The image shows two mathematical formulas labeled $f_1$ and $f_2$. These formulas are related to statistical analysis, particularly involving a comparison of observed values $R_t$ and target or expected values $T_t$, over a series of $n$ observations.

Formula $f_1$:

$f_1 = \left\{ \frac{\sum_{t=1}^n |R_t - T_t|}{\sum_{t=1}^n R_t} \right\} \cdot 100$

Explanation:

• $R_t$: Represents the observed value at time $t$.
• $T_t$: Represents the target or expected value at time $t$.
• $n$: Total number of observations.

The formula $f_1$ calculates the mean absolute error (MAE) between the observed and target values as a percentage of the total observed values. Here’s how the calculation works:

1. Difference Calculation: For each time $t$, calculate the absolute difference $|R_t - T_t|$.
2. Summation: Sum up all these absolute differences across all $n$ observations.
3. Normalize: Divide the sum of absolute differences by the sum of all observed values $\sum_{t=1}^n R_t$.
4. Percentage: Multiply the result by 100 to express the error as a percentage.

Formula $f_2$:

$f_2 = 50 \cdot \log \left\{ 1 + \left( \frac{1}{n} \sum_{t=1}^n (R_t - T_t)^2 \right)^{0.5} \cdot 100 \right\}$

Explanation:

• This formula involves calculating the root mean square error (RMSE) between the observed and target values, and then applying a logarithmic transformation to it.

1. Square the Differences: For each $t$, calculate the squared difference $(R_t - T_t)^2$.
2. Mean of Squares: Compute the mean of these squared differences by dividing by the number of observations $n$.
3. Root Mean Square Error: Take the square root of the mean of squared differences, which gives the RMSE.
4. Scale and Logarithm: Multiply by 100 to scale the RMSE, then add 1 and take the logarithm (base 10) of the result, and finally, multiply by 50 to get the final value.

Summary:

• $f_1$ gives the mean absolute error as a percentage of the total observed values.
• $f_2$ gives a logarithmic measure of the root mean square error.

These formulas are likely used in assessing the accuracy of predictions in some context, with $f_1$ focusing on absolute deviations and $f_2$ emphasizing squared deviations and a logarithmic scale.

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Do you want more detailed steps on how to perform these calculations or have any other questions?

Here are five related questions to explore:

1. How does the Mean Absolute Error (MAE) differ from the Root Mean Square Error (RMSE)?
2. In what situations would you prefer to use $f_1$ over $f_2$?
3. What are the implications of the logarithmic transformation in $f_2$?
4. How do these formulas behave if the observed values $R_t$ are consistently close to the target values $T_t$?
5. Can these metrics be applied to other types of data beyond time-series?

Tip: When working with error metrics, always consider the context and the sensitivity of the error to outliers, as some metrics may amplify larger errors more than others.