Which correlation coefficient is the most appropriate to compare 2 time series? I want to compare the variation of one variable for 2 regions, have regional data for the last 30 years. Is Pearson correlation ok or should I rely on Kendall's tau b or Spearman's rho and why? I tried to google it and analyse what I found, but I'm still not sure.
Solved – Comparing time series: Pearson correlation, Kendall’s tau b or Spearman’s rho
autocorrelationcorrelationcross correlationdata visualizationtime series
Related Solutions
This paper discusses the contexts where you can and can't use a normal approximation for Tau. According to Wikipedia, it also looks like the validity of normal/Z depends on how your version of Tau handles ties. My sense is that it's probably safer not to assume that it's Gaussian, especially with relatively low sample sizes.
I couldn't think of a reason why Kendall's Tau wouldn't be compatible with the bootstrap, but I wasn't 100% sure. So I looked it up:
Here's a paper by Brad Efron, the inventor of the bootstrap, that uses it for Tau (Section 5).
Here's a paper that spends some time discussing the bootstrap in the context of Tau (mostly Section 4).
Looks like you shouldn't have any serious problems using the bootstrap for tau.
- Is there a way to test if my data is monotonic prior to Spearman's rho / Kendall's tau correlation calculations?
You could plot the data and look for a non-monotone shape.
Also, you could fit a generalized additive model (GAM) which estimates nonparametric functions of the predictor variables. This can be done in the mgcv package in R.
For example:
require(mgcv)
set.seed(123)
n <- 100
x <- runif(n,-5,5)
y <- x^2 + rnorm(n,0,4)
plot(x,y, col="red")
which produces:
Note that
> cor.test(x, y, method = "kendall")
sample estimates:
tau
-0.01454545
> cor.test(x, y, method = "spearman")
sample estimates:
rho
-0.005664566
So, both Spearman's rho and Kendall's tau are not helpful.
Now, if we run a GAM, we get
> summary(m0 <- gam(y~s(x)))
.
.
.
Approximate significance of smooth terms:
edf Ref.df F p-value
s(x) 8.277 8.861 46.72 <2e-16 ***
.
.
.
With edf>1 there is evidence of non-linearity in the data, which doesn't prove that the association is non-monotonic, but nevertheless suggests that it might be.
Is it possible to decompose my dataset into monotonic sections, to analyse them separately?
Yes ! Sticking with the same dataset, we can do:
x1 <- x[x<0]
y1 <- y[x<0]
x2 <- x[x>=0]
y2 <- y[x>=0]
cor.test(x1, y1, method = "kendall")
cor.test(x1, y1, method = "spearman")
which gives:
sample estimates:
tau
-0.5878084
sample estimates:
rho
-0.7905983
and this handles the first segment of the data, then:
cor.test(x2, y2, method = "kendall")
cor.test(x2, y2, method = "spearman")
which gives:
sample estimates:
tau
0.7446809
sample estimates:
rho
0.9155874
So here we can see a strong negative association in the first segment and a strong positive association in the second.
- Is there any equivalent to Spearman's rho test (or Kendall's tau) that accounts for multiple monotonic components?
Not that I am aware of.
Best Answer
For time series some version of Pearson correlation is most used, in the form of the autocorrelation function (for one series, correlated with itself at various lags) and the cross-correlation function (for two series) likewise. They are correct when all conditional expectation are linear.
If you suspect that may not be the case, you should start with some visualization of the two series! I have not seen any detailed descriptive analysis of two time series, that would be rather interesting ... In R you could play with the function
coplotand you could make scatterplot matrices, replacing what would be one number in each of the two functions above (autocorrelation, crosscorrelation) with a scatterplot. You could also look into copulas used with time series.