Choosing the Right Polynomial Order for Data Fitting in MATLAB

Which order polynomial would provide the smoothest fit?

In MATLAB, a 1001 order polynomial could theoretically provide the most accurate fit for the data. However, this may lead to overfitting which can affect the model's accuracy on new data, hence a balance must be struck between smoothness and model generalizability.

Final answer: Option D (1001) is correct.

Explanation:

In the context of MATLAB programming where x = rand(1001, 1), we are generating a vector of 1001 random values. The question pertains to creating a polynomial fit to this data using polynomial regression. When it comes to the smoothest fit, in theory, a polynomial of higher degree would fit the data points more closely, so a 1001 order polynomial could potentially provide the most accurate fit.

However, from a practical point of view, using such a high order polynomial for fit could lead to overfitting. Overfitting means that your model will work well with the provided data but may fail to predict new values accurately. Therefore, choosing the polynomial order often requires a compromise between the smoothness and the generalizability of the model.

It will be better to start with a low degree polynomial (like 1 or 10) and then increase it gradually while assessing the fit and the prediction accuracy. Using a method such as cross-validation can help determine the degree of polynomial that generalizes best to unseen data.