Вивчайте MSE, RMSE, and MAE

To evaluate regression models, you need reliable ways to measure how close the model's predictions are to the actual values. Three of the most widely used metrics for this purpose are Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE). These metrics quantify the average difference between predicted values ( $\hat{y}_i$ ) and true values ( $y_i$ ) across all samples.

Mean Squared Error (MSE):

The MSE calculates the average of the squares of the errors:
$\mathrm{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$
Here, $n$ is the number of samples, $y_i$ is the true value, and $\hat{y}_i$ is the predicted value for the $i$ -th sample.
Root Mean Squared Error (RMSE):

The RMSE is simply the square root of the MSE, which brings the error metric back to the original unit of the output variable:
$\mathrm{RMSE} = \sqrt{\mathrm{MSE}} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}$
Mean Absolute Error (MAE):

The MAE computes the average of the absolute differences between predicted and actual values:
$\mathrm{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|$

Each of these metrics uses the concept of error — the difference between an observed value and its prediction — to summarize model performance.


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from sklearn.metrics import mean_squared_error, mean_absolute_error
import numpy as np

# Example true values and predictions
y_true = np.array([3.0, -0.5, 2.0, 7.0])
y_pred = np.array([2.5, 0.0, 2.1, 7.8])

# Calculate Mean Squared Error
mse = mean_squared_error(y_true, y_pred)

# Calculate Root Mean Squared Error
rmse = mean_squared_error(y_true, y_pred, squared=False)

# Calculate Mean Absolute Error
mae = mean_absolute_error(y_true, y_pred)

print(f"MSE: {mse:.3f}")
print(f"RMSE: {rmse:.3f}")
print(f"MAE: {mae:.3f}")

When choosing between MSE, RMSE, and MAE, you must understand how each metric responds to large errors, also known as outliers. Both MSE and RMSE penalize larger errors more severely because errors are squared before averaging. This means that a single large error can significantly increase the MSE and RMSE values, making these metrics highly sensitive to outliers. RMSE is often preferred when you want an error metric in the same unit as the target variable and still want to emphasize large errors.

On the other hand, MAE treats all errors equally by taking the absolute value of each error, making it more robust to outliers. MAE gives a more direct interpretation of the average error magnitude, but it does not highlight large errors as strongly as MSE or RMSE.

Guidance on metric selection:

Use MSE or RMSE if you want to penalize large errors more and are concerned about the impact of outliers;
Use MAE if you want a straightforward measure of average error that is less sensitive to outliers;
Always consider the context and business impact of errors when selecting a metric for your regression problem.

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Mean Squared Error (MSE):

The MSE calculates the average of the squares of the errors:
$\mathrm{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$
Here, $n$ is the number of samples, $y_i$ is the true value, and $\hat{y}_i$ is the predicted value for the $i$ -th sample.
Root Mean Squared Error (RMSE):

The RMSE is simply the square root of the MSE, which brings the error metric back to the original unit of the output variable:
$\mathrm{RMSE} = \sqrt{\mathrm{MSE}} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}$
Mean Absolute Error (MAE):

The MAE computes the average of the absolute differences between predicted and actual values:
$\mathrm{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|$

Each of these metrics uses the concept of error — the difference between an observed value and its prediction — to summarize model performance.


              12345678910111213141516171819
            
from sklearn.metrics import mean_squared_error, mean_absolute_error
import numpy as np

# Example true values and predictions
y_true = np.array([3.0, -0.5, 2.0, 7.0])
y_pred = np.array([2.5, 0.0, 2.1, 7.8])

# Calculate Mean Squared Error
mse = mean_squared_error(y_true, y_pred)

# Calculate Root Mean Squared Error
rmse = mean_squared_error(y_true, y_pred, squared=False)

# Calculate Mean Absolute Error
mae = mean_absolute_error(y_true, y_pred)

print(f"MSE: {mse:.3f}")
print(f"RMSE: {rmse:.3f}")
print(f"MAE: {mae:.3f}")

Guidance on metric selection:

Use MSE or RMSE if you want to penalize large errors more and are concerned about the impact of outliers;
Use MAE if you want a straightforward measure of average error that is less sensitive to outliers;
Always consider the context and business impact of errors when selecting a metric for your regression problem.

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Дякуємо за ваш відгук!

Секція 2. Розділ 2