Returns the Double (Holt) exponential smoothing out-of-sample forecast estimate.

## Syntax

**DESMTH**(

**X**,

**Order**,

**Alpha**,

**Beta**,

**Optimize**,

**T**,

**Return Type**)

**X** is the univariate time series data (a one-dimensional array of cells (e.g. rows or columns)).

**Order** is the time order in the data series (i.e. the first data point's corresponding date (earliest date=1 (default), latest date=0)).

Order | Description |
---|---|

1 | ascending (the first data point corresponds to the earliest date) (default) |

0 | descending (the first data point corresponds to the latest date) |

**Alpha** is the level smoothing factor (alpha should be between zero(0) and one(1) (exclusive)). If missing or omitted, 0.333 value is used..

**Beta** is the trend smoothing factor (beta should be between zero(0) and one(1) (exclusive)). If missing or omitted, 0.333 value is used.

**Optimize** is a flag (True/False) for searching and using optimal value of the smoothing factor. If missing or omitted, optimize is assumed False.

**T** is the forecast time/horizon beyond the end of X. If missing, a default value of 0 (Latest or end of X) is assumed.

**Return Type**is a number that determines the type of return value: 0 (or missing) = Forecast, 1=Alpha, 2=Beta, 3=level component (series), 4=trend component (series), 5=one-step forecasts (series).

Return Type | Description |
---|---|

0 or omitted | Forecast value |

1 | Level smoothing parameter (alpha) |

2 | Trend smoothing parameter (beta) |

3 | level component (series) |

4 | trend component (series) |

5 | one-step forecasts (series) |

## Remarks

- The time series is homogeneous or equally spaced.
- The time series may include missing values (e.g. #N/A) at either end.
- The double exponential smoothing is best applied to time series that exhibit prevalent additive (non-exponential) trend, but do not exhibit seasonality.
- The recursive form of the Holt’s double exponential smoothing equation is expressed as follows:

$$ \begin{array}{l} \hat{F}_t(m)=S_t+m\times b_t\\ \\ S_{t\succ 1}=\alpha \times X_t | (1-\alpha)(S_{t-1} + b_{t-1})\\ b_{t\succ 1}=\beta \times (S_t - S_{t-1})+(1-\beta)b_{t-1} \end{array} $$

Where:

- $X_t$ is the value of the time series at time t.
- $S_{t}$ is a smoothed estimate of the value of the time series X at the end of period t.
- $b_{t}$ is a smoothed estimate of average growth at the end of period t.
- $\alpha$ is the level smoothing coefficient.
- $\beta$ is the trend smoothing coefficient.
- $\hat{F}_t(m)$ is the m-step-ahead forecast values for $X$ from time t.

- In DESMTH, we compute two simple, but interdependent, exponential series: level and trend. They are inter-dependent in sense that both components must be updated each period.
- The smoothing coefficient $\alpha$ is again used to control speed of adaptation to local level but a second smoothing constant $\beta$ is introduced to control the degree of a local trend carried through to multi-step-ahead forecast periods.
- For $\alpha = \beta$ , then Holt’s double exponential smoothing is equivalent to Brown’s linear exponential smoothing method.
- For $\beta = 0$ and the start value for trend ($b_1$ ) is also set to zero(0), the Holt’s double exponential smoothing produces the same forecasts as Brown’s simple exponential smoothing.
- The DESMTH 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.
- 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.
- This method requires two starting values ($S_1,b_1$)to start the recursive updating of the equation. In NumXL, we set those values as follows:
- $S_1$ is set to the in-sample mean, and for a very short time series, it is set as the value of the first observation.

$$ S_{1}=\left\{\begin{array}{l} X_1\\ \frac{\sum_{t=1}^N X_t}{N} \end{array}\right. \begin{array}{r} N \leq 4\\ N \gt 4 \end{array} $$ - $b_1$ is set to the slope of regression trend line. If not enough observations are available, then $b_1$ is set to zero(0).

$$ b_{1}=\left\{\begin{array}{l} 0\\ \mathrm{Reg.\,Slope} \end{array}\right. \begin{array}{r} N \leq 4\\ N \gt 4 \end{array} $$

- $S_1$ is set to the in-sample mean, and for a very short time series, it is set as the value of the first observation.
- Starting from NumXL version 1.63, the DESMTH has a built-in optimizer to find the best value of ($\alpha,\beta$) that minimize the SSE (loss function ($U(.)$)) for the one-step forecast calculated in-sample.

$$ \begin{array}{l} U(\alpha,\beta)=\mathrm{SSE}=\sum_{t=1}^{N-1}(X_{t+1}-\hat{F}_t(1))^2\\ \min_{\alpha,\beta \in (0,1)} U(\alpha,\beta) \end{array} $$ - For initial values, the NumXL optimizer will use the input value of (alpha,beta) (if available) in the minimization problem, and the initial values for the two-smoothing series ($S_1, b_1$ ) are computed from the input data.
- Starting from NumXL version 1.65. the DESMTH function return the found optimal value for (alpha,beta), and the corresponding one-step smoothing series of level, trend and forecast calculated in-sample.
- The time series must have at least four (4) observation with non-missing values to use the built-in optimizer.
- NumXL implements the spectral projected gradient (SPG) method for finding the minima with a boxed boundary.
- The SPG requires loss function value and the gradient ($\nabla$). NumXL implements the exact derivative formula (vs. numerical approximation) for performance purposes.

$$ \begin{array}{l} \nabla U = \frac{\partial U}{\partial \alpha} \vec{e_\alpha} + \frac{\partial U}{\partial \beta} \vec{e_\beta}\\ \\ \frac{\partial U}{\partial \alpha} = -2\times\sum_{t=1}^{N-1}(X_{t+1}-\hat{F}_t(1))\times \frac{\partial \hat{F}_t}{\partial \alpha}\\ \frac{\partial U}{\partial \beta} = -2\times\sum_{t=1}^{N-1}(X_{t+1}-\hat{F}_t(1))\times \frac{\partial \hat{F}_t}{\partial \beta}\\ \\ \frac{\partial \hat{F}_t}{\partial \alpha}=\frac{\partial S_t}{\partial \alpha}+\frac{\partial b_t}{\partial \alpha}\\ \frac{\partial \hat{F}_t}{\partial \beta}=\frac{\partial S_t}{\partial \beta}+\frac{\partial b_t}{\partial \beta}\\ \\ \frac{\partial S_t}{\partial \alpha}=X_t + (1-\alpha)(\frac{\partial S_{t-1}}{\partial \alpha}+ \frac{\partial b_{t-1}}{\partial \alpha})-(S_{t-1}+b_{t-1})\\ \frac{\partial S_t}{\partial \beta}=(1-\alpha)(\frac{\partial S_{t-1}}{\partial \beta}+ \frac{\partial b_{t-1}}{\partial \beta})\\ \\ \frac{\partial b_t}{\partial \alpha}= \beta \times (\frac{\partial S_t}{\partial \alpha} -\frac{\partial S_{t-1}}{\partial \alpha}) + (1-\beta) \frac{\partial b_{t-1}}{\partial \alpha}\\ \frac{\partial b_t}{\partial \beta} = (S_t-S_{t-1})+\beta(\frac{\partial S_t}{\partial \beta} -\frac{\partial S_{t-1}}{\partial \beta})+(1-\beta)\frac{\partial b_{t-1}}{\partial \beta} -b_{t-1} \\ \end{array} $$ - Internally, during the optimization, NumXL computes recursively both the smoothed time series, levels, trends, and the in-sample 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.

- The SPG requires loss function value and the gradient ($\nabla$). NumXL implements the exact derivative formula (vs. numerical approximation) for performance purposes.
- In general, the SSE function in DESMTH yields a continuous smooth convex monotone curve, that SPG minimizer almost always finds an optimal solution in a very few iterations.

## Examples

## 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

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