To calculate the simple exponential smoothing forecast for the 4th week (Week 4) using the given time series values and the smoothing constant \(\alpha = 0.4\), we need to follow these steps:
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Identify the time series values:
- Week 1: 6
- Week 2: 4
- Week 3: 3
- Week 4: 5
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Initialize the forecast: The first forecast (for Week 2) is typically the actual value of Week 1. \[ F_1 = V_1 = 6 \]
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Apply the formula for exponential smoothing: The formula for simple exponential smoothing is: \[ F_t = \alpha V_{t-1} + (1 - \alpha) F_{t-1} \] where:
- \(F_t\) is the forecast for the current period \(t\),
- \(V_{t-1}\) is the actual value from the previous period,
- \(F_{t-1}\) is the forecast from the previous period.
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Calculate forecasts:
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Forecast for Week 2 (\(F_2\)): \[ F_2 = \alpha V_1 + (1 - \alpha) F_1 = 0.4 \cdot 6 + 0.6 \cdot 6 = 6 \]
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Forecast for Week 3 (\(F_3\)): \[ F_3 = \alpha V_2 + (1 - \alpha) F_2 = 0.4 \cdot 4 + 0.6 \cdot 6 = 1.6 + 3.6 = 5.2 \]
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Forecast for Week 4 (\(F_4\)): \[ F_4 = \alpha V_3 + (1 - \alpha) F_3 = 0.4 \cdot 3 + 0.6 \cdot 5.2 = 1.2 + 3.12 = 4.32 \]
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So, the simple exponential smoothing forecast for the 4th week is approximately 4.32.