Calculation of RMSE and $R^2$ to compare five numerical methods for determining Weibull parameters

Question

I have calculated Weibull parameters namely, shape parameter and scale parameter using five numerical methods: graphical, moment, maximum likelihood, empirical and energy pattern factor methods. Since I need to compare these methods to find out best method, I need to calculate RMSE and $R^2$ for each method.

But even if I know the formulas for RMSE and $R^2$, I am not being able calculate as I am confused and a bit unfamiliar with the terms used in those formulas. Please refer the following research journal and I want to clearly understand about 'Prediction performance of Weibull distribution model' section of the journal.

My queries regarding 'statistical tests' are:

1. How do you calculate 'frequency of weibull' or sometimes it says 'predicted data' ?
2. To calculate RMSE and $R^2$, do we use frequency of the data ( observed and predicted) or actual data (e.g wind speed, temperature)?

Answer 1

Let's decipher formulas in the paper:

$$RMSE = \left[ \frac{1}{N}\sum^N_{i=1} (y_i - x_i)^2 \right]^{1/2}$$

this one is pretty clear: RMSE is a square root of averaged square difference between $y_i$ and $x_i$ where $y_i$ is an observed frequency and $x_i$ is a frequency predicted by the model.

The formula for $R^2$ they provide is more complicated:

$$R^2 = \frac{ \sum^N_{i=1} (y_i - z_i)^2 - \sum^N_{i=1} (y_i - x_i)^2 }{ \sum^N_{i=1} (y_i - z_i)^2 } $$

It is ambiguous since authors describe its parameters as "Where: $y_i$ is the actual data, $x_i$ is the predicted data using the Weibull distribution, $z$ is the predicted data using the Weibull distribution, $N$ is the number of observations", so $x$ and $z$ seem to mean the same thing and it would not make sens. However, they quote other paper they wrote that describes $z$ as an mid value of $y$.