# Estimating event probability from historical time series with clear seasonality

I would like to predict the average number of days in a year for which two conditions are true:

• daily average temperature is below zero celsius
• the day was preceded by at least four days with daily average temperature below zero celsius

I've historical daily average temperature data for the location available for about 10 years. My initial approach was to use the one sided Chebyshev inequality which can be used to approximate a probability if the distribution is not known. However in this application I am interested in the probability of a special condition, can I use the Chebyshev inequality for a dummy time series as well? I.e.: 1 if condition is fullfilled, otherwise 0, --> the dataset would therefore look something like 0,0,0,0,0,1,1,0,0,0,0,0,1,1,1,1,1,1,etc.

How would you approach a problem like that from a different angle, the data clearly has seasonality, is there any distribution which I could use to have a better estimate than Chebyshev?

• Welcome to the site! I tried to make title a bit more informative about the nature of the problem.
– user88
Apr 11, 2011 at 21:10
• I'm missing something. If we are predicting "the average number of days in a year" -- why would this necessarily be seasonal? Annual sunspot numbers are seasonal because that's an 11 year cycle, but it's not obvious to me that a seasonal component would be necessary here. Apr 12, 2011 at 4:51
• Actually, you are right. Since I am not interested in predicting whether an actual day in future might fulfill the conditions I guess I can aggregate the data, however then I will only have 10 observations regarding the number of days in the past (i.e.: 10,14,12,...), which is quite a small sample. I will check if I can get more data, and maybe a simple a Gaussian distribution might enough.
– Keek
Apr 12, 2011 at 8:27

I think the joint distribution of temperature data on successive days could be reasonably modelled using a multi-variate Gaussian (Gaussian distributions are often used in statistical downscaling of temperature). What I would try would be to regress the mean and covariance matrix of the temperature time series on sine and cosine components of the day of year (to deal with the seasonality). The details on how to do that are given in a paper by Peter Williams, Williams uses a neural network, but I would start off with just a linear model. This will give you what climatologists would call a "weather generator" (of sorts). Using this you could generate as many synthetic time series as you want with the appropriate statistical properties, from which you could estimate the probabilities you require directly. You would need to estimate the window over which temperatures were usefully correllated - which may be quite high in winter due to blocking patterns (for the U.K. anyway). A bit baroque I suppose, but it would be the thing I would try!

• Linear model? Every meteorologist will tell you that everything they do is just a small adjustment to "tomorrow will be the same" ;-)
– user88
Apr 11, 2011 at 21:15
• Thanks for the hint, I've not much experience with multivariate Gaussian distributions, but I will definitely try that approach as well. Right know I am more interested in a quick & dirty solution, which does not have to be very sophisticated and zbicyclist's comment reminded me that I can ignore the seasonality entirely.
– Keek
Apr 12, 2011 at 8:38
• @mbq ;o) That is indeed the baseline meterologists need to beat! Apr 13, 2011 at 7:26

I know little about meteorology, so my following assumptions may be wrong: today's temperature is similar to yesterday's and the day before yesterday's (maybe more days going back), and also similar to temperature a year age, two years ago, three years ago, etc.

If these assumptions got reinforcement I would use an ARMA model using days -1, -2, … and -365, -365*2, -365*3, … as predictors of today's temperature, and maybe a few days looking back in the moving average terms. (You can imagine many variants of this model.)

After fitting the model I would make a large number of model based simulations predicting the temperatures for each of the following 365 days, and count the cases satisfying the two conditions.

• For temp in meteorology I beleive using seasonal lag (at -365 days) is unnatural: the similaity is because of a seasonal climate, if 365 days ago where hotter, say, than the clima expected value, there is no reason to beleave today also to he hotter Mar 2, 2019 at 0:47