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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.)

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What is temperature from a sensor? Does it give a true reading in the sun or in the shade? If in the shade how much shade, and what happens it the sun reflects off of a window into that shade where the sensor is? What happens if the air temperature is a mixture of different temperatures and the wind is blowing? From these questions it should be obvious that an instantaneous temperature reading from a single sensor is, in effect, a mini-micro climate, so the question then becomes what type of temperature reading does one take as desirable? Is it the temperature at one foot off the ground? At 6 feet? At 60 feet?

Essentially, the problem becomes one of defining how one wants to extract a representative temperature value. As a passing cloud can lower the temperature during the day and increase it at night, there is no absolute solution to the 'what is my local temperature' problem unless we first define what 'local' means. This is a bit like having it rain on one side of the street and not the other, and what local means might be a variable rather than a fixed concept.

Having said that, when one fits a curve in Fourier space, one can use a filter to eliminate high frequency data, and if one does so, the values are then smoothed. The question then becomes now much of the lost high frequency data is noise, and how much is signal, and that in turn depends on how many sensors are being used in which localities and whether or not the localities are being actively temperature mapped or passively assigned into regions, whether there is any vertical dimension to the localities, and probably other factors.

In sum, without defining exactly what the question is, there is no particular answer, and in this answer my intent is to stimulate the formulation of certain bounds upon the question so that an answer becomes more tractable.

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