Let \(\maxage\) be the start of the open interval in a population age distribution, assumed to be a multiple of five. The first step in Feeney's method is to redistribute the excess population at ages a multiple of five to the neighbouring age intervals of four years. This is done iteratively by first setting \begin{eqnarray} \fpop{x}&=&\pop{}{x}\\ \fpos{x}&=&\pop{4}{x+1} \end{eqnarray} for \(x\) a multiple of five in the range \(0\le x \lt\maxage\) and then repeating the update \begin{eqnarray} \fpop{x}&\leftarrow&\fpop{x}-(\Delta_x-1)(\fpos{x-5}+\fpos{x})\\ \fpos{x}&\leftarrow&(\Delta_x+\Delta_{x+5}-1)\fpos{x} \end{eqnarray} until convergence. Here \begin{equation} \Delta_x=\frac{8}{9}\left[\frac{\fpos{x-5}+\fpop{x}+\fpos{x}}{\fpos{x-5}+\fpos{x}}\right] \quad\mathrm{for}\quad 0\lt x\lt\maxage \end{equation} and \begin{equation} \Delta_0=\Delta_{\maxage}=1 . \end{equation} The second step is to form the smoothed age distribution at five-year intervals using the interpolation \begin{eqnarray} \spop{5}{0}&=&\pop{5}{0}\\ \spop{5}{x}&=&3\fpop{x}+2\fpop{x+5},\quad 0\lt x\lt\maxage-5\\ \spop{5}{\maxage-5}&=&\pop{5}{\maxage-5}\\ \spop{\infty}{\maxage}&=&\pop{\infty}{\maxage} \end{eqnarray} The third step is to adjust each value by a common factor to recover the original population total. The five-year intervals are then split using Sprague.
The age distribution is abridged into five-year intervals and then split using Sprague.
The population is smoothed at single-year intervals using a running weighted average \begin{equation} \spop{}{x}=\sum_{k=-\window}^\window \weight{\window}{k}\pop{}{x+k}/\sum_{k=-\window}^\window \weight{\window}{k} \end{equation} The window width \(\window=\min(5,|x|,|\maxage-x|)\). The default window weights are symmetric in \(k\) with values
| \(k\) | \(\quad 0\quad\) | \(\quad 1\quad\) | \(\quad 2\quad\) | \(\quad 3\quad\) | \(\quad 4\quad\) | \(\quad 5\quad\) |
|---|---|---|---|---|---|---|
| \(\weight{5}{k}\) | 5 | 4 | 3 | 2 | 1 | 0 |
| \(\weight{4}{k}\) | 5 | 4 | 3 | 2 | 1 | |
| \(\weight{3}{k}\) | 4 | 3 | 2 | 1 | ||
| \(\weight{2}{k}\) | 3 | 2 | 1 | |||
| \(\weight{1}{k}\) | 2 | 1 | ||||
| \(\weight{0}{k}\) | 1 |
The default weights can be changed by appending extrainfo strings Weights1, Weights2, Weights3, Weights4, and Weights5 to the buildmethod value. For example to specify \(\weight{2}{k}\) use WEIGHTED_AVERAGE:"Weights2=\(\weight{2}{-2}\) | \(\weight{2}{-1}\)|\(\weight{2}{0}\)|\(\weight{2}{1}\)|\(\weight{2}{2}\)".
The age distribution is abridged into five-year intervals. For each mid panel ten-year interval the smoothed five-year populations are \begin{eqnarray} \spop{5}{x+5}&=&\pop{10}{x}\left[1+\left(\frac{\pop{10}{x-10}}{\pop{10}{x+10}}\right)^{1/4}\right]^{-1}\\ \spop{5}{x}&=&\pop{10}{x}-\spop{5}{x+5} \end{eqnarray} The five-year intervals are then split using Sprague.
The age distribution is abridged into five-year intervals. For each mid panel ten-year interval the smoothed five-year populations are \begin{eqnarray} \spop{5}{x}&=&\pop{10}{x}/2+\left(\pop{10}{x-10}-\pop{10}{x+10}\right)/16\\ \spop{5}{x+5}&=&\pop{10}{x}-\spop{5}{x} \end{eqnarray} The five-year intervals are then split using Karup-King.
The age distribution is abridged into five-year intervals. In the first ten-year interval the smoothed five-year populatons are \begin{eqnarray} \spop{5}{x+5}&=&(8\,\pop{10}{x}+5\,\pop{10}{x+10}-\pop{10}{x+20})/25,\\ \spop{5}{x}&=&\pop{10}{x}-\spop{5}{x+5}. \end{eqnarray} For the mid ten-year intervals the smoothed five-year populations are \begin{eqnarray} \spop{5}{x+5}&=&(-\pop{10}{x-10}+11\,\pop{10}{x}+2\,\pop{10}{x+10})/24\\ \spop{5}{x}&=&\pop{10}{x}-\spop{5}{x+5} \end{eqnarray} For the last ten-year interval the smoothed five-yeaer populations are \begin{eqnarray} \spop{5}{x}&=&(-\pop{10}{x-20}+5\,\pop{10}{x-10}+8\,\pop{10}{x})/24\\ \spop{5}{x+5}&=&\pop{10}{x}-\spop{5}{x} \end{eqnarray} The five-year intervals are then split using Sprague.
The age distribution is abridged into five-year intervals. In the mid-panel ten-year populations are smoothed using \begin{equation} \spop{10}{x}=(\pop{10}{x-10}+2\,\pop{10}{x}+\pop{10}{x+10})/4. \end{equation} The ten-year intervals are then divided into five-year intervals using Arriaga's light smoothing method.
The age distribution is abridged into five-year intervals and then smoothed using \begin{equation} \spop{5}{x}=(-\pop{5}{x-10}+4\,\pop{5}{x-5}+10\,\pop{5}{x}+4\,\pop{5}{x+5}-\pop{5}{x+10})/16 \end{equation} for \(10\le x\le \maxage-10\). The five-year intervals are then split using Sprague.