SESMTH - (Brown's) Simple Exponential Smoothing

1. The time series is homogeneous or equally spaced.
2. The time series may include missing values (e.g. #N/A) at either end.
3. The simple exponential smoothing is best applied to time series that do not exhibit a prevalent trend and do not exhibit seasonality.
4. The recursive form of the simple exponential smoothing equation is expressed as follows:
   
   $$ S_{t \succ 1}= \alpha\times X_t + (1-\alpha)\times S_{t-1} $$ $$ \hat{F}_t(m)=S_t $$
   Where:
   
   • $X_t$ is the value of the time series at time t.
   • $S_t$ is the smoothed level.
   • $\alpha$ is the smoothing factor/coefficient for level.
   • $\hat{F}_t(m)$ is the m-step-ahead forecast values for $X$ from time t.
5. For $\alpha = 1$, the simple exponential smoothing is equivalent to the naïve no change extrapolation (NCE) method, or simply a random walk.
6. For $\alpha = 0$, the forecast will be a constant taking its values from the starting value for Level.
7. In general, the smoothing coefficient $\alpha$ is used to control the speed of adaptation to the local level.
8. The SESMTH calculate a point forecast. There is no probabilistic model assumed for the simple exponential smoothing, so we can’t derive a statistical confidence interval for the computed values.
9. In practice, the Mean Squared Error (MSE) for prior out-of-sample forecast values are often used as a proxy for the uncertainty (i.e. variance) in the most recent forecast value.
10. This method requires a starting value for Level (i.e. $S_t$) to start the recursive updating of the equation. In NumXL, the starting value is set to the mean of the first four observations, and for a short time series, it is set as the value of the first observation.
    
    $$ S_1=\left\{\begin{array}{l} X_1\\ \frac{\sum_{t=1}^4 X_t}{4} \end{array}\right. \begin{array}{r} N \leq 4\\ N \gt 4 \end{array} $$
11. Starting from NumXL version 1.63, the SESMTH has a built-in optimizer to find the best value of $\alpha$ that minimize the SSE (loss function ($U(.)$)) for the one-step forecast calculated in-sample.
    
    $$ \begin{array}{l} U(\alpha)=\mathrm{SSE}=\sum_{t=1}^{N-1}(X_{t+1}-\hat{F}_t(1))^2\\ \min_{\alpha \in (0,1)} U(\alpha) \end{array} $$
12. For initial values, the NumXL optimizer will use the input value of alpha (if available) in the minimization problem, and the initial level (i.e. $S_1$) is computed from the input data.
13. Starting from NumXL version 1.65. the SESMTH function returns the found optimal value for alpha, and the corresponding one-step smoothing forecast calculated in-sample series.
14. The time series must have at least three (3) observation with non-missing values to use the built-in optimizer.
15. NumXL implements the spectral projected gradient (SPG) method for finding the minima with a boxed boundary.
    • The SPG requires loss function value and the 1st derivative. NumXL implements the exact derivative formula (vs. numerical approximation) for performance purposes.
      
      $$ \begin{array}{l} \frac{\partial U}{\partial \alpha}=-2\times\sum_{t=1}^{N-1}(X_{t+1}-\hat{F}_t(1))\times \frac{\partial \hat{F}_t}{\partial \alpha} \\ \\ \frac{\partial \hat{F}_t}{\partial \alpha}=\frac{\partial S_t}{\partial \alpha} \\ \\ \frac{\partial S_t}{\partial \alpha}=X_{t+1}+(1-\alpha)\times \frac{\partial S_{t-1}}{\partial \alpha}-S_{t-1} \end{array} $$
    • Internally, during the optimization, NumXL computes recursively both the in-sample levels and the in-sample level derivatives, which are used for the loss function and its derivative
    • The SPG is an iterative (recursive) method, and it is possible that the minima can’t be found the within allowed number of iterations and/or tolerance. In this case, NumXL will not fail, instead NumXL uses the best alpha found so far
    • The SPG has no provision to detect or avoid local minima trap. There is no guarantee of global minima.
16. In most cases, the SSE function yields a continuous smooth convex monotone curve, that SPG minimizer almost always finds an optimal solution in a very few iterations

Example 1:

Files Examples

Related Links

• Wikipedia - Exponential smoothing

References

• Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
• Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
• D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906