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 xc(t1), xc(t2), …, xc(tK) be the normalized values in years t1, t2, …, tK. These years need not be consecutive due to missing data. An aggregate value xc for indicator c is computed by exponential smoothing, using the a weighted average
\[{x_c} = \frac{{{x_c}{\rm{(}}{t_K}{\rm{)}} + {x_c}{\rm{(}}{t_{K - 1}}{\rm{)}}{\beta ^{{t_K} - {t_{K - 1}}}} + \ldots + {x_c}{\rm{(}}{t_1}{\rm{)}}{\beta ^{{t_K} - {t_1}}}}}{{1 + {\beta ^{{t_K} - {t_{K - 1}}}} + \ldots + {\beta ^{{t_K} - {t_1}}}}}\]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
\[{\left[ {{x_c}{\rm{(}}{t_1}{\rm{)}} - {{\hat x}_c}{\rm{(}}{t_1}{\rm{)}}} \right]^2} + \ldots + {\left[ {{x_c}{\rm{(}}{t_K}{\rm{)}} - {{\hat x}_c}{\rm{(}}{t_K}{\rm{)}}} \right]^2}\]The quantity \({\hat x_c}{\rm{(}}{t_k}{\rm{)}}\) is the weighted average of indicator data prior to year tk, and is given by
\[\begin{array}{*{20}{l}}{{{\hat x}_c}{\rm{(}}{t_1}{\rm{)}} = 0}\\{{{\hat x}_c}{\rm{(}}{t_{k + 1}}{\rm{)}} = \frac{{{x_c}{\rm{(}}{t_k}{\rm{)}} + {x_c}{\rm{(}}{t_{k - 1}}{\rm{)}}{\beta ^{{t_k} - {t_{k - 1}}}} + \ldots + {x_c}{\rm{(}}{t_1}{\rm{)}}{\beta ^{{t_k} - {t_1}}}}}{{1 + {\beta ^{{t_k} - {t_{k - 1}}}} + \ldots + {\beta ^{{t_k} - {t_1}}}}},\;k = 1, \ldots ,K - 1}\end{array}\]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 xc is imputed using an approach to described in a separate section.