# How should I understand smoothing in functional data analysis from a modelling perspective (specifically for temperature data)?

The specifics of this question are that I am looking at daily maximum and minimum temperature from the GHCND data set obtained from NOAA's API and I view the temperatures observed at days in a year as a single function where the input $$t$$ is day of the year and the output $$f(t)$$ is some local maximum or minimum temperature. When using the fda package in R, the first step is a smoothing step, representing the data as coefficients of a finite number basis of functions (I prefer the Fourier basis, to capture the cyclical nature of temperature). In the book accompanying fda, Functional Data Analysis with R and MATLAB (Ramsay et. al., 2009), this step not only smooths the data and reduces its dimensionality but also allows for, say, analyzing the dynamics of functional data by taking derivatives of the resulting (often very smooth) functions. These coefficients are estimated often via least-squares regression, at least in fda.

But I'm bothered because I feel like I should understand what, from a modelling perspective, the smoothing step does and what we just found, especially since the smooth curves are then treated as equivalent to the original data. The smoothing step does more than just interpolate the data; it seems to change the data. Ramsay et. al. even will compute confidence regions for smoothed curves, but in the case of temperature data I don't know what those confidence regions are trying to locate; after all, it seems the temperature measured at a site at a particular time is what it is, and I'm not convinced we're just trying to remove random error.

And yet smoothing the data and working with the smoothed data also seems to make good sense, especially since we can study the dynamics of smoothed data via derivatives.

So can someone explain how smoothing should be interpreted from a modelling perspective? For example, in temperature data, am I differentiating a short-term climate effect from "noisy" weather?

(I should also add that I know little about climate science; I'm a PhD student studying statistics and my advisor works a lot in functional data analysis. He wants me to look at this climate data set and start getting a sense of what questions are asked in this area that relate to functional data analysis.)