A counterexample to decomposing climate shifts and trends by weather types

The literature contains a sizable number of publications where weather types are used to decompose climate shifts or trends into contributions of frequency and mean of those types. They are all based on the product rule, that is, a transformation of a product of sums into a sum of products, the latter providing the decomposition. While there is nothing to argue about the transformation itself, its interpretation as a climate shift or trend decomposition is bound to fail. While the case of a climate shift may be viewed as an incomplete description of a more complex behaviour, trend decomposition indeed produces bogus trends, as demonstrated by a synthetic counterexample with well‐defined trends in type frequency and mean. Consequently, decompositions based on that transformation, be it for climate shifts or trends, must not be used. Example of a process with two weather types of given frequency and mean (top) and corresponding single trends (bottom).