Data weighting on semi log scale

[raul_l]

Hi

I have to perform a fitting procedure on a semi logarithmic scale but I'm not sure how to weigh the experimental data points. I'm studying the decay of the luminescence of a certain type of crystals and the function I'm using has the form

$$I = \frac{1}{\tau} e^{-\frac{t}{\tau}} \frac{-Li_{2}(-erf(\sqrt{\frac{t}{\tau}})}{erf(\sqrt{\frac{t}{\tau}})}$$

where $$Li_{2}$$ and erf are the dilogarithm and error functions. (for simplicity I've omitted some constants). Basically, it's more or less a pure exponential with some nonlinearities at the beginning.

My adviser said that I should try to weigh each data point with the inverse of its value (or the square of inverse). Otherwise I get imprecise values for some of the parameters that depend on the smaller values of the data points. But I'm not sure if this is the right way to do this. It just doesn't feel right.

I was wondering if there are any general rules/guidelines for choosing the correct weights on a semi logarithmic scale.

 

[Redbelly98]

Your advisor is correct. Alternatively, you could take the log of the measured values and do a fit to them, with equal weighting.

 

[Mene]

Hello,

Thank you for sharing your question and the details of your study. Weighting of data points on a semi logarithmic scale can be a bit tricky, but there are some general guidelines that can help guide your decision.

Firstly, it is important to understand the purpose of data weighting. In general, data weighting is used to give more importance to certain data points over others in a fitting procedure. This is often done to account for experimental uncertainties or to emphasize certain trends in the data.

On a semi logarithmic scale, the weight of a data point can be thought of as its relative contribution to the overall fit. Since the y-axis is logarithmic, the weight of a data point will be inversely proportional to its value on the y-axis. This means that data points with smaller values will have a higher weight and thus a greater influence on the fit.

Based on this understanding, your adviser's suggestion of using the inverse of the data point as the weight seems reasonable. This would give more weight to the smaller values, which may help to improve the precision of your fit parameters.

However, it is also important to consider the specific behavior of your data and the function you are fitting. In your case, you mentioned that there are nonlinearities at the beginning of the decay. This means that the data points at the beginning of the curve may not follow the same trend as the rest of the data. In this case, it may be more appropriate to manually adjust the weights of these data points to give them less influence on the fit.

In summary, there are no hard and fast rules for choosing the correct weights on a semi logarithmic scale. It ultimately depends on the specific characteristics of your data and the function you are fitting. However, considering the purpose of data weighting and the behavior of your data can help guide your decision. I would also recommend discussing your approach with your adviser and potentially consulting with other experts in your field for their insights. Best of luck with your study!