This arithmetic smoothing technique aims to eliminate irregularities of the population pyramid by averaging values in a moving window of user-defined width.
mav(Value, Age, n = 3, OAG = TRUE, tails = FALSE)
| Value | numeric. A vector of demographic counts in single age groups. |
|---|---|
| Age | integer. A vector of ages corresponding to the lower integer bound of the counts. |
| n | integer. A single number, (often 3 or 5), indicating the number of years taken to smooth the population distribution by single ages. |
| OAG | logical. Whether or not the top age group is open. Default |
| tails | logical. If set to |
Vector with the smoothed demographic counts.
The moving window is applied symmetrically. By default (tails = FALSE) data endpoints are imputed with NAs in output: the is nothing under 0 or over the highest closed age group to average with. The open age group is not used in averaging, and it is returned as-is. Age intervals are assumed uniform. This function could be used with either single or 5-year age groups.
If tails is set to TRUE, then tails have been imputed using moving averages with successively smaller values of n, the cascade method.
Roger G, Waltisperger D, Corbille-Guitton C (1981). Les structures par sexe et âge en Afrique. GDA, Paris, France.
Pop <-c(303583,390782,523903,458546,517996,400630,485606,325423,471481,189710, 385442,143205,270890,145105,138078,157444,153035,91566,247160,73115, 384222,83551,198555,111347,129851,477510,149272,100814,178465,50684, 577167,51878,97788,55544,58011,393200,85048,51131,80336,31246, 454698,34864,51810,31146,26618,228718,38504,23616,40836,15589, 339158,21349,26997,17590,17513,119763,22704,12336,17635,8485, 323263,9535,13906,9063,8294,90459,9817,6376,8884,3773,160609) Age <- 0:70 # final age group assumed open mav(Pop, n = 3, Age = Age)#> 0 1 2 3 4 5 6 #> NA 406089.333 457743.667 500148.333 459057.333 468077.333 403886.333 #> 7 8 9 10 11 12 13 #> 427503.333 328871.333 348877.667 239452.333 266512.333 186400.000 184691.000 #> 14 15 16 17 18 19 20 #> 146875.667 149519.000 134015.000 163920.333 137280.333 234832.333 180296.000 #> 21 22 23 24 25 26 27 #> 222109.333 131151.000 146584.333 239569.333 252211.000 242532.000 142850.333 #> 28 29 30 31 32 33 34 #> 109987.667 268772.000 226576.333 242277.667 68403.333 70447.667 168918.333 #> 35 36 37 38 39 40 41 #> 178753.000 176459.667 72171.667 54237.667 188760.000 173602.667 180457.333 #> 42 43 44 45 46 47 48 #> 39273.333 36524.667 95494.000 97946.667 96946.000 34318.667 26680.333 #> 49 50 51 52 53 54 55 #> 131861.000 125365.333 129168.000 21978.667 20700.000 51622.000 53326.667 #> 56 57 58 59 60 61 62 #> 51601.000 17558.333 12818.667 116461.000 113761.000 115568.000 10834.667 #> 63 64 65 66 67 68 69 #> 10421.000 35938.667 36190.000 35550.667 8359.000 6344.333 NA #> 70 #> 160609.000if (FALSE) { odds <- seq(3, 11, by = 2) nwindows <- sapply(odds, mav, Value = Pop, Age = Age, OAG = TRUE, tails = FALSE) cols <- gray(seq(.8, 0, length = 5)) lwds <- seq(3, 1, length = 5) plot(Age,Pop, col = "red", xlab = "Age", ylab = "The counts", pch=16, main = "Moving average windows and extreme heaping") matplot(Age,nwindows,type='l',col=cols, lwd=lwds, add=TRUE,lty=1) legend("topright", lty=1, col = cols, lwd = lwds, legend = paste0("n=",seq(3,11,by=2))) } # For cascading smoothing on the tails: mav(Pop, Age, tails = TRUE)#> 0 1 2 3 4 5 6 7 #> 303583.00 402262.50 457743.67 500148.33 459057.33 468077.33 403886.33 427503.33 #> 8 9 10 11 12 13 14 15 #> 328871.33 348877.67 239452.33 266512.33 186400.00 184691.00 146875.67 149519.00 #> 16 17 18 19 20 21 22 23 #> 134015.00 163920.33 137280.33 234832.33 180296.00 222109.33 131151.00 146584.33 #> 24 25 26 27 28 29 30 31 #> 239569.33 252211.00 242532.00 142850.33 109987.67 268772.00 226576.33 242277.67 #> 32 33 34 35 36 37 38 39 #> 68403.33 70447.67 168918.33 178753.00 176459.67 72171.67 54237.67 188760.00 #> 40 41 42 43 44 45 46 47 #> 173602.67 180457.33 39273.33 36524.67 95494.00 97946.67 96946.00 34318.67 #> 48 49 50 51 52 53 54 55 #> 26680.33 131861.00 125365.33 129168.00 21978.67 20700.00 51622.00 53326.67 #> 56 57 58 59 60 61 62 63 #> 51601.00 17558.33 12818.67 116461.00 113761.00 115568.00 10834.67 10421.00 #> 64 65 66 67 68 69 70 #> 35938.67 36190.00 35550.67 8359.00 6979.25 8884.00 160609.00if (FALSE) { # Compare nwindows_tails <- sapply(odds, mav, Value = Pop, Age = Age, OAG = TRUE, tails = TRUE) colnames(nwindows) <- odds colnamaes(nwindows_tails) <- odds # NA triangles are completed with # successively smaller ns. head(nwindows) head(nwindows_tails) tail(nwindows) tail(nwindows_tails) }