# goodnessOfFit

Goodness of fit between test and reference data

## Syntax

fit = goodnessOfFit(x,xref,cost_func)

## Description

fit = goodnessOfFit(x,xref,cost_func) returns the goodness of fit between the data, x, and the reference, xref using a cost function specified by cost_func.

## Input Arguments

 x Test data. x is an Ns-by-N matrix, where Ns is the number of samples and N is the number of channels. x can also be a cell array of multiple test data sets. x must not contain any NaN or Inf values. xref Reference data. xref must be of the same size as x. xref can also be a cell array of multiple reference sets. In this case, each individual reference set must be of the same size as the corresponding test data set. xref must not contain any NaN or Inf values. cost_func Cost function to determine goodness of fit. cost_func must be one of the following strings: 'MSE' — Mean square error:$fit=\frac{{‖x-xref‖}^{2}}{Ns}$where, Ns is the number of samples, and ‖ indicates the 2-norm of a vector. fit is a scalar value.'NRMSE' — Normalized root mean square error:$fit\left(i\right)=1-\frac{‖xref\left(:,i\right)-x\left(:,i\right)‖}{‖xref\left(:,i\right)-mean\left(xref\left(:,i\right)\right)‖}$where, ‖ indicates the 2-norm of a vector. fit is a row vector of length N and i = 1,...,N, where N is the number of channels.NRMSE costs vary between -Inf (bad fit) to 1 (perfect fit). If the cost function is equal to zero, then x is no better than a straight line at matching xref.'NMSE' — Normalized mean square error:$fit\left(i\right)=1-{‖\frac{xref\left(:,i\right)-x\left(:,i\right)}{xref\left(:,i\right)-mean\left(xref\left(:,i\right)\right)}‖}^{2}$where, ‖ indicates the 2-norm of a vector. fit is a row vector of length N and i = 1,...,N, where N is the number of channels.NMSE costs vary between -Inf (bad fit) to 1 (perfect fit). If the cost function is equal to zero, then x is no better than a straight line at matching xref.

## Output Arguments

 fit Goodness of fit between test and reference data. For a single test data set and reference pair, fit is returned as a: Scalar if cost_func is MSE.Row vector of length N if cost_func is NRMSE or NMSE. N is the number of channels. If x and/or xref are cell arrays, then fit is an array containing the goodness of fit values for each test data and reference pair.

## Examples

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### Calculate Goodness of Fit of Between Estimated and Measured Data

Obtain the measured output.

yref = z1.y;

z1 is an iddata object containing measured input/output data. z1.y is the measured output.

Obtain the estimated output.

sys = tfest(z1,2);
y_sim = sim(sys,z1(:,[],:));

sys is a second-order transfer function estimated using the measured input/output data. y is the output estimated using sys and the measured input.

Calculate the goodness of the fit between the measured and estimated outputs.

cost_func = 'NRMSE';
y = y_sim.y;
fit = goodnessOfFit(y,yref,cost_func);

The goodness of fit is calculated using the normalized root mean square error as the cost function.

Alternatively, you can use compare to calculate the goodness of fit:

opt = compareOptions('InitialCondition','z');
compare(z1,sys,opt);