This weekend I’m a bit busy again, however Roman M has done an interesting post on combining surface station anomalies.
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Combining Stations (Plan B)
In his post on combining station monthly temperature series, Tamino proposes a method for calculating an “offset” for each station. This offset is intended to place the series for the stations all on at the same level. The reason that makes such an accommodation necessary is the fact that there are often missing values in a given record – sometimes for complete years at a stretch. Simple averages in such a situation can provide a distorted view of what the overall picture looks like.
In his proposed “optimal” method, Tamino suggests a procedure based on least square methods to calculate a set of coefficients for shifting a station record up or down before calculating a combined series for the set of stations. The starting point is a sum of squares (which for calculational and discussion purposes I have rewritten in a slightly different form):
Here,
i (and j) = 1, 2, …, r, = the station identifiers for a set of r stations
t = 1, 2, …, N, = the time (i.e. year and month) of the observation
xi(t) = the observed temperature at station i at time t
µi = the offset value which is subtracted from station i
δi(t) = indicator of whether xi(t) is missing (0) or available (1)
In essence, the Sum of Squares measures the squared differences between the “adjusted” temperatures of every pair of available stations at each time with each pair weighted equally. The estimated offsets minimize this sum. The mathematically adept person will recognize that the solution for the individual µ’s is not unique, however the differences between them are. Tamino chooses to set µ1 = 0 to choose a specific solution.
Read the rest here