Modelling the shape of a distribution / proportions based on a predictor I'm a moderate beginner in stats, so this may or may not be something you can sensibly answer.
In my data I have two parameters for each timeline, the number of events in a normalized timeline and the location of these events on the timeline. I have noticed a trend by which when the number of events is low, they tend to cluster towards the beginning or end of the time-line. 
The location of these events on the timeline can be seen as a proportion of the events that take place in that data bin (say 10%). Eventually these proportions will add up to one for each timeline.
I am now looking to build a model of this relation. A quick and coarse way would be to model the proportions at each of these 10 bins among the timeline. And the data seems to allow for such modelling even with a linear regression (after some log transformations).
And thus I could have a model for each of the bins (10 models altogether) and predict the values on the basis of that. At the same time, this leaves aside the very strict dependency on the 10 predicted values, that they are just about the proportions and will always add up to 1.
Another option that I tried was to look at the distance from the edges as a function of the number of events, and the model seemed to give good results, but it also is not ideal, as on low numbers the events seem to be clustered more towards the end rather then beginning, and it is generally leaving out a lot of information.
The best case scenario, I would like to model the distribution of the events on the timeline as predicted by the number of events (and by looking at the data, the connection seems to be there. An impression of the data can be seen from figures below, I can also share a reproducible example if it helps. Thanks! I will also appreciate if you can point me to similar issues that have already been solved.
Figures: The left image is an average for low number of events, the right one for high number of events.


 A: As vectors of proportion are defined on the bounded support $[0, 1]$, it is natural to represent them with probability distributions that are also defined over this bounded support. A natural choice is then to use the Dirichlet distribution or some more sophisticated distribution such as the generalized Dirichlet or the Beta-Liouville distribution (for which the Dirichlet distribution is a special case).
When a simple distribution is not enough to model the data, a mixture of these distributions is used. A mixture of distribution is defined as:
$$ P_{mix}(x) = \sum_{i=1}^M w_i p_i(x) \; ,$$
where the $w_i$'s are the weights (or mixing coefficients) of each component and $p_i$'s are single probability distributions (for instance Dirichlet distributions). The sum of the $w_i$'s is equal to 1 and $M$ represents the number of components (distributions) in the mixture.
The parameters of the mixture (i.e., the mixing coefficients and the parameters of each distribution) can be computed used an Expectation-Maximization algorithm or a method of moments (the latter can actually be used as an initialization of the former, which is in general more accurate but computationally heavier).
A reference work presenting a method to estimate the finite mixtures of Dirichlet: Unsupervised learning of finite mixture model based on the Dirichlet distribution and its application, by Bouguila, Ziou and Vaillancourt in IEEE Transaction on Image Processing (2004)
Equivalent publications exist for the generalized Dirichlet and Beta-Liouville mixture models.
When working with time-series of vectors of proportions for which the dynamical component has to be model, one choice can be to use hidden Markov models based on these same distributions.
For the Dirichlet-based HMM, all the equations for the estimation of the model are given in the following research report: Dynamical Dirichlet mixture model, by Chen, Barber and Odobez, IDIAP-RR 2007-2. Equivalent research papers have been published for HMMs based on the generalized Dirichlet and the Beta-Liouville distributions.
