Optimization¶
The methods used for the actual optimization
Bayesian¶
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class
F_UNCLE.Opt.Bayesian.Bayesian(simulations, models, opt_keys=None, name=u'Bayesian', *args, **kwargs)¶ A class for performing Bayesian inference on a model given data
Attributes
- Attributes:
simulations(list): Each element is a tuple with the following elements
- A simulation
- Experimental results
model(PhysicsModel): The model under consideration sens_matrix(nump.ndarray): The (nxm) sensitivity matrix
- n model degrees of freedom
- m total experiment DOF
- [i,j] sensitivity of model response i to experiment DOF j
Options These can be set as keyword arguments in a call to self.__init__
Name Type Def Min Max Units Description outer_atol (float) 1E-6 0.0 1.0 Absolute tolerance on change in likelihood for outer loop convergence outer_rtol (float) 1E-4 0.0 1.0 Relative tolerance on change in likelihood for outer loop convergence maxiter (int) 6 1 100 Maximum iterations for convergence of the likelihood constrain (bool) True None None Flag to constrain the optimization precondition (bool) True None None Flag to scale the problem debug (bool) False None None Flag to print debug information verb (bool) True None None Flag to print stats during optimization Note
The options outer_atol and prior_weight are deprecated and should be used for debugging purposes only
Methods
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__call__(comm=None)¶ Determines the best candidate EOS function for the models
Parameters: comm (MPI.intracom) – An MPI communicator for sensitities Returns: length 3, elements are: - (Bayesian): A copy of self with the optimal models
- (list): is of solution history elements are:
- (np.ndarray) Log likelihood, (nx1) where n is number of iterations
- (np.ndarray) model dof history (nxm) where n is iterations and m is the model dofs
- (dict) sensitivity matrix of all experiments to
- the model, keys are simulation keys
- (dict) fisher information matrix for all experiments
- keys are simulation keys
Return type: (tuple)
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__init__(simulations, models, opt_keys=None, name=u'Bayesian', *args, **kwargs)¶ Instantiates the Bayesian analysis
Parameters: Keyword Arguments: name (str) – Name for the analysis.(‘Bayesian’)
Returns: None
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_check_inputs(models, simulations)¶ Checks that the values for model and simulation are valid
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_get_constraints()¶ Get the constraints on the model
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_get_model_pq()¶ Gets the quadratic optimization matrix contributions from the prior
Parameters: None – Returns: elements are - (np.ndarray): p, a nxn matrix where n is the model DOF
- (np.ndarray): q, a nx1 matrix where n is the model DOF
Return type: (tuple)
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_get_sens(initial_data=None, comm=None)¶ Gets the sensitivity of the simulated experiment to the EOS
The sensitivity matrix is the attribute self.sens_matrix which is set by this method
Note
This method is specific to the model type under consideration. This implementation is only for spline models of EOS
Parameters: None – Keyword Arguments: initial_data – The results of each simulation with the current best model. Each element in the list corresponds tho the output from a __call__ to each element in the self.simulations list Returns: None
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_get_sim_pq(initial_data, sens_matrix)¶ Gets the QP contributions from the model
Note
This method is specific to the model type under consideration. This implementation is only for spline models of EOS
Parameters: - initial_data (list) – A list of the initial results from the simulations in the same order as in the sim list
- sens_matrix (np.ndarray) – The sensitivity matrix
Returns: Elements are:
- (np.ndarray): P, a nxn matrix where n is the model DOF
- (np.ndarray): q, a nx1 matrix where n is the model DOF
Return type: (tuple)
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_local_opt(initial_data, sens_matrix)¶ Solves the quadratic problem for maximization of the log likelihood
Parameters: - initial_data (list) – The initial data corresponding to simulations from sims
- sens_matrix (np.array) – The sensitivity matrix about the model
Returns: The step direction for greatest improvement in log likelihood
Return type: (np.ndarray)
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_on_str()¶ Print method for Bayesian model called by Struct.__str__
Parameters: None – Returns: String describing the Bayesian object Return type: (str)
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static
fisher_decomposition(fisher_dct, simid, models, mkey, tol=0.001)¶ Calculate a spectral decomposition of the fisher information matrix
Parameters: Keyword Arguments: tol (float) – Eigenvalues less than tol are ignored
Returns: Elements are:
(list): Eigenvalues greater than tol
(np.ndarray): nxm array.
- n is number of eigenvalues greater than tol
- m is model dof
(np.ndarray): nxm array
- n is the number of eigenvalues greater than tol
- m is an arbitrary dimension of independent variable
(np.ndarray): vector of independent variable
Return type: (tuple)
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get_data()¶ Generates a set of data at each experimental data-point
For each pair of simulations and experiments, generates the the simulation response, aligning it with the experimental data
Parameters: None – Returns: List of lists for experiment comparison data - independent value
- dependent value of interest
- the summary data (3rd element) of sim
Return type: (list)
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get_hessian(initial_data=None, simid=None)¶ Gets the Hessian (matrix of second derivatives) of the simulated experiments to the EOS
\[\begin{split}H(f_i) = \begin{smallmatrix} \frac{\partial^2 f_i}{\partial \mu_1^2} & \frac{\partial^2 f_i}{\partial \mu_1 \partial \mu_2} & \ldots & \frac{\partial^2 f_i}{\partial \mu_1 \partial \mu_n}\\ \frac{\partial^2 f_i}{\partial \mu_2 \partial \mu_1} & \frac{\partial^2 f_i}{\partial \mu_2^2} & \ldots & \frac{\partial^2 f_i}{\partial \mu_2 \partial \mu_n}\\ \frac{\partial^2 f_i}{\partial \mu_n \partial \mu_1} & \frac{\partial^2 f_i}{\partial \mu_n \partial \mu_2} & \ldots & \frac{\partial^2 f_i}{\partial \mu_n^2} \end{smallmatrix}\end{split}\]\[H(f) = (H(f_1), H(f_2), \ldots , H(f_n))\]where
\[\begin{split}f \in \mathcal{R}^m \\ \mu \in \mathcal{R}^m\end{split}\]
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get_opt_dof()¶ Returns a vector of the model DOF for the optimization
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model_log_like()¶ Gets the log likelihood of the self.model given that model’s prior
Returns: Log likelihood of the model Return type: (float) \[\log(p(f|y))_{model} = -\frac{1}{2}(f - \mu_f)\Sigma_f^{-1}(f-\mu_f)\]
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plot_convergence(hist, axes=None, linestyles=[u'-k'], labels=[])¶ Parameters: hist (tuple) – Convergence history, elements 0. (list): MAP history 1. (list): DOF history
Keyword Arguments: - axes (plt.Axes) – The axes on which to plot the figure, if None, creates a new figure object on which to plot.
- linestyles (list) – Strings for the linestyles
- labels (list) – Strings for the labels
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plot_fisher_data(fisher_data, axes=None, fig=None, linestyles=[], labels=[])¶ Parameters: fisher_dat (tuple) – Data from the fisher_decomposition function see docscring for definition
Keyword Arguments: - axes (plt.Axes) – Ignored
- fig (plt.Figure) – A valid figure to plot on
- linestyles (list) – A list of valid linestyles Ignored
- labels (list) – A list of labels Ignored
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shape()¶ Gets the dimensions of the problem
Returns: The (n, m) dimensions of the problem - n is the total degrees of freedom of all the model responses
- m is the degrees of freedom of the model
Return type: (tuple)
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sim_log_like(initial_data)¶ Gets the log likelihood of the simulations given the data
Parameters: - model (PhysicsModel) – The model under investigation
- initial_data (list) – A list of the initial data for the simulations Each element in the list is the output from a __call__ to the corresponding element in the self.simulations list
Returns: Log likelihood of the simulation given the data
Return type: (float)
\[\log(p(f|y))_{model} = -\frac{1}{2} (y_k - \mu_k(f))\Sigma_k^{-1}(y_k-\mu_k(f))\]