I'm comparing concentration $C$ of a contaminant in the same spatial region at two time point 2000 and 2010 with sample size of $N_{2000}$ = 51 and $N_{2010}$ = 26 (not all the samples are from the same location), mean of $\mu(C)_{2000}$ = 47 and $\mu(C)_{2010}$ = 27 (determined by block kriging of all point observations) and variance of $V(C)_{2000}$ = 89 and $V(C)_{2010}$ = 68 (kriging variance). To determine if there has been any significant change over the last 10 years, we first need to determine the variance of change in the area:

$V(\Delta C) = V(C)_{2000} + V(C)_{2010} - V(C)_{2000,2010}$

where, $V(\Delta C)$ is the variance of change over time; and $V(C)_{2000,2010}$ is the covariance between the two temporal samples. Does anyone know how to determine the $V(C)_{2000,2010}$ term in the above equation?

  • $\begingroup$ Do you expect covariance in the measurement error? Or do you expect covariance in the concentration over locations? $\endgroup$ – Gijs May 10 '17 at 10:46
  • $\begingroup$ I expect covariance in the concentration over locations $\endgroup$ – ToNoY May 10 '17 at 14:35
  • $\begingroup$ Fit a model to the combined 2000 and 2010 data. Use a dummy variable to indicate time (2000 vs. 2010) and fit the spatial trend, stratified by the time dummy variable. If you use something like a generalized additive model, you can estimate the covariance using the inverse fisher information (e.g. using vcov in R). A simpler solution applies if you think the covariance is positive--you could just ignore it in the formula used and the resulting inference will be conservative. $\endgroup$ – gammer May 12 '17 at 5:38
  • $\begingroup$ Would bootstrapping (over cells/positions of the kriged grid) be possible with the software/code you are using ? $\endgroup$ – keepAlive May 13 '17 at 18:59
  • $\begingroup$ I think bootstrapping is possible, your answer is promising but would it be possible to create a working example? It doesn't need to be comprehensive (e.g. doing for thousands of cells, etc.), something that you can reproduce with some numbers assigned to your drawing by hand would be sufficient. Also, no need to derive the covariance function by hand assuming most software/programs make those complicated calculations easier but it would be easier if you could provide the final answers for your calculations for the steps you've mentioned. $\endgroup$ – ToNoY May 15 '17 at 15:54


not all the samples are from the same location

is equivalent to

almost all the samples are from the same location

or put differently that $2000$ and $2010$ spatial supports broadly intersect (as in $\rm{Fig.1}$), the approach below is applyable.

enter image description here

What "Broadly" means ? Whether focusing only on the intersection does or does not change your question of reasearch. E.g. if the intersection is restricted to a too small, say, urban area, and that you were originally interested in the overall metropolitan area, your question of reasearch would be changed and what follows will thus not suit you.

Focusing only on the intersection of the two years spatial support

You can build a $2D$ kriging-interpolated continuum on which you can then project a grid whose peremiter consists of the convex envelope of the set of points belonging to the intersection described above (and shown in $\rm{Fig.1}$). The nodes of the so-projected grid are going to be used as "individuals" within the bootstrapping proccess. Like so (see $\rm{Fig.2}$), each node will have year-$2000$ and year-$2010$ concentrations attached.

enter image description here

The steps are:

  1. Get, for each boostrap resample $b=1, ..., k$ and for all of its individuals (nodes) $i=1, ..., n$ (duplicated positions $\iff$ not all positions from the original sample, since the resample size is still $n$): $C_{b,i,2000}$, $C_{b,i,2010}$ and $C_{b,i,2000} \times C_{b,i,2010}$
  2. Compute the three sample means of each resample $b=1, ..., k$ as follows

    $\forall b, \overline{C_{b,2000}} = \frac{1}{n}\sum_{i=1}^n C_{b,i,2000}$

    $\forall b, \overline{C_{b,2010}} = \frac{1}{n}\sum_{i=1}^n C_{b,i,2010}$

    $\forall b, \overline{C_{b,2000} \times C_{b,2010}}= \frac{1}{n}\sum_{i=1}^n C_{b,i,2000} \times C_{b,i,2010}$

Once you are provided with these three empirical bootstrap distributions of sample means, you may want to compute the corresponding three empirical bootstrap means

  1. Compute $\hat{E}(C_{2000})$, $\hat{E}(C_{2010})$ and $\hat{E}(C_{2000} \times C_{2010})$, as follows:

    $\hat{E}(C_{2000}) = \frac{1}{k} \sum_{b=1}^k \overline{C_{b,2000}}$

    $\hat{E}(C_{2010}) = \frac{1}{k} \sum_{b=1}^k \overline{C_{b,2010}}$

    $\hat{E}(C_{2000} \times C_{2010}) = \frac{1}{k} \sum_{b=1}^k \overline{C_{b,2000} \times C_{b,2010}}$

And finally (also reusing your notation for covariance)

  1. Compute $V(C)_{2000,2010} = cov(C_{2000},C_{2010}) = \hat{E}(C_{2000} \times C_{2010}) - \hat{E}(C_{2000})\hat{E}(C_{2010})$

And do not forget to set $k$ as close as possible to $\frac{(2n-1)!}{n!(n-1)!}$ (which is very likely to be computationally expensive).

Naturally, I could have skipped step 2, computing directly the three empirical bootstrap means, e.g. $\hat{E}(C_{2000}) = \frac{1}{kn} \sum_{b=1}^k \sum_{i=1}^nC_{b,i,2000}$
The critical points in this approach are (i) how such it is possible for the practician to bootstrap over his grid nodes and (ii) how such she can access data related to each grid node and process them (computing and storing results)
My sources are theoretical: using the two definitions of bootstrapping and covariance.

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  • $\begingroup$ I do not provide any sort of working example. Simply because as you surely know, CV is not a place for programming. A contrario, your question will very likely be migrated on Stack Overflow. $\endgroup$ – keepAlive May 13 '17 at 15:08
  • $\begingroup$ Working examples can overcome ambiguous or unclear descriptions. In this case, since you haven't explained how any of your four "gets" are to be computed, either an expanded account of your solution or a worked example would seem to be essential for communicating it reliably to your readers. $\endgroup$ – whuber May 13 '17 at 19:16
  • $\begingroup$ @Whuber. Actually, the critical points in my answer are (i) how such it is possible for the OP to bootstrap over his grid nodes and (ii) how such he can access data related to each grid node. Furthermore, when visiting OP's SO-profile, I saw that he has programming capabilties (in R) which predispose him to easily develop the above computing algorythme. Also, I would be glad to know if what I explain makes sense to you. $\endgroup$ – keepAlive May 13 '17 at 20:16
  • $\begingroup$ I'm afraid I don't follow your answer. With this question, as with almost all spatial analyses of this sort, it's important to disclose the model of spatio-temporal variability you might have in mind. Without a model about all one could do is perform a traditional calculation of covariance, but (a) you don't seem to do that and (b) there is no reason to suppose there is any overlap of sample supports at all. $\endgroup$ – whuber May 13 '17 at 21:39

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