Annual aggregation

If there are multiple results within one year in a particular monitoring programme, the GMP methodology specifies, that the arithmetic mean should be used for computing the central tendency (mean). From the nature of most of POPs in the air, whose distribution of values inclines to be lognormal, the use of geometric mean (or median as a non-parametric method) appears to be an eligible option, but hides a risk of inappropriate data treatment. Try to compare all the central tendencies in our example.

Annual aggregation

The process of annual aggregation represents nothing else than searching for one representative value to substitute 1-year segment of the time series. On one hand this process omits and hides certain characteristics of the time series (especially the within-year fluctuations), but on the other hand it also allows to compute some more extensive statistics not to be affected by details - especially the long-term time trends.

As in the previous examples, the selection of the best function used for aggregation of more measurements into one number depends on the distribution of the values. If we search for the central tendency, the most often used mean (intended also by the GMP methodology):

is suitable for normal (symmetric in general) distributions, while geometric mean:

is better in case of lognormal distribution. The nonparametric central tendency is represented by median:

.

Nevertheless, not only the central tendency is important for characterization of the one year period of measurements. We can use other functions to get the information about spread and distribution of the values. E.g., maximum tell us about the highest concentration through the year, which could be of high importance, or the number of individual measurement describes the design of pollutant monitoring. Therefore, there is an option to input your own function into the R example.

It is simple to compute an annual aggregation using the genanaggr function from the genasis R package, which allows using arbitrary aggregation function from the R language. There are two additional arguments allowing data filtering, which are not included in our interactive example: conditions of usability of (concentration) values inside individual years for the aggregation. First, the minn argument, default set up to 4, which restricts the minimal number of measurements during each year as a criterion of representativeness and second, the gap argument, default set up to 3, which restricts the maximal time gap betwen consequents measurements (and start/end of the year) in scale of average time gap between them as a criterion of equidistance:

genanaggr(flu,dat)

[1] 483.6099 461.4400 398.4615 237.4217

genanaggr(flu,dat,gap=2,minn=12)

[1] 483.6099 461.4400 398.4615 237.4217

As you can see, the results are the same, resulting from the fact, that the example data are equidistant and have at least 12 measurements each year.

We can also use another function than implicit set mean:

genanaggr(flu,dat,method="gmean")

[1] 178.5820 267.7369 233.9670 153.0955

Due to the well known AG unequality, the geometric mean (gmean) is lower than the arithmetic one. It is possible to use any other function including own defined functions inside the genanaggr function.