To calculate the Mean Squared Error (MSE) using the average of past values method, we predict the value for each week based on the average of all previous weeks' values. Here is the step-by-step process:

Data:

Week	Value
1	19
2	12
3	15
4	10
5	18
6	14

Steps:

1. For Week 1, we cannot predict since no past values are available.
2. For Week 2, predict using the average of Week 1.
3. For Week 3, predict using the average of Weeks 1 and 2.
4. For Week 4, predict using the average of Weeks 1, 2, and 3.
5. For Week 5, predict using the average of Weeks 1, 2, 3, and 4.
6. For Week 6, predict using the average of Weeks 1, 2, 3, 4, and 5.

The formula for MSE is:

$\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (\text{Actual Value}_i - \text{Predicted Value}_i)^2$

Where $n$ is the number of predictions (in this case, $n = 5$).

Let’s compute this step-by-step.

Predictions and Errors:

Week	Value	Predicted Value	Error (Value - Predicted)	Squared Error
2	12	$\frac{19}{1} = 19$	$12 - 19 = -7$	$49$
3	15	$\frac{19 + 12}{2} = 15.5$	$15 - 15.5 = -0.5$	$0.25$
4	10	$\frac{19 + 12 + 15}{3} = 15.33$	$10 - 15.33 = -5.33$	$28.44$
5	18	$\frac{19 + 12 + 15 + 10}{4} = 14$	$18 - 14 = 4$	$16$
6	14	$\frac{19 + 12 + 15 + 10 + 18}{5} = 14.8$	$14 - 14.8 = -0.8$	$0.64$

MSE Calculation:

$\text{MSE} = \frac{49 + 0.25 + 28.44 + 16 + 0.64}{5} = \frac{94.33}{5} = 18.87$

Final Answer:

The Mean Squared Error (MSE) is 18.87.

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Would you like further details or an example of another method?

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5 Related Questions:

1. How does the average of past values method compare to other forecasting methods?
2. Can you explain how to calculate the Root Mean Squared Error (RMSE) for this data?
3. How does the choice of the window size (number of past values) affect the MSE?
4. What other methods can be used to improve forecasting for this time series data?
5. How can we handle missing data in a time series for calculating MSE?

Tip:

Always visualize your time series data to better understand patterns and trends before applying forecasting methods.