Hi @bashtage ,

Persuant to #24, I did a quick draft of additional support for IV-like formula in formulaic (in addition to the multi-part formula that was already implemented). There are some bugs and rough edges, but would you mind taking a look and adding any suggestions? I'm also not sure whether this should be a plugin or part of the default stack, so your thoughts there would be helpful too. All naming/etc is in draft status, so you can feel free to suggest improvements there.

Suppose you wanted to model some data using IV. With these patches you could write:

>>> from formulaic import Formula
>>> Formula("y ~ x1 + x2 + [ x3 + x4 ~ z1 + z2]")
.lhs:
    y
.rhs:
    root:
        1 + x1 + x2 + x3_hat + x4_hat
    .deps:
        [0]:
            .lhs:
                x3 + x4
            .rhs:
                1 + z1 + z2

The resulting formula could then be parsed by the consumer of the formula to do the right things.

If you end up using an interaction term, or later multiplying, formulaic still does the right thing.

>>> formulaic.Formula("y ~ x0 + [ x1:x2 ~ z1 + z2 ] : x3")
.lhs:
    y
.rhs:
    root:
        1 + x0 + x1:x2_hat:x3
    .deps:
        [0]:
            .lhs:
                x1:x2
            .rhs:
                1 + z1 + z2

The x1:x2_hat is considered one factor, and looked up by name.

Note that this could also (with a small amount of effort) also be used for double ML (if we add a delta transform/operator), and more general things like:

>>> formulaic.Formula("y ~ x1 + x2 + [ x2 + x3 ~ z1 + z2 ] + [ x4 ~ z3 + [z4 ~ a1 + a2 ] ]")
.lhs:
    y
.rhs:
    root:
        1 + x1 + x2 + x2_hat + x3_hat + x4_hat
    .deps:
        [0]:
            .lhs:
                x2 + x3
            .rhs:
                1 + z1 + z2
        [1]:
            .lhs:
                x4
            .rhs:
                root:
                    1 + z3 + z4_hat
                .deps:
                    [0]:
                        .lhs:
                            z4
                        .rhs:
                            1 + a1 + a2

Though this does stress credulity a bit.

Lastly, I plan to add some utility methods to Formulaic to allow easy recursive iteration over the formula to assist with the evaluation of dependencies and updating of the dataframe as you go up the tree. This might even be able to be integrated into the high-level tooling, if so desired, with the user passing a dep_data_resolver hook of some description.

closes: #24