# LeastSquaresResults object

Results object containing estimation results from least-squares regression

## Description

The LeastSquaresResults object is a superclass of two results objects: NLINResults object and OptimResults object. These objects contain estimation results from fitting a SimBiology® model to data using sbiofit with any supported algorithm.

If sbiofit uses the nlinfit estimation algorithm, the results object is the NLINResults object. If sbiofit uses any other supporting algorithm, then the results object is an OptimResults object. See the sbiofit function for the list of supported algorithms.

## Method Summary

 boxplot(LeastSquaresResults,OptimResults,NLINResults) Create box plot showing the variation of estimated SimBiology model parameters fitted(LeastSquaresResults,OptimResults,NLINResults) Return simulation results of SimBiology model fitted using least-squares regression plot(LeastSquaresResults,OptimResults,NLINResults) Compare simulation results to the training data, creating a time-course subplot for each group plotActualVersusPredicted(LeastSquaresResults,OptimResults,NLINResults) Compare predictions to actual data, creating a subplot for each response plotResidualDistribution(LeastSquaresResults,OptimResults,NLINResults) Plot the distribution of the residuals plotResiduals(LeastSquaresResults,OptimResults,NLINResults) Plot residuals for each response, using time, group, or prediction as x-axis predict(LeastSquaresResults,OptimResults,NLINResults) Simulate and evaluate fitted SimBiology model random(LeastSquaresResults,OptimResults,NLINResults) Simulate SimBiology model, adding variations by sampling error model summary(LeastSquaresResults,OptimResults,NLINResults) Plot a summary figure that contains estimated values and estimation statistics

## Properties

 GroupName Categorical variable representing the name of the group associated with the results, or [] if the 'Pooled' name-value pair argument was set to true when you ran sbiofit. Beta Table of estimated parameters where the jth row represents the jth estimated parameter βj. It contains transformed values of parameter estimates if any parameter transform is specified.Standard errors of these parameter estimates (StandardError) are calculated as: sqrt(diag(COVB)). It can also contain the following variables:Bounds — the values of transformed parameter bounds that you specified during fittingCategoryVariableName — the names of categories or groups that you specified during fittingCategoryValue — the values of category variables specified by CategoryVariableName This table contains one row per distinct parameter value. ParameterEstimates Table of estimated parameters where the jth row represents the jth estimated parameter βj. This table contains untransformed values of parameter estimates.Standard errors of these parameter estimates (StandardError) are calculated as: sqrt(diag(CovarianceMatrix)).It can also contain the following variables:Bounds — the values of parameter bounds that you specified during fittingCategoryVariableName — the names of categories or groups that you specified during fittingCategoryValue — the values of category variables specified by CategoryVariableNameThis table contains sets of parameter values that are identified for each individual or group. J Jacobian matrix of the model, with respect to an estimated parameter, that is, $J\left(i,j,k\right)={\frac{\partial {y}_{k}}{\partial {\beta }_{j}}|}_{{t}_{i}}$ where ti is the ith time point, βj is the jth estimated parameter in the transformed space, and yk is the kth response in the group of data. COVB Estimated covariance matrix for Beta, which is calculated as: COVB = inv(J'*J)*MSE. CovarianceMatrix Estimated covariance matrix for ParameterEstimates, which is calculated as: CovarianceMatrix = T'*COVB*T, where T = diag(JInvT(Beta)). JInvT(Beta) returns a Jacobian matrix of Beta which is inverse transformed accordingly if you specified any transform to estimated parameters.For instance, suppose you specified the log-transform for an estimated parameter x when you ran sbiofit. The inverse transform is: InvT = exp(x), and its Jacobian is: JInvT = exp(x) since the derivative of exp is also exp. R Residuals matrix where Rij is the residual for the ith time point and the jth response in the group of data. LogLikelihood Maximized loglikelihood for the fitted model. AIC Akaike Information Criterion (AIC), calculated as AIC = 2*(-LogLikelihood + P), where P is the number of parameters. BIC Bayes Information Criterion (BIC), calculated as BIC = -2*LogLikelihood + P*log(N), where N is the number of observations, and P is the number of parameters. DFE Degrees of freedom for error, calculated as DFE = N-P, where N is the number of observations and P is the number of parameters. MSE Mean squared error. SSE Sum of squared (weighted) errors or residuals. Weights Matrix of weights with one column per response and one row per observation. EstimatedParameterNames Cell array of character vectors specifying estimated parameter names. ErrorModelInfo Table describing the error models and estimated error model parameters. It has one row per error model.The ErrorModelInfo.Properties.RowsNames property identifies which responses the row applies to.The table contains three variables: ErrorModel, a, and b. The ErrorModel variable is categorical. The variables a and b can be NaN when they do not apply to a particular error model. There are four built-in error models. Each model defines the error using a standard mean-zero and unit-variance (Gaussian) variable e, the function value f, and one or two parameters a and b. In SimBiology, the function f represents simulation results from a SimBiology model. 'constant': $y=f+ae$'proportional': $y=f+b|f|e$'combined': $y=f+\left(a+b|f|\right)e$'exponential': $y=f\ast \mathrm{exp}\left(ae\right)$ EstimationFunction Name of the estimation function. DependentFiles File names to include for deployment.

### Note

Loglikelihood, AIC, and BIC properties are empty for LeastSquaresResults objects that were obtained before R2016a.

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