Spatialize: A Python/C++ library for Ensemble Spatial Analysis (ESA)

An open source library for spatial analysis that combines the simplicity of basic methods with the power of geostatistical tools.

Overview

Spatialize implements Ensemble Spatial Analysis (ESA), which encompasses two complementary approaches: Ensemble Spatial Interpolation (ESI) and Ensemble Spatial Simulation (ESS). These novel methods address the limitations of traditional geostatistical approaches by leveraging ensemble learning techniques.

ESI works by generating multiple estimates for each target location by creating different spatial partitions of the sample data and applying an interpolation algorithm within each local subset. These local estimates are then aggregated to produce robust predictions. ESS extends this framework to provide stochastic simulation capabilities.

Designed to bridge the gap between expert and non-expert users of geostatistics, Spatialize provides automated tools that eliminate the need for manual spatial analysis and extensive domain expertise.

Main features:

• Automated Spatial Estimation: Minimal user intervention required
• Stochastic Modelling & Ensemble Learning: Robust, scalable and suitable for large datasets
• Uncertainty Quantification: Provides both point estimates and empirical posterior distributions
• Flexible Data Support: Works with both gridded and non-gridded data
• Hyperparameter Optimization: Built-in grid search with cross-validation
• High Performance: C++ core with Python interface

Installation

The source code is currently hosted on GitHub at: https://github.com/alges/spatialize

Direct installers for the latest released version are available at the Python Package Index (PyPI).

PyPI

pip install spatialize

System Requirements

• Python 3.10+
• Compatible with Linux, macOS, and Windows

Dependencies

• NumPy: Powerful n-dimensional arrays and numerical computing tools
• pandas: Fast, powerful, flexible and easy to use open source data analysis and manipulation tool
• Matplotlib: Visualization with Python
• scikit-learn: Machine Learning in Python
• SciPy: Fundamental algorithms for scientific computing in Python

Core Concepts

Local Interpolators

• IDW (Inverse Distance Weighting): Simple yet powerful with configurable distance exponent
• Kriging: Geostatistical method with multiple variogram models (spherical, exponential, cubic and gaussian)
• Adaptive IDW: Automatically optimizes IDW parameters (exponent, anisotropy) per partition cell via leave-one-out validation — no manual tuning required

Note: Adaptive IDW is not a separate function — pass local_interpolator="adaptiveidw" to esi_griddata() / esi_nongriddata().

Local Classifiers

• knn_pca: Adaptive anisotropic k-NN, the default classifier for categorical ESI
• scikit-learn: Wraps any fitted scikit-learn estimator (e.g. SVM, Random Forest, Decision Tree)

Partition Methods

• Mondrian Forests: Uses recursive, axis-aligned partitions (supports up to 5D)
• Voronoi Forests: Uses Voronoi diagram-based partitions (supports up to 2D)

Quick Start

Here are a few examples to get you started. For further examples, please check the spatialize examples repository or the spatialize documentation.

Basic Gridded Data Estimation

import numpy as np
from spatialize.gs.esi import esi_griddata

# Generate sample data
def func(x, y):		# a kind of "cubic" function
    return x * (1 - x) * np.cos(4 * np.pi * x) * np.sin(4 * np.pi * y ** 2) ** 2

points = np.random.random((100, 2))
values = func(points[:, 0], points[:, 1])

# Define the estimation grid
grid_x, grid_y = np.mgrid[0:1:50j, 0:1:50j]

# Perform ESI estimation
result = esi_griddata(points, values, (grid_x, grid_y),
		      local_interpolator="idw",
		      p_process="mondrian",
		      n_partitions=300,
		      alpha=0.8,
		      exponent=1.0
		      )

# Get results
estimation = result.estimation()
precision = result.precision()

# Quick visualization
result.quick_plot()

Non-gridded Data Estimation

from spatialize.gs.esi import esi_nongriddata

# Define target locations
target_points = np.random.random((50, 2))

# Perform estimation, using Kriging as local interpolator
result = esi_nongriddata(points, values, target_points,
		         local_interpolator="kriging",
		         model="spherical",
		         nugget=0.1,
		         range=10.0,
		         sill=1.0
		         )

Automated Hyperparameter Search

from spatialize.gs.esi import esi_hparams_search

# Search for optimal parameters
search_result = esi_hparams_search(points, values, (grid_x, grid_y),
			           local_interpolator="idw",
			           griddata=True,
			           k=10,
			           exponent=[1.0, 2.0, 3.0, 4.0],
			           alpha=[0.7, 0.8, 0.9],
			           n_partitions=[100, 300, 500]
			           )

# Perform estimation using best parameters found
best_result = esi_griddata(points, values, (grid_x, grid_y),
			   local_interpolator="idw",
			   best_params_found=search_result.best_result()
			   )

# Visualize search results
search_result.plot_cv_error()

Adaptive ESI

from spatialize.gs.esi import esi_griddata

# Adaptive IDW optimizes exponent and anisotropy per partition cell automatically,
# so no exponent/alpha-per-axis tuning is required
result = esi_griddata(points, values, (grid_x, grid_y),
		      local_interpolator="adaptiveidw",
		      n_partitions=200,
		      alpha=0.7
		      )

result.quick_plot()

Categorical ESI

import numpy as np
from spatialize.gs.cat_esi import cat_esi_nongriddata

cat_points = np.array([[0.1, 0.2], [0.5, 0.6], [0.8, 0.1]])
cat_values = np.array(['A', 'B', 'A'])
cat_targets = np.array([[0.3, 0.3], [0.7, 0.7]])

result = cat_esi_nongriddata(cat_points, cat_values, cat_targets,
			     classifier="knn_pca",
			     n_partitions=300,
			     alpha=0.8
			     )

print(result.estimation())  # predicted categories
print(result.precision())   # per-location agreement ratio

Ensemble Spatial Simulation (ESS)

from spatialize.gs.ess import ess_sample
from spatialize.empirical import FittedModelFactory

# ess_sample draws posterior simulations from an existing ESI ensemble
sim_result = ess_sample(esi_result=result,
			n_sims=1000,
			fitted_model_factory=FittedModelFactory(
				point_model_name="kde",
				kernel="tophat"
			)
			)

License

Apache-2.0

Citing Spatialize

Please refer to the following articles when publishing work relating to this library or the ESI model:

@article{spatialize2026,
	author  = {Navarro, Felipe and Ega{\~n}a, {\'A}lvaro F. and Ehrenfeld, Alejandro and Garrido, Felipe and Valenzuela, Mar{\'i}a Jes{\'u}s and S{\'a}nchez-P{\'e}rez, Juan F. },
	title   = {Spatialize v1.0: a Python/C++ library for ensemble spatial interpolation},
	journal = {Geoscientific Model Development},
	year    = {2026},
	volume  = {19},
	number  = {10},
	pages   = {4633--4660},
	doi     = {https://doi.org/10.5194/gmd-19-4633-2026},
	url     = {https://gmd.copernicus.org/articles/19/4633/2026/},
	issn    = {}
	}

@article{
	title = {Spatial distributional estimation via ensemble spatial analysis},
	journal = {AIMS Mathematics},
	volume = {10},
	number = {11},
	pages = {26351-26388},
	year = {2025},
	issn = {2473-6988},
	doi = {10.3934/math.20251159},
	url = {https://www.aimspress.com/article/doi/10.3934/math.20251159},
	author = {Alvaro F. Ega{\~n}a and Gonzalo D{\'i}az and Felipe Navarro and Mohammad Maleki and Juan F. S{\'a}nchez-P{\'e}rez},
	keywords = {geostatistics, computational geostatistics, generative geostatistics, non-linear geostatistics, distributional geostatistics, geostatistical simulation, empirical copula, data-driven methods},
	}

@article{AdaptiveESI2025,
	author  = {Ega{\~n}a, {\'A}lvaro F. and Valenzuela, María Jesús and Maleki, Mohammad and S{\'a}nchez-P{\'e}rez, Juan F. and Díaz, Gonzalo},
	title   = {Adaptive ensemble spatial analysis},
	journal = {Scientific Reports},
	year    = {2025},
	volume  = {15},
	number  = {1},
	pages   = {26599},
	doi     = {10.1038/s41598-025-08844-z},
	url     = {https://doi.org/10.1038/s41598-025-08844-z},
	issn    = {2045-2322}
	}

@article{ESI2021,
	author  = {Ega{\~n}a, {\'A}lvaro F. and Navarro, Felipe and Maleki, Mohammad and Grand{\'o}n, Francisca and Carter, Francisco and Soto, Fabi{\'a}n},
	title   = {Ensemble Spatial Interpolation: A New Approach to Natural or Anthropogenic Variable Assessment},
	journal = {Natural Resources Research},
	volume  = {30},
	number  = {5},
	pages   = {3777--3793},
	year    = {2021},
	doi     = {https://doi.org/10.1007/s11053-021-09860-2},
	url     = {https://link.springer.com/article/10.1007/s11053-021-09860-2}
	}