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What is the probability that the weather tomorrow will be equal to the weather today (e.g. more specifically, if we know there had been more than 1mm precipitation during one day, what is the conditional probability that there will be more than 1mm precipitation also the next day)? Can you guide me to some references?

I want to use this example in a lecture to illustrate to physics students the concept of probability in forecasting.

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  • $\begingroup$ What you are talking about is called "persistence forecast" and it is the first guest at the weather tomorrow. You can check: ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fcst/mth/prst.rxml $\endgroup$ – arkaia Oct 15 '18 at 13:32
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    $\begingroup$ @arkaia first guess or first guest ? :) $\endgroup$ – gansub Oct 15 '18 at 14:05
  • $\begingroup$ Funny. brain fart $\endgroup$ – arkaia Oct 15 '18 at 15:22
  • $\begingroup$ It's the first guest to the forecasting party! $\endgroup$ – JeopardyTempest Oct 15 '18 at 15:23
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    $\begingroup$ This would make an interesting study. There's plenty of data out there: opendata.stackexchange.com/questions/10154 -- if anyone's seriously interested (and can't find an existing study), feel free to contact me, I'd be interested in participating (contact info in profile) $\endgroup$ – Barry Carter Oct 15 '18 at 19:54
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As mentioned, the term you are looking for is persistence. Generally its strength varies wildly depending on what variable you are looking at, the location, and by whether you also further select other factors like season and other meteorological variables.

I would expect daily precipitation persistence is a relatively bad predictor, given that precipitation is often either a short-term event driven by very transient dynamics like frontal passage, or sees a fairly random spatial distribution over small areas driven by small perturbations and nonlinear interactions (thus typically leading to rain PoPs regularly being more useful than binary forecasts).

Persistence is often used as a basis to compute forecast skill... but it seems to be be reported in research using more sophisticated summarization metrics like skill scores than straight probabilities.

However, with a little searching I did at least find A Markov Chain Model for the Probability of Precipitation Occurrence in Intervals of Various Length by James E. Caskey, Jr., which did at least have a table along the lines you've asked... for Denver, Colorado between 1949-1958, using 0.01" (0.254 mm) as the condition:

enter image description here

For those struggling to figure the values out, in this table when $\rho_1$ is similar to $r_1$, utility of precipitation persistence is low.

Colorado is a mountainous area, so I could imagine large-scale wind direction aloft probably proves more significant than many areas. For a location like Florida in the summer or Oklahoma in the spring, I'd imagine the skill is lower as persisting variables are less important. Still, because humidity often varies more slowly, regularly controlled by air masses and large scale wind directions, which often are directed by large synoptic weather features, there will likely be at least some difference in the conditional probabilities in all locations/seasons.

Don't know if that data is useful enough for your lecture purposes, but hopefully it offers a branch to seek into. You may be able to dig up a few more similar tables by searching on through research journal results in places like the AMS or, if need and desire be, could calculate such variables from station data available at sites like NCEI.

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  • $\begingroup$ What you write is as always nice but my first view was: mathematical approach-> 0! to what is understood as == on programming. There is not going to be equal quantities -you can fall in subatomical measures there is not that "groundhod day"-.But I understand persistance is one of that index between 0 and 1 comming from probability and there is some theory behind this in meteo $\endgroup$ – user12525 Oct 15 '18 at 17:08
  • $\begingroup$ @Universal_learner I can't say I'm quite sure what you're saying here... but yes, over large data sets, the two percentages would become more and more equal if there is no forecast skill for persistence (the "random noise" contribution/confidence interval shrinks). But because each day isn't entirely independent to the next, there is some skill in persistence (in fact that winter data suggests it's about 3.5 times as likely to precipitate there the day after a wet day compared to a dry one, pretty remarkable!) $\endgroup$ – JeopardyTempest Oct 15 '18 at 17:23
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    $\begingroup$ It would be better if I write in spanish and translator sorry :( meaning my first thougth was probability is zero. The closest you wil be is 0 rainfall 0 clouds but irradiation changes cause of seasonality. I must don't understand well the question. What I learned is there is a meteo index called persistance. For that I say there must be physics/mathematic theory behind more than "probabilities of an equal next day"; measure of the daily persistance on meteo patterns looks to be the origin of that variable called persistance. I much appreciate your divulgation. $\endgroup$ – user12525 Oct 15 '18 at 17:28
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Pearson publish a data set for A-level students (high school) that includes weather data from various locations. While the data set is (deliberately) flawed [one task for students is to identify flaws in the data set] it is "real" data and questions of persistence can be investigated with it.

For example, in Heathrow, May-October 2015, there were 80 days on which it rained and forty days on which it rained, given it had rained the previous day. So at this time the probability of rain was about 0.43 and the conditional probability is about 0.50. So forecasting the weather based only on persistence gives a rather poor forecast.

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