Вивчайте Evaluating Forecast Accuracy | Implementing ARIMA for Forecasting

Evaluating the accuracy of your ARIMA model's forecasts is essential for understanding its performance and making decisions based on its predictions. Three widely used metrics for this purpose are Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE). Each metric provides a different perspective on forecast errors and can influence how you interpret your model's results.

Mean Absolute Error (MAE) measures the average magnitude of the errors in a set of forecasts, without considering their direction. It is calculated as the mean of the absolute differences between the predicted and actual values. The formula for MAE is:

\text{MAE} = \frac{1}{n} \sum_{i=1}^{n} \left| y_i - \hat{y}_i \right|

where $y_i$ are the actual values, $\hat{y}_i$ are the predicted values, and $n$ is the number of predictions. MAE is easy to interpret and gives equal weight to all errors, making it suitable when all deviations are equally important.

Root Mean Squared Error (RMSE) is similar to MAE but gives more weight to larger errors by squaring the differences before averaging and then taking the square root. The formula for RMSE is:

\text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 }

RMSE is sensitive to outliers and penalizes large errors more than MAE. This makes it useful when large errors are particularly undesirable in your application.

Mean Absolute Percentage Error (MAPE) expresses forecast accuracy as a percentage, making it scale-independent and easy to interpret across different datasets. The formula for MAPE is:

\text{MAPE} = \frac{100\%}{n} \sum_{i=1}^{n} \left| \frac{y_i - \hat{y}_i}{y_i} \right|

MAPE can be especially helpful for communicating forecast accuracy to non-technical audiences. However, it can be problematic when actual values are very close to zero, leading to extremely large or undefined percentage errors.

Each metric has its strengths and weaknesses. MAE is straightforward and robust to outliers, RMSE emphasizes large errors, and MAPE provides relative error in percentage terms. The choice of metric should align with the specific goals and characteristics of your forecasting scenario.

Note

Choosing the right metric depends on your business context. If large errors are especially costly, RMSE may be most appropriate. If you need a simple, interpretable measure, MAE is often preferred. When comparing models across different scales, MAPE's percentage-based approach can be advantageous—just be cautious with series that include values near zero.


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import numpy as np

# Example actual and predicted values
actual = np.array([100, 120, 130, 115, 140])
predicted = np.array([110, 118, 128, 120, 135])

# Calculate MAE
mae = np.mean(np.abs(actual - predicted))

# Calculate RMSE
rmse = np.sqrt(np.mean((actual - predicted) ** 2))

# Calculate MAPE (avoiding division by zero)
mape = np.mean(np.abs((actual - predicted) / actual)) * 100

print(f"MAE: {mae:.2f}")
print(f"RMSE: {rmse:.2f}")
print(f"MAPE: {mape:.2f}%")

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\text{MAE} = \frac{1}{n} \sum_{i=1}^{n} \left| y_i - \hat{y}_i \right|