Answer:
Explanation:
Let's calculate the measures of forecast error using both the naive (most recent value) method and the average of historical data.
Given time series data:
Week: 1 2 3 4 5 6
Value: 18 14 16 11 17 15
Naive Method:
For the naive method, the forecast for the next period is the most recent value. So, the forecast for Week 7 is 15.
Historical Data (Average) Method:
The forecast for Week 7 using the average of historical data is the average of values in Weeks 1 to 6:
\[ \text{Forecast for Week 7} = \frac{18 + 14 + 16 + 11 + 17 + 15}{6} \]
\[ \text{Forecast for Week 7} = \frac{91}{6} \approx 15.17 \]
Now, let's calculate the measures of forecast error:
Mean Absolute Error (MAE):
\[ MAE = \frac{1}{n} \sum_{i=1}^{n} |Y_i - \hat{Y}_i| \]
Naive Method:
\[ MAE_{\text{Naive}} = |15 - 15| = 0 \]
Historical Data Method:
\[ MAE_{\text{Historical}} = \frac{1}{6} (|18-15| + |14-15| + |16-15| + |11-15| + |17-15| + |15-15|) \]
\[ MAE_{\text{Historical}} = \frac{1}{6} (3 + 1 + 1 + 4 + 2 + 0) = \frac{11}{6} \approx 1.83 \]
Mean Squared Error (MSE):
\[ MSE = \frac{1}{n} \sum_{i=1}^{n} (Y_i - \hat{Y}_i)^2 \]
Naive Method:
\[ MSE_{\text{Naive}} = (15 - 15)^2 = 0 \]
Historical Data Method:
\[ MSE_{\text{Historical}} = \frac{1}{6} ( (18-15)^2 + (14-15)^2 + (16-15)^2 + (11-15)^2 + (17-15)^2 + (15-15)^2 ) \]
\[ MSE_{\text{Historical}} = \frac{1}{6} (9 + 1 + 1 + 16 + 4 + 0) = \frac{31}{6} \approx 5.17 \]
Mean Absolute Percentage Error (MAPE):
\[ MAPE = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{Y_i - \hat{Y}_i}{Y_i} \right| \times 100 \]
Naive Method:
\[ MAPE_{\text{Naive}} = \left| \frac{15 - 15}{15} \right| \times 100 = 0 \]
Historical Data Method:
\[ MAPE_{\text{Historical}} \approx \frac{1}{6} (16.67 + 7.14 + 6.25 + 36.36 + 11.76 + 0) \]
\[ MAPE_{\text{Historical}} \approx \frac{78.18}{6} \approx 13.03 \]
Comparison:
- The naive method has lower errors in MAE, MSE, and MAPE compared to the historical data method.
- Therefore, the naive method provides more accurate forecasts based on the given measures of forecast error.