What is Mean Absolute Error (MAE)?

MAE is defined as the average of absolute difference between forecasted values and true values. Where y’ is forecasted value and y is the true value. n is total number of values in the test set. MAE tells us how big of an error we can expect from the forecast on an average2. The error values are in the original units of forecasted values and MAE = 0 indicates that there is no error in the forecasted values. The lower the MAE value, the better the model is; a value of zero means there is no error in the forecast. In other words, when comparing multiple models, the model with the lowest MAE is considered better. However, MAE does not indicate the relative size of the error and it becomes difficult to differentiate big errors from small errors. It can be used with other metrics (see Root Mean Square Error below) to determine if the errors are larger. Additionally, MAE can gloss over problems to do with low data volume; see the last two metrics in this article to address that issue.

What is Mean Absolute Percentage Error (MAPE)

MAPE is defined as the percentage of the average of absolute difference between forecasted values and true values, divided by true value. Where y’ is forecasted value and y is the true value. n is total number of values in the test set. Because of the in denominator, it works best with data without zeros and extreme values. If this value is extremely small or large, the MAPE value also takes an extreme value. The lower the MAPE, the better the model is. Keep in mind that MAPE works best with data without zeros and extreme values. Like MAE, MAPE also understates the impact of large but infrequent errors due to extreme values. To deal with this problem, Mean Square Error can be used. This metric can gloss over problems to do with low data volume; see the last two metrics in this article to deal with that situation.

What is Mean Squared Error (MSE)?

MSE is defined as the average of squares of the error3. It is also defined as the metric that assesses the quality of forecasting model or predictor. MSE also incorporates both the variance (the spread of predicted values from each other) and bias (the distance of predicted value from its true value). Where y’ is forecasted value and y is the true value. n is total number of values in the test set. MSE is almost always positive and values closer to zero are better. Because of the square term (as seen in the formula above), this metric penalizes large errors or outliers more compared to the small errors. The closer to zero MSE is, the better. While it solves the extreme value and zero problem of MAE and MAPE, it might be disadvantageous in some cases. This metric can overlook issues when dealing with low data volume; see Weighted Absolute Percentage Error and Weighted Mean Absolute Percentage Error to deal with that.

What is Root Mean Squared Error (RMSE)?

This metric is an extension of MSE and is defined as the square root of mean square error. Where y’ is forecasted value and y is the true value. n is total number of values in the test set. Like MSE, this metric also penalizes larger errors more. This metric is also always positive and lower values are better. One advantage of this calculation is that the RMSE value is in the same unit as the forecasted value. This makes it easier to understand compared to MSE. RMSE can also be compared to MAE to determine whether the forecast contains large but infrequent errors. The larger the difference between RMSE and MAE the more inconsistent the error size.2 This metric can gloss over problems to do with low data volume; see the last two metrics in this article to address that issue.

What is Normalized Root Mean Squared Error (NRMSE)?

NRMSE is an extension of RMSE and is calculated by normalizing RMSE. There are two popular methods for normalizing RMSE: using mean or using the range of the true values (difference of minimum and maximum values). Where ymax is the maximum true value and ymin is the minimum true value. NRMSE is often used to compare different datasets or forecasting models that have different scales (units and gross revenue, for example). The smaller the value is, the better the performance of the model. This metric can overlook issues when dealing with low data volume; see Weighted Absolute Percentage Error and Weighted Mean Absolute Percentage Error to address this.

What is Weighted Mean Absolute Percentage Error (WMAPE)?

WMAPE is an extension of WAPE and is defined as the weighted mean absolute percentage error. Where y’ is forecasted value and y is the true value. n is total number of values in the test set. The lower the value of WMAPE, the better the performance of the model. This metric is helpful for low volume data where each observation has different priority when evaluating forecasting models. Observations with higher priority have a higher weight value. The larger the error in high priority forecast values, the larger the WMAPE value is.

Error Metrics: How to Evaluate Your Forecasting Models

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