# The Effect of Data Transformation on Singular Spectrum Analysis for Forecasting
## 5 Data Analysis
R packages \"Rssa\" [36, 37, 38] and \"nparLD\" [39] are employed to implement SSA forecasting and the nonparametric repeated measure factorial test, respectively. We apply SSA to three versions of a dataset: a dataset without any transformation, a standardised dataset, and a log-transformed dataset. For each of the three datasets, we obtain RMSFE from out-of-sample forecasting at forecast horizons \(h=1,3,6,12\). It is noteworthy that our aim in this paper is to examine the effect of transformation in SSA forecasting. Thus, we consider the best forecast based on the RMSFE of the last 12 data points regardless of whether the forecast is from the recurrent or vector-based approach. We also know that the window length \(m\), the number of components \(k\), and the forecasting methods (recurrent and vector) affect the forecasting outcome. Thus, we adopt a computationally intensive approach by considering combinations of \(m\) and \(k\), and methods that provide the minimum RMSFE for the out-of-sample forecast for the last 12 data points. The RMSFEs obtained from the computationally intensive approach are given in Tables A3-A6. Figure 1: A selection of nine real time series. Given that the best forecasting results are achieved by util ising a computationally intensive approach, we seek to identify the factors that can affect the RMSFE. In order to address this, we employ statistical tests described in Section 4.2. For each of the series with RMSFE reported in Tables 12-13, we examine the characteristics of the time series by employing a statistical test, as described in Section 4.2. At this stage, we are ready with the inputs for nonparametric repeated measure factorial test to conduct testing on the treatment effect (data transformation) under different characteristics of these time series. Results obtained from the Wald type tests are provided in Table 2.  
Based on the Wald-type test results in Table 2, we may conclude that, at the \(\alpha=0.05\) significance level,  
1. normality does not affect SSA forecasting performance;
2. stationarity affects SSA forecasting performance in long-term forecasting (h = 12) but not at shorter horizons;
3. skewness and sampling (observation) frequency affect SSA forecasting performance;
4. transformation does not affect SSA forecasting performance, but the interaction between sampling frequencies and transformation is significant, which means the SSA performance is affected by transformation at some sampling frequencies. The above findings are important in the practice for several reasons. First, in the real world, it is well known that most time series do not meet the assumption of normality. However, as the effect of normality and its interactions with transformations are not significant, when faced with normally distributed data, our findings indicate that there is no impact on the forecasting accuracy of SSA with or without data transformations. Furthermore, these findings also indicate that data transformations do not improve the forecast accuracy in non-normal data either. Secondly, we find that, when series are stationary, it affects the long-term forecasting accuracy of SSA. However, when generating short-term forecasts, the forecasting accuracy of SSA is not affected by stationarity. Thirdly, in reality, as most time series are skewed and increasingly found at varying frequencies (especially following the emergence of Big Data), these findings show that forecasters should remember that varying skewness and frequency of data are features indicative of the need for careful exploration of the use of SSA as the forecasts are sensitive to these features. In general, transformations are not required when forecasting with SSA, as there is no evidence of transformations impacting the SSA forecasting performance; however, there could be a significant impact at certain sampling frequencies. This indicates that, when modelling data with different frequencies, the sensitivity of SSA forecasts to such frequencies could potentially be controlled by transforming the input data.