and reveal combinations of parameters that are either structurally identifiable or +non-identifiable based on the underlying model formulation. + +.. note:: + + Detailed descriptions and example code for this method will be added in a future update. + +Profile Likelihood +------------------ + +Profile likelihood analysis evaluates parameter estimability by systematically varying one parameter while +re-optimizing the remaining parameters to maintain consistency with the model and observed data. This approach +is closely related to likelihood ratio–based uncertainty quantification and provides a robust characterization +of practical identifiability through the shape of the likelihood surface. In addition, it can reveal structural +non-identifiability when flat or unbounded profiles indicate parameter combinations that are not uniquely +determined by the model formulation, particularly in nonlinear systems. + +.. note:: + + Detailed descriptions and example code for this method will be added in a future update. + +Multi-start Initialization +-------------------------- + +Multi-start initialization assesses parameter estimability by exploring a range of initial guesses. Because +parameter estimation problems are often nonlinear and may exhibit multiple local minima, different initializations +can lead to different parameter estimates. By solving the estimation problem from multiple starting points, one can +evaluate the robustness of the solution and identify potential issues related to non-convexity or non-identifiability. +Consistent convergence to a unique solution across initializations suggests that the parameters are structurally +identifiable within the model formulation, whereas sensitivity to initialization indicates potential estimability +issues. + +.. note:: + + Detailed descriptions and example code for this method will be added in a future update. \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/parmest/index.rst b/doc/OnlineDocs/explanation/analysis/parmest/index.rst index 4616a2134d4..54e85a67071 100644 --- a/doc/OnlineDocs/explanation/analysis/parmest/index.rst +++ b/doc/OnlineDocs/explanation/analysis/parmest/index.rst @@ -27,4 +27,6 @@ Index of parmest documentation graphics.rst examples.rst parallel.rst + objective.rst + estimability.rst api.rst diff --git a/doc/OnlineDocs/explanation/analysis/parmest/objective.rst b/doc/OnlineDocs/explanation/analysis/parmest/objective.rst new file mode 100644 index 00000000000..ccce0eb8ac2 --- /dev/null +++ b/doc/OnlineDocs/explanation/analysis/parmest/objective.rst @@ -0,0 +1,8 @@ +.. _objectivesection: + +Objective Options +================= + +.. note:: + + Detailed descriptions and example code for the objective options in parmest will be added in a future update. \ No newline at end of file diff --git a/doc/OnlineDocs/explanation/analysis/parmest/overview.rst b/doc/OnlineDocs/explanation/analysis/parmest/overview.rst index 7f6f23c9140..7a3f7538fde 100644 --- a/doc/OnlineDocs/explanation/analysis/parmest/overview.rst +++ b/doc/OnlineDocs/explanation/analysis/parmest/overview.rst @@ -12,10 +12,12 @@ Functionality in parmest includes: * Model-based parameter estimation using experimental data * Covariance matrix estimation -* Bootstrap resampling for parameter estimation +* Bootstrap resampling for uncertainty quantification * Confidence regions based on single or multi-variate distributions -* Likelihood ratio +* Likelihood ratio test * Leave-N-out cross validation +* Regularization for objective function improvement +* Multi-start initialization optimization * Parallel processing Background