Because differenced series are series of differences, the reconstruction is simply a cumulative sum of the input values. Which values are summed together is dependent on the lag of the differenced series – for example, if the series was differenced with a lag of 1, then reconstruction each reconstructed value would be a sum of the previous reconstructed value and the corresponding value in the difference series. If the series was differenced with a lag of 2, then it would be a sum of the reconstructed value 2 steps prior, with the corresponding value, and so on. If the series was differenced multiple times, the previous steps would occur that many times, using the partially reconstructed series as input.. It is important to note that to completely reconstruct the original series, the initial values of the original series must be given. If the series was differenced with lag 1, then the initial value of the original series must be present for a full reconstruction. With a lag of 2, the initial 2 values must be present, and so on. If the series was differenced multiple times, then the initial values of the intermediate steps must be given.. Reconstruction can be done without initial values, but the output series will not be identical to the original series, merely similar in characteristics.
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