Question

Using the data below, what is the value of RMSE?
Week Time Series Value Forecast
1 6
2 3 3.00
3 5 6.00
4 2 5.00
5 8 2.00

Answer 1

To calculate the Root Mean Square Error (RMSE) of the forecasted values, we first need to identify the observed values (Time Series Values) and the corresponding forecasted values. The RMSE is calculated using the following formula:

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

where:
- \(y_i\) are the observed values,
- \(\hat{y}_i\) are the forecasted values,
- \(n\) is the number of observations.

From the data provided:

| Week | Time Series Value (y_i) | Forecast (\hat{y}_i) |
|------|-------------------------|----------------------|
| 1 | 6 | |
| 2 | 3 | 3.00 |
| 3 | 5 | 6.00 |
| 4 | 2 | 5.00 |
| 5 | 8 | 2.00 |

Now, we will only consider weeks with available forecast values (weeks 2, 3, 4, and 5):

1. **Week 2**: \(y_2 = 3\), \(\hat{y}_2 = 3.00\)
Error for Week 2: \((3 - 3)^2 = 0\)

2. **Week 3**: \(y_3 = 5\), \(\hat{y}_3 = 6.00\)
Error for Week 3: \((5 - 6)^2 = 1\)

3. **Week 4**: \(y_4 = 2\), \(\hat{y}_4 = 5.00\)
Error for Week 4: \((2 - 5)^2 = 9\)

4. **Week 5**: \(y_5 = 8\), \(\hat{y}_5 = 2.00\)
Error for Week 5: \((8 - 2)^2 = 36\)

Now, we will sum these squared errors:

\[
\text{Total Squared Error} = 0 + 1 + 9 + 36 = 46
\]

The number of observations (n) is 4 (Week 2, Week 3, Week 4, Week 5).

Now we can calculate RMSE:

\[
\text{RMSE} = \sqrt{\frac{1}{4} \cdot 46} = \sqrt{11.5} \approx 3.39
\]

Therefore, the RMSE is approximately **3.39**.