Error

whitson⁺ facilitates regression on parameters to minimize the sum of squares of weighted residuals in the context of observed data and corresponding predictions. The objective function in is defined as:

$$\begin{equation} \chi^2 = \sum_{i=1}^{N} (r_i w_i)^2 \end{equation}$$

Here, $r_i$ represents the RMS residual error between observed measurement $i$ and its corresponding prediction, while $w_i$ is the user-assigned weighting factor for that residual. Ideally, these weighting factors should be inversely proportional to the standard deviations of the residuals. Minimizing $\chi^2$ in provides the maximum likelihood estimation of the model parameters, assuming independent and normally distributed residuals.

The default weighting factors are 1, while they can also be set to 0 (no importance), and 10 (high importance). Reported is a root-mean-square (RMS) residual defined as:

$$\begin{equation} \bar{r} = \sqrt{\frac{\chi^2}{\sum_{i=1}^{N} (w_i)^2}} \end{equation}$$

This metric is related to $\chi^2$ but is more easily interpreted. The residuals are calculated as relative percentages using the formula:

$$\begin{equation} r_i = \frac{100(y_{i,\text{hist}} - y_{i,\text{sim}})}{y_{i,\text{ref}}} \end{equation}$$

where $y_{i,\text{hist}}$ is the historical production value, $y_{i,\text{sim}}$ is the simulated value, and $y_{i,\text{ref}}$ is the reference value for observation $i$. The reference value is the nonzero weighted value in the series with the largest magnitude.