rtrendproportions

Determine total trend given unit trend and proportion

I have data (below table) on trend (% over 10 years) in units of a whole. I also know the proportion of the whole each unit makes up. I want to do three things.

  1. Calculate overall trend given that information
  2. Calculate what the trend needs to be in each unit, given current proportions, if I want the overall trend to change to 2% over the next 10 yrs
  3. Calculate what the trend needs to be in Unit 12 to reach a goal of 2% over the next 10 yrs if if all other unit trends (and proportions) stay the same

I thought there might be formulas for these, but I haven't found one (I asked the Mathematics community and got no response here). Maybe it's not that simple, and I really need to code it out so that I can manipulate annual indices until I get the appropriate trend. Any packages out there that might do that in R?

Unit Proportion of total 10yrT
1 0.073 0.29
2 0.009 1.13
3 0.003 0.29
4 0.003 -0.46
5 0.315 -1.77
6 0.013 -0.26
7 0.007 1.54
8 0.003 4.57
9 0.227 -2.35
10 0.202 -1.45
11 0.008 -0.13
12 0.139 -0.79
Solution

  • 1) If you are asking how to calculate a weighted mean in R it is the weighted.mean function. Using dat defined reproducibly in the Note at the end

    prop <- dat[[2]]
trend <- dat[[3]]
wm <- weighted.mean(trend, prop); wm
## [1] -1.43993

    2) weighted.mean is linear in the first argument and the weighted mean of a constant is that constant so:

    weighted.mean(trend + x, prop) 
= weighted.mean(trend, prop) + weighted.mean(x, prop)
= weighted.mean(trend, prop) + x
= wm + x

    and if that is to equal to 2 then x = 2 - wm

    x <- 2 - wm; x
## [1] 3.43993

weighted.mean(trend + x, prop) # check
## [1] 2

    3) Now if e12 is a vector of 11 zeros followed by a scalar 1 and y is a scalar then again by linearity

    weighted.mean(trend + y * e12, prop)
= weighted.mean(trend, prop) + y * weighted.mean(e12, prop)
= wm + y * weighted.mean(e12, prop)

    and if that is to equal 2 then solving we have y = (2 - wm)/weighted.mean(e12, prop) so

    e12 <- c(numeric(11), 1); e12
##  [1] 0 0 0 0 0 0 0 0 0 0 0 1

y <- (2 - wm)/weighted.mean(e12, prop); y
## [1] 24.79719

weighted.mean(trend + y * e12, prop) # check
## [1] 2

    4) Alternately use uniroot to solve these equations instead of using elementary algebra.

    uniroot(\(x) weighted.mean(trend + x, prop) - 2, c(-100, 100))
## $root
## [1] 3.43993
## ...snip...

uniroot(\(y) weighted.mean(trend + y * e12, prop) - 2, c(-100, 100))
## $root
## [1] 24.79719
## ...snip...

    Note

    dat <- data.frame(
  Unit = 1:12,
  `Proportion of total` = c(0.073, 0.009, 0.003, 0.003, 0.315, 0.013, 
     0.007, 0.003, 0.227, 0.202, 0.008, 0.139),
  `10yrT` = c(0.29, 1.13, 0.29, -0.46, -1.77, -0.26, 1.54, 4.57, -2.35, 
     -1.45, -0.13, -0.79),
  check.names = FALSE
)