(1) Component analysis
Regardless of mathematical technique and the complexity of statistical models applied in the time series prediction, component analysis mainly focuses on four key components of the time series, namely level, trend, seasonality and noises (Mentzer & Moon, 2005). The goal of the statistical analysis is to determine the level, trend and seasonality features of a time series.
Level is usually described with different forms of average, which reflect the central tendency of the historical time series data. The simplest measure of level is the arithmetic mean. If the time series is relatively stable over a certain period and only havs small movements around a fixed point, as shown in Figure 3.1, the future development can be forecasted using the arithmetic mean of the historical data:
Fn+1= ∑nt=1St /n (3.1) Where,
Fn+1: forecast value for the next period;
St: historical data at the period t;
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Figure 3.1: Level only time series
The advantage of this measure is that it is usually quite straightforward to calculate and it also removes the effect of fluctuations. However, when the time series is not stable with small level changes, the performance of the average-based forecast can be quite poor (Mentzer & Moon, 2005). In order to dampen out the prediction errors resulting from the instability and fluctuation of the time series, moving average was introduced, which is the average of the time series data in a previous fixed period (Jones, 1966).
Fn+1 = ∑nt=n+1−iSt/i (3.2)
Where,
i: the selected periods which is also known as the selected ‘window’.
Compared to simple average which basically gives the same weights to all the historical data, moving average only considers the data captured in the selected window as the indicator for the forecast. Apparently, moving average based prediction can better reflect the recent development of the time series. However, as the selection of the size of the window is quite subjective, there is no absolute benchmark to assess whether the selected window is optimal or not. When the window size becomes larger, the moving average gets closer to the simple arithmetic average. On the contrary, when only one period is chosen for the moving average calculation, it basically assumes the last period as the best forecast for the next period (Mentzer & Moon, 2005). In order to address the issues of moving average and establish a more systematic way of assigning weightings to the previous periods, a technical method, such as exponential smoothing which gives more emphasis on the most recent data (Jones, 1966), is introduced. 0 1 2 3 0 2 4 6 8
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Ft+1 = αSt+ (1 − α)Ft (3.3)
Where,
St: actual data at the period t;
Ft: forecast value for the period t;
α: weighting for the estimation (between 0 and 1).
When the time series does not stay at its current level and switches to other levels continuously, trend is brought into the prediction model to reflect the level changes. In general, trend can be considered as the difference between the two levels:
Tt= Lt− Lt−1 (3.4)
Where,
Tt: trend at the period t;
Lt: level at the period t.
Similar to the estimation of level, exponential smoothing can also be applied to estimate trend (Mentzer & Moon, 2005):
Tt+1= β(Lt− Lt−1) + (1 − β)Tt (3.5)
Where,
Tt+1: forecast trend for the next period;
Lt− Lt−1: actual trend at the period t; Tt: forecast trend in the period t;
β : weighting for estimation (between 0 and 1).
Apart from level and trend, some time series also reveal significant seasonal patterns. For instance, the subsidence of the subgrade in plateau during winter and summer can vary materially. There are two main methods to adjust for seasonality, namely additive seasonal adjustment and multiplicative seasonal adjustment:
Ft+1= St+ SAt−Z (Additive) (3.6)
Ft+1= St⁄SAt−Z (Multiplicative) (3.7)
Where,
Ft+1: forcast seasonality for the next period;
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Z: the length of the seasonal cycle.
Combining the estimation of level, trend and seasonality together, the forecast of the time series can be made as follows:
Ft+1= α(St⁄SAt−Z) + (1 − α)(Lt−1+ Tt) (3.8)
Where,
Ft+1: forcast data for the next period;
St/SAt-Z: actual data at period t;
Lt-1+Tt: forcast data for the period t.
(2) Auto-regressive moving average
Among various statistical models which were built up based on the rationale introduced above, auto-regressive moving average (ARMA) is one of the most prevailing methodologies. It is a combination of two models including the auto- regressive (AR) model and moving average (MA) model. The AR model assumes that the prediction of the time series can be produced based on a finite number of previous observations:
St= c + ∑pn=1γnSt−n+ εt (3.9)
Where,
St: forcast value for the period t;
c: a constant;
p: number of historical periods involved, also known as the order of the AR model;
γ: parameters;
St-n: actual data for the period t-n;
εt: white noise at the period t, which is stationary and has expected value of 0.
Similarly, a MA model tries to estimate the future movements of the time series based on the average of the time series together with random noises during selected previous periods. The number of periods selected is known as the order of the MA model.
St = μ + εt+ ∑qn=1θnεt−n (3.10)
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μ: the mean of the time series (a constant);
εt: white noise at period t, which is stationary and has expected value of 0;
q: number of historical periods involved, also known as the order of the MA model;
θ: parameters;
εt−n: white noise at period t-n, which is stationary and has expected value of 0;
When considering both the AR and MA in the time series prediction model, it becomes auto-regressive moving average model (ARMA). The model ARMA (p, q) can be structured as follows (Brockwell & Davis, 2002):
St= c + εt+ ∑n=1p γnSt−n+ ∑qn=1θnεt−n (3.11)
In the equation above, the AR part represents the impact of historical time points on the current period, while the MA part can be considered as random fluctuations / shocks in the past. The ARMA model is particularly suitable for a time series which is not only influenced by historical movements but also hit by unexpected shocks. The most critical part for an ARMA model is to decide its orders, i.e. the values of p and q respectively. Since the ARMA model was firstly introduced by Wold in 1938, which described ARMA model theoretically (Wold, 1938), mathematicians have developed various methodologies to calculate and choose the order of the model. The most popular method was introduced by Box and Jenkins (Box & Jenkins, 1970) in 1970, which significantly facilitated the utilisation of ARMA model and made it feasible to apply the ARMA model to real world data. The main elements of Box and Jenkins approach are (Makridakis & Hibon, 1997):
• Proposing methods of transforming non-stationary time series to stationary series
• Using autocorrelation and partial autocorrelation to determine optimal values of p and q for the model
• Providing computer programme-based modules to estimate the parameters of the model
• Examining and optimising both the values of p and q as well as the parameters The ARMA model constructed through the Box-Jenkins methodology is known as the auto-regressive integrated moving average (ARIMA) model, which has become one
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of the most popular prediction models for time series. The ARIMA model can be interpreted as ARIMA (p, d, q), where p and q have the same definition as in the ARMA model. The additional d represents the number of differencing that the time series needs to make in order to be stationary.