{"id":2,"date":"2014-09-05T04:41:07","date_gmt":"2014-09-05T03:41:07","guid":{"rendered":"http:\/\/localhost\/wordpress-4.0\/wordpress\/?page_id=2"},"modified":"2025-12-12T12:53:02","modified_gmt":"2025-12-12T11:53:02","slug":"algorithm","status":"publish","type":"page","link":"https:\/\/project.dke.maastrichtuniversity.nl\/ssd\/algorithm\/","title":{"rendered":"Algorithm"},"content":{"rendered":"

Singular Spectrum Decomposition (SSD)<\/strong>\u00a0algorithm is highly based on the Singular Spectrum Analysis (SSA) <\/strong>algorithm, which has been shown suitable for short and noisy time series analysis. SSA is a principal component analysis-based nonparametric spectral estimation method, which uses no prior knowledge about the biology and physics of the system. The main\u00a0advantage of SSA is that its components are not necessarily harmonic functions and can capture highly non-harmonic oscillatory shapes. One main disadvantage is that both the window length, or embedding dimension, and the principal components to choose for reconstructing a specific component series have to be selected manually.\u00a0Those two processes have been automated in the SSD algorithm.<\/p>\n

First step of SSD algorithm is finding the trajectory matrix<\/strong> X<\/strong>. With respect to standard way of building it, a new approach is proposed.
\nGiven a time series x(n) of length N and an embedding dimension M, an ( M x N ) matrix X\u00a0<\/strong>is generated such that its i-th\u00a0row (where i = 1,…,M)\u00a0\u00a0is obtained as:
\n\"Hence:\u00a0\"X= [ x_{{1}}^{T}, x_{{2}}^{T},\ldots , x_{{M}}^{T}]^{T}\"<\/p>\n

For example, given the time series\u00a0\"x(n)= \left \{ 1,2,3,4,5 \right \}\", and an embedding dimension M=3, then the matrix X<\/strong> is:<\/p>\n

  <\/span>   <\/span>\"\[ \ X = \left| \begin{array}{ccccc}<\/p>\n

In the SSA algorithm\u00a0matrix\u00a0X<\/strong>\u00a0 would\u00a0correspond to the 3 x 3 left block of the above matrix:<\/p>\n

  <\/span>   <\/span>\"\[ \ X = \left| \begin{array}{ccc}<\/p>\n

Thanks to this new definition of the trajectory matrix in the SSD method, the oscillatory content of\u00a0X<\/strong> is enhanced and useful properties for the decrease of energy of the residual are provided.<\/p>\n

Next, the diagonal averaging procedure needs to be adapted to this new definition of trajectory matrix. In order to carry out the average along the i-th\u00a0cross-diagonal of\u00a0X<\/strong>, the wrapped part of the right hand block must be correctly appended to the top right of the left hand block.\u00a0With this approach every cross-diagonal has length\u00a0M. An example is shown below:<\/p>\n

  <\/span>   <\/span>\"\[ \ X = \left| \begin{array}{ccccc}<\/p>\n

However, care has to be taken in case of a sizable trend in the series. To avoid that situation at first iteration, a test for sizable trend\u00a0is performed.\u00a0In case the sizable trend is detected,\u00a0then a standard trajectory matrix is generated to guarantee reliable estimation of the trend. The\u00a0X<\/strong> found by that method is used in all the following iterations.<\/p>\n

Another vital step is\u00a0choosing the embedding dimension<\/strong>. The following criterion has been adopted for the automated choice of M at iteration j:<\/p>\n