Why is interpolation said to have no unique solution?

A sigmoidal curve is chosen because the physical mechanism of a wind turbine dictates that power output does not increase linearly with wind speed. A minimum wind speed is required to start the blades rotating, and at higher speeds, the generator reaches its maximum electrical output.

Conditions: The relationship between wind speed and power output is being modeled.; The physical constraints of the wind turbine (minimum cut-in speed and maximum rated power) are considered.

In the bivariate formulation, interpolation requires the fitted function to match the measured values exactly at every given data point, expressed as $z_i = F(x_i, y_i)$. Approximation relaxes this exact fitting by allowing a deviation or error term $e(x_i, y_i)$, expressed as $z_i = F(x_i, y_i) + e(x_i, y_i)$.

Conditions: Given m points $(x_i, y_i, z_i)$ with $i=1,m$.; Find a function $F(x,y)$.; Interpolation requires exact matching at the sample points.; Approximation allows an error term $e(x_i, y_i)$.

To compare different curve fitting approaches, you should use statistical metrics such as R-squared, Adjusted R-squared, Root Mean Square Error (RMSE), and Sum of Squared Errors (SSE). R-squared measures the proportion of variance explained by the model, while RMSE and SSE measure the average magnitude of the errors, with lower values indicating a better fit.

Conditions: Two or more curve fitting models are being evaluated.; The models are fitted to the same dataset.

In the MATLAB Curve Fitter app, you select the predictor variable for the x-axis and the response variable for the y-axis. For the wind turbine example, wind speed is selected as the independent variable (predictor) on the x-axis, and power output is selected as the dependent variable (response) on the y-axis.

Conditions: The data are available in MATLAB.; The user wants to visualize and fit a relationship between two specific variables.

You can force your curve fit to respect physical limits by formulating it as a constrained optimization problem using the Optimization Toolbox. Define the model coefficients as optimization variables and assign specific lower and upper bounds to each coefficient.

Conditions: Requires an optimization solver like lsqnonlin.; Objective function must be defined based on the model equation (e.g., minimizing sum of squared errors).; Physical or mathematical constraints on coefficients are known.