Evaluating forecast accuracy is crucial in time series analysis. It helps determine the reliability of forecasting models and compare different methods. By calculating measures like Mean Absolute Error, Root Mean Squared Error, and Mean Absolute Percentage Error, analysts can assess model performance.

These accuracy measures have unique strengths and limitations. MAE is easy to interpret, RMSE emphasizes larger errors, and MAPE allows for scale-independent comparisons. Choosing the right measure depends on the specific forecasting problem, data characteristics, and decision-maker preferences.

Evaluating Forecast Accuracy

Importance of forecast accuracy evaluation

• Determines reliability and effectiveness of forecasting models helps select the most appropriate model for a given time series (e.g., ARIMA, exponential smoothing)
• Compares performance of different forecasting methods identifies the best approach for a specific dataset (e.g., univariate vs. multivariate models)
• Identifies areas for improvement in the forecasting process reveals potential issues such as overfitting or underfitting
• Enables informed decision-making based on the forecasted values supports strategic planning and resource allocation (e.g., inventory management, budgeting)
• Monitors performance of the chosen model over time detects changes in the underlying patterns or relationships in the data
• Allows adjusting model parameters or exploring alternative methods when necessary ensures the model remains accurate and relevant as new data becomes available

Mean Absolute Error calculation

• Quantifies average absolute difference between forecasted and actual values provides a measure of the average magnitude of the errors
• Formula: $MAE = \frac{1}{n} \sum_{t=1}^{n} |y_t - \hat{y}_t|$
  • $n$: number of observations
  • $y_t$: actual value at time $t$
  • $\hat{y}_t$: forecasted value at time $t$
• Expressed in the same units as the original data facilitates interpretation and communication of the results (e.g., dollars, units sold)
• Lower MAE indicates better forecast accuracy smaller average absolute difference between forecasted and actual values
• Less sensitive to outliers compared to RMSE reduces the impact of extreme errors on the overall accuracy measure

Root Mean Squared Error computation

• Quantifies average squared difference between forecasted and actual values emphasizes larger errors due to the squared term
• Formula: $RMSE = \sqrt{\frac{1}{n} \sum_{t=1}^{n} (y_t - \hat{y}_t)^2}$
  • $n$: number of observations
  • $y_t$: actual value at time $t$
  • $\hat{y}_t$: forecasted value at time $t$
• Expressed in the same units as the original data allows for direct comparison with the scale of the data
• Lower RMSE indicates better forecast accuracy smaller average squared difference between forecasted and actual values
• More sensitive to outliers compared to MAE gives more weight to large errors, which may be desirable in some applications (e.g., financial risk management)

Mean Absolute Percentage Error interpretation

• Expresses average absolute error as a percentage of the actual values provides a scale-independent measure of forecast accuracy
• Formula: $MAPE = \frac{100%}{n} \sum_{t=1}^{n} |\frac{y_t - \hat{y}_t}{y_t}|$
  • $n$: number of observations
  • $y_t$: actual value at time $t$
  • $\hat{y}_t$: forecasted value at time $t$
• Allows comparison of forecast accuracy across different datasets or time series useful when dealing with data on different scales (e.g., sales of different products)
• Lower MAPE indicates better forecast accuracy smaller average percentage error between forecasted and actual values
• Can be undefined or misleading when actual values are close to or equal to zero requires caution when interpreting MAPE for time series with small or zero values

Comparison of accuracy measures

• MAE strengths:
  • Easy to interpret due to being expressed in the same units as the original data
  • Less sensitive to outliers compared to RMSE
• MAE limitations:
  • Does not provide information about the direction of the errors (over- or under-prediction)
  • May not be suitable for comparing forecast accuracy across different scales
• RMSE strengths:
  • Provides a quadratic loss function, which is differentiable and useful for optimization
  • Sensitive to larger errors, making it useful when the cost of large errors is higher
• RMSE limitations:
  • More sensitive to outliers compared to MAE
  • The squared term makes the interpretation less intuitive compared to MAE
• MAPE strengths:
  • Scale-independent, allowing for comparison of forecast accuracy across different datasets or time series
  • Expresses errors as percentages, which can be more intuitive for some users
• MAPE limitations:
  • Can be undefined or misleading when the actual values are close to or equal to zero
  • May not be suitable for time series with small or zero values
• The choice of the appropriate accuracy measure depends on the specific requirements of the forecasting problem, the nature of the data, and the preferences of the decision-makers