[forecast]: allow for "batched" forecasting

[gofford]

Description

mlforecast currently allows for recursive single-model forecasts, or direct multi-model forecasts where a single model is trained to predict a particular data point in the horizon.

A third option here is a middle ground between recursive and direct. In this option, multiple models can be trained but each is responsible for predicting a "batch" of the forecast horizon recursively. This is useful for long forecasting horizons with high resolution data.

Use case

Consider the case were I wish to predict the next 13 weeks of sales at a daily resolution. My options here are to have either 1 recursive model, which will have degrading performance with increasing horizon due to compounded error, or 91 individual forecasting models where each model trains a particular day ahead. Neither of these are ideal.

I can aggregate to weekly level, but I lose the daily resolution which is important for tracking the impact of things like promotions and daily events.

An alternative model that I have used in the past would be to have 13 models, where each model predicts a sequence of 7 steps (1 week).

[jmoralez]

Hey @gofford, thanks for the suggestion. Just to be clear, what you'd like to have is train model1 to predict 1 step ahead and use it to predict the first seven steps ahead, train model2 to predict 8 steps ahead and use it to predict the next seven steps ahead and so on?

[gofford]

Hey @jmoralez, pretty much! Although I'm not 100% sure that it's only a 1-step ahead forecast. When I've built similar (tree-based) forecasters manually I've trained and validated against the fit to the week as a whole, rather than a single point. I think we're talking about the same thing though.

[jmakov]

Is there any update on this?

[gofford]

I had forgotten that I had requested this but I would be keen to get an update too.

[jmoralez]

@gofford can you provide an example? I'm still not entirely sure on how the models are trained and how the features are updated in each step.

[gofford]

@jmoralez My models have set this up such that the test set is the size of the desired batch, and models are trained with increasing subsets of input lags.

Hopefully a diagram helps. Let's say we want a 4 week forecast at daily resolution, where each model is 1 week of data.

Key: 
    -: training data
    *: test data 

1st model - use up to lag1
|-------|-------|-------|-------|-------|*******|

2nd model: - use up to lag7
|-------|-------|-------|-------|       |*******|

3rd model: - use up to lag14
|-------|-------|-------|       |       |*******|

4th model: - use up to lag21
|-------|-------|       |       |       |*******|

At prediction time, each model is applied to the full dataset. Yielding models that predict over successive sets of future points.

Key:
    -: input
    a, b, c, d: predicted

1st model - effectively predicts 7 steps ahead
|-------|-------|-------|-------|-------|-------|aaaaaaa|

2nd model: - 8-14 steps ahead
|-------|-------|-------|-------|-------|-------|       |bbbbbbb|

3rd model: - 15-21 steps ahead
|-------|-------|-------|-------|-------|-------|       |       |ccccccc|

4th model: - 22-28 steps ahead
|-------|-------|-------|-------|-------|-------|       |       |       |ddddddd|

Which, when combined gives you a batched forecast, where each batch is a week of data at daily resolution.

|-------|-------|-------|-------|-------|-------|aaaaaaa|bbbbbbb|ccccccc|ddddddd|

There might be some formal concerns of instability and continuity at the boundary but in my practical experience it's not a major problem.

[gofford]

@jmoralez does this up? Let me know if you need any more information

[jmoralez]

Would it be something like this?

import numpy as np
from mlforecast import MLForecast
from sklearn.base import clone
from sklearn.linear_model import LinearRegression
from utilsforecast.data import generate_series

max_horizon = 4  # total horizon
steps_per_model = 2  # number of steps each model predicts recursively

series = generate_series(2)
fcst = MLForecast(
    models=[],
    freq='D',
    lags=[1],
)
X, y = fcst.preprocess(series, max_horizon=max_horizon, return_X_y=True)
base_model = LinearRegression()

# train each model on its corresponding target
# first model is trained on the target at t, second at t+2
models = []
for i in range(0, max_horizon, steps_per_model):
    mask = ~np.isnan(y[:, i])
    models.append(clone(base_model).fit(X[mask], y[mask, i]))

# each model predicts steps_per_model recursively
# and we update the target with its predictions
preds = []
with fcst.ts._maybe_subset(None):
    fcst.ts._predict_setup()
    for model in models:
        with fcst.ts._backup():
            for _ in range(steps_per_model):
                new_x = fcst.ts._get_features_for_next_step()
                step_preds = model.predict(new_x)
                preds.append(step_preds)
                fcst.ts._update_y(step_preds)

# result
fcst.make_future_dataframe(h=max_horizon).assign(pred=np.vstack(preds).ravel(order='F'))