Summary:

The Universal Temperature Index is a measure of thermal comfort or discomfort perceived by humans, and is estimated by a environmental model from the measured values of air temperature, radiation, humidity etc. The model is complex to run, and so polynomial approximations for a quick, albeit not totally accurate, estimations have been developed. The standard polynomial approximation incurs in errors that are deemed too large. The present study presents another approximation method, based on orthogonal polynomial regression that seems to provide more accurate results.

Recommendation: The manuscript is well written and the study seems to have not technical flaws. For that standpoint I have very few comments. However, I do have a more general question on the motivation of the study, which I think the authors should address or justify more thoroughly

Main point

1) The manuscript mentions another alternative method, namely interpolation from an available look-up table that contains about 100 thousand values. This is also the approach recommended by Bröde (2021a). The manuscript argues that the storage of 100 thousand values makes the calculation cumbersome, but I clearly disagree. This storage would amount to roughly 1 MB of data, which is a very small space. Intuitively, I would argue that an interpolation of that table can produce very accurate values with a simple spline or linear algorithm. So the question arises as what would be the advantages of the algorithm presented in this manuscript relative to the look-up table interpolation.

I am not arguing that the study is not valuable, as it presents a possible way of producing more accurate estimation of the index, but the reader would ask themselves if it really worth the effort.

Bröde (2021a) argues that " This chapter provides hints and guidelines on how to handle these issues, and especially encourages the application of the hardly used look-up table approach, which will help avoiding many, if not all concerns related to UTCI calculation via the regression polynomial"

Minor points

2) The labels in Figure 1 are too small. This also the case to a lesser degree in other Figures. Figure 3 is ok, so I would recommend to homogenize the font size in all figures.

3) In table 2, the reader has to infer which is the train loss and the test loss. It seems that the train loss is the upper number, but this could be indicated more explicitly. It seems that the train loss numbers require to be wrapped by a []

The article presents a new approximation for the Universal Thermal Climate Index by applying sparse regression with orthogonal polynomials to the Fiala thermo-physiological model. The proposed approach improves predictive accuracy and numerical stability over the existing standards (particularly in extrapolation), while maintaining comparable computational efficiency.

The manuscript is very well written and well illustrated .

I understand that the techniques (sparse model discovery) used for the regression are standard and no novelty is presented in that front. But I would add at least the kind of equations that these methods are aiming at. This will help interpreting the results (e.g. Table 1 or Figure 3, Why for a given polynomial degree the number of parameters change?).

Is the proposed function a linear combination of Legendre polynomials? Are Ta, va, Tr−Ta and rH its input variables?

Please present the shape of the polynomial basis expansions that you are fitting.



Minor comments

- In the introduction, when it is explained that the water vapor is not included, I would add that the relative humidity is included to account for its effect.



- "Training is conducted on only 20% of the available data, while performance is assessed on the remaining 80%"

Are these sets taken randomly?

Is this needed for a better fitting?

Is the number of points a limitation of the regression method?