r - 为什么在R中使用ccf()和cor()会得到不同的结果？

Question

为什么在cor()和ccf()之间得到不同的相关结果？

library(xts)
> set.seed(123)
> ts1 = xts(1:100, as.POSIXlt(1366039619, tz="", origin="1970-01-01") + rnorm(100, 0, 3))
> ts2 = xts(1:100, as.POSIXlt(1366039619, tz="", origin="1970-01-01") + rnorm(100, 0, 3))

> as.vector(ccf(as.integer(ts1[,1]), as.integer(ts2[,1]), lag.max =10, plot =F, na.action=na.pass)$acf)
 [1] -0.13747975 -0.00747975 -0.09497750 -0.01031203 -0.07564956  0.19881488 -0.11353135  0.01673867  0.12900690  0.00059706 -0.09642964  0.20852985  0.02476448  0.00126913 -0.03467147 -0.04284728 -0.05561356
[18]  0.08875188  0.01587159 -0.04449745  0.01002100

> sapply(seq(-10, 10), function(x, ts1, ts2) { cor(ts1[,1], lag(ts2[,1], x), use="complete.obs") }, ts1, ts2)
 [1] -0.154055651 -0.008411318 -0.104222576 -0.011595184 -0.082495425  0.210464976 -0.118454928  0.018112365  0.132716811  0.000694595 -0.096429643  0.209312640  0.025156993  0.001450175 -0.035451383
[16] -0.043902825 -0.057842616  0.093863686  0.017485161 -0.047042779  0.011511559

> sapply(seq(-10, 10), function(x, ts1, ts2) { cor(ts1[,1], lag(ts2[,1], x), use="complete.obs") }, ts1, ts2) - as.vector(ccf(as.integer(ts1[,1]), as.integer(ts2[,1]), lag.max =10, plot =F, na.action=na.pass)$acf)
 [1] -0.0165759032546357876203 -0.0009315701778466996610 -0.0092450780124607306876 -0.0012831523310935632337 -0.0068458595845764941279  0.0116500945970494651505 -0.0049235745757881255180
 [8]  0.0013736907995123247284  0.0037099107611970050247  0.0000975349354166987759 -0.0000000000000000277556  0.0007827869094209904954  0.0003925162566637135919  0.0001810479989895477041
[15] -0.0007799161627975795263 -0.0010555407353524254299 -0.0022290547145371181204  0.0051118107350296843050  0.0016135741880074876142 -0.0025453295798825298357  0.0014905566679348520448

更新

由于ccf()使用acf()，因此可以将差异减小为:

> as.vector(acf(c(42, 5, 65437, 23), plot=F, lag.max=1)$acf)
[1]  1.000000 -0.416954
> cor(c(42, 5, 65437, 23), c(NA, 42, 5, 65437), use="pairwise.complete.obs")
[1] -0.500218
> cor(c(42, 5, 65437, 23), c(5, 65437, 23, NA), use="pairwise.complete.obs")
[1] -0.500218

Answer 1

在示例中，cor和acf之间有一些区别。让我们选择一个更易于管理(并且已经贬低)的示例:

x = c(-2,-1,0,1,2)
acf(x, plot = F, lag.max = NULL)
# Autocorrelations of series ‘x’, by lag
#   0    1    2    3    4 
# 1.0  0.4 -0.1 -0.4 -0.4 

这是acf如何到达的，例如对于lag=2:

acf_lag_2 = sum(x*c(x[c(-1,-2)],NA,NA), na.rm = T) /
            sqrt(sum(x*x)*sum(x*x))

将此与您的cor构造将做的对比:

cor(x, c(0,1,2,NA,NA), use="pairwise.complete.obs") # = cor(c(-2,-1,0), c(0,1,2)) = 1

cor_lag_2 = sum((c(-2,-1,0)+1)*(c(0,1,2)-1)) /   # recall cor needs to demean both vectors
            sqrt(sum(c(-1,0,1)*c(-1,0,1))*sum(c(-1,0,1)*c(-1,0,1)))

因此，acf从一开始就只进行一次贬值，并在整个过程中使用它进行归一化，而cor将针对每个滞后分别进行归一化和贬值。