LeastSquaresResults

Results object containing estimation results from least-squares regression

Description

The LeastSquaresResults contains estimation results from fitting a SimBiology^® model to data using sbiofit with any supported algorithm.

Creation

Use the sbiofit function to create a LeastSquaresResults object that is a superclass of two results objects: NLINResults object and OptimResults object.

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.

Properties

GroupName — Name of the group associated with the results

categorical | empty array

Name of the group associated with the results, specified as a categorical. If the 'Pooled' name-value pair argument was set to true when you ran sbiofit, then GroupName is returned as an empty array or [].

Beta — Table of estimated parameters

table

Table of estimated parameters, specified as a table. The jth row of the table represents the jth estimated parameter β_j. 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

table

Table of estimated parameters, specified as a table. The jth row of the table represents the jth estimated parameter β_j. 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 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 sets of parameter values that are identified for each individual or group.

J — Jacobian matrix of the model

array

Jacobian matrix of the model, specified as an array. The Jacobian matrix with respect to an estimated parameter is

$$J(i,j,k)={\frac{\partial y_k}{\partial {\beta }_j}|}_{t_i}$$

where t_i is the ith time point, β_j is the jth estimated parameter in the transformed space, and y_k is the kth response in the group of data.

COVB — Estimated covariance matrix for Beta

matrix

Estimated covariance matrix for Beta, specified as a matrix. This matrix is calculated as: COVB = inv(J'*J)*MSE.

CovarianceMatrix — Estimated covariance matrix for ParameterEstimates

matrix

Estimated covariance matrix for ParameterEstimates, specified as a matrix. This matrix 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

matrix

Residuals matrix, specified as a matrix. R_ij is the residual for the ith time point and the jth response in the group of data.

LogLikelihood — Maximized loglikelihood for the fitted model

scalar

Maximized loglikelihood for the fitted model, specified as a scalar.

AIC — Akaike Information Criterion (AIC)

scalar

Akaike Information Criterion (AIC), specified as a scalar. The AIC is calculated as AIC = 2*(-LogLikelihood + P), where P is the number of parameters.

BIC — Bayes Information Criterion (BIC)

scalar

Bayes Information Criterion (BIC), specified as a scalar. The BIC is 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

scalar

Degrees of freedom for error (DFE), specified as a scalar. The DFE is calculated as DFE = N-P, where N is the number of observations and P is the number of parameters.

MSE — Mean squared error

scalar

Mean squared error, specified as a scalar.

SSE — Sum of squared (weighted) errors or residuals

scalar

Sum of squared (weighted) errors or residuals, specified as a scalar.

Weights — Matrix of weights

matrix

Matrix of weights, specified as a matrix with one column per response and one row per observation.

Data — Data used for fitting

groupedData object

Data used for fitting, specified as a groupedData object.

In most cases, this Data property contains a copy of groupedData specified as the input data in the sbiofit call or the Data property of a fitproblem object. One exception is that the Data property of unpooled fit results objects contain only the subset of data for the individual group used for fitting.

EstimatedParameterNames — Estimated parameter names

cell array of character vectors

Estimated parameter names, specified as a cell array of character vectors.

ErrorModelInfo — Error models and estimated error model parameters

table

Error models and estimated error model parameters, specified as a table.

• The table 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': $$y=f+b\left|f\right|e$$

• 'combined': $$y=f+(a+b\left|f\right|)e$$

• 'exponential': $$y=f\ast \exp (ae)$$

EstimationFunction — Name of the estimation function

character vector

Name of the estimation function, specified as a character vector.

DependentFiles — File names to include for deployment

cell array of character vectors

File names to include for deployment, specified as a cell array of character vectors.

Note

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

Object Functions

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

Version History

Introduced in R2014a