In a time series forecasting model, the demands for five time periods were 10, 13, 15, 18 and 22. A linear regression fit resulted in an equation F = 6.9 + 2.9t where F is the forecast for period t. The sum of the absolute deviations for the five data is
2.2
Calculating the Sum of Absolute Deviations in Time Series Forecasting
This problem requires us to calculate the sum of the absolute deviations between the actual demands and the forecasts generated by a linear regression model for a given time series.
Understanding the Data and the Forecast Model
We are given the actual demands for five time periods:
- Period 1 (t=1): Demand = 10
- Period 2 (t=2): Demand = 13
- Period 3 (t=3): Demand = 15
- Period 4 (t=4): Demand = 18
- Period 5 (t=5): Demand = 22
The linear regression forecast model is given by the equation: \(F = 6.9 + 2.9t\), where \(F\) is the forecast for time period \(t\).
Steps to Calculate Sum of Absolute Deviations
To find the sum of the absolute deviations, we need to follow these steps:
- Calculate the forecast (\(F\)) for each time period (\(t\)) using the given equation.
- Calculate the deviation (Error) for each period: Deviation = Actual Demand - Forecast.
- Calculate the absolute deviation for each period: |Deviation|.
- Sum up the absolute deviations for all five periods.
Calculating Forecasts and Deviations
Let's calculate the forecast, deviation, and absolute deviation for each period:
- Period 1 (t=1): Forecast \(F_1 = 6.9 + 2.9 \times 1 = 6.9 + 2.9 = 9.8\). Deviation = Actual Demand - Forecast = \(10 - 9.8 = 0.2\). Absolute Deviation = \(|0.2| = 0.2\).
- Period 2 (t=2): Forecast \(F_2 = 6.9 + 2.9 \times 2 = 6.9 + 5.8 = 12.7\). Deviation = Actual Demand - Forecast = \(13 - 12.7 = 0.3\). Absolute Deviation = \(|0.3| = 0.3\).
- Period 3 (t=3): Forecast \(F_3 = 6.9 + 2.9 \times 3 = 6.9 + 8.7 = 15.6\). Deviation = Actual Demand - Forecast = \(15 - 15.6 = -0.6\). Absolute Deviation = \(|-0.6| = 0.6\).
- Period 4 (t=4): Forecast \(F_4 = 6.9 + 2.9 \times 4 = 6.9 + 11.6 = 18.5\). Deviation = Actual Demand - Forecast = \(18 - 18.5 = -0.5\). Absolute Deviation = \(|-0.5| = 0.5\).
- Period 5 (t=5): Forecast \(F_5 = 6.9 + 2.9 \times 5 = 6.9 + 14.5 = 21.4\). Deviation = Actual Demand - Forecast = \(22 - 21.4 = 0.6\). Absolute Deviation = \(|0.6| = 0.6\).
We can summarize these values in a table:
| Period (t) | Actual Demand | Forecast (F) | Deviation (Demand - F) | Absolute Deviation (|Deviation|) |
|---|---|---|---|---|
| 1 | 10 | 9.8 | 0.2 | 0.2 |
| 2 | 13 | 12.7 | 0.3 | 0.3 |
| 3 | 15 | 15.6 | -0.6 | 0.6 |
| 4 | 18 | 18.5 | -0.5 | 0.5 |
| 5 | 22 | 21.4 | 0.6 | 0.6 |
Summing the Absolute Deviations
The final step is to sum the absolute deviations calculated for each period:
Sum of Absolute Deviations = \(0.2 + 0.3 + 0.6 + 0.5 + 0.6\)
Sum of Absolute Deviations = \(2.2\)
Therefore, the sum of the absolute deviations for the five data points is 2.2.
Important Questions from Forecasting
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Select the correct option :
Which of the following is a technique used for forecasting?
