When fitting a linear model with polynomial basis functions, which of the following has no impact on the model complexity?

Question

BlackTom AI · Accepted Answer

The question asks about what does not affect the model complexity when fitting a linear model with polynomial basis functions.

First, consider what determines complexity in this setting. In a linear model using polynomial basis functions, the expressiveness or capacity of the model is governed by the number of basis functions (which is tied to the degree of the polynomial and the input dimensionality) and any regularization or sparsity constraints. In other words, complexity is primarily about how many parameters there are and how flexible the basis is, not about the particular numeric values observed in the training data.

Now, evaluate the (implicit) options. If one option states that the values of the input variables have no impact on complexity, that aligns with the idea that merely changing the data points (the specific x values) does not change how many basis functions you included or how many parameters you estimate. It may affect the fit, conditioning, or numerical stability, but not the model’s capacity or the number of parameters.

By contrast, options that would claim complexity is affected by factors like the magnitude of input values, the ordering of data, or the particular samples used would be incorrect in the strict sense of model capacity. The polynomial basis size is determined by the degree and interaction terms, not by the particular values the inputs take.

Therefore, the statement that 'the values of the input variables' have no impact on model complexity is consistent with the theoretical view that complexity is a function of the chosen basis (degree, interactions) and regularization, rather than the observed data values themselves.

In summary, the core concept is that complexity is about the structure and number of parameters, which are set before or independently of the specific observed x values in the training set. The data values influence the fitted coefficients and numerical behavior, but not the number of basis functions or the capacity of the model.

FIT5201 - MUM S1 2025 Quiz 1 2025 S1

When fitting a linear model with polynomial basis functions, which of the following has no impact on the model complexity?

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