To calculate the Root Mean Square Error (RMSE), we need to follow these steps:
- Calculate the error for each week: The error is the difference between the actual time series value and the forecast value.
- Square each error: This helps in eliminating negative values and emphasizes larger errors.
- Calculate the mean of the squared errors: We sum the squared errors and divide by the number of observations.
- Take the square root of the mean of the squared errors: This gives us the RMSE.
Let's proceed with the provided data:
Step 1: Calculate Errors
For each week, we calculate the error as follows:
\[ \text{Error} = \text{Time Series Value} - \text{Forecast} \]
Using the values from the data provided:
-
Week 1:
- Time Series Value = 6
- Forecast = Not provided (we can assume it's 6 based on typical forecasting where Week 1 might be equal to its value)
- Error = \(6 - 6 = 0\)
-
Week 2:
- Time Series Value = 3
- Forecast = 6.00
- Error = \(3 - 6 = -3\)
-
Week 3:
- Time Series Value = 5
- Forecast = 7.00
- Error = \(5 - 7 = -2\)
-
Week 4:
- Time Series Value = 2
- Forecast = 3.00
- Error = \(2 - 3 = -1\)
-
Week 5:
- Time Series Value = 8
- Forecast = 6.00
- Error = \(8 - 6 = 2\)
Step 2: Square each error
Now we square each of the errors calculated:
- Week 1: \(0^2 = 0\)
- Week 2: \((-3)^2 = 9\)
- Week 3: \((-2)^2 = 4\)
- Week 4: \((-1)^2 = 1\)
- Week 5: \(2^2 = 4\)
Step 3: Calculate the mean of the squared errors
Next, we sum these squared errors and divide by the number of observations (in this case, 5).
\[ \text{Mean Squared Error} = \frac{0 + 9 + 4 + 1 + 4}{5} = \frac{18}{5} = 3.6 \]
Step 4: Take the square root of the mean of the squared errors
Finally, we take the square root of the mean squared error to obtain the RMSE.
\[ \text{RMSE} = \sqrt{3.6} \approx 1.897 \]
Result
Thus, the value of the RMSE is approximately 1.897.