Kriging

Introduction

Kriging is a sophisticated geostatistical method used in memories-dev for optimal spatial interpolation. It provides the best linear unbiased prediction (BLUP) of intermediate values in spatial data.

Mathematical Foundation

The basic equation for Kriging estimation at an unsampled location \(s_0\) is:

\[\hat{Z}(s_0) = \sum_{i=1}^n \lambda_i Z(s_i)\]
where:

  • \(\hat{Z}(s_0)\) is the predicted value at location \(s_0\)

  • \(\lambda_i\) are the Kriging weights

  • \(Z(s_i)\) are the observed values at sampled locations

The weights \(\lambda_i\) are determined by solving the Kriging system:

\[\begin{split}\begin{bmatrix} \gamma(h_{11}) & \gamma(h_{12}) & \cdots & \gamma(h_{1n}) & 1 \\ \gamma(h_{21}) & \gamma(h_{22}) & \cdots & \gamma(h_{2n}) & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \gamma(h_{n1}) & \gamma(h_{n2}) & \cdots & \gamma(h_{nn}) & 1 \\ 1 & 1 & \cdots & 1 & 0 \end{bmatrix} \begin{bmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ \lambda_n \\ \mu \end{bmatrix} = \begin{bmatrix} \gamma(h_{10}) \\ \gamma(h_{20}) \\ \vdots \\ \gamma(h_{n0}) \\ 1 \end{bmatrix}\end{split}\]
where:

  • \(\gamma(h)\) is the semivariogram function

  • \(h_{ij}\) is the distance between points i and j

  • \(\mu\) is the Lagrange multiplier

Implementation ———— Here’s how to use Kriging in memories-dev: .. code-block:: python

from memories.spatial import Kriging

# Initialize Kriging with parameters kriging = Kriging(

variogram_model=”exponential”, anisotropy_scaling=1.0, anisotropy_angle=0.0

)

# Fit the model kriging.fit(

coordinates=sample_points, # Shape: (n_samples, n_dims) values=sample_values, # Shape: (n_samples,) variogram_parameters={

“sill”: 1.0, “range”: 100.0, “nugget”: 0.1

}

)

# Make predictions with uncertainty predictions, variances = kriging.predict(

coordinates=prediction_points, # Shape: (n_predictions, n_dims) return_variance=True

)

Variogram Models

memories-dev supports several variogram models:

  1. Exponential Model

\[\gamma(h) = c_0 + c_1\left(1 - \exp\left(-\frac{h}{a}\right)\right)\]

  1. Spherical Model

\[\begin{split}\gamma(h) = \begin{cases} c_0 + c_1\left(\frac{3h}{2a} - \frac{h^3}{2a^3}\right) & \text{for } h \leq a \\ c_0 + c_1 & \text{for } h > a \end{cases}\end{split}\]

  1. Gaussian Model

\[\gamma(h) = c_0 + c_1\left(1 - \exp\left(-\frac{h^2}{a^2}\right)\right)\]
where:

  • \(c_0\) is the nugget effect

  • \(c_1\) is the sill

  • \(a\) is the range

  • \(h\) is the lag distance

Visualization ———– .. mermaid:

    A3[Variogram Parameters]
end

subgraph "Kriging Process"
    B1[Variogram Fitting]
    B2[Weight Calculation]
    B3[Interpolation]
end

subgraph "Output"
    C1[Predictions]
    C2[Prediction Variance]
    C3[Kriging Maps]
end

A1 --> B1
A2 --> B1
A3 --> B1
B1 --> B2
B2 --> B3
B3 --> C1
B3 --> C2
C1 --> C3
C2 --> C3

Performance Considerations

  1. Computational Complexity

The computational complexity of Kriging is: - Variogram fitting: O(n²) - Weight calculation: O(n³) - Prediction: O(n) per prediction point

where n is the number of sample points.

  1. Memory Requirements

Memory usage scales with: - Sample points: O(n²) for the Kriging matrix - Prediction points: O(m) where m is the number of prediction points

  1. Optimization Strategies

memories-dev implements several optimization strategies:

# Use local kriging to reduce computation
kriging.fit(
    coordinates=sample_points,
    values=sample_values,
    max_points=100  # Use only nearest 100 points
)

# Enable parallel processing
kriging.predict(
    coordinates=prediction_points,
    n_jobs=-1  # Use all available cores
)

Validation Methods

  1. Cross-Validation

# Perform leave-one-out cross-validation
scores = kriging.cross_validate(
    coordinates=sample_points,
    values=sample_values,
    method="loo"
)

# Calculate validation metrics
rmse = scores["rmse"]
mae = scores["mae"]
r2 = scores["r2"]

  1. Validation Plots

# Generate validation plots
kriging.plot_validation(
    actual=actual_values,
    predicted=predicted_values,
    variance=prediction_variance
)

Example Applications

  1. Elevation Interpolation

# Interpolate elevation data
elevation_kriging = Kriging(
    variogram_model="spherical",
    coordinates_type="geographic"
)

elevation_map = elevation_kriging.fit_predict(
    coordinates=elevation_points,
    values=elevation_values,
    grid_size=(100, 100)  # Output resolution
)

  1. Environmental Monitoring

# Monitor air quality
pollution_kriging = Kriging(
    variogram_model="gaussian",
    anisotropy_scaling=1.5  # Account for wind direction
)

pollution_map, uncertainty = pollution_kriging.fit_predict(
    coordinates=sensor_locations,
    values=pollution_levels,
    return_variance=True
)

References

  1. Cressie, N. (1990). “The Origins of Kriging”. Mathematical Geology, 22(3), 239-252.

  2. Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer.

  3. Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press.