Zero-value problems of the logarithmic mean divisia index decomposition method

Recently, the Logarithmic Mean Divisia Index (LMDI) approach to energy decomposition has been espoused as the preferred indexing method. Whilst the LMDI method provides perfect decomposition, and is time-reversal invariant, its strategy to handle zero-values is not necessarily robust. In order to overcome this problem, it has been recommended to substitute a small value δ=10-10–10-20δ=10-10–10-20 for any zero values in the underlying data set, and allow the calculation to proceed as usual. The decomposition results are said to converge as δδ approaches zero. However, we show that under this recommended procedure the LMDI can produce significant errors if applied in the decomposition of a data set containing a large number of zeroes and/or small values. To overcome this problem, we recommend using the analytical limits of LMDI terms in cases of zero values. These limits can be substituted for entire computational loops, so that in addition to providing the correct decomposition result, this improved procedure also drastically reduces computation times.