One of the earlier applications of SSA in combination with forecasting was done by Kepenne and Ghil (1993). The data was first prefiltered by applying multichannel SSA (M-SSA), where after the time series was forecasted with the maximum entropy method (MEM). This approach allowed predictabilities of the subannual variability in atmospheric angular momentum of up to a month. The combination of these two techniques proved to be very successful, due to the nature of SSA (and therefore M-SSA) to remove variations in the data, resulting in a smooth time series and the good reputation of MEM of being able to predict smooth time series quite accurately.
Lisi et al. (1995) saw the opportunity to combine nonlinear dynamical system and artificial intelligence theory to perform forecasting of time series. They first performed adaptive noise reduction of the data by using an algorithm based on SSA and then did the forecasting by means of standard feed forward neural prediction models. They repeated the SSA of the series for a whole spectrum of retained components, calculating the normalized mean square prediction error of each forecast of the reconstruction of the original signal. The number of principal components corresponding to the smallest normalized mean square prediction error was then retained. They found that their approach was very successful for relatively short and very noisy time series, both regarding short- and long-term predictions.
The work from the above study was continued in (Lisi and Medio, 1997) where the predictability of the exchange rate was investigated, in reference to a model that implies future prices are unpredictable if the information set that are used for the predictions is the past prices. The validity of this model was taken into question and attempts were made to provide forecasts for exchange rate time series. Lisi and Medio mentioned that existing tests for nonlinearity had many shortcomings, in that they failed to generate consensus despite their relatively weak hypothesis tests. The probable reason for this was given as the lack of robustness of the tests and the differences in their power functions.
The aim of the paper from Lisi and Medio was to test a hypothesis they had labelled the Nonlinear Hypothesis. This hypothesis stated that the data does contain some structure and this structure can be exploited for short-term predictions. It was also assumed that the data have been generated by a nonlinear generating system with some additional noise added. This hypothesis was tested by combining a number of techniques from the dynamical
systems theory in a novel way. This combination of techniques essentially resulted in SSA and multi-channel SSA (MSSA). The method used for the prediction of the economic time series is the nearest neighbour method, which is based on the principle that similar states over a small enough time interval will have similar successors. The successors of the past states are therefore used to predict the successors of the future similar states.
The data series that Lisi and Medio used to test their hypothesis, were the monthly spot exchange rates of seven major foreign currencies and it was attempted to do one-step ahead forecasting. These data sets were divided into training and validation data sets, with the predictions being done on both unfiltered data and data that have been filtered using MSSA, as well as a trivial prediction using the random walk hypothesis. Two-channel SSA was used for the filtering and when the prediction of the validation data was then attempted, the second currency was used as an auxiliary currency. The mean square prediction error, as well as the mean absolute prediction error was used to quantify the comparison between the random walk prediction and the two local linear or nearest neighbour prediction models. Their results indicated that the time series could successfully be modelled on the short term and that the local linear predictions on the filtered data outperformed that of the unfiltered data.
Masulli et al. (2001) presented a constructive approach to time series learning and the forecasting of individual rainfall intensities series. The specific method used was a decompositive ensemble method that was based on SSA. This method extended the constructive approach to the learning of discontinuous and/or intermittent signals.
A multi-layer perceptron neural network was used for the modelling, where the embedding dimension was determined by the Global False Nearest Neighbours method and the time lag of the input was the first minimum of the average mutual information of the signal (Abarbanel, 1996). Even though (Ormerod and Campbell, 1997) had serious misgivings about the value of SSA in a predictive environment, the application of SSA to prediction can be supported by the argument from (Masulli et al., 2001) that, since the principal components are filtered versions of the signal and are typically band-limited, they should behave more regular than the raw series and would hence be more predictable.
Computational costs were reduced during the predictions by combining the reconstructed components of similar explained variance into reconstructed waves and then predicting these waves. The prediction of the original series was then recovered as the sum of those of all the individual series components.
It was found that by using a constructive methodology, efficient predictors could be designed, even for complex signal such as those that are discontinuous or intermittent. The ensemble method combined an unsupervised and a supervised step. The unsupervised step was the decomposition where the original signal was decomposed with the aid of SSA into
reconstructed waves. The supervised step was the designing and learning of the MLP predictors for each of the reconstructed waves, with the aid of suggestions from dynamical systems theory.
Singular spectrum analysis was also combined with the even more advanced technique of genetic algorithms to forecast the solar cycle (Orfila et al., 2002). The results that were obtained from this study was in good agreement with known behaviour of the solar cycle and it would therefore seem as if SSA can also be used to reconstruct data to be forecasted with genetic algorithms.
2.4.2 Change point detection
The detection of changes in time series is another area that has been the focus of a great amount of research. In (Moskvina, 2001, Moskvina and Schmidt, 2003, Moskvina and Zhigljavsky, 2003) work was done into the application of SSA to change-point detection. It was found that the sequential application of SSA could be used to detect change-points in time series. The change-point detection algorithm is based on the idea that if the mechanism generating a certain time series changes at a specific moment in time, the distance between the subspace spanned by the eigenvectors and the vectors after the change point will increase.
Moskvina identified four parameters, in addition to the normal SSA parameters (see chapter 3), that should be optimized for the change-point detection algorithm. These are the section of the time series on which SSA will be applied, two parameters concerning the test sample of which the closeness is evaluated and the threshold value by which the significance level is
determined. It was found that this technique based on singular spectrum analysis performed well when compared numerically to other change-point detection techniques.
This technique for change point detection has also been applied by Choi et al. (2002), in an effort to facilitate the timely detection of changes in traffic loads. The emphasis of their work was rather on determining suitable sampling techniques and the change point detection algorithm based on SSA was just used to validate the sampling techniques.