Here for instance is that temperature data for Moscow: Sometimes it’s most of the story, and often it’s the most obvious part, yet there can be more, or less, than meets the eye. If we write the book of a time series in polynomials it is the first chapter, and is most responsive to the longest time scale behavior. There is certainly value in knowing the linear trend, one can’t deny its utility, it tells us about the long est-term trend. Some reserve the word “trend” for the linear trend. When the background level is colder we’re more likely to get cold extremes, and when it’s hotter we’ll get more extreme heat. If it changes while the nature of the fluctuations remains the same, the probability of record-setting extremes will of course change. By “trend value” I mean exactly that: the background level at a given moment. What we’re really after is the background level against which temperature variations have their sway. But whatever its pattern, we usually identify the longer-term pattern of change with the trend. It might have wiggled around a lot but not really gone anywhere until some new factor came into play. It might be increasing, or decreasing, or it might not be changing at all. The longer-term is certainly something worth knowing about. Any given time series, say July average temperature in the Moscow region since 1881, might exhibit both short-term (more brief) and long-term (more lasting) patterns of change.
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