1) If the Quantile method were the best approach for all situations, it would be universally used. Clearly, it is not. Every method comes with its own set of assumptions, range of applicability, strengths, and limitations.

2) I understand that the Quantile method does not assume an underlying statistical distribution, unlike least squares regression, which relies on specific assumptions and is better suited for certain scenarios. In our case, the Quantile method is generally more appropriate because it handles large, non-Gaussian deviations from the trend effectively.

3) However, the issue is that we are not dealing with purely statistical deviations. The data also exhibits distinct, time-dependent behaviors. For example, even with limited data, we can observe that peaks decay exponentially over time.

4) The Quantile method works by dividing the data using a specific scheme. It assigns more weight to data below a given quantile (in terms of deviations from the model) and less weight to data above it.

5) When the Quantile method interpolates the highest quantiles, it essentially finds a compromise between modeling the decay of the peaks and capturing the general power law trend, which dominates due to smaller deviations. Nevertheless, the exponential decay of the peaks is partially accounted for.

6) The outcome is a set of power laws with varying slopes. While this result is mathematically correct given how the Quantile algorithm operates, it poses significant challenges in terms of interpretation.

7) This kind of data is far from the typical examples found in textbooks, and as such, it demands careful examination and thoughtful analysis.

I’d like to hear your interpretation of the different power laws produced by the Quantile regression. I value your perspective and am keen to understand how you make sense of these results.