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I have two time series, for example:

a = c(2, 1, 2, 1, 2, 1, 2)
b = c(NA, NA, 1, 2, 1, 2, 1)
ccf(a, b, na.action=na.omit, plot=FALSE)

The results of ccf shows the following:

Autocorrelations of series ‘X’, by lag

    -3     -2     -1      0      1      2      3 
 0.400 -0.567  0.800 -1.000  0.800 -0.567  0.400 

When lag equals 0, the ccf values is -1. However, I can't figure out why the value is 0.8 (lag = -1) and -0.567 (lag = -2).

I've read the link from Why do I get different results using ccf() and cor() in R?. But it is based on acf and doesn't contains NAs.

How to calculate it when it contains NA?

Specifically, what is the formula when calculate when lag = -2 in this toy example ?

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2 Answers 2

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As the simple correlation coefficient between the lagged series from the sample gives biased estimation of the population correlation coefficient $\rho_{ij} \left( t \right)$, an unbiased estimator should be applied.

If you take a look at the built in help (?ccf), there is a reference there to the book Venables, W. N. and Ripley, B. D. (2002): Modern Applied Statistics with S. Fourth Edition. Springer-Verlag. On page 390 you can find the estimation formula for ccf:

$$c_{ij}\left( t \right) = \frac{1}{n} \sum_{s = \max \left( 1, -t \right)}^{\min\left( n - t, n \right)}{\left( X_i \left( s + t \right) - \overline{X_i} \right) \left( X_j\left( s \right) - \overline{X_j} \right)}, \qquad r_{ij}\left( t \right) = \frac{c_{ij}\left( t \right)}{\left| c_{ij}\left( 0 \right) \right|}$$

(Actually $r_{ij} \left( t \right)$ is not there, but it can be easily deducted from acf functions $r_t$. The latter is $r_t = \frac{c_t}{c_0}$ there, without the absolute value in the denominator, as $c_0$ is always positive in case of acf, but it is obviously needed in case of ccf (think about $r_{ij} \left( 0 \right) = -1$ as the case with a and b in this question).

As

a <- c(2, 1, 2, 1, 2, 1, 2)
b <- c(NA, NA, 1, 2, 1, 2, 1)
ccf(a, b, na.action=na.omit, plot=FALSE)

is equivalent with

a <- c(2, 1, 2, 1, 2)
b <- c(1, 2, 1, 2, 1)
ccf(a, b, plot=FALSE)

with the result

Autocorrelations of series ‘X’, by lag

    -3     -2     -1      0      1      2      3 
 0.400 -0.567  0.800 -1.000  0.800 -0.567  0.400 

you can check the calculations applying the above formulas 'manually' with the next R code:

a <- c(2, 1, 2, 1, 2)
b <- c(1, 2, 1, 2, 1)
n <- length(a)
c_0 <- abs(1 / n * sum((a - mean(a)) * (b - mean(b))))
for (t in -3:3) {
  if (t <= 0) {
    c_t <- 1 / n * sum((a[1:(n + t)] - mean(a)) * (b[(1 - t):n] - mean(b)))
  } else {
    c_t <- 1 / n * sum((a[(1 + t):n] - mean(a)) * (b[1:(n - t)] - mean(b)))
  }
  r_t <- c_t / c_0
  print(r_t)
}

with results

[1] 0.4
[1] -0.5666667
[1] 0.8
[1] -1
[1] 0.8
[1] -0.5666667
[1] 0.4
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  • $\begingroup$ Thank you for your answers. I still have two questions. (1) How to calculate when using na.action = na.pass in ccf in this toy example? According to your answer, na.omit ignore the information of the first two value in a. When using ccf(a, b, na.action=na.pass, plot=FALSE) , the ccf returns -0.99 (lag = 0), 0.825 (lag = -1). (2) Why does the simple correlation coefficient between the lagged series from the sample give biased estimation of the population correlation coefficient ? Thank you in advance. $\endgroup$
    – Xi Wang
    Commented Apr 11, 2020 at 7:57
  • $\begingroup$ I think these are separate questions, so it is not the best idea to discuss them in the comments, but some quick hints: (1) from ccf help using na.pass "means that the estimate computed may well not be a valid autocorrelation sequence, and may contain missing values". So I think it's better to omit NAs. To get more insight, I think you have to check the source code of ccf. (2) Check this: stats.stackexchange.com/questions/220961/… $\endgroup$
    – oszkar
    Commented Apr 11, 2020 at 9:04
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There may be a mistake in the formula from @oszkar's answer. You can see this by modifying b from the example above. The code

a <- c(2, 1, 2, 1, 2)
b <- c(1, 2, -1, -2, 1)
n <- length(a)
c_0 <- abs(1 / n * sum((a - mean(a)) * (b - mean(b))))
for (t in -3:3) {
    if (t <= 0) {
        c_t <- 1 / n * sum((a[1:(n + t)] - mean(a)) * (b[(1 - t):n] - mean(b)))
    } else {
        c_t <- 1 / n * sum((a[(1 + t):n] - mean(a)) * (b[1:(n - t)] - mean(b)))
    }
    r_t <- c_t / c_0
    print(r_t)
}

yields

[1] -3.4
[1] 2.9
[1] 0.2
[1] 1
[1] 0.2
[1] -3.1
[1] 0.6

It may not make sense to have a correlation outside of [-1, 1]. The ccf in R:

ccf(a, b, 3, plot = F)$acf

gives

, , 1

            [,1]
[1,] -0.37777778
[2,]  0.32222222
[3,]  0.02222222
[4,]  0.11111111
[5,]  0.02222222
[6,] -0.34444444
[7,]  0.06666667

The form for $r_{ij}(t)$ used in R is actually $$r_{ij}(t)=\dfrac{c_{ij}(t)}{\sqrt{\sigma_i^2\sigma_j^2}}$$ where $$\sigma_i^2=\frac{1}{n} \sum_{s=1}^{n}\left[X_i(s)-\overline{X}_i\right]^2.$$ That is, the denominator is the square root of the product of the "population variance" of the two time series. With this formula you can see that the results below match with the results from ccf.

a <- c(2, 1, 2, 1, 2)
b <- c(1, 2, -1, -2, 1)
n <- length(a)
c_0 <- sqrt(sum((a - mean(a)) ^ 2 / n) * sum((b - mean(b)) ^ 2 / n))
for (t in -3:3) {
    if (t <= 0) {
        c_t <- 1 / n * sum((a[1:(n + t)] - mean(a)) * (b[(1 - t):n] - mean(b)))
    } else {
        c_t <- 1 / n * sum((a[(1 + t):n] - mean(a)) * (b[1:(n - t)] - mean(b)))
    }
    r_t <- c_t / c_0
    print(r_t)
}

gives

[1] -0.3777778
[1] 0.3222222
[1] 0.02222222
[1] 0.1111111
[1] 0.02222222
[1] -0.3444444
[1] 0.06666667
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  • $\begingroup$ This is a good observation -- but please don't leap to the conclusion that the other answer has a mistake. There is more than one way to define the ccf. This is analogous to different perspectives on the acf, as I explain at stats.stackexchange.com/a/81764/919. If the series is so short that different formulas give appreciably different results, then it's probably too short to be relying on any of those formulas! $\endgroup$
    – whuber
    Commented Apr 3, 2023 at 1:25
  • 1
    $\begingroup$ @whuber Thanks for the message! I understand that there might be different formulas for the ccf with different perspectives. But the formula oszakr used can produce cross-correlation outside [-1, 1], so I have modified my answer to reflect on this. $\endgroup$
    – T. J.
    Commented Apr 3, 2023 at 3:14

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