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In case of creating an index using standardised variables(between 0 and 1), how do we deal with the zeroes(while using Geometric mean for aggregation)? it could be the case that only one indicator is zero while other indicators are high? Anyone has any idea how do they deal with for instance in the case of calculating the HDI?

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Assuming "HDI" is the UN Human Development Index (first Google hit), defined by Wikipedia as something like

$$\mathrm{HDI}=\sqrt[3]{\mathrm{Longevity}\times\mathrm{Education}\times\mathrm{Income}}$$

then $\mathrm{HDI}=0$ if any of the three contributing indices is $0$. This makes sense, in terms of what HDI is trying to measure (e.g. a certain minimum standard is required in all three categories to count as "developed").

In Matlab, this result would happen automatically for your case, as log(0)=-Inf and your indices are all non-negative and finite, giving log_HDI=-Inf and exp(log_HDI)=0. I cannot say what would happen on other platforms. (I think Matlab follows the IEEE standard here. I have seen some platforms that define their own non-IDE types, so you should check.)

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This is cross-posted from Mathematics:

There is an obscure but useful paper that derives the geometric mean for negative and zero values.

Let $\{x_1, \ldots, x_{n_+}, x_{n_+ + 1}, \ldots, x_n\} = \{\textbf{x} \in \mathbb R_{\ge 0}^n\}$ be a sequence of $n$ independent random variables where $\{x_1, \ldots, x_{n_+}\} = \{\textbf{x}_+ \in \mathbb R_{> 0}^{n_+}\}$.

Let $G_+ = \big(\prod_{i = 1}^{n_+} x_i \big)^{\frac{1}{n_+}}$ be the geometric mean of $\textbf{x}_+$.

Let $w_+ = \frac{n_+}{n}$ be the fraction of values that are non-zero, thus $1 - w_+ = w_0$ the fraction of values that are zero.

Let $G = w_+ \times G_+ + (1-w_+) \times 0 = w_+\times G_+ + w_0 \times 0= w_+G_+$ be the tri-geometrical geometric mean, which is the weighted sum of $G_+$ and the geometric mean of the zeroes, which is by definition zero.

The paper goes on to show that $G$ is approximately log-normally distributed. The term "tri-geometrical" refers to the fact that it is a special case of three weighted geometric means, one for negative values, one for positive, one for zero.

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