## Exponential smoothing

Annual indicator data are often unavailable or imprecise. Moreover, past
environmental pressures have significant cumulative effects. To deal with
these issues, present and past indicator data are combined into a single value
using exponentially weighted sums. Suppose that *K* measurements of
indicator *c* are available for some country. Let *x _{c}(t_{1}),
x_{c}(t_{2}), …, x_{c}(t_{K}*) be the
normalized values in years

*t*. These years need not be consecutive due to missing data. An aggregate value

_{1}, t_{2}, …, t_{K}*x*for indicator

_{c}*c*is computed by exponential smoothing, using the a weighted average

in which older observations are assigned geometrically decreasing weights
with parameter *β* ∈ [0, 1]. The smoothing parameter *β* is
chosen so as to minimize the mean squared error

The quantity \({\hat x_c}{\rm{(}}{t_k}{\rm{)}}\) is the weighted average of indicator data *prior to* year *t _{k}*,
and is given by

It should be noted that the weights *β* differ among countries
as well as among indicators. If no indicator data are available for some country,
a value *x _{c}* is imputed using an approach to described in
a separate section.