# The Effect of Data Transformation on Singular Spectrum Analysis for Forecasting
## 5 Data Analysis
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Model** & **Factor** & \multicolumn{4}{c}{**P-Value**} \\  & & **h = 1** & **h = 3** & **h = 6** & **h = 12** \\ \hline RMSFE \(\sim\) Tr + Skew & Skew & **0.0037** & **0.0043** & **0.0056** & **0.0131** \\  + Tr \(\times\) Skew & Tr & 0.0718 & 0.1447 & 0.4186 & 0.2098 \\  & Tr \(\times\) Skew & 0.4177 & 0.5106 & 0.2120 & 0.1482 \\ \hline RMSFE \(\sim\) Tr + Stationarity & Stationarity & 0.0997 & 0.053 & 0.0501 & **0.0248** \\  + Tr \(\times\) Stationarity & Tr & 0.2351 & 0.3754 & 0.7607 & 0.5276 \\  & Tr \(\times\) Stationarity & 0.5160 & 0.6808 & 0.7678 & 0.3792 \\ \hline RMSFE \(\sim\) Tr + Normality & Normality & 0.5052 & 0.5320 & 0.4954 & 0.5820 \\  + Tr \(\times\) Normality & Tr & 0.0747 & 0.1152 & 0.5849 & 0.4892 \\  & Tr \(\times\) Normality & 0.2492 & 0.3576 & 0.4042 & 0.4549 \\ \hline RMSFE \(\sim\) Tr + Freq & Freq & **0.0000** & **0.0000** & **0.0000** & **0.0000** \\  + Tr \(\times\) Freq & Tr & 0.0841 & 0.1194 & 0.1355 & 0.1143 \\  & Tr \(\times\) Freq & **0.0000** & **0.0000** & **0.0000** & **0.0000** \\ \hline RMSFE \(\sim\) Tr & Tr & 0.4271 & 0.6740 & 0.9535 & 0.4860 \\ \hline \hline \end{tabular} Here, Freq, Skew, and Tr represent frequency, skewness, and transformation, respectively. Bold values show the significant effects at the \(\alpha=0.05\) significance level.  
\end{table} 
Table 2: Wald-type test results. Sampling frequencies under investigation are hourly (F H), daily (F D), weekly (F W), monthly (F M), and annual (F A). When the relative effect plots in Figures 3 and 4 are compared with the effect plots in Figure 2, we can evaluate how the hourly (F H), weekly (F W), quarterly (F Q), and annual (F A) sampling frequencies are affecting the forecasting performance of SSA. Moreover, the change in shape of the transform's relative effects (e.g., see the difference between the shapes of \"F Q\" and \"F H\" lines in Figures 3 and 4) suggests an interaction between transformation and sampling frequency. We analyse the results by forecasting horizon. It can be seen in Figure 3 that, in very short-term forecasting (\(h=1\)), the standardisation produces a comparatively large RMSFE in quarterly frequencies, while the log transformation reports a slightly larger RMSFE at daily, quarterly, hourly, and annual frequencies. This indicates that users should certainly avoid transforming data with quarterly frequencies when forecasting at \(h=1\) step ahead with SSA. In the short-term forecasting horizon (\(h=3\)) (see Figure 3), the smallest RMSFE belongs to standardisation for monthly frequencies, while standardisation has the largest RMSFE at quarterly frequencies. In mid- and long-term forecasting horizons (\(h=6\) and \(12\)), which are visible in Figure 4, the following can be seen. At \(h=6\) steps ahead, standardisation produces the lowest RMSFE at monthly sampling frequencies, whilst it has the largest RMSFE in quarterly and weekly time series data. The log transformation produces higher RMSFEs at daily, hourly, and annual frequencies. Accordingly, the only instance when standardisation could produce better forecasts with SSA at this horizon is when faced with monthly data. At \(h=12\) steps ahead, standardisation leads to better forecasts at daily frequencies, whilst log transformations can provide better forecasts with SSA at weekly frequencies. Finally, these findings indicate that standardisation should only be used to transform data when forecasting with SSA at \(h=12\) steps ahead at the daily frequency, at \(h=3\) or \(h=6\) steps ahead when dealing with a monthly frequency, and at \(h=1\) step ahead when forecasting data with monthly or weekly frequencies. At the same time, standardisation should not be employed when forecasting quarterly data at any frequency, as it worsens the forecasting accuracy by comparatively larger margins. Interestingly, log transformations are only suggested when dealing with forecasting weekly data at \(h=6\) or \(h=12\) steps ahead.