FPLS: Functional Partial Least Squares

A Python package for functional regression using Partial Least Squares (PLS), implementing the methods developed in:

Babii, A., Carrasco, M., & Tsafack, I. (2025). "Functional Partial Least-Squares: Adaptive Estimation and Inference." Journal of the American Statistical Association.

Overview

The fpls package provides a clean, scikit-learn-style API for fitting functional PLS regression models with adaptive component selection. The core algorithm uses a conjugate gradient method to solve the functional PLS problem, with integrals approximated via Riemann sums on uniform grids.

Key Features:

• FunctionalPLS estimator with .fit(), .predict(), and .fit_predict() methods
• Adaptive early stopping for automatic component selection
• Support for both NumPy arrays and Pandas DataFrames
• Visualization tools for plotting coefficient functions

Installation

Install library:

pip install fpls

Quick Start

Example: Temperature Effects on Crop Yields

This example demonstrates FPLS on a real-world climate impact study. Using 70 years of county-level data east of 100th meridian, 1950–2020, we analyze how temperature exposure affects corn and soybean yields. The outcome variable is log-yield (bushels per acre), and the functional regressor is the distribution of temperature exposure during the crop growing season, measured in growing degree-days and discretized into 1°C temperature bins.

from fpls import load_example_data, fit_fpls, plot_coefficient_function
import matplotlib.pyplot as plt

# Load corn and soybean data
X_corn, y_corn, s = load_example_data("corn")
X_soy, y_soy, _ = load_example_data("soybeans")

# Fit FPLS models with 10 components
coef_corn, _ = fit_fpls(X_corn, y_corn, m_max=10, ds=1.0)
coef_soy, _ = fit_fpls(X_soy, y_soy, m_max=10, ds=1.0)

# Extract coefficients using m components
m = 4
beta_corn = coef_corn[:, m]
beta_soy = coef_soy[:, m]

Visualizing Results

# Create a figure with two subplots
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Plot corn on the first subplot
plot_coefficient_function(
    s, beta_corn,
    title="Impact of Temperature on Corn Yield",
    xlabel="Temperature (°C)",
    ylabel="Log Yield (Bushels)",
    color="#2E86AB",
    ax=ax1
)

# Plot soybeans on the second subplot
plot_coefficient_function(
    s, beta_soy,
    title="Impact of Temperature on Soybeans Yield",
    xlabel="Temperature (°C)",
    ylabel="Log Yield (Bushels)",
    color="#E85D04",
    ax=ax2
)

fig.savefig('docs/images/crop_yield_comparison.png', dpi=300, bbox_inches='tight')
plt.show()

Output:

The coefficient functions reveal how different temperature ranges affect crop yields, with the functional approach capturing nonlinearities of temperature effects at different parts of the distribution.

API Reference

Main Classes

FunctionalPLS(m_max=10, grid_size=None)

• .fit(X, y, ds=None) - Fit the model
• .predict(X, n_components=None) - Make predictions
• .fit_predict(X, y, ds=None, n_components=None) - Fit and predict
• .coef_ - Fitted coefficients (shape: n_features × (m_max+1))

Functions

fit_fpls(X, y, m_max=10, ds=None)

• Convenience function returning (coef, ds) tuple

select_components(X, y, m_max=10, ds=None, tau=1.01, delta=0.1, xi=0.01)

• Adaptive early stopping to select optimal number of components

Visualization:

• plot_coefficient_function(s, beta, ...) - Plot single coefficient function
• plot_comparison(s_list, beta_list, titles, ...) - Compare multiple functions
• load_example_data(dataset) - Load example datasets ('corn' or 'soybeans')

Utilities:

• create_uniform_grid(start, end, n_points) - Create discretization grid
• compute_mse(y_true, y_pred) - Mean squared error
• compute_r2(y_true, y_pred) - R-squared
• center_functional_data(X) - Center by mean function
• frisch_waugh_residualize(Z, X_controls) - Residualize out controls

Examples

See the examples/ directory for complete worked examples, including:

• Basic functional regression on simulated data
• Crop yield analysis

Citation

If you use this package in your research, please cite:

@article{babii2023functional,
  title={Functional Partial Least-Squares: Adaptive Estimation and Inference},
  author={Babii, Andrii and Carrasco, Marine and Tsafack, Idriss},
  journal={Journal of the American Statistical Association},
  year={2025}
}

Authors

• Andrii Babii
• Marine Carrasco
• Idriss Tsafack