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By looking at a lag plot, it seems that my dataset is random (no autocorrelation whatsoever). Should this be sufficient to infer independence of observations?

As a matter of fact, I know that the observations are dependent and hence randomness should not guarantee that. But intuitively it feels otherwise.

Given that randomness does not determine independence. Is there some statistical measure to test independence of observations?

edit: Please note that the data is not of timeseries nature, it is the average bandwidth usage of a customer across the month.

enter image description here

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  • $\begingroup$ Define what you mean by randomness in this context. $\endgroup$ – Glen_b Mar 20 '14 at 13:36
  • $\begingroup$ By randomness I mean that the variable does not show any autocorrelation. From inspecting lag plots, I find no patterns. $\endgroup$ – tool.ish Mar 20 '14 at 13:41
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    $\begingroup$ I'm not so much worried yet about the fact that doing a lag plot makes no sense. My issue is far more fundamental. Your data are either observed over time (a time series) or they aren't. You have asserted that they aren't a time series. If your data are not "over time" how does one lag them? One would have to move a series by one in a dimension it doesn't even posses! $\endgroup$ – Glen_b Mar 21 '14 at 6:57
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    $\begingroup$ Unless the index represents some meaningful ordering, it's no more to the point than any random set of integers. Does the index represent some relevant kind of ordering? $\endgroup$ – Glen_b Mar 21 '14 at 8:26
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    $\begingroup$ There is no basis for ordering this data that I used for lagging - a random set of integers would be equivalent. Since it is univariate, the only plausible ordering I can think of is a sort on its values - which probably defeats the whole purpose of doing a lag-plot. $\endgroup$ – tool.ish Mar 21 '14 at 9:05
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Randomness is not the same as independence. There may be a small dependency in there but it appears to be lost in the noise. You can try various measures of independence (googleable). Pearson's correlation coefficient is the simplest one and will work best if there is a small linear correlation. But by looking at that, I can tell that you won't find a value that inconsistent with zero but you can still go ahead and try.

There are other ones such as Brownian Covariance Distance that works for any kind of relation, not just linear, and it is good when you have to blindly check correlation of a large number of pairs of variables (for example in automated financial trading) especially when there is obvious non-linear structure.

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  • $\begingroup$ P.S. This looks like growth rates of companies or something similar like ROE. I've made these plots before and found very little correlation. Past performance is not a good indicator of future return. $\endgroup$ – Dave31415 Mar 20 '14 at 15:15
  • $\begingroup$ I agree with the statement that "Randomness is not the same as independence". By your reference to pearson's coefficient, i'm assuming you are talking about "independence of variables" right? The question is with respect to "independence of observations". $\endgroup$ – tool.ish Mar 21 '14 at 10:08
  • $\begingroup$ You'll have to explain more then. Are you interested in whether the points are clustered? Or is there some other hidden variables that we are not seeing on the plot. Brownian Covariance Distance still might be of interest to you. $\endgroup$ – Dave31415 Mar 21 '14 at 13:38
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there is no one statistical measure of independence, it's impossible to test. the way people go about is to make assumptions about certain aspect of independence and test it. for instance, look at Engle's ARCH test, it tests for heteroscedasticity in time series, i.e. that variance doesn't change.

another example of dependence: $var(y_t)\sim y_{t-1}$, which would be difficult to detect visually in the scatter plot you have shown.

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