1Blurb:: 2Response type suitable for calibration or least squares 3 4Description:: 5Responses for a calibration study are specified using \c 6calibration_terms and optional keywords for weighting/scaling, data, 7and constraints. In general when calibrating, Dakota automatically 8tunes parameters \f$ \theta \f$ to minimize discrepancies or residuals 9between the model and the data: 10 11\f[ R_{i} = y^{Model}_i(\theta) - y^{Data}_{i}. \f] 12 13Note that the problem specification affects what must be returned to 14Dakota in the \ref interface-analysis_drivers-fork-results_file : 15 16\li If calibration data <em>is not specified</em>, then each of the 17 calibration terms returned to Dakota through the \ref interface is a 18 residual \f$ R_{i} \f$ to be driven toward zero. 19 20\li If calibration data <em>is specified</em>, then each of the 21 calibration terms returned to Dakota must be a response \f$ 22 y^{Model}_i(\theta) \f$, which Dakota will difference with the data 23 in the specified data file. 24 25<b> Constraints </b> 26 27(See general problem formulation at \ref 28responses-objective_functions.) The keywords \ref 29responses-calibration_terms-nonlinear_inequality_constraints and \ref 30responses-calibration_terms-nonlinear_equality_constraints specify the 31number of nonlinear inequality constraints \em g, and nonlinear 32equality constraints \em h, respectively. When interfacing to 33external applications, the responses must be returned to %Dakota in 34this order in the \ref interface-analysis_drivers-fork-results_file : 35<ol> <li>calibration terms</li> <li>nonlinear inequality 36constraints</li> <li>nonlinear equality constraints</li> </ol> 37 38An optimization problem's linear constraints are provided to the 39solver at startup only and do not need to be included in the data 40returned on every function evaluation. Linear constraints are 41therefore specified in the \ref variables block through the \ref 42variables-linear_inequality_constraint_matrix \f$A_i\f$ and \ref 43variables-linear_equality_constraint_matrix \f$A_e\f$. 44 45Lower and upper bounds on the design variables \em x are also 46specified in the \ref variables block. 47 48<b> Problem Transformations</b> 49 50Weighting or scaling calibration terms is often appropriate to account 51for measurement error or to condition the problem for easier solution. 52Weighting or scaling transformations are applied in the following 53order: 54 55<ol> 56<li> When present, observation error variance \f$ \sigma_i \f$ or full 57 covariance \f$ \Sigma\f$, optionally specified through \c 58 experiment_variance_type, is applied to residuals first: 59 60 \f[ R^{(1)}_i = \frac{R_{i}}{\sigma_{i}} = \frac{y^{Model}_i(\theta) - 61 y^{Data}_{i}}{\sigma_{i}} \textrm{, or} \f] 62 63 \f[ 64 R^{(1)} = \Sigma^{-1/2} R = \Sigma^{-1/2} \left(y^{Model}(\theta) - 65 y^{Data}\right), \f] 66 resulting in the typical variance-weighted least squares formulation 67 \f[ \textrm{min}_\theta \; R(\theta)^T \Sigma^{-1} R(\theta) \f] 68</li> 69<li> Any active scaling transformations are applied next, e.g., for 70 characteristic value scaling: 71 72 \f[ R^{(2)}_i = \frac{R^{(1)}_i}{s_i} \f] 73</li> 74<li> Finally the optional weights are applied in a way that preserves 75 backward compatibility: 76 77 \f[ R^{(3)}_i = \sqrt{w_i}{R^{(2)}_i} \f] 78 79 so the ultimate least squares formulation, e.g., in a scaled and 80 weighted case would be 81 82 \f[ f = \sum_{i=1}^{n} w_i \left( \frac{y^{Model}_i - 83 y^{Data}_i}{s_i} \right)^2 \f] 84</li> 85</ol> 86 87<em>Note that specifying observation error variance and weights are mutually 88exclusive in a calibration problem.</em> 89 90Topics:: 91Examples:: 92Theory:: 93 94%Dakota calibration terms are typically used to solve problems of 95parameter estimation, system identification, and model 96calibration/inversion. Local least squares calibration problems are 97most efficiently solved using special-purpose least squares solvers 98such as Gauss-Newton or Levenberg-Marquardt; however, they may also be 99solved using any general-purpose optimization algorithm in %Dakota. 100While Dakota can solve these problems with either least squares or 101optimization algorithms, the response data sets to be returned from 102the simulator are different when using \ref 103responses-objective_functions versus \ref responses-calibration_terms. 104 105Least squares calibration involves a set of residual 106functions, whereas optimization involves a single objective function 107(sum of the squares of the residuals), i.e., \f[f = \sum_{i=1}^{n} 108R_i^2 = \sum_{i=1}^{n} \left(y^{Model}_i(\theta) - y^{Data}_{i} \right)^2 \f] 109where \e f is the objective function and the set of \f$R_i\f$ 110are the residual functions, most commonly defined as the difference between a model response and data. Therefore, function values and derivative 111data in the least squares case involve the values and derivatives of 112the residual functions, whereas the optimization case involves values 113and derivatives of the sum of squares objective function. This means that 114in the least squares calibration case, the user must return each of 115\c n residuals 116separately as a separate calibration term. Switching 117between the two approaches sometimes requires different simulation 118interfaces capable of returning the different granularity of response 119data required, although %Dakota supports automatic recasting of 120residuals into a sum of squares for presentation to an optimization 121method. Typically, the user must compute the difference between the 122model results and the observations when computing the residuals. 123However, the user has the option of specifying the observational data 124(e.g. from physical experiments or other sources) in a file. 125 126Faq:: 127See_Also:: responses-objective_functions, responses-response_functions 128