# Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models

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## Abstract

**:**

## 1. Introduction

_{24,168}model is a model with an additive trend and multiplicative seasonality; the model is adjusted by using the first-order autocorrelation error, and it has two seasonalities: One daily (subscript 24) and one weekly (subscript 168). The equations defining the AMC

_{24,168}model are described in Equations (5)–(9). The general formulae of all possible models depending on the trend and seasonal type can be found in Appendix A:

## 2. Related Work

_{0}) is calculated as the slope between the average of the last available full period q and the first, as indicated by Equation (10), where the only seasonality is of length s.

_{0}) is obtained once the series trend has been eliminated. It consists in first obtaining the trend of the series and then the level, as seen in Equation (11):

## 3. Materials and Methods

#### 3.1. Level Methods

#### 3.2. Trend Methods

#### 3.3. Seasonality Calculation Methods

1 | 2 | … | q | |

1 | ${X}_{1}/{A}_{1}^{\left(i\right)}$ | ${X}_{{s}_{i}+1}/{A}_{2}^{\left(i\right)}$ | ${X}_{{s}_{i}+1}/{A}_{2}^{\left(i\right)}$ | |

… | ||||

${s}_{i}$ | ${X}_{{s}_{i}}/{A}_{1}^{\left(i\right)}$ | ${X}_{2{s}_{i}}/{A}_{2}^{\left(i\right)}$ | ${X}_{{m}_{1}{s}_{i}+1}/{A}_{{m}_{i}}^{\left(i\right)}$ |

## 4. Results

#### 4.1. Trend Analysis

#### 4.2. Level Analysis

#### 4.3. Seasonality Analysis

_{24,168}model have been used. Therefore, the initial values for the 24- and 168-hour length seasonality are shown. Figure 6 shows the distribution of the initial values for these seasonalities using the simple method. Similarly, Figure 7 reflects this same distribution but using the Winters method.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Trend | Trend | |||||

None | Additive | Damped Additive | Multiplicative | Damped Multiplicative | ||

Seasonality | None | ${S}_{t}=\alpha {X}_{t}+\left(1-\alpha \right){S}_{t-1}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha {X}_{t}+\left(1-\alpha \right)\left({S}_{t-1}+{T}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}-{S}_{t-1}\right)+\left(1-\gamma \right){T}_{t-1}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}+k{T}_{t}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha {X}_{t}+\left(1-\alpha \right)\left({S}_{t-1}+\varphi {T}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}-{S}_{t-1}\right)+\left(1-\gamma \right)\varphi {T}_{t-1}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}+{T}_{t}{\displaystyle \sum}_{n=1}^{k}{\varphi}^{n}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha {X}_{t}+\left(1-\alpha \right)\left({S}_{t-1}{R}_{t-1}\right)$ ${R}_{t}=\gamma \left({S}_{t}/{S}_{t-1}\right)+\left(1-\gamma \right){R}_{t-1}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}{R}_{t}^{k}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha {X}_{t}+\left(1-\alpha \right)\left({S}_{t-1}{R}_{t-1}^{\varphi}\right)$ ${R}_{t}=\gamma \left({S}_{t}/{S}_{t-1}\right)+\left(1-\gamma \right){R}_{t-1}^{\varphi}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}{R}_{t}^{{\displaystyle \sum}_{j=1}^{k}{\varphi}^{j}}+{\phi}_{AR}^{k}{\epsilon}_{t}$ |

Additive | ${S}_{t}=\alpha ({X}_{t}-{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}}^{\left(i\right)})+\left(1-\alpha \right){S}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\left({X}_{t}-{S}_{t}-{\displaystyle {\displaystyle \sum}_{j\ne i}}{I}_{t-{s}_{j}}^{\left(j\right)}\right)+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}+{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha ({X}_{t}-{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}}^{\left(i\right)})+\left(1-\alpha \right)\left({S}_{t-1}+{T}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}-{S}_{t-1}\right)+\left(1-\gamma \right){T}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\left({X}_{t}-{S}_{t}-{\displaystyle {\displaystyle \sum}_{j\ne i}}{I}_{t-{s}_{j}}^{\left(j\right)}\right)+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}+k{T}_{t}+{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha ({X}_{t}-{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}}^{\left(i\right)})+\left(1-\alpha \right)\left({S}_{t-1}+\varphi {T}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}-{S}_{t-1}\right)+\left(1-\gamma \right)\varphi {T}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\left({X}_{t}-{S}_{t}-{\displaystyle {\displaystyle \sum}_{j\ne i}}{I}_{t-{s}_{j}}^{\left(j\right)}\right)+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}+{T}_{t}{{\displaystyle \sum}}_{n=1}^{k}{\varphi}^{n}+{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha ({X}_{t}-{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}}^{\left(i\right)})+\left(1-\alpha \right)\left({S}_{t-1}{R}_{t-1}\right)$ ${R}_{t}=\gamma \left({S}_{t}/{S}_{t-1}\right)+\left(1-\gamma \right){R}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\left({X}_{t}-{S}_{t}-{\displaystyle {\displaystyle \sum}_{j\ne i}}{I}_{t-{s}_{j}}^{\left(j\right)}\right)+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}{R}_{t}^{k}+{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\alpha ({X}_{t}-{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}}^{\left(i\right)})+\left(1-\alpha \right)\left({S}_{t-1}{R}_{t-1}^{\varphi}\right)$ ${R}_{t}=\gamma \left({S}_{t}/{S}_{t-1}\right)+\left(1-\gamma \right){R}_{t-1}^{\varphi}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\left({X}_{t}-{S}_{t}-{\displaystyle {\displaystyle \sum}_{j\ne i}}{I}_{t-{s}_{j}}^{\left(j\right)}\right)+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}{R}_{t}^{{{\displaystyle \sum}}_{j=1}^{k}{\varphi}^{j}}+{\displaystyle {\displaystyle \sum}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+$ | |

Damped Multiplicative | ${S}_{t}=\frac{\alpha {X}_{t}}{{{\displaystyle \prod}}_{i}{I}_{t-{s}_{i}}^{\left(i\right)}}+\left(1-\alpha \right){S}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\frac{{X}_{t}}{{S}_{t}{{\displaystyle \prod}}_{j\ne i}{I}_{t-{s}_{j}}^{\left(j\right)}}+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)={S}_{t}{\displaystyle {\displaystyle \prod}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\frac{\alpha {X}_{t}}{{{\displaystyle \prod}}_{i}{I}_{t-{s}_{i}}^{\left(i\right)}}+\left(1-\alpha \right)\left({S}_{t-1}+{T}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}-{S}_{t-1}\right)+\left(1-\gamma \right){T}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\frac{{X}_{t}}{{S}_{t}{{\displaystyle \prod}}_{j\ne i}{I}_{t-{s}_{j}}^{\left(j\right)}}+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)=\left({S}_{t}+k{T}_{t}\right){\displaystyle {\displaystyle \prod}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\frac{\alpha {X}_{t}}{{{\displaystyle \prod}}_{i}{I}_{t-{s}_{i}}^{\left(i\right)}}+\left(1-\alpha \right)\left({S}_{t-1}+\varphi {T}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}-{S}_{t-1}\right)+\left(1-\gamma \right)\varphi {T}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\frac{{X}_{t}}{{S}_{t}{{\displaystyle \prod}}_{j\ne i}{I}_{t-{s}_{j}}^{\left(j\right)}}+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)=\left({S}_{t}+{T}_{t}{{\displaystyle \sum}}_{n=1}^{k}{\varphi}^{n}\right){\displaystyle {\displaystyle \prod}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\frac{\alpha {X}_{t}}{{{\displaystyle \prod}}_{i}{I}_{t-{s}_{i}}^{\left(i\right)}}+\left(1-\alpha \right)\left({S}_{t-1}{R}_{t-1}\right)$ ${T}_{t}=\gamma \left({S}_{t}/{S}_{t-1}\right)+\left(1-\gamma \right){R}_{t-1}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\frac{{X}_{t}}{{S}_{t}{{\displaystyle \prod}}_{j\ne i}{I}_{t-{s}_{j}}^{\left(j\right)}}+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)=\left({S}_{t}{R}_{t}^{k}\right){\displaystyle {\displaystyle \prod}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ | ${S}_{t}=\frac{\alpha {X}_{t}}{{{\displaystyle \prod}}_{i}{I}_{t-{s}_{i}}^{\left(i\right)}}+\left(1-\alpha \right)\left({S}_{t-1}{R}_{t-1}^{\varphi}\right)$ ${R}_{t}=\gamma \left({S}_{t}/{S}_{t-1}\right)+\left(1-\gamma \right){R}_{t-1}^{\varphi}$ ${I}_{t}^{\left(i\right)}={\delta}^{\left(i\right)}\frac{{X}_{t}}{{S}_{t}{{\displaystyle \prod}}_{j\ne i}{I}_{t-{s}_{j}}^{\left(j\right)}}+\left(1-{\delta}^{\left(i\right)}\right){I}_{t-{s}_{i}}^{\left(i\right)}$ ${\widehat{X}}_{t}\left(k\right)=\left({S}_{t}{R}_{t}^{{{\displaystyle \sum}}_{j=1}^{k}{\varphi}^{j}}\right){\displaystyle {\displaystyle \prod}_{i}}{I}_{t-{s}_{i}+k}^{\left(i\right)}+{\phi}_{AR}^{k}{\epsilon}_{t}$ |

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**Figure 2.**Means and Fisher’s LSD (least significant differences) diagram of the MAPE obtained against the nHWT models considered.

**Figure 3.**Mean graphs for the MAPE according to the initialization methods analyzed. Left column: Additive trend methods, right column: Multiplicative. From top to bottom: Level, trend, and seasonality. The levels used are those described in Table 4.

**Figure 4.**Distribution of initial values obtained for the trend in additive trend models $AM{C}_{24,168}$ and $AA{C}_{24,168}$.

**Figure 5.**Distribution of the initial values obtained for the level in the additive trend models $AM{C}_{24,168}$ y $AA{C}_{24,168}$.

**Figure 6.**Distribution of the initial seasonal indices for the AMC

_{24,168}model using the simple method. In the left part, the seasonality of 24 hours is used, and in the right part the seasonality of 168 hours is used. The central line represents the average of the values, regardless of the time of the year, while the shaded area represents the variability with ± standard deviation.

**Figure 7.**Distribution of the initial seasonal indices for the AMC

_{24,168}model using the Winters method. In the left part, the seasonality of 24 hours is used, and in the right part the seasonality of 168 hours is used. The central line represents the average of the values, regardless of the time of the year, while the shaded area represents the variability with ± standard deviation.

**Table 1.**Multiple seasonal Holt–Winters models’ nomenclature according to the trend and seasonal method.

Trend | Seasonal | |||||
---|---|---|---|---|---|---|

AR (1) Adjusted | Non-Adjusted | |||||

None | Additive | Multiplicative | None | Additive | Multiplicative | |

None | NNC | NAC | NMC | NNL | NAL | NML |

Additive | ANC | AAC | AMC | ANL | AAL | AML |

Damped Additive | dNC | dAC | dMC | dNL | dAL | dML |

Multiplicative | MNC | MAC | MMC | MNL | MAL | MML |

Damped Multiplicative | DNC | DAC | DMC | DNL | DAL | DML |

**Table 2.**Proposed methods and the method published in [25] for obtaining the initial value of the level in multiple-seasonal methods.

Denomination | Calculation |
---|---|

Moving average adapted from Holt [3] | ${S}_{0}=\frac{1}{{S}_{m}}\left[\frac{{X}_{1+{s}_{m}}-{X}_{1}}{{s}_{m}}+\frac{{X}_{2+{s}_{m}}-{X}_{2}}{{s}_{m}}+\dots +\frac{{X}_{2{s}_{m}}-{X}_{{s}_{m}}}{{s}_{m}}\right]$ |

First value [25] | ${\mathrm{S}}_{0}={\mathrm{X}}_{1}$ |

First period’s average adapted from Winters [4] | ${S}_{0}=\frac{{{\displaystyle \sum}}_{1}^{{s}_{m}}{X}_{i}}{{s}_{m}}$ |

Denomination | Calculation |
---|---|

Newbold [26] | ${T}_{0}=0$ (additive) ${T}_{0}=1$ (multiplicative) |

Taylor [6] | ${T}_{0}=\frac{\left(\frac{{\overline{D}}_{1}-{\overline{D}}_{2}}{{s}_{m}}+\frac{{{\displaystyle \sum}}_{1}^{2{s}_{m}}\nabla {D}_{12}}{2{s}_{p}}\right)}{2}$ |

Overall and two periods method. Multiplicative. | ${T}_{0}=\frac{{{\displaystyle \sum}}_{i=1,\dots ,q}{T}_{0i}^{\prime}}{q}$, with ${T}_{0i}^{\prime}=\{\begin{array}{ccc}\frac{{\overline{D}}_{q}-{\overline{D}}_{1}}{q}& \mathit{i}\mathit{f}& {s}_{i}={s}_{m}\\ \frac{{{\displaystyle \sum}}_{1}^{{s}_{i}}\nabla {D}_{i}}{{s}_{i}}& \mathit{i}\mathit{f}& rest\end{array}$ |

Overall and two periods method. Multiplicative. | ${T}_{0}={\displaystyle \prod}_{1}^{{n}_{s}}{T}_{0i}^{\prime}$, with ${T}_{0i}^{\prime}=\{\begin{array}{ccc}\frac{{\overline{D}}_{q}-{\overline{D}}_{1}}{q}& \mathit{i}\mathit{f}& {s}_{i}={s}_{m}\\ \frac{{{\displaystyle \sum}}_{1}^{{s}_{i}}\nabla {D}_{i}}{{s}_{i}}& \mathit{i}\mathit{f}& rest\end{array}$ |

Factor | Type | Initialization Methods | |
---|---|---|---|

New Methods | Older | ||

Level | Controllable | Moving average, average | First value [25], Taylor [6] |

Trend | Controllable | Overall, two periods | Newbold [26], Taylor [6] |

Seasonality | Controllable | Normal, Winters, NIST | Simple [6] |

**Table 5.**Analysis of the variance for the factors studied using the models with multiplicative trend.

Source | Sum of Squares | df | Mean Square | F-Ratio | p-Value | |
---|---|---|---|---|---|---|

Main effects | ||||||

A | Level | 283.89 | 3 | 94.63 | 6.66 | 0.0002 |

B | Trend | 2.96E6 | 3 | 984124. | 69240.76 | 0.0000 |

C | Seasonality | 1099.63 | 3 | 366.52 | 25.79 | 0.0000 |

D | Model | 19.36 | 1 | 19.36 | 1.36 | 0.2432 |

E | Season | 91.29 | 3 | 30.43 | 2.14 | 0.0931 |

Interactions | ||||||

AB | 876.46 | 9 | 97.3843 | 6.85 | 0.0000 | |

BC | 300.32 | 9 | 33.3691 | 2.35 | 0.0124 | |

BE | 296.07 | 9 | 32.8968 | 2.31 | 0.0138 | |

CD | 189.57 | 3 | 63.1903 | 4.45 | 0.0040 | |

Residuals | 28483.0 | 2004 | 14.2131 | |||

Total | 2.98E6 | 2047 |

**Table 6.**Analysis of the variance for the factors studied using the models with additive trend, without adjustment.

Source | Sum of Squares | df | Mean Square | F-ratio | p-Value | |
---|---|---|---|---|---|---|

Main effects | ||||||

A | Level | 0.8485 | 3 | 0.282849 | 5.59 | 0.0008 |

B | Trend | 1.3228 | 3 | 0.440938 | 8.71 | 0.0000 |

C | Seasonality | 1864.54 | 3 | 621.514 | 12276.07 | 0.0000 |

D | Model | 73.2721 | 1 | 73.2721 | 1447.26 | 0.0000 |

E | Season | 1.1902 | 3 | 0.396725 | 7.84 | 0.0000 |

Interactions | ||||||

AE | 1.19074 | 9 | 0.132305 | 2.61 | 0.0053 | |

BD | 1.1674 | 3 | 0.389135 | 7.59 | 0.0000 | |

CD | 299.458 | 3 | 99.8192 | 1971.62 | 0.0000 | |

Residuals | 102.218 | 2019 | 0.0506281 | |||

Total | 2345.21 | 2047 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Trull, O.; García-Díaz, J.C.; Troncoso, A.
Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models. *Mathematics* **2020**, *8*, 268.
https://doi.org/10.3390/math8020268

**AMA Style**

Trull O, García-Díaz JC, Troncoso A.
Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models. *Mathematics*. 2020; 8(2):268.
https://doi.org/10.3390/math8020268

**Chicago/Turabian Style**

Trull, Oscar, Juan Carlos García-Díaz, and Alicia Troncoso.
2020. "Initialization Methods for Multiple Seasonal Holt–Winters Forecasting Models" *Mathematics* 8, no. 2: 268.
https://doi.org/10.3390/math8020268