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6 changes: 4 additions & 2 deletions nbs/docs/models/Holt.ipynb
Original file line number Diff line number Diff line change
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"\n",
"The components in the ETS taxonomy have clear interpretations: level shows average value per time period, trend reflects the change in the value, while seasonality corresponds to periodic fluctuations (e.g. increase in sales each January). Based on the the types of the components above, it is theoretically possible to devise 30 ETS models with different types of error, trend and seasonality. Figure 1 shows examples of different time series with deterministic (they do not change over time) level, trend, seasonality and with the additive error term.\n",
"\n",
"![\"Figure 1: Time series corresponding to the additive error ETS models\"](https://openforecast.org/adam/Svetunkov--2022----ADAM_files/figure-html/ETSTaxonomyAdditive-1.png)\n",
"![\"Figure 1: Time series corresponding to the additive error ETS models\"](https://openforecast.org/adam/Svetunkov--2023----Forecasting-and-Analytics-with-the-Augmented-Dynamic-Adaptive-Model--ADAM-_files/figure-html/ETSTaxonomyAdditive-1.png)\n",
"*Figure 4.1: Time series corresponding to the additive error ETS models*\n",
"\n",
"Things to note from the plots in Figure.1:\n",
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"cell_type": "markdown",
"metadata": {},
"source": [
"![](https://openforecast.org/adam/Svetunkov--2022----ADAM_files/figure-html/ETSTaxonomyMultiplicative-1.png)\n",
"![](https://openforecast.org/adam/Svetunkov--2023----Forecasting-and-Analytics-with-the-Augmented-Dynamic-Adaptive-Model--ADAM-_files/figure-html/ETSTaxonomyMultiplicative-1.png)\n",
"*Figure 2: Time series corresponding to the multiplicative error ETS models*\n",
"\n",
"The graphs in Figure 2 show approximately the same idea as the additive case, the main difference is that the error variance increases with increasing data level; this becomes clearer in ETS(M,A,N) and ETS(M,M,N) data. This property is called heteroskedasticity in statistics, and Hyndman et al. (2008) argue that the main benefit of multiplicative error models is to capture this characteristic."
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"\n",
"\n",
"Table 1: Additive error ETS models\n",
"\n",
"| | Nonseasonal\t|Additive\t|Multiplicative|\n",
"|----|-----------|-----------|--------------|\n",
"|No trend|$\\begin{aligned} &y_{t} = l_{t-1} + \\epsilon_t \\\\ &l_t = l_{t-1} + \\alpha \\epsilon_t \\end{aligned}$ |$\\begin{aligned} &y_{t} = l_{t-1} + s_{t-m} + \\epsilon_t \\\\ &l_t = l_{t-1} + \\alpha \\epsilon_t \\\\ &s_t = s_{t-m} + \\gamma \\epsilon_t \\end{aligned}$ |$\\begin{aligned} &y_{t} = l_{t-1} s_{t-m} + \\epsilon_t \\\\ &l_t = l_{t-1} + \\alpha \\frac{\\epsilon_t}{s_{t-m}} \\\\ &s_t = s_{t-m} + \\gamma \\frac{\\epsilon_t}{l_{t-1}} \\end{aligned}$|\n",
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"\n",
"\n",
"Table 2: Multiplicative error ETS models\n",
"\n",
"| |Nonseasonal\t |Additive\t|Multiplicative|\n",
"|------|-------------|----------|--------------|\n",
"|No trend| $\\begin{aligned} &y_{t} = l_{t-1}(1 + \\epsilon_t) \\\\ &l_t = l_{t-1}(1 + \\alpha \\epsilon_t) \\end{aligned}$ | $\\begin{aligned} &y_{t} = (l_{t-1} + s_{t-m})(1 + \\epsilon_t) \\\\ &l_t = l_{t-1} + \\alpha \\mu_{y,t} \\epsilon_t \\\\ &s_t = s_{t-m} + \\gamma \\mu_{y,t} \\epsilon_t \\end{aligned}$ | $\\begin{aligned} &y_{t} = l_{t-1} s_{t-m}(1 + \\epsilon_t) \\\\ &l_t = l_{t-1}(1 + \\alpha \\epsilon_t) \\\\ &s_t = s_{t-m}(1 + \\gamma \\epsilon_t) \\end{aligned}$|\n",
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