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Objective functions define how the discrepancy between model predictions and experimental data is quantified during parameter fitting.

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Cost Function Formulations

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Residual Form (Array)

The residual vector represents the difference between model predictions and data: r(x)=ymodel(x)−ydatar(x) = y_{\text{model}}(x) - y_{\text{data}} where xx represents the parameter vector.

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Canonical Form (Scalar)

The canonical cost function aggregates the residuals into a single scalar value: Φ(x)∝r(x)⊤r(x)=∑iNri(x)2\Phi(x) \propto r(x)^\top r(x) = \sum_i^N r_i(x)^2 This sum-of-squares formulation is fundamental to least-squares optimization.

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Common Cost Functions

The basic sum of squared residuals:ΦSSE(x)=∑iNri2=r⊤r\Phi_{\text{SSE}}(x) = \sum_i^N r_i^2 = r^\top rThis can be represented in both array and scalar forms, making it compatible with all optimization algorithms.

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Maximum Likelihood Estimation (MLE)

Under the assumption of independent, identically distributed Gaussian errors, the MLE cost function is: rMLE(x)=ymodel−ydatar_{\text{MLE}}(x) = y_{\text{model}} - y_{\text{data}} ΦMLE(x)=∑iNri2=r⊤r\Phi_{\text{MLE}}(x) = \sum_i^N r_i^2 = r^\top r This is equivalent to the sum squared error formulation, providing a probabilistic interpretation of least-squares fitting.
For most optimization algorithms, the default sum-squared-error formulation provides the best compatibility and performance. Use MSE or RMSE when you need interpretable, scale-independent metrics.
To pair an objective with a cost function in a data fit, see Pipelines → Data Fitting → Objective Functions.