Which regression can be apply to these data?

Question

I have a data table comprising a number of independent and dependent variables

Independent variables: $x_1,x_2, \ldots, x_n$. Variables are, as usual, numerical, some or them can be categorical

Dependent variables: $y_1, y_2, \ldots, y_m$, all numerical. They are results of some follow-up measurements and satisfy a monotony condition: $y_1 \geqslant y_2 \geqslant \ldots \geqslant y_m$ (in fact, measurements are performed at non-equidistant instants $t_1, t_2, \ldots, t_m$)

Number of cases: say, $N$

I would like to find the following dependency: $(x_1,x_2, \ldots, x_n) \rightarrow Y(t),\ t= t_1, t_2, \ldots, t_m$ and estimate principal questions of regression analysis (importance of variables etc)

Answer 1

To learn the mapping and ascertain feature importances, I think the following could work.

The categorical variables would be converted to a numerical representation. Fit a random forest regressor to learn the mapping between $X$ and $Y$. Then, use permutation importance testing to determine which independent variables contribute the most to the $R^2$ averaged over $Y$.

For this particular task, the $y_i$ are related by their relative rank. I have suggested RandomForestRegressor because it naturally handles multivariate regression (i.e. it doesn't assume independence between the $y_i$ targets), and it requires less feature preprocessing. In general, you can arrange any model into either a univariate setup (treating each $y_i$ as a separate regression task), or a multivariate one.

If you require a more mathematical/statistical characterisation, consider asking on the stats StackExchange CrossValidated as well.