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 

$\delta =10^{-10}–10^{-20}$

 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 

$\delta$

 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.

n recent articles by Ang and colleagues (Ang et al., 1998, Ang et al., 2003; Ang, 2004a, Ang, 2004b), the Logarithmic Mean Divisia Index (LMDI) approach to energy decomposition analysis has been espoused as the preferred indexing method. Whilst the LMDI 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 

$\delta =10^{-10}–10^{-20}$

 for any zero values in the underlying data set, and allow the calculation to proceed as usual (Ang et al., 1998; Ang, 2004b, p. 4). The decomposition results are said to converge as 

$\delta$

 approaches zero (Ang and Choi, 1997, p. 68).

In this article we show that this procedure can lead to significant errors in decompositions of data sets containing a large number of zeroes and/or small values.1 In the following, we first briefly review the LMDI methodology, then demonstrate the zero-value problem, and finally offer a remedy.

The Logarithmic Mean Divisia Index (LMDI) decomposition—at least under the recommended procedure—may be theoretically zero-value robust, but cannot handle zero values if the data set to be decomposed contains a large number of zeroes and/or small values. Substituting a small value 

$\delta =10^{-10}–10^{-20}$

 for any zero values, and allowing the calculation to proceed as usual can lead to significant errors.

To overcome this problem, we recommend using the limits published by Ang et al. (1998) for the eight possible cases of zero values (see Table 1, right column). These limits can be substituted for entire computational loops representing multiple summations as in Eqs. (5), (6), (7). In addition to providing the correct decomposition result, this improved procedure also drastically reduces computation times for the LMDI, especially when zero values are numerous and/or when the function to be decomposed is a product of many factors and/or large matrices.

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