Simple exponentail smoothing (SES)
or exponential moving average
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a technique for smoothing time series data using exponential window function
- exponential functions are used to assign exponentially decreasing weights over time
easily learned and applied procedure for determining based on prior assumptions
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such as seasonality
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raw data sequence is represented by \(\{ x_t\}\)
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beginning at time \(t=0\)
- which may be regarded as a best estimate of what the next value of \(x\) will be
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beginning at time \(t=0\)
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when the sequence of observations begins at time \(t=0\)
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the simplest form of exponential smoothing is given below:
\(s_0=x_0\) \(s_t=\alpha x_t + (1-\alpha)s_{t-1}, t>0\)
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\(/alpha\) is the smoothing factor
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and \(0 <\alpha<1\)
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if \(s_{t-1}\) is substituted into \(s_t\) continously
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so that the formula of \(s_t\) is fully expressed in terms of \(\{x_t\}\)
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then exponentially decaying weighting factors
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on each raw data \(x_t\) is revealed
- showing how exponential smoothing is named
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on each raw data \(x_t\) is revealed
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then exponentially decaying weighting factors
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so that the formula of \(s_t\) is fully expressed in terms of \(\{x_t\}\)
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if \(s_{t-1}\) is substituted into \(s_t\) continously
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and \(0 <\alpha<1\)
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\(/alpha\) is the smoothing factor
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SES is not able to predict what would be observed at \(t+m\) based on the raw data up to \(t\) , while the double exponential smoothing and triple exponential smoothing can be used for the prediction due to the presence of \(b_t\) as the sequence of best estimates of the linear trend
finding \(\alpha\) and \(l\)
since all fitted values are function of \(\alpha\) and \(l\)
- then SSE \(= \sum_{t=1}^T (y_t - \hat{y}_t)^2\) is also a function of \(\alpha\) and \(l\) only
a common strategy in many models is to fund \(\alpha\) , \(l\) such that SSE gets minimized
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this optimization problem will be solved by R
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the method for choosing \(\alpha\) must be decided by the modeler
- the method of least squares might be used to determine the value of \(\alpha\) for which the sum of the quantities \((s_t-x_{t+1})^2\) is minimized
The unknown parameters and the initial values for any exponential smoothing method can be estimated by minimizing the sum of squared errors (SSE)
Time constant
the time constant of an exponential moving average is the amount of time
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for that smoothed response of a unit step function
- to reach \(1-1/e \approx 63.2\%\) of the final signal
the relationship between this time constant, \(\tau\) , and the smoothing factor, \(\alpha\) , is given by the formula below:
\(\alpha = 1 - e^{-\Delta T / \tau}\) , thus \(\tau = - \frac{\Delta T}{ln(1-\alpha)}\)
- where \(\Delta T\) is the sampling time interval of the discrete time implementation
if sampling time is fast compared to time constant then by using Taylor expansion of the exponential function , \(\alpha \approx \frac{\Delta T}{\tau}\) , thus \(\tau \approx \frac{\Delta T}{\alpha}\)
Practice
let \(l = \hat{y}_1\) be the first fitted value \(t=1 : \hat{y}_2 = \alpha y_1 + (1-\alpha)\hat{y}_1 = \alpha y_1 + (1 - \alpha)l\)
\(t=2 : \hat{y}_3 = \alpha y_2 + (1-\alpha)\hat{y}_2 = \alpha y_2 + (1 - \alpha)(\alpha y_1 + (1 - \alpha)l) = \alpha y_2 + \alpha(1-\alpha) y_1 + (1 - \alpha)^2 l\)
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