Introducing correlation structures via a rolling average

Now that we’ve introduced random, systematic and structured random errors, let’s move on to explore the concept of correlation structures. In order to do so, let’s consider the simple case of an unweighted rolling average of three measured values.

The figure below shows 9 measured values, labelled  x_0 to x_8 , and 7 averages, x_a to x_I , each of which is calculated using three consecutive measured values. For example, x_c  is the mean of x_2, x_3 and x_4 .

Notice that each of the measured values, x_0 to x_8 , contributes to more than one of the averages. In turn, the errors on each of the measured values will affect the value of more than one of the averages. For example, each of the measured values x_1 and x_2 is used in the calculation of both \bar{x}_a and \bar{x}_b . The result of this is that the errors in \bar{x}_a and \bar{x}_b are correlated. On the other hand, notice that there is no error correlation between \bar{x}_h and \bar{x}_b , because they do not have any measured values in common

If we now focus on \bar{x}_d , we can visualise the number of measured values in common with the other averages by looking at the graph below, which shows the fraction of common measured values that \bar{x}_d shares with the other averages we introduced in the diagram above.

We can use this graph to infer something about the error correlation between \bar{x}_d and our other averages: the ‘closer’ the two averages, the greater the number of measured values in common, and the greater the error correlation. Conversely, two averages that are ‘further away’ have less values in common and weaker error correlation, and averages that are sufficiently far apart have no error correlation.

In other words, the error correlation drops linearly relative to a particular average value, giving a triangular correlation structure. In general, then, for rolling averages of n measured values the correlation structure takes the form of a triangle with peak 1 and full base 2n , as shown in the diagram below.

So, we’ve now seen that a rolling average has a triangular correlation structure. Other processes give rise to other correlation structures, and we’ll look at some examples of these in the next recipe. However, before we do, let’s move on to turn our attention towards common correlation dimensions encountered in Earth observation.