Statistical model for quantities that add up to 1

Question

I want to create a model for quantities $z$ that live in a probability simplex, that is, they are nonnegative and always add up to 1:

$$ S = \left\{z \in \mathbb{R}^{k} : z_1 + \dots + z_{k} = 1, z_i \ge 0 \text{ for } i = 1, \dots, k \right\},$$ where $k$ is relatively small, let's say $k \leq 5$.

Additionally, I know these values come from a black box process where I only know some details about it, e.g. an input array of the form: $(x_1, \ldots, x_k)$ with additional features $(\theta_1, \ldots, \theta_k)$ where again, $(x_1, \ldots, x_k)$ are nonnegative and add up to one (notice it's also of dimension $k$).

For modelling purposes let's assume we got samples of all these arrays ($x$, $z$ and $\theta$) as time series (always all values measured at the same time), and additionally we know that:

1. There's a dependence of the $z$ array from the $x$'s and $\theta$'s, but with a lag, that is, $z_t$ depends on the previous time values of $x$ and $\theta$. This lag is not constant over time but it can't grow indefinitely either, let's say it's a maximum of $N$ previous timestamps where $N$ < 10

2. The $z$ and $x$ quantities are relative rates, not really probabilities

3. The output $z$ is somehow a mixture of the input $x$ of previous timestamps, e.g. $z_i^T$ will depend strongly on the respective $x_i^{t}$ for recent values $T-N \leq t \leq T-1$, and their associated parameters inside $\theta_i^{t}$ (that correspond to the $i$t-h quantity). But it won't be necessarily inside their convex combination, i.e. it could happen that the $z_i^{T}$ lies outside the minimum and maximum values of the recent $x_i^{t}$ values

I think that taking ideas from the following fields can be useful:

1. Graphical models - with $k$ (or $k-1$) output nodes
2. Neural networks - use a multilayer perceptron using an idea similar to the softmax activation function
3. Generalized additive models (create something similar to the multinomial logit model but with a response variable that isn't a classification response but probabilities)
4. Multivariate time series - but I'm not familiar with ways to impose the bounds that the series are between 0 and 1 and also the add-up-to-1 condition into the modelling framework

And I wonder if there's already some literature or known model for this kind of processes (nonnegative continuous variables that add up to a fixed quantity), but my search hasn't brought results yet. Do you have any pointers to relevant modelling approaches?

Answer 1

As others have written, this is compositional data analysis. We have a compositional-data tag, so searching for threads carrying, e.g., this tag and the "time-series" one may be helpful.

One paper on forecasting compositional time series is Snyder et al. (2017, IJF). Essentially, the idea is to transform the original compositional time series, then forecast the transformed data, and finally back-transform the forecasts in a way that respects the compositional nature. Thus, this approach should also work with other forecasting methods, like neural networks, or anything that uses lags or causal effects.