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Firstly, how we can define seasons - Winter, Spring, Summer and Autumn? There are many possibilities, let's choose average twenty-four hours temperature $\bar t$.
Next we set $t_1$ and $t_2$ (usually $0^\circ C$ and $6^\circ C$).
Some meteorologists define the winter as period when $\bar{t} < t_1$ steadily (stably).
So the summer is period, when $\bar{t} > t_2$ stably.
Between $t_1$ and $t_2$ are placed transitional time (for my purposes no matter spring or fall).

There is hypothesis that in Central Europe (and in most of Russia) transitional seasons will be shorter and shorter in the future.
Well meteorology is a weird science for me so let's use analytical statistics! With Wolfram Mathematica it's possible to get a lot of weather data. Here is example of the analysis: for Moscow 1994-95 (and zoomed period Feb-Jun'94)

enter image description here enter image description here

And here is the question: how to find duration of transitional seasons correctly enough?

Days with $t_1 < \bar{t} < t_2$ painted green. But duration of all green days is not correct solution, because many of them are thaws and frosts, not transition!

So I am trying to find an algorithmic definition of transitional seasons duration.

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    $\begingroup$ You need to rigidly define the hypothesis before trying to answer it. As it stands you have a hypothesis relating to 'transitional seasons', but then when you attempt to answer that you find the way you've define that to be imprecise. You now want to find a suitable way to define transition, after looking at the data. Stop. Any analysis you subsequently do is worthless, since you can now (consciously or not) tweak your post hoc definition of 'transitional season' to either support or refute the hypothesis. There is already enough climate pseudo-science that does this. Please don't add to it. $\endgroup$ Commented Feb 13, 2013 at 6:17
  • $\begingroup$ Why don't just define seasons as they are standardly defined, using solstices and equinoxes? That is (for the Northern emisphere): spring 21 March - 21 June, summer 21 June - 23 September, autumn 23 September - 21 December, winter 21 December - 21 March. $\endgroup$
    – nico
    Commented Feb 13, 2013 at 6:20
  • $\begingroup$ @nico: Because the standard definitions don't allow lesobrod to test their hypothesis about season lengths changing. What they are missing is some clear and reliable criteria as to what will mark the start and end of "stable" and "transitional". And, albeit with less harsh of a tone than Bogdanovist, that is what I suggest they do before moving forward. There are many possible ways to define this... and there is probably not a "correctly enough" way to recommend. $\endgroup$ Commented Feb 13, 2013 at 6:23
  • $\begingroup$ @drknexus: I think it is important to have those. He would have a null hypothesis that the seasons would start and end at those dates, then fix some precise baseline parameters for temperature for each season (e.g. taken from a certain arbitrarily chosen reference period). He can then determine whether there is a deviation from those temperature parameter over the course of the standard seasons. It is clear that the approach shown in the images is very naïve as it does only consider each temperature value and not its neighborood. $\endgroup$
    – nico
    Commented Feb 13, 2013 at 6:34
  • $\begingroup$ @nico: Yeah, but that approach denies the premise of his question. Mind you, it might be appropriate to deny, but I prefer to grant people their premises when possible. Your approach would get him in the ball park of answering his question, but it would be a through a back-door rather than directly addressing his hypothesis of changing "season" duration. I think it is probably reasonable to try to create a function that generates the temperature shapes he is talking about and then fit that function to the yearly data and compare parameter estimates across years. $\endgroup$ Commented Feb 13, 2013 at 6:42

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I’m going to answer on my own question.
In fact, main factor for climate is daily average insolation. On 100 km above Earth it's value depends only from latitude and position of day in the year (orange curve). From this point, we have no any "stable" seasons, just smooth variations of warm and cold. For convenience, the period with curve above the middle may be "summer", and below -"winter".
Daily average temperature (blue points) is depends on insolation (but on many others factors too).
In Tashkent (pic.1) it is close to insolation curve behavior, but in Lisbon 2001 (pic 2) looks chaotic enough (all data are rescaled for comparison).
Tashkent climate curve Lisbon climate curve

These pictures suggest me an idea. Let's call “transition“ day when
daily average temperature crosses year average temperature line.
In ideal continental climate there should be only two such days in the year.
In oceanic climate every third may be transition day. So I'd like to suggest measure for "seasonality" of the weather. It's just number of transition days on the year, divided by 365.
Here are examples of this measure for Khabarovsk and London during last decades: enter image description here enter image description here
For me it's clear that such measure makes sense, but I can't prove it, using strict stats methods.

And now we can define what are spring and fall days, regardless of the phenologist’ assumptions and using only year average temperature. Let's partition 365 days on samplings by 5-10 days (length of sampling is the only free parameter!) and calculate seasonality measure for every sampling.
Here are another two examples.
enter image description here enter image description here

I see spring and fall seasons absolutely clear, regardless of the "vegetations temperature", "hormone level" etc. Exact duration of transition seasons is dependent from length of sampling (but general qualia stay the same) and this is really statistic question.
That’s all, so I’ll be very glad to see some corrections and opinions.

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