1Blurb::
2(Experimental) Post-calibration calculation of model discrepancy correction
3
4Description::
5The goal of parameter calibration is to minimize the difference between
6experimental data, \f$d(x)\f$, and model observations, \f$M(\theta, x)\f$,
7where \f$\theta\f$ are the model parameters and \f$x\f$ is a configuration
8variable, such as temperature or pressure. However, it is not uncommon that, at
9the conclusion of parameter calibration, the agreement between experimental
10data and the calibrated model is not "close enough." This is often due to
11model form or structural error. In this case, the canonical equation
12
13\f[
14d (x) = M (\theta, x) + \varepsilon
15\f]
16
17is replaced by one that also includes model discrepancy \f$\delta(x)\f$,
18
19\f[
20d (x) = M (\theta, x) + \delta(x) + \varepsilon.
21\f]
22
23In the Dakota implementation, the calculation of \f$\delta(x)\f$ is performed
24after the model parameters \f$\theta\f$ are calibrated. For each observable
25\f$d_i\f$, the discrepancy
26
27\f[
28\delta_i(x_j) = d_i(x_j) - M_i(\theta^*, x_j)
29\f]
30
31is calculated for each value \f$x_j\f$ of the configuration variable, where
32\f$\theta^*\f$ is the optimal parameter value obtained during the
33calibration. For scalar responses, the model discrepancy is only a function of
34the configuration variables, and there is one discrepancy regression model for
35each observable \f$d_{i}\f$.
36This set of discrepancy
37models may be specified to be either Gaussian process or polynomial regression
38models of constant, linear, or quadratic order, and each model is fit to the
39calculated discrepancy values. See the
40\ref method-bayes_calibration-model_discrepancy-discrepancy_type
41command for more details regarding these options.
42For field responses, the model discrepancy is
43a function of the configuration variables as well as the independent field
44coordinates (such as time or space), and there is one discrepancy regression
45model for each field.
46In this case, the discrepancy models are Gaussian process models. The
47calculation of model discrepancy has not been tested for cases in which
48responses are mixed scalar and field responses.
49
50
51
52The user
53may then specify new or "prediction" configurations at which the corrected
54model \f$M(\theta^*, x_{new}) + \delta(x_{new})\f$ should be calculated, using
55one of the \c num_prediction_configs, \c prediction_configs, or
56\c import_prediction_configs keywords. If none of these keywords is specified,
57the number of prediction configurations is set to 10 for scalar responses.
58The corresponding prediction variances are also calculated, according to
59
60\f[
61\Sigma_{total}(x) = \Sigma_{\delta}(x) + \sigma^2_{exp}I.
62\f]
63
64Here, \f$\Sigma_{\delta}(x)\f$ is the (co)variance calculated from the
65Gaussian process or polynomial regression model, and \f$\sigma^2_{exp}\f$ is
66the maximum variance, taken over all configuration variables for that
67observation. In the case of field responses, the default prediction
68configurations are set equal to the input configurations, and the variance
69information contains only the variance calculated from the Gaussian process
70correction model. Further details can be found in the Dakota User's and Theory
71Manuals.
72
73<b> Usage Tips </b>
74
75For field responses, the keyword \ref
76responses-calibration_terms-field_calibration_terms-read_field_coordinates
77<i>must</i> be specified when computing the model discrepancy. See \ref
78responses-calibration_terms-field_calibration_terms for more information
79regarding options for calibration with field responses.
80
81<b> Expected Output </b>
82
83The resulting values of
84\f$\delta(x_{new})\f$ will be exported to the file specified using
85\c export_discrepancy_file or to the default file
86\c dakota_discrepancy_tabular.dat. The values of the corrected model at the
87specified prediction configurations will be exported to the file specified
88using \c export_corrected_model_file or to the default file
89\c dakota_corrected_model_tabular.dat, and the corresponding prediction
90variances will be exported to \c dakota_discrepancy_variance_tabular.dat
91or the file specified with \c export_corrected_variance_file.
92
93Examples::
94Extensive examples can be found in the Dakota User's and Theory Manuals.
95The input files below illustrate the options available to \c model_discrepancy
96
97\verbatim
98model_discrepancy
99  discrepancy_type  gaussian_process
100  num_prediction_configs  11
101  export_discrepancy_file  "discrepancy_values.txt"
102  export_corrected_model_file  "corrected_model.txt"
103  export_corrected_variance_file  "prediction_variance.txt"
104\endverbatim
105
106\verbatim
107model_discrepancy
108  discrepancy_type  polynomial
109    correction_order  constant
110  prediction_configs  1  1.5  2  2.5  3
111\endverbatim
112
113\verbatim
114model_discrepancy
115  discrepancy_type  polynomial
116    correction_order  linear
117  import_prediction_configs "prediction_configs.txt"
118\endverbatim
119
120
121