Hello

I am surprised that the correlation plot is so symmetric around the identity. This means that if you exchange yesterday and today axis, the plot is almost the same.  In turn, this means that the dynamics is symmetric : there is very weak time orientation !

Here is a suggestion for another way to create the model using bivariate probability distribution :

You could maybe model the random variable (yesterday, today) by an inverted Clayton copula (also called HRT) with suitable marginals CDFs. 
The model CDF will be of the form C[ F[ tT ] , G[ yT ] ]. Then, to get the conditional law of probability you have to cut the bivariate PDF model (i.e. make the value constant ) at a particular value of the yesterday temperature yT. So you get a univariate, PDF fonction c[ F[ tT ] , G[ yT=cste ] ] , to be rescaled by probability of yT, which is the probabilty density fonction you are looking for.

If it works this approach could be much more light and general than modelling individual conditionnal probabilities by a familly of distributions. 

What is nice with copula is that they usually have a simple symbolic expression, so if the margins also have one, you get a symbolic closed form of your model.


Nicolas

Hi Anton

2) I don't agree , the cloud of points plot is a Poincaré section of some periodic dynamic signal, and the symmetry is not mandatory.
For example consider :which gives a kind of multispiral plot.

3) I did some trials will the Clayton copula, not so sure now it would give the best fit. But quite sure that the copula approach is worth to try.
The Clayton copula has higher correlation for the head of marginals but  Mathematica does not have the inverted Clayton availaible. The Gumbel correlates both head and tail ends, and it is easy to use it in Mathematica.

Here is the idea with the Gumbel :
(I am not used to do fitting of data, so this just gives an idea and this is  far to be a good fit)
 which gives for 0° a plot which shows that the most probable temperature is higher, this is coherent with your results

The overall behaviour of the distribution : symmetry in the middle range, and skew trend in the high temperatures seems to be correct,
nevertheless the model is very approximative and far to be as much accurate than with your quantile regression package.