How can I calculate proportion of variance of an aggregate time-series accounted for by the variance of a sub-series? As I'm not sure what the correct terminology is for explaining my question, I'll illustrate it with an example:
I have a time-series (say, for example, annual alcohol consumption in the USA) that is exactly made up of the sums of several sub-series (continuing the example, annual consumption of spirits, beer and wine). The variation in one of these component time-series is much greater than in the others, meaning that it dominates the variation in the aggregate series. For example:
n.obs <- 20
set.seed(1990)
x1 <- cumsum(rnorm(n.obs, mean = 0, sd = 1)) + 20
x2 <- cumsum(rnorm(n.obs, mean = 0, sd = 1)) + 20
x3 <- cumsum(rnorm(n.obs, mean = 0, sd = 1)) + 20
x4 <- cumsum(rnorm(n.obs, mean = 0, sd = 8)) + 100
y <- x1 + x2 + x3 + x4
As you can see, the variation in y
is dominated by the variation in x4
:
How can I quantify the extent to which the variation of y
is accounted for by the variation in x4
?
My initial thought was to calculate the variance in y-x4
compared to the variance in y
(i.e. 1 - (var(y - x4) / var(y))
) but if I do this for all sub-series the values do not sum to one as I would expect.