Absolute error is the magnitude of the difference between a measured or predicted value and its reference value. Taking the absolute value removes the sign, so overestimates and underestimates both contribute positive errors. It measures closeness to the reference, not measurement repeatability by itself.

Absolute error and Mean Absolute Error explained

How to Calculate the Absolute Error

The formula for absolute error is:

\[Absolute Error = |Observed Value – Actual Value| \]

or

\[ \Delta x = |x_i – x| \]

where:

  • \( \Delta x \) is the absolute error,
  • \( x_i \) is the actual value,
  • \( x \) is the measured value.

The use of the absolute value ensures that the error magnitude is accurately reflected, regardless of its direction. As an example, let’s say the actual temperature for a particular day is 82°F, and the predicted temperature generated by the model is 86°F. The absolute error would be:

\[ | 82- 86 | = 4\]

What Is Absolute Accuracy Error?

The signed error is \(e=x_e-x_t\), where \(x_e\) is the experimental measurement and \(x_t\) is the reference value. The absolute error is \(|e|=|x_e-x_t|\). The absolute-value bars are essential when reporting an error magnitude.

The Mean Absolute Error (MAE)

The mean absolute error (MAE) represents the average of all absolute errors in a set of measurements​​​​. MAE is a useful metric for evaluating the accuracy of predictions or forecasts, particularly in cases where the data has 1-dimensional characteristics. It provides a clear indication of the average magnitude of error and offers a straightforward way to compare the performance of different models or data categories.

The mean absolute error can then be calculated as the average of all absolute errors, using the formula:

\[ MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i| \]

Here, \(n\) is the number of paired observations, \(y_i\) is the actual or reference value for observation i, and \(\hat{y}_i\) is its prediction. For repeated measurements of one fixed reference value, all the \(y_i\) may be the same.

Application in Different Fields, Comparison with Other Error Measures

MAE finds applications in various domains, such as time series analysis, where it serves as a common measure of forecast error, and in remote sensing, where it helps in understanding both quantity and allocation disagreements​​​​.

While similar to other error metrics like Mean Squared Error (MSE), MAE stands out for its simplicity and direct interpretability. Unlike MSE, which squares the differences, potentially exaggerating large errors, MAE provides a more balanced and linear view of the error magnitudes​​.

MAE Practical Examples

Consider a scenario involving multiple temperature measurements. The MAE in this case would be the average of the absolute differences between each recorded temperature and the true temperature, offering a clear picture of the overall accuracy of the measurements.

\[ \text{Actual Values: } [ 10, 15, 12, 18, 20 ] \] \[ \text{Predicted Values: } [ 12, 15, 10, 20, 18 ] \]

That means:

\[ i=1 … | 10 – 12 | = 2 \] \[ i=2 … | 15 – 15 | = 0 \] \[ i=3 … | 12 – 10 | = 2 \] \[ i=4 … | 18 – 20 | = 2 \] \[ i=5 … | 20 – 18 | = 2 \]

To calculate the Mean Absolute Error (MAE), you now sum up all the absolute errors (2 + 0 + 2 + 2 +2) and then divide by the number of observations (5).

\[ MAE = \frac{2 + 0 + 2 + 2 + 2 }{5} = \frac{8}{5} = 1.6 \]

The MAE in this case is 1.6. This value represents the average magnitude of the errors between the actual and predicted values, without considering their direction. In other words, on average, the predictions deviate from the actual values by 1.6 units.

This is a measure of the accuracy of the predictions: the lower the MAE, the more accurate the predictions. An MAE of 1.6 suggests that the predictions are fairly close to the actual values, with an average error of 1.6 units per prediction.