Advanced Features

The ReVAR algorithm by default trains the Long-Range AR model on a prediction subspace spanning the top principal coefficients of the training data. This reduces the spatial dimensionality of the data and thus reduces the number of prediction weights in the Long-Range AR model, which can be necessary to prevent overfitting. For this reason, the ReVAR class first uses the method pre_processing to find this prediction subspace and uses LongRangeAR with this specific prediction structure.

However, this prediction subspace is far from the only way to reduce the number of prediction weights in Long-Range AR. For many data sets, other options may also be effective.

For this reason, the LongRangeAR class is flexible to different prediction structures. That is, you can specify which vector components to use for predicting every other vector component. We call this choice of spatial prediction structure the Prediction Window.

This section covers how to use built-in prediction window structures, define your own custom structure, and discover structure automatically.

Built-in Prediction Window Structures

The package provides standard prediction window structures for common data types. These are passed to the model via the prediction_window_structure argument.

1. Distance-Based Structure Use this for data with a spatial or logical order (e.g., sensors along a line). It enforces a “neighbor” relationship, assuming periodic boundaries (circular wrapping).

from aomodel.prediction_window_structure import DistanceWindowStructure

# Each component is predicted by neighbors within distance 1
structure = DistanceWindowStructure(distance=1)

2. Prediction Subspace This is the default structure used by ReVAR. Use this for hierarchical data (like PCA modes). It models connections only among the top vector components.

from aomodel.prediction_window_structure import SubspaceWindowStructure

# Only the first 5 components predict each other
structure = SubspaceWindowStructure(subspace_dimension=5)

3. Full-Window Structure If the data has a low spatial dimensionality, the linear predictive model will likely benefit from including all possible vector component pairs inside of the prediction window.

from aomodel.prediction_window_structure import FullWindowStructure

# Include all possible vector component pairs
structure = FullWindowStructure()

In the case that you don’t want certain vector components to be predicted (e.g., low-variance PCA modes), you can specify which vector components are assigned prediction weights. Vector components which are not assigned prediction weights will be fully modeled by the spatial distribution of the residuals \(\boldsymbol{\xi}_n\).

from aomodel.prediction_window_structure import FullWindowStructure
import numpy as np

vector_dimensionality = 10                  # Spatial dimension of the data is 10.
predicted_components = np.arange(0, 5)      # Only predict the top 5 vector components.

# Use all 10 components to predict the top 5
structure = FullWindowStructure(predicted_components=predicted_components)

Defining Custom Structures

If your data has a complex topology (e.g., a specific sensor grid that isn’t linear), you can define a custom prediction mask.

Option A: Explicit Boolean Mask You can pass a boolean matrix directly using ExplicitWindowStructure. The mask must have shape (Vector Dimensionality, Vector Dimensionality). A True value at mask[i, j] indicates that component j is used to predict component i. In this case, a non-zero prediction weight is assigned to this index.

import numpy as np
from aomodel.prediction_window_structure import ExplicitWindowStructure

# Create a custom mask for 3 components
mask = np.array([
    [True,  False, True],  # Comp 0 predicted by 0 and 2
    [False, True,  False], # Comp 1 predicted by 1 only
    [True,  True,  True]   # Comp 2 predicted by everyone
])

model = LongRangeAR(
    vector_dimensionality=3,
    prediction_window_structure=ExplicitWindowStructure(mask)
)

Option B: Subclassing For reusable logic, you can subclass PredictionWindowStructure and implement the get_mask method.

from aomodel.prediction_window_structure import PredictionWindowStructure

class MyCustomGrid(PredictionWindowStructure):
    def get_mask(self, vector_dimensionality):
        mask = ... # Your logic here
        return mask

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Disclaimer: Approved for public release; distribution is unlimited. Public Affairs release approval # AFRL-2026-1309.