# 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 fitting`CategoryVariableName` — the names of categories or groups that you specified during fitting`CategoryValue` — 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 fitting`CategoryVariableName` — the names of categories or groups that you specified during fitting`CategoryValue` — the values of category variables specified by `CategoryVariableName`This 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.