## 2 Forecasting Models  
### State Space Models  
Exponential smoothing methods have been used with success to generate easily reliable forecasts for a wide range of time series since the 1950s [12]. In these methods, forecasts are calculated using weighted averages where the weights decrease exponentially as observations come from further in the past--the smallest weights are associated with the oldest observations. The most common representation of these methods is the component form. Component form representations of exponential smoothing methods comprise a forecast equation and a smoothing equation for each of the components included in the method. The components that may be included are the level component, the trend component and the seasonal component. By considering all of the combinations of the trend and seasonal components, fifteen exponential smoothing methods are possible. Each method is usually labeled by a pair of letters (TS) defining the type of 'Trend' and 'Seasonal' components. The possibilities for each component are: \(\text{Trend}=\{\text{N},\text{A},\text{A}_{\text{d}},\text{M},\text{M}_{\text{ d}}\}\) and \(\text{Seasonal}=\{\text{N},\text{A},\text{M}\}\). For example, \((\text{N},\text{N})\) denotes the simple exponential smoothing method, \((\text{A},\text{N})\) denotes [PERSON]'s linear method, \((\text{A}_{\text{d}},\text{N})\) denotes the additive damped trend method, \((\text{A},\text{A})\) denotes the additive Holt-Winters method and \((\text{A},\text{M})\) denotes the multiplicative Holt-Winters method, to mention the most popular For illustration, denoting the time series by \(y_{1},y_{2},\ldots,y_{n}\) and the forecast of \(y_{t+h}\), based on all of the data up to time \(t\), by \(\hat{y}_{t+h|t}\), the component form for the method \((\text{A},\text{A})\) is [13; 14]:  
\[\hat{y}_{t+h|t} = l_{t}+hb_{t}+s_{t-m+h_{m}^{+}}, \tag{1}\] \[l_{t} = \alpha\left(y_{t}-s_{t-m}\right)+\left(1-\alpha\right)\left(l_{t- 1}+b_{t-1}\right),\] (2) \[b_{t} = \beta^{*}\left(l_{t}-l_{t-1}\right)+\left(1-\beta^{*}\right)b_{t -1},\] (3) \[s_{t} = \gamma\left(y_{t}-l_{t-1}-b_{t-1}\right)+\left(1-\gamma\right)s_ {t-m}, \tag{4}\]  
where \(m\) denotes the period of the seasonality, \(l_{t}\) denotes an estimate of the level (or the smoothed value) of the series at time \(t\), \(b_{t}\) denotes an estimate of the trend (slope) of the series at time \(t\), \(s_{t}\) denotes an estimate of the seasonality of the series at time \(t\) and \(\hat{y}_{t+h|t}\) denotes the point forecast for \(h\) periods ahead where \(h_{m}^{+}=\lfloor\left(h-1\right)\text{ mod }m\rfloor+1\) (which ensures that the estimates of the seasonal indices used for forecasting come from the final year of the sample (the notation \(\lfloor u\rfloor\) means the largest integer not greater than \(u\)). The initial states \(l_{0},b_{0},s_{1-m},\ldots,s_{0}\) and the smoothing parameters \(\alpha,\beta^{*},\gamma\) are estimated from the observed data. The smoothing parameters \(\alpha,\beta^{*},\gamma\) are constrained between \(0\) and \(1\) so that the equations can be interpreted as weighted averages. Details about all of the other methods may be found in [PERSON] et al. [13]. To be able to generate prediction (or forecast) intervals and other properties,Hyndman et al. [9] (amongst others) developed a statistical framework for all exponential smoothing methods. In this statistical framework, each stochastic model, referred to as a state space model, consists of a measurement (or observation) equation that describes the observed data, and state (or transition) equations that describe how the unobserved components or states (level, trend, seasonal) change over time. For each exponential smoothing method, [PERSON] et al. [9] describe two possible state space models, one corresponding to a model with additive random errors and the other corresponding to a model with multiplicative random errors, giving a total of 30 potential models.