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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.

Learn how to calculate Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) in statistical analysis with detailed formulas and explanations. Explore the differences between MAE and RMSE, and understand the implications of logarithmic transformation in error metrics. Suitable for advanced learners and professionals.

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