Time series evaluation metrics in ML Python

Introduction

We use time series evaluation metrics to check how well our model predicts future values based on past data. This helps us know if our model is good or needs improvement.

When forecasting daily sales for a store to plan inventory.

When predicting temperature changes for weather forecasts.

When estimating stock prices for investment decisions.

When monitoring energy consumption to optimize usage.

When analyzing website traffic to prepare for high demand.

Syntax

ML Python

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

# y_true: actual values
# y_pred: predicted values

mae = mean_absolute_error(y_true, y_pred)
mse = mean_squared_error(y_true, y_pred)
rmse = np.sqrt(mse)

mean_absolute_error (MAE) measures average absolute difference between actual and predicted values.

mean_squared_error (MSE) squares the differences before averaging, penalizing bigger errors more.

root mean squared error (RMSE) is the square root of MSE, giving error in original units.

Examples

Calculates MAE for small lists of actual and predicted values.

ML Python

mae = mean_absolute_error([3, 5, 2], [2.5, 5, 2.1])
print(mae)

Calculates MSE for the same values, showing squared error average.

ML Python

mse = mean_squared_error([3, 5, 2], [2.5, 5, 2.1])
print(mse)

Calculates RMSE, which is easier to interpret as it is in the same scale as the data.

ML Python

rmse = np.sqrt(mean_squared_error([3, 5, 2], [2.5, 5, 2.1]))
print(rmse)

Sample Model

This program compares actual and predicted time series values using MAE, MSE, and RMSE to evaluate prediction accuracy.

ML Python

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

# Actual values of a time series
actual = [100, 150, 200, 250, 300]
# Predicted values from a model
predicted = [110, 140, 210, 240, 310]

# Calculate MAE
mae = mean_absolute_error(actual, predicted)
# Calculate MSE
mse = mean_squared_error(actual, predicted)
# Calculate RMSE
rmse = np.sqrt(mse)

print(f"MAE: {mae:.2f}")
print(f"MSE: {mse:.2f}")
print(f"RMSE: {rmse:.2f}")

Important Notes

Lower values of MAE, MSE, and RMSE mean better predictions.

RMSE is more sensitive to large errors than MAE.

Always compare metrics on the same dataset to judge model improvements.

Summary

Time series evaluation metrics help measure prediction errors.

MAE, MSE, and RMSE are common and easy to use.

Use these metrics to improve and trust your forecasting models.

Practice

1. Which metric measures the average absolute difference between predicted and actual values in time series forecasting?

easy

A. Mean Squared Error (MSE)

B. Mean Absolute Error (MAE)

C. Root Mean Squared Error (RMSE)

D. R-squared (Coefficient of Determination)

Solution

1. Step 1: Understand the definition of MAE

   MAE calculates the average of the absolute differences between predicted and actual values, showing average error size.

2. Step 2: Compare with other metrics

   MSE and RMSE square errors, while R-squared measures variance explained, not average error.

3. Final Answer:

   Mean Absolute Error (MAE) -> Option B

4. Quick Check:

   Average absolute difference = MAE [OK]

Hint: MAE uses absolute differences, no squaring involved [OK]

Common Mistakes:

• Confusing MAE with MSE or RMSE
• Thinking R-squared measures error size
• Assuming RMSE is the same as MAE

2. Which of the following is the correct formula for Root Mean Squared Error (RMSE) given errors \(e_i = y_i - \hat{y}_i\) for \(n\) points?

easy

A. RMSE = \(\sum_{i=1}^n e_i^2\)

B. RMSE = \(\frac{1}{n} \sum_{i=1}^n |e_i|\)

C. RMSE = \(\frac{1}{n} \sum_{i=1}^n e_i\)

D. RMSE = \(\sqrt{\frac{1}{n} \sum_{i=1}^n e_i^2}\)

Solution

1. Step 1: Recall RMSE formula

   RMSE is the square root of the average of squared errors, so it must include squaring, averaging, then square root.

2. Step 2: Check each option

   RMSE = \(\sqrt{\frac{1}{n} \sum_{i=1}^n e_i^2}\): \(\sqrt{\frac{1}{n} \sum_{i=1}^n e_i^2}\) matches the formula exactly. RMSE = \(\sum_{i=1}^n e_i^2\) misses averaging and root. RMSE = \(\frac{1}{n} \sum_{i=1}^n |e_i|\) is MAE. RMSE = \(\frac{1}{n} \sum_{i=1}^n e_i\) is mean error (not squared).

3. Final Answer:

   RMSE = \(\sqrt{\frac{1}{n} \sum_{i=1}^n e_i^2}\) -> Option D

4. Quick Check:

   RMSE = sqrt(mean squared errors) [OK]

Hint: RMSE = square root of average squared errors [OK]

Common Mistakes:

• Forgetting to take square root
• Using absolute errors instead of squared
• Not dividing by number of points

3. Given actual values \([3, 5, 2, 7]\) and predicted values \([2, 5, 4, 8]\), what is the Mean Squared Error (MSE)?

medium

A. 1.5

B. 1.25

C. 2.0

D. 0.75

Solution

1. Step 1: Calculate errors and square them

   Errors: 3-2=1, 5-5=0, 2-4=-2, 7-8=-1. Squared errors: 1, 0, 4, 1.

2. Step 2: Compute average of squared errors

   Sum = 1+0+4+1=6. Average = 6/4 = 1.5.

3. Final Answer:

   1.5 -> Option A

4. Quick Check:

   Sum squared errors / count = 1.5 [OK]

Hint: Square errors, sum, then divide by count [OK]

Common Mistakes:

• Using absolute errors instead of squared
• Forgetting to average over all points
• Mixing predicted and actual values

4. Identify the error in this Python code calculating MAE for time series predictions:

def mae(actual, predicted):
    errors = [a - p for a, p in zip(actual, predicted)]
    return sum(errors) / len(errors)

medium

A. Use multiplication instead of subtraction in errors

B. Divide by sum of errors instead of length

C. Errors should be absolute values before summing

D. No error, code is correct

Solution

1. Step 1: Analyze error calculation

   The code calculates errors as differences but does not take absolute values, which MAE requires.

2. Step 2: Understand MAE definition

   MAE is mean of absolute errors, so errors must be wrapped with abs() before summing.

3. Final Answer:

   Errors should be absolute values before summing -> Option C

4. Quick Check:

   MAE needs absolute errors [OK]

Hint: MAE sums absolute errors, not raw differences [OK]

Common Mistakes:

• Skipping absolute value in error calculation
• Dividing by wrong denominator
• Confusing MAE with MSE

5. You have two forecasting models evaluated on the same dataset. Model A has MAE=2.5 and RMSE=3.0, Model B has MAE=2.0 and RMSE=3.5. Which model is generally better and why?

hard

A. Model A, because lower RMSE means fewer large errors

B. Model B, because higher RMSE indicates better fit

C. Model B, because lower MAE means better average error

D. Model A, because MAE and RMSE must be equal for best model

Solution

1. Step 1: Interpret MAE and RMSE values

   Model B has lower MAE but higher RMSE, meaning it has better average error but more large errors. Model A has lower RMSE, indicating fewer large errors.

2. Step 2: Decide which metric matters more

   RMSE penalizes large errors more, so lower RMSE often means more reliable predictions without big mistakes.

3. Final Answer:

   Model A, because lower RMSE means fewer large errors -> Option A

4. Quick Check:

   Lower RMSE means fewer big errors [OK]

Hint: Lower RMSE means fewer big errors; prefer it if large errors matter [OK]

Common Mistakes:

• Choosing model with lower MAE ignoring RMSE
• Thinking higher RMSE is better
• Expecting MAE and RMSE to be equal