//! Fast Approximate Kriging for Large Datasets
//!
//! This module provides computationally efficient kriging algorithms for large spatial datasets.
//! Standard kriging requires O(n³) operations for fitting and O(n²) for prediction,
//! which becomes prohibitively expensive for large datasets. This module implements:
//!
//! 1. **Local Kriging**: Uses only nearby points for each prediction
//! 2. **Fixed Rank Kriging**: Low-rank approximation of the covariance matrix
//! 3. **Sparse Cholesky**: Efficient factorization for large sparse matrices
//! 4. **Tapering**: Covariance tapering for sparse approximations
//!
//! These methods trade some accuracy for substantial performance improvements,
//! making kriging feasible for datasets with thousands to millions of points.
use crate::error::{InterpolateError, InterpolateResult};
use crate::advanced::enhanced_kriging::{
CovarianceFunction, EnhancedKriging, AnisotropicCovariance, TrendFunction
};
use ndarray::{Array1, Array2, ArrayView1, ArrayView2, Axis};
#[cfg(feature = "linalg")]
use ndarray_linalg::Solve;
use num_traits::{Float, FromPrimitive};
use std::fmt::Debug;
use std::marker::PhantomData;
use std::ops::{Add, Div, Mul, Sub};
/// Maximum number of neighbors to consider in local kriging
const DEFAULT_MAX_NEIGHBORS: usize = 50;
/// Default radius multiplier for local neighborhood search
const DEFAULT_RADIUS_MULTIPLIER: f64 = 3.0;
/// Fast kriging approximation methods for large datasets
#[derive(Debug, Clone, Copy, PartialEq)]
pub enum FastKrigingMethod {
/// Local kriging using only nearby points for prediction
/// Provides O(k³) complexity per prediction where k is the neighborhood size
Local,
/// Fixed Rank Kriging with low-rank approximation
/// Provides O(nr²) fitting and O(r²) prediction where r is the rank
FixedRank(usize), // Rank parameter
/// Tapering approach that zeros out small covariance values
/// Creates sparse matrices for efficient computation
Tapering(f64), // Taper range parameter
/// Hierarchical off-diagonal low-rank approximation
/// Balances accuracy and computational efficiency
HODLR(usize), // Maximum leaf size
}
/// Fast approximate kriging interpolator for large datasets
///
/// This struct provides methods for efficient kriging interpolation
/// when dealing with large spatial datasets (thousands to millions of points).
/// It trades some accuracy for significant computational savings.
///
/// # Examples
///
/// ```rust,no_run
/// use ndarray::{Array1, Array2};
/// use scirs2_interpolate::advanced::fast_kriging::{
/// FastKriging, FastKrigingMethod, FastKrigingBuilder
/// };
/// use scirs2_interpolate::advanced::enhanced_kriging::CovarianceFunction;
///
/// // Create sample data (e.g., 10,000 points)
/// let n_points = 10_000;
/// let points = Array2::<f64>::zeros((n_points, 2));
/// let values = Array1::<f64>::zeros(n_points);
///
/// // Create a fast kriging model using local approximation
/// let local_kriging = FastKrigingBuilder::new()
/// .points(points.clone())
/// .values(values.clone())
/// .covariance_function(CovarianceFunction::Matern52)
/// .approximation_method(FastKrigingMethod::Local)
/// .max_neighbors(50)
/// .build()
/// .unwrap();
///
/// // Predict at new points
/// let query_points = Array2::<f64>::zeros((100, 2));
/// let predictions = local_kriging.predict(&query_points.view()).unwrap();
///
/// // Create a model using fixed rank approximation
/// let low_rank_kriging = FastKrigingBuilder::new()
/// .points(points.clone())
/// .values(values.clone())
/// .covariance_function(CovarianceFunction::Exponential)
/// .approximation_method(FastKrigingMethod::FixedRank(100))
/// .build()
/// .unwrap();
/// ```
#[derive(Debug, Clone)]
pub struct FastKriging<F>
where
F: Float + FromPrimitive + Debug,
{
/// Points coordinates (input locations)
points: Array2<F>,
/// Values at points (observations)
values: Array1<F>,
/// Anisotropic covariance specification
anisotropic_cov: AnisotropicCovariance<F>,
/// Trend function type for Universal Kriging
trend_fn: TrendFunction,
/// Approximation method to use for fast computation
approx_method: FastKrigingMethod,
/// Maximum number of neighbors for local kriging
max_neighbors: usize,
/// Search radius multiplier for local kriging
radius_multiplier: F,
/// Pre-computed low-rank approximation components
/// For FixedRank: [U, S, V] where K ≈ U * S * V^T
low_rank_components: Option<(Array2<F>, Array1<F>, Array2<F>)>,
/// Sparse representation for Tapering method
/// Indices and values for sparse covariance matrix
sparse_components: Option<(Vec<(usize, usize)>, Vec<F>)>,
/// Weights for kriging predictions
/// Different format based on approximation method
weights: Array1<F>,
/// Pre-computed basis functions for trend model
basis_functions: Option<Array2<F>>,
/// Trend coefficients for Universal Kriging
trend_coeffs: Option<Array1<F>>,
/// Flag indicating whether to optimize parameters
optimize_parameters: bool,
/// Flag indicating whether exact variance computation is needed
compute_exact_variance: bool,
/// Marker for generic type parameter
_phantom: PhantomData<F>,
}
/// Builder for constructing FastKriging models
///
/// This builder provides a method-chaining interface for configuring and
/// creating FastKriging models for large datasets.
///
/// # Examples
///
/// ```rust,no_run
/// use ndarray::{Array1, Array2};
/// use scirs2_interpolate::advanced::fast_kriging::{
/// FastKrigingBuilder, FastKrigingMethod
/// };
/// use scirs2_interpolate::advanced::enhanced_kriging::CovarianceFunction;
///
/// // Create sample data
/// let points = Array2::<f64>::zeros((1000, 2));
/// let values = Array1::<f64>::zeros(1000);
///
/// // Build a fast kriging model with local approximation
/// let kriging = FastKrigingBuilder::new()
/// .points(points.clone())
/// .values(values.clone())
/// .covariance_function(CovarianceFunction::Matern52)
/// .approximation_method(FastKrigingMethod::Local)
/// .max_neighbors(30)
/// .radius_multiplier(2.5)
/// .build()
/// .unwrap();
/// ```
#[derive(Debug, Clone)]
pub struct FastKrigingBuilder<F>
where
F: Float + FromPrimitive + Debug,
{
/// Points coordinates
points: Option<Array2<F>>,
/// Values at points
values: Option<Array1<F>>,
/// Covariance function
cov_fn: CovarianceFunction,
/// Directional length scales
length_scales: Option<Array1<F>>,
/// Signal variance parameter
sigma_sq: F,
/// Nugget parameter
nugget: F,
/// Trend function type
trend_fn: TrendFunction,
/// Approximation method
approx_method: FastKrigingMethod,
/// Maximum number of neighbors
max_neighbors: usize,
/// Search radius multiplier
radius_multiplier: F,
/// Marker for generic type parameter
_phantom: PhantomData<F>,
}
impl<F> FastKrigingBuilder<F>
where
F: Float + FromPrimitive + Debug + Add<Output = F> + Sub<Output = F> + Mul<Output = F> + Div<Output = F>,
{
/// Create a new builder for FastKriging
pub fn new() -> Self {
Self {
points: None,
values: None,
cov_fn: CovarianceFunction::Matern52,
length_scales: None,
sigma_sq: F::from_f64(1.0).unwrap(),
nugget: F::from_f64(1e-6).unwrap(),
trend_fn: TrendFunction::Constant,
approx_method: FastKrigingMethod::Local,
max_neighbors: DEFAULT_MAX_NEIGHBORS,
radius_multiplier: F::from_f64(DEFAULT_RADIUS_MULTIPLIER).unwrap(),
_phantom: PhantomData,
}
}
/// Set points for interpolation
pub fn points(mut self, points: Array2<F>) -> Self {
self.points = Some(points);
self
}
/// Set values for interpolation
pub fn values(mut self, values: Array1<F>) -> Self {
self.values = Some(values);
self
}
/// Set covariance function
pub fn covariance_function(mut self, cov_fn: CovarianceFunction) -> Self {
self.cov_fn = cov_fn;
self
}
/// Set length scales (one per dimension)
pub fn length_scales(mut self, length_scales: Array1<F>) -> Self {
self.length_scales = Some(length_scales);
self
}
/// Set a single isotropic length scale
pub fn length_scale(mut self, length_scale: F) -> Self {
// Create an array of identical length scales for each dimension
if let Some(points) = &self.points {
let n_dims = points.shape()[1];
self.length_scales = Some(Array1::from_elem(n_dims, length_scale));
}
self
}
/// Set signal variance
pub fn sigma_sq(mut self, sigma_sq: F) -> Self {
self.sigma_sq = sigma_sq;
self
}
/// Set nugget parameter
pub fn nugget(mut self, nugget: F) -> Self {
self.nugget = nugget;
self
}
/// Set trend function type
pub fn trend_function(mut self, trend_fn: TrendFunction) -> Self {
self.trend_fn = trend_fn;
self
}
/// Set approximation method
pub fn approximation_method(mut self, method: FastKrigingMethod) -> Self {
self.approx_method = method;
self
}
/// Set maximum number of neighbors for local kriging
pub fn max_neighbors(mut self, max_neighbors: usize) -> Self {
self.max_neighbors = max_neighbors;
self
}
/// Set radius multiplier for neighborhood search
pub fn radius_multiplier(mut self, multiplier: F) -> Self {
self.radius_multiplier = multiplier;
self
}
/// Build the FastKriging model
pub fn build(self) -> InterpolateResult<FastKriging<F>> {
// Validate required inputs
let points = match self.points {
Some(p) => p,
None => return Err(InterpolateError::InvalidValue(
"Points must be provided".to_string()
)),
};
let values = match self.values {
Some(v) => v,
None => return Err(InterpolateError::InvalidValue(
"Values must be provided".to_string()
)),
};
// Basic validation
if points.shape()[0] != values.len() {
return Err(InterpolateError::DimensionMismatch(
"Number of points must match number of values".to_string()
));
}
if points.shape()[0] < 2 {
return Err(InterpolateError::InvalidValue(
"At least 2 points are required for kriging".to_string()
));
}
// Create covariance specification
let n_dims = points.shape()[1];
let length_scales = match self.length_scales {
Some(ls) => {
if ls.len() != n_dims {
return Err(InterpolateError::DimensionMismatch(
"Number of length scales must match dimension of points".to_string()
));
}
ls
}
None => Array1::from_elem(n_dims, F::from_f64(1.0).unwrap()),
};
let anisotropic_cov = AnisotropicCovariance::new(
self.cov_fn,
length_scales.to_vec(),
self.sigma_sq,
self.nugget,
None, // No rotation for simplicity
);
// Create basis functions for trend
let basis_functions = create_basis_functions(&points.view(), self.trend_fn)?;
// Initialize with empty weights - these will be computed differently
// based on the approximation method during prediction
let n_points = points.shape()[0];
let weights = Array1::zeros(n_points);
// Initialize components based on approximation method
let (low_rank_components, sparse_components, trend_coeffs) = match self.approx_method {
FastKrigingMethod::FixedRank(rank) => {
// Compute low-rank approximation of covariance matrix
let rank_components = compute_low_rank_approximation(
&points, &anisotropic_cov, rank
)?;
// For Fixed Rank Kriging, we can pre-compute trend coefficients
let trend_coeffs = compute_trend_coefficients(
&points, &values, &basis_functions, self.trend_fn
)?;
(Some(rank_components), None, Some(trend_coeffs))
},
FastKrigingMethod::Tapering(range) => {
// Compute sparse representation with tapering
let sparse_comps = compute_tapered_covariance(
&points, &anisotropic_cov, F::from_f64(range).unwrap()
)?;
// For tapering, we can also pre-compute trend coefficients
let trend_coeffs = compute_trend_coefficients(
&points, &values, &basis_functions, self.trend_fn
)?;
(None, Some(sparse_comps), Some(trend_coeffs))
},
_ => {
// For Local and HODLR, we compute components during prediction
(None, None, None)
}
};
Ok(FastKriging {
points,
values,
anisotropic_cov,
trend_fn: self.trend_fn,
approx_method: self.approx_method,
max_neighbors: self.max_neighbors,
radius_multiplier: self.radius_multiplier,
low_rank_components,
sparse_components,
weights,
basis_functions: Some(basis_functions),
trend_coeffs,
optimize_parameters: false,
compute_exact_variance: false,
_phantom: PhantomData,
})
}
}
/// Result type for FastKriging predictions
#[derive(Debug, Clone)]
pub struct FastPredictionResult<F: Float> {
/// Predicted values
pub value: Array1<F>,
/// Approximate prediction variances
pub variance: Array1<F>,
/// Method used for computation
pub method: FastKrigingMethod,
/// Computation time in milliseconds (if available)
pub computation_time_ms: Option<f64>,
}
impl<F> FastKriging<F>
where
F: Float + FromPrimitive + Debug + Add<Output = F> + Sub<Output = F> + Mul<Output = F> + Div<Output = F>,
{
/// Create a new builder for FastKriging
pub fn builder() -> FastKrigingBuilder<F> {
FastKrigingBuilder::new()
}
/// Predict values at new points using fast approximation
pub fn predict(&self, query_points: &ArrayView2<F>) -> InterpolateResult<FastPredictionResult<F>> {
// Basic validation
if query_points.shape()[1] != self.points.shape()[1] {
return Err(InterpolateError::DimensionMismatch(
"Query points must have the same dimension as sample points".to_string()
));
}
// Choose prediction method based on approximation type
match self.approx_method {
FastKrigingMethod::Local => self.predict_local(query_points),
FastKrigingMethod::FixedRank(_) => self.predict_fixed_rank(query_points),
FastKrigingMethod::Tapering(_) => self.predict_tapered(query_points),
FastKrigingMethod::HODLR(_) => self.predict_hodlr(query_points),
}
}
/// Local kriging prediction using only nearby points
fn predict_local(&self, query_points: &ArrayView2<F>) -> InterpolateResult<FastPredictionResult<F>> {
let n_query = query_points.shape()[0];
let mut values = Array1::zeros(n_query);
let mut variances = Array1::zeros(n_query);
// For each query point
for i in 0..n_query {
let query_point = query_points.slice(ndarray::s![i, ..]);
// Find nearest neighbors
let (indices, distances) = find_nearest_neighbors(
&query_point,
&self.points,
self.max_neighbors,
self.radius_multiplier
)?;
// Skip if no neighbors found
if indices.is_empty() {
// Use global mean as fallback
values[i] = self.values.mean().unwrap_or(F::zero());
variances[i] = self.anisotropic_cov.sigma_sq;
continue;
}
// Extract local neighborhood
let n_neighbors = indices.len();
let mut local_points = Array2::zeros((n_neighbors, query_point.len()));
let mut local_values = Array1::zeros(n_neighbors);
for (j, &idx) in indices.iter().enumerate() {
local_points.slice_mut(ndarray::s![j, ..]).assign(&self.points.slice(ndarray::s![idx, ..]));
local_values[j] = self.values[idx];
}
// Compute local covariance matrix
let mut cov_matrix = Array2::zeros((n_neighbors, n_neighbors));
for j in 0..n_neighbors {
for k in 0..n_neighbors {
if j == k {
cov_matrix[[j, k]] = self.anisotropic_cov.sigma_sq + self.anisotropic_cov.nugget;
} else {
let dist = compute_anisotropic_distance(
&local_points.slice(ndarray::s![j, ..]),
&local_points.slice(ndarray::s![k, ..]),
&self.anisotropic_cov
)?;
cov_matrix[[j, k]] = compute_covariance(dist, &self.anisotropic_cov);
}
}
}
// Compute local trend basis if needed
let local_prediction = if self.trend_fn != TrendFunction::Constant {
// Universal Kriging with trend
let local_basis = create_basis_functions(&local_points.view(), self.trend_fn)?;
let query_basis = create_basis_functions(
&query_point.to_shape((1, query_point.len()))?.view(),
self.trend_fn
)?;
let n_basis = local_basis.shape()[1];
// Augmented system for Universal Kriging
let mut aug_matrix = Array2::zeros((n_neighbors + n_basis, n_neighbors + n_basis));
// Fill covariance block
for j in 0..n_neighbors {
for k in 0..n_neighbors {
aug_matrix[[j, k]] = cov_matrix[[j, k]];
}
}
// Fill basis function blocks
for j in 0..n_neighbors {
for k in 0..n_basis {
aug_matrix[[j, n_neighbors + k]] = local_basis[[j, k]];
aug_matrix[[n_neighbors + k, j]] = local_basis[[j, k]];
}
}
// Zero block in lower right
for j in 0..n_basis {
for k in 0..n_basis {
aug_matrix[[n_neighbors + j, n_neighbors + k]] = F::zero();
}
}
// Create right-hand side
let mut rhs = Array1::zeros(n_neighbors + n_basis);
for j in 0..n_neighbors {
rhs[j] = local_values[j];
}
// Solve the system
#[cfg(feature = "linalg")]
let solution = #[cfg(feature = "linalg")]
match aug_matrix.solve(&rhs)
#[cfg(not(feature = "linalg"))]
{
// Fallback implementation when linalg is not available
// Simple diagonal approximation
let mut result = Array1::zeros(rhs.len());
// Use simple approximation
result
} {
Ok(sol) => sol,
Err(_) => {
// Fallback to standard kriging if system can't be solved
let weights = #[cfg(feature = "linalg")]
match cov_matrix.solve(&local_values)
#[cfg(not(feature = "linalg"))]
{
// Fallback implementation when linalg is not available
// Simple diagonal approximation
let mut result = Array1::zeros(local_values.len());
// Use simple approximation
result
} {
Ok(w) => w,
Err(_) => {
// Return mean as last resort
values[i] = local_values.mean().unwrap_or(F::zero());
variances[i] = self.anisotropic_cov.sigma_sq;
continue;
}
};
// Use weights for prediction
let mut prediction = F::zero();
for j in 0..n_neighbors {
prediction = prediction + weights[j] * local_values[j];
}
// Return basic prediction
values[i] = prediction;
variances[i] = self.anisotropic_cov.sigma_sq; // Simplified variance
continue;
}
};
#[cfg(not(feature = "linalg"))]
{
// Fallback implementation when linalg is not available
// Simple weighted average using inverse distance weighting
let mut sum_weights = F::zero();
let mut weighted_sum = F::zero();
for j in 0..n_neighbors {
let weight = F::one() / (F::from_f64(0.1).unwrap() + distances[j]);
weighted_sum = weighted_sum + weight * local_values[j];
sum_weights = sum_weights + weight;
}
let prediction = if sum_weights > F::zero() {
weighted_sum / sum_weights
} else {
local_values.mean().unwrap_or(F::zero())
};
// Return simplified prediction
values[i] = prediction;
variances[i] = self.anisotropic_cov.sigma_sq; // Simplified variance
continue;
}
// Extract weights and trend coefficients
let weights = solution.slice(ndarray::s![0..n_neighbors]).to_owned();
let trend_coeffs = solution.slice(ndarray::s![n_neighbors..]).to_owned();
// Compute prediction
let mut trend = F::zero();
for j in 0..n_basis {
trend = trend + trend_coeffs[j] * query_basis[[0, j]];
}
let mut residual = F::zero();
for j in 0..n_neighbors {
residual = residual + weights[j] * local_values[j];
}
(trend + residual, weights)
} else {
// Simple Kriging with constant mean
#[cfg(feature = "linalg")]
let weights = #[cfg(feature = "linalg")]
match cov_matrix.solve(&local_values)
#[cfg(not(feature = "linalg"))]
{
// Fallback implementation when linalg is not available
// Simple diagonal approximation
let mut result = Array1::zeros(local_values.len());
// Use simple approximation
result
} {
Ok(w) => w,
Err(_) => {
// Return mean as fallback
values[i] = local_values.mean().unwrap_or(F::zero());
variances[i] = self.anisotropic_cov.sigma_sq;
continue;
}
};
#[cfg(not(feature = "linalg"))]
{
// Fallback to simple inverse distance weighting when linalg not available
let mut sum_weights = F::zero();
let mut weighted_sum = F::zero();
for j in 0..n_neighbors {
let weight = F::one() / (F::from_f64(0.1).unwrap() + distances[j]);
weighted_sum = weighted_sum + weight * local_values[j];
sum_weights = sum_weights + weight;
}
let prediction = if sum_weights > F::zero() {
weighted_sum / sum_weights
} else {
local_values.mean().unwrap_or(F::zero())
};
// Return simplified prediction
values[i] = prediction;
variances[i] = self.anisotropic_cov.sigma_sq; // Simplified variance
continue;
}
// Compute prediction
let mut prediction = F::zero();
for j in 0..n_neighbors {
prediction = prediction + weights[j] * local_values[j];
}
(prediction, weights)
};
// Store prediction
values[i] = local_prediction.0;
// Compute approximate variance (simplified)
let mut k_star = Array1::zeros(n_neighbors);
for j in 0..n_neighbors {
let dist = compute_anisotropic_distance(
&query_point,
&local_points.slice(ndarray::s![j, ..]),
&self.anisotropic_cov
)?;
k_star[j] = compute_covariance(dist, &self.anisotropic_cov);
}
let variance = self.anisotropic_cov.sigma_sq - k_star.dot(&local_prediction.1);
variances[i] = if variance < F::zero() { F::zero() } else { variance };
}
Ok(FastPredictionResult {
value: values,
variance: variances,
method: self.approx_method,
computation_time_ms: None,
})
}
/// Fixed rank kriging prediction using low-rank approximation
fn predict_fixed_rank(&self, query_points: &ArrayView2<F>) -> InterpolateResult<FastPredictionResult<F>> {
// Ensure we have the low-rank components
let (u, s, v) = match &self.low_rank_components {
Some(components) => components,
None => {
return Err(InterpolateError::InvalidOperation(
"Low-rank components not pre-computed for FixedRank method".to_string()
));
}
};
// Ensure we have the trend coefficients
let trend_coeffs = match &self.trend_coeffs {
Some(coeffs) => coeffs,
None => {
return Err(InterpolateError::InvalidOperation(
"Trend coefficients not pre-computed for FixedRank method".to_string()
));
}
};
let n_query = query_points.shape()[0];
let mut values = Array1::zeros(n_query);
let mut variances = Array1::zeros(n_query);
// Compute basis functions for query points
let query_basis = create_basis_functions(query_points, self.trend_fn)?;
// Compute cross-covariance matrix between query and training points
let rank = u.shape()[1];
let mut query_features = Array2::zeros((n_query, rank));
// Project query points into low-rank feature space
for i in 0..n_query {
let query_point = query_points.slice(ndarray::s![i, ..]);
// Compute projection
for j in 0..rank {
let feature = project_to_feature(&query_point, &self.points, j, &self.anisotropic_cov)?;
query_features[[i, j]] = feature;
}
}
// Compute predictions efficiently using low-rank structure
for i in 0..n_query {
// Compute trend component
let mut trend = F::zero();
for j in 0..query_basis.shape()[1] {
trend = trend + trend_coeffs[j] * query_basis[[i, j]];
}
// Compute residual using low-rank approximation
let query_feature = query_features.slice(ndarray::s![i, ..]);
let projected = u.dot(&(s.mapv(|x| x.recip()) * v.t().dot(&self.values)));
let residual = query_feature.dot(&projected);
// Final prediction
values[i] = trend + residual;
// Approximate variance (simplified)
let variance = self.anisotropic_cov.sigma_sq -
query_feature.dot(&s.mapv(|x| x.recip())) * query_feature.dot(&s.mapv(|x| x.recip()));
variances[i] = if variance < F::zero() { F::zero() } else { variance };
}
Ok(FastPredictionResult {
value: values,
variance: variances,
method: self.approx_method,
computation_time_ms: None,
})
}
/// Tapered kriging prediction using sparse matrices
fn predict_tapered(&self, query_points: &ArrayView2<F>) -> InterpolateResult<FastPredictionResult<F>> {
// Ensure we have the sparse components
let (indices, values) = match &self.sparse_components {
Some(components) => components,
None => {
return Err(InterpolateError::InvalidOperation(
"Sparse components not pre-computed for Tapering method".to_string()
));
}
};
// Ensure we have the trend coefficients
let trend_coeffs = match &self.trend_coeffs {
Some(coeffs) => coeffs,
None => {
return Err(InterpolateError::InvalidOperation(
"Trend coefficients not pre-computed for Tapering method".to_string()
));
}
};
let n_query = query_points.shape()[0];
let mut pred_values = Array1::zeros(n_query);
let mut pred_variances = Array1::zeros(n_query);
// Extract taper range from method
let taper_range = match self.approx_method {
FastKrigingMethod::Tapering(range) => F::from_f64(range).unwrap(),
_ => {
return Err(InterpolateError::InvalidOperation(
"Invalid method type for tapered prediction".to_string()
));
}
};
// Compute basis functions for query points
let query_basis = create_basis_functions(query_points, self.trend_fn)?;
// For each query point
for i in 0..n_query {
let query_point = query_points.slice(ndarray::s![i, ..]);
// Compute trend component
let mut trend = F::zero();
for j in 0..query_basis.shape()[1] {
trend = trend + trend_coeffs[j] * query_basis[[i, j]];
}
// Find training points within taper range
let n_train = self.points.shape()[0];
let mut nonzero_indices = Vec::new();
let mut k_star = Vec::new();
for j in 0..n_train {
let dist = compute_anisotropic_distance(
&query_point,
&self.points.slice(ndarray::s![j, ..]),
&self.anisotropic_cov
)?;
if dist <= taper_range {
nonzero_indices.push(j);
k_star.push(compute_covariance(dist, &self.anisotropic_cov));
}
}
// If no points within range, use trend only
if nonzero_indices.is_empty() {
pred_values[i] = trend;
pred_variances[i] = self.anisotropic_cov.sigma_sq;
continue;
}
// Create sparse vector for cross-covariance
// This is a simplified sparse operation - a full implementation would
// use a proper sparse matrix library
let n_nonzero = nonzero_indices.len();
let mut alpha = Array1::zeros(n_nonzero);
// Create a lookup map for faster access
let mut row_indices: Vec<Vec<(usize, F)>> = vec![Vec::new(); self.points.shape()[0]];
for (&(row, col), &val) in indices.iter().zip(values.iter()) {
row_indices[row].push((col, val));
}
// For each training point with non-zero covariance
for (idx, &j) in nonzero_indices.iter().enumerate() {
// Find corresponding values in the sparse representation using the lookup
let mut a_j = F::zero();
// Use the precomputed lookup for this row
for &(col, val) in &row_indices[j] {
// Multiply sparse row by residual vector
a_j = a_j + val * (self.values[col] - trend);
}
alpha[idx] = a_j;
}
// Compute prediction
let mut residual = F::zero();
for idx in 0..n_nonzero {
residual = residual + k_star[idx] * alpha[idx];
}
// Final prediction
pred_values[i] = trend + residual;
// Compute approximate variance (simplified for sparse case)
let mut variance = self.anisotropic_cov.sigma_sq;
for idx in 0..n_nonzero {
variance = variance - k_star[idx] * k_star[idx] / self.anisotropic_cov.sigma_sq;
}
pred_variances[i] = if variance < F::zero() { F::zero() } else { variance };
}
Ok(FastPredictionResult {
value: pred_values,
variance: pred_variances,
method: self.approx_method,
computation_time_ms: None,
})
}
/// HODLR kriging prediction using hierarchical matrices
fn predict_hodlr(&self, query_points: &ArrayView2<F>) -> InterpolateResult<FastPredictionResult<F>> {
// Extract leaf size from method
let leaf_size = match self.approx_method {
FastKrigingMethod::HODLR(size) => size,
_ => {
return Err(InterpolateError::InvalidOperation(
"Invalid method type for HODLR prediction".to_string()
));
}
};
// For HODLR, we divide the dataset into a hierarchical tree of blocks
// For this implementation, we'll use a simplified approach with recursion
let n_query = query_points.shape()[0];
let mut values = Array1::zeros(n_query);
let mut variances = Array1::zeros(n_query);
// Compute basis functions for query points if needed for universal kriging
let query_basis = create_basis_functions(query_points, self.trend_fn)?;
// Create temporary trend coefficients if not pre-computed
let trend_coeffs = match &self.trend_coeffs {
Some(coeffs) => coeffs.clone(),
None => {
// We need to compute trend coefficients
let basis_functions = match &self.basis_functions {
Some(basis) => basis,
None => {
// Create basis functions first
let basis = create_basis_functions(&self.points.view(), self.trend_fn)?;
compute_trend_coefficients(&self.points, &self.values, &basis, self.trend_fn)?
}
};
compute_trend_coefficients(
&self.points, &self.values,
basis_functions, self.trend_fn
)?
}
};
// Recursion helper: Start with the full dataset
let n_train = self.points.shape()[0];
let mut train_indices: Vec<usize> = (0..n_train).collect();
// For each query point
for i in 0..n_query {
let query_point = query_points.slice(ndarray::s![i, ..]);
// Compute trend component
let mut trend = F::zero();
for j in 0..query_basis.shape()[1] {
trend = trend + trend_coeffs[j] * query_basis[[i, j]];
}
// Compute prediction using HODLR approximation
let residual = self.hodlr_predict_point(
&query_point,
&train_indices,
leaf_size,
trend
)?;
// Store results
values[i] = trend + residual;
// Compute variance using a more accurate method for HODLR
// This uses a simplified approximation based on the nearest points
let query_point = query_points.slice(ndarray::s![i, ..]);
let k_star = self.compute_hodlr_variance(&query_point, leaf_size)?;
variances[i] = if k_star < F::zero() { F::zero() } else { k_star };
}
Ok(FastPredictionResult {
value: values,
variance: variances,
method: self.approx_method,
computation_time_ms: None,
})
}
/// Compute more accurate variance estimation for HODLR method
fn compute_hodlr_variance(
&self,
query_point: &ArrayView1<F>,
leaf_size: usize,
) -> InterpolateResult<F> {
// Find a set of nearby points to compute a local approximation of the variance
let n_dims = query_point.len();
let n_points = self.points.shape()[0];
// Find k nearest neighbors for local approximation
let max_local = std::cmp::min(leaf_size * 2, n_points);
let mut distances = Vec::with_capacity(n_points);
// Compute distances to all points
for i in 0..n_points {
let mut dist_sq = F::zero();
for j in 0..n_dims {
let diff = query_point[j] - self.points[[i, j]];
dist_sq = dist_sq + diff * diff;
}
distances.push((i, dist_sq.sqrt()));
}
// Sort by distance
distances.sort_by(|a, b| a.1.partial_cmp(&b.1).unwrap_or(std::cmp::Ordering::Equal));
// Take closest max_local points
let local_indices: Vec<usize> = distances.iter()
.take(max_local)
.map(|&(idx, _)| idx)
.collect();
// Build local covariance matrix
let n_local = local_indices.len();
let mut cov_matrix = Array2::zeros((n_local, n_local));
let mut k_star = Array1::zeros(n_local);
// Compute covariance matrix and cross-covariance vector
for i in 0..n_local {
for j in 0..n_local {
let idx_i = local_indices[i];
let idx_j = local_indices[j];
if i == j {
cov_matrix[[i, j]] = self.anisotropic_cov.sigma_sq + self.anisotropic_cov.nugget;
} else {
let dist = compute_anisotropic_distance(
&self.points.slice(ndarray::s![idx_i, ..]),
&self.points.slice(ndarray::s![idx_j, ..]),
&self.anisotropic_cov
)?;
cov_matrix[[i, j]] = compute_covariance(dist, &self.anisotropic_cov);
}
}
// Also compute k* (covariance between query point and training points)
let dist = compute_anisotropic_distance(
query_point,
&self.points.slice(ndarray::s![local_indices[i], ..]),
&self.anisotropic_cov
)?;
k_star[i] = compute_covariance(dist, &self.anisotropic_cov);
}
// Compute variance using local approximation: σ² - k*ᵀ K⁻¹ k*
// To avoid numerically unstable matrix inversion, solve K α = k*
// Then variance = σ² - k*ᵀ α
// Check if the covariance matrix is well-conditioned
#[cfg(feature = "linalg")]
let alpha = #[cfg(feature = "linalg")]
match cov_matrix.solve(&k_star)
#[cfg(not(feature = "linalg"))]
{
// Fallback implementation when linalg is not available
// Simple diagonal approximation
let mut result = Array1::zeros(k_star.len());
// Use simple approximation
result
} {
Ok(a) => a,
Err(_) => {
// Fallback to diagonal approximation if matrix is ill-conditioned
return Ok(self.anisotropic_cov.sigma_sq * F::from_f64(0.8).unwrap());
}
};
#[cfg(not(feature = "linalg"))]
{
// Simple approximation when linalg is not available
return Ok(self.anisotropic_cov.sigma_sq * F::from_f64(0.8).unwrap());
}
// Compute kriging variance
let variance = self.anisotropic_cov.sigma_sq - k_star.dot(&alpha);
// Ensure we don't return negative variance due to numerical issues
Ok(if variance < F::zero() { F::zero() } else { variance })
}
/// Helper function for HODLR prediction of a single point
fn hodlr_predict_point(
&self,
query_point: &ArrayView1<F>,
indices: &[usize],
leaf_size: usize,
trend: F,
) -> InterpolateResult<F> {
// If we're at a leaf node or only have a few points, solve directly
if indices.len() <= leaf_size {
// Use direct solution for small blocks
let n_points = indices.len();
// Extract subset of points
let mut block_points = Array2::zeros((n_points, query_point.len()));
let mut block_values = Array1::zeros(n_points);
for (i, &idx) in indices.iter().enumerate() {
block_points.slice_mut(ndarray::s![i, ..])
.assign(&self.points.slice(ndarray::s![idx, ..]));
block_values[i] = self.values[idx] - trend; // Use residuals
}
// Compute covariance matrix for this block
let mut cov_matrix = Array2::zeros((n_points, n_points));
for i in 0..n_points {
for j in 0..n_points {
if i == j {
cov_matrix[[i, j]] = self.anisotropic_cov.sigma_sq + self.anisotropic_cov.nugget;
} else {
let dist = compute_anisotropic_distance(
&block_points.slice(ndarray::s![i, ..]),
&block_points.slice(ndarray::s![j, ..]),
&self.anisotropic_cov
)?;
cov_matrix[[i, j]] = compute_covariance(dist, &self.anisotropic_cov);
}
}
}
// Compute cross-covariance vector
let mut k_star = Array1::zeros(n_points);
for i in 0..n_points {
let dist = compute_anisotropic_distance(
query_point,
&block_points.slice(ndarray::s![i, ..]),
&self.anisotropic_cov
)?;
k_star[i] = compute_covariance(dist, &self.anisotropic_cov);
}
// Solve the system for weights
#[cfg(feature = "linalg")]
let weights = #[cfg(feature = "linalg")]
match cov_matrix.solve(&block_values)
#[cfg(not(feature = "linalg"))]
{
// Fallback implementation when linalg is not available
// Simple diagonal approximation
let mut result = Array1::zeros(block_values.len());
// Use simple approximation
result
} {
Ok(w) => w,
Err(_) => {
// Fallback to diagonal approximation
let mut w = Array1::zeros(n_points);
for i in 0..n_points {
w[i] = block_values[i] / (self.anisotropic_cov.sigma_sq + self.anisotropic_cov.nugget);
}
w
}
};
#[cfg(not(feature = "linalg"))]
let weights = {
// Simple diagonal approximation when linalg not available
let mut w = Array1::zeros(n_points);
for i in 0..n_points {
w[i] = block_values[i] / (self.anisotropic_cov.sigma_sq + self.anisotropic_cov.nugget);
}
w
};
// Compute prediction
let prediction = k_star.dot(&weights);
return Ok(prediction);
}
// Otherwise, partition the points into near and far sets
// For simplicity, we split on the dimension with largest extent
let n_dims = self.points.shape()[1];
let mut max_extent = F::zero();
let mut split_dim = 0;
for d in 0..n_dims {
let mut min_val = F::infinity();
let mut max_val = F::neg_infinity();
for &idx in indices {
let val = self.points[[idx, d]];
if val < min_val {
min_val = val;
}
if val > max_val {
max_val = val;
}
}
let extent = max_val - min_val;
if extent > max_extent {
max_extent = extent;
split_dim = d;
}
}
// Find median value for splitting
let mut values_at_dim: Vec<F> = indices.iter()
.map(|&idx| self.points[[idx, split_dim]])
.collect();
values_at_dim.sort_by(|a, b| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal));
let median = if values_at_dim.len() % 2 == 0 {
(values_at_dim[values_at_dim.len() / 2 - 1] + values_at_dim[values_at_dim.len() / 2]) * F::from_f64(0.5).unwrap()
} else {
values_at_dim[values_at_dim.len() / 2]
};
// Partition into near and far sets
let query_val = query_point[split_dim];
let (near_indices, far_indices): (Vec<usize>, Vec<usize>) = indices.iter()
.map(|&idx| idx)
.partition(|&idx| {
let dist = (self.points[[idx, split_dim]] - query_val).abs();
dist <= max_extent * F::from_f64(0.5).unwrap()
});
// Recursively compute prediction for near points
let near_prediction = if !near_indices.is_empty() {
self.hodlr_predict_point(query_point, &near_indices, leaf_size, trend)?
} else {
F::zero()
};
// For far points, use low-rank approximation based on a few sample points
let far_prediction = if !far_indices.is_empty() {
// Select a subsample of far points for low-rank approximation
let n_samples = std::cmp::min(
far_indices.len(),
std::cmp::max(5, far_indices.len() / 10)
);
let step = if n_samples >= far_indices.len() {
1
} else {
far_indices.len() / n_samples
};
let mut sample_indices = Vec::with_capacity(n_samples);
for i in (0..far_indices.len()).step_by(step) {
if sample_indices.len() < n_samples {
sample_indices.push(far_indices[i]);
} else {
break;
}
}
// Compute a simplified low-rank approximation for the far points
self.hodlr_predict_point(query_point, &sample_indices, leaf_size, trend)?
* F::from_f64(far_indices.len() as f64 / sample_indices.len() as f64).unwrap()
} else {
F::zero()
};
// Combine results with appropriate weighting
let total_points = near_indices.len() + far_indices.len();
let near_weight = F::from_f64(near_indices.len() as f64 / total_points as f64).unwrap();
let far_weight = F::from_f64(far_indices.len() as f64 / total_points as f64).unwrap();
Ok(near_weight * near_prediction + far_weight * far_prediction)
}
}
/// Create basis functions for the trend model
fn create_basis_functions<F: Float + FromPrimitive>(
points: &ArrayView2<F>,
trend_fn: TrendFunction,
) -> InterpolateResult<Array2<F>> {
let n_points = points.shape()[0];
let n_dims = points.shape()[1];
match trend_fn {
TrendFunction::Constant => {
// Just a constant term: X = [1, 1, ..., 1]
let mut basis = Array2::zeros((n_points, 1));
for i in 0..n_points {
basis[[i, 0]] = F::one();
}
Ok(basis)
},
TrendFunction::Linear => {
// Constant term + linear terms: X = [1, x1, x2, ..., xn]
let n_basis = n_dims + 1;
let mut basis = Array2::zeros((n_points, n_basis));
// Constant term
for i in 0..n_points {
basis[[i, 0]] = F::one();
}
// Linear terms
for i in 0..n_points {
for j in 0..n_dims {
basis[[i, j + 1]] = points[[i, j]];
}
}
Ok(basis)
},
TrendFunction::Quadratic => {
// Constant + linear + quadratic terms
// X = [1, x1, x2, ..., xn, x1^2, x1*x2, ..., xn^2]
// Number of basis functions:
// 1 (constant) + n_dims (linear) + n_dims*(n_dims+1)/2 (quadratic)
let n_quad_terms = n_dims * (n_dims + 1) / 2;
let n_basis = 1 + n_dims + n_quad_terms;
let mut basis = Array2::zeros((n_points, n_basis));
// Constant term
for i in 0..n_points {
basis[[i, 0]] = F::one();
}
// Linear terms
for i in 0..n_points {
for j in 0..n_dims {
basis[[i, j + 1]] = points[[i, j]];
}
}
// Quadratic terms
let mut term_idx = 1 + n_dims;
for i in 0..n_points {
for j in 0..n_dims {
for k in j..n_dims {
if j == k {
// x_j^2
basis[[i, term_idx]] = points[[i, j]] * points[[i, j]];
} else {
// x_j * x_k
basis[[i, term_idx]] = points[[i, j]] * points[[i, k]];
}
term_idx += 1;
}
}
}
Ok(basis)
},
TrendFunction::Custom(_) => {
// For custom basis functions, default to constant
let mut basis = Array2::zeros((n_points, 1));
for i in 0..n_points {
basis[[i, 0]] = F::one();
}
Ok(basis)
},
}
}
/// Compute trend coefficients using least squares
fn compute_trend_coefficients<F: Float + FromPrimitive>(
points: &Array2<F>,
values: &Array1<F>,
basis_functions: &Array2<F>,
trend_fn: TrendFunction,
) -> InterpolateResult<Array1<F>> {
// Basic least squares: β = (X'X)^(-1) X'y
let xtx = basis_functions.t().dot(basis_functions);
let xty = basis_functions.t().dot(values);
#[cfg(feature = "linalg")]
match xtx.solve(&xty) {
Ok(coeffs) => Ok(coeffs),
Err(_) => {
// Fallback to simple mean for constant trend
let mut coeffs = Array1::zeros(basis_functions.shape()[1]);
coeffs[0] = values.mean().unwrap_or(F::zero());
Ok(coeffs)
}
}
#[cfg(not(feature = "linalg"))]
{
// When linalg is not available, use simple mean for constant trend
let mut coeffs = Array1::zeros(basis_functions.shape()[1]);
coeffs[0] = values.mean().unwrap_or(F::zero());
Ok(coeffs)
}
}
/// Find the k nearest neighbors to a query point
fn find_nearest_neighbors<F: Float + FromPrimitive>(
query_point: &ArrayView1<F>,
points: &Array2<F>,
max_neighbors: usize,
radius_multiplier: F,
) -> InterpolateResult<(Vec<usize>, Vec<F>)> {
let n_points = points.shape()[0];
let n_dims = points.shape()[1];
// Calculate distances
let mut distances = Vec::with_capacity(n_points);
for i in 0..n_points {
let mut dist_sq = F::zero();
for j in 0..n_dims {
let diff = query_point[j] - points[[i, j]];
dist_sq = dist_sq + diff * diff;
}
distances.push((i, dist_sq.sqrt()));
}
// Sort by distance
distances.sort_by(|a, b| {
a.1.partial_cmp(&b.1).unwrap_or(std::cmp::Ordering::Equal)
});
// Determine search radius based on max_neighbors
let radius = if distances.len() > max_neighbors {
distances[max_neighbors - 1].1 * radius_multiplier
} else {
// If we have fewer points than max_neighbors, use all points
F::infinity()
};
// Select points within radius, up to max_neighbors
let mut indices = Vec::with_capacity(max_neighbors);
let mut selected_distances = Vec::with_capacity(max_neighbors);
for (idx, dist) in distances {
if dist <= radius && indices.len() < max_neighbors {
indices.push(idx);
selected_distances.push(dist);
}
}
Ok((indices, selected_distances))
}
/// Compute anisotropic distance between two points
fn compute_anisotropic_distance<F: Float + FromPrimitive>(
p1: &ArrayView1<F>,
p2: &ArrayView1<F>,
anisotropic_cov: &AnisotropicCovariance<F>,
) -> InterpolateResult<F> {
let n_dims = p1.len();
if n_dims != anisotropic_cov.length_scales.len() {
return Err(InterpolateError::DimensionMismatch(
"Number of length scales must match dimension of points".to_string()
));
}
// Simple anisotropic distance (no rotation)
let mut sum_sq = F::zero();
for i in 0..n_dims {
let diff = (p1[i] - p2[i]) / anisotropic_cov.length_scales[i];
sum_sq = sum_sq + diff * diff;
}
Ok(sum_sq.sqrt())
}
/// Evaluate the covariance function
fn compute_covariance<F: Float + FromPrimitive>(
r: F,
anisotropic_cov: &AnisotropicCovariance<F>,
) -> F {
match anisotropic_cov.cov_fn {
CovarianceFunction::SquaredExponential => {
// σ² exp(-r²)
anisotropic_cov.sigma_sq * (-r * r).exp()
}
CovarianceFunction::Exponential => {
// σ² exp(-r)
anisotropic_cov.sigma_sq * (-r).exp()
}
CovarianceFunction::Matern32 => {
// σ² (1 + √3r) exp(-√3r)
let sqrt3_r = F::from_f64(3.0).unwrap().sqrt() * r;
anisotropic_cov.sigma_sq * (F::one() + sqrt3_r) * (-sqrt3_r).exp()
}
CovarianceFunction::Matern52 => {
// σ² (1 + √5r + 5r²/3) exp(-√5r)
let sqrt5_r = F::from_f64(5.0).unwrap().sqrt() * r;
let factor = F::one()
+ sqrt5_r
+ F::from_f64(5.0).unwrap() * r * r / F::from_f64(3.0).unwrap();
anisotropic_cov.sigma_sq * factor * (-sqrt5_r).exp()
}
CovarianceFunction::RationalQuadratic => {
// σ² (1 + r²/(2α))^(-α)
let alpha = anisotropic_cov.extra_params;
let r_sq_div_2a = r * r / (F::from_f64(2.0).unwrap() * alpha);
anisotropic_cov.sigma_sq * (F::one() + r_sq_div_2a).powf(-alpha)
}
}
}
/// Compute low-rank approximation of the covariance matrix
fn compute_low_rank_approximation<F: Float + FromPrimitive>(
points: &Array2<F>,
anisotropic_cov: &AnisotropicCovariance<F>,
rank: usize,
) -> InterpolateResult<(Array2<F>, Array1<F>, Array2<F>)> {
// In a full implementation, you would use Nyström method or randomized SVD
// For this simplified example, we'll compute a small sample covariance matrix
let n_points = points.shape()[0];
let max_sample = std::cmp::min(rank * 2, n_points);
// Use a subset of points
let mut sample_indices = Vec::with_capacity(max_sample);
let step = n_points / max_sample;
for i in 0..max_sample {
sample_indices.push(i * step);
}
let mut sample_cov = Array2::zeros((max_sample, max_sample));
// Compute sample covariance matrix
for i in 0..max_sample {
for j in 0..max_sample {
let idx_i = sample_indices[i];
let idx_j = sample_indices[j];
if i == j {
sample_cov[[i, j]] = anisotropic_cov.sigma_sq + anisotropic_cov.nugget;
} else {
let dist = compute_anisotropic_distance(
&points.slice(ndarray::s![idx_i, ..]),
&points.slice(ndarray::s![idx_j, ..]),
anisotropic_cov
)?;
sample_cov[[i, j]] = compute_covariance(dist, anisotropic_cov);
}
}
}
// Compute SVD of sample covariance
let (u, s, vt) = #[cfg(feature = "linalg")]
match sample_cov.svd(true, true)
#[cfg(not(feature = "linalg"))]
{
// Fallback for SVD when linalg is not available
return Err(InterpolateError::UnsupportedOperation(
"SVD requires the linalg feature to be enabled".to_string()
));
} {
Ok((u, s, vt)) => (u, s, vt),
Err(_) => {
return Err(InterpolateError::ComputationError(
"SVD computation failed for low-rank approximation".to_string()
));
}
};
// Truncate to desired rank
let actual_rank = std::cmp::min(rank, s.len());
let u_r = u.slice(ndarray::s![.., 0..actual_rank]).to_owned();
let s_r = s.slice(ndarray::s![0..actual_rank]).to_owned();
let v_r = vt.slice(ndarray::s![0..actual_rank, ..]).t().to_owned();
Ok((u_r, s_r, v_r))
}
/// Compute tapered covariance representation
fn compute_tapered_covariance<F: Float + FromPrimitive>(
points: &Array2<F>,
anisotropic_cov: &AnisotropicCovariance<F>,
taper_range: F,
) -> InterpolateResult<(Vec<(usize, usize)>, Vec<F>)> {
let n_points = points.shape()[0];
let mut indices = Vec::new();
let mut values = Vec::new();
// Pre-allocate capacity for indices and values (estimating size)
let estimated_nnz = n_points + (n_points * n_points / 4); // Rough estimate
indices.reserve(estimated_nnz);
values.reserve(estimated_nnz);
// Compute sparse representation with tapering more efficiently
for i in 0..n_points {
// Diagonal element
let diagonal_value = anisotropic_cov.sigma_sq + anisotropic_cov.nugget;
indices.push((i, i));
values.push(diagonal_value);
// Off-diagonal elements (lower triangular)
for j in 0..i { // Iterate only through lower triangular part
// Compute distance
let dist = compute_anisotropic_distance(
&points.slice(ndarray::s![i, ..]),
&points.slice(ndarray::s![j, ..]),
anisotropic_cov
)?;
// Apply tapering
if dist <= taper_range {
let value = compute_covariance(dist, anisotropic_cov);
// Add entries in sorted order (row, col) to allow for faster lookups
indices.push((i, j));
values.push(value);
indices.push((j, i));
values.push(value);
}
}
}
Ok((indices, values))
}
/// Project a point onto a feature in the feature space
fn project_to_feature<F: Float + FromPrimitive>(
query_point: &ArrayView1<F>,
points: &Array2<F>,
feature_idx: usize,
anisotropic_cov: &AnisotropicCovariance<F>,
) -> InterpolateResult<F> {
let n_points = points.shape()[0];
// In a full implementation, this would use the eigenvectors
// For this simplified example, we'll use landmark points
// Use a landmark point as basis for the feature
let landmark_idx = feature_idx % n_points;
let landmark = points.slice(ndarray::s![landmark_idx, ..]);
// Project by computing covariance with landmark
let dist = compute_anisotropic_distance(
query_point,
&landmark,
anisotropic_cov
)?;
let projection = compute_covariance(dist, anisotropic_cov);
Ok(projection)
}
/// Creates a new FastKriging model with local approximation
///
/// This convenience function provides a simpler interface for creating
/// a FastKriging model using the local approximation method, which is
/// suitable for most large datasets.
///
/// # Arguments
///
/// * `points` - Coordinates of sample points (n_points × n_dimensions)
/// * `values` - Values at sample points (n_points)
/// * `cov_fn` - Covariance function to use
/// * `length_scale` - Isotropic length scale parameter
/// * `max_neighbors` - Maximum number of neighbors to use
///
/// # Returns
///
/// A FastKriging interpolator with local approximation method
///
/// # Example
///
/// ```rust,no_run
/// use ndarray::{Array1, Array2};
/// use scirs2_interpolate::advanced::fast_kriging::{
/// make_local_kriging, CovarianceFunction
/// };
///
/// // Create sample data
/// let points = Array2::<f64>::zeros((1000, 2));
/// let values = Array1::<f64>::zeros(1000);
///
/// // Create a local kriging model
/// let kriging = make_local_kriging(
/// &points.view(),
/// &values.view(),
/// CovarianceFunction::Matern52,
/// 1.0, // length_scale
/// 50 // max_neighbors
/// ).unwrap();
///
/// // Make a prediction
/// let query_point = Array2::<f64>::zeros((1, 2));
/// let pred = kriging.predict(&query_point.view()).unwrap();
/// ```
pub fn make_local_kriging<F: Float + FromPrimitive + Debug + Add<Output = F> + Sub<Output = F> + Mul<Output = F> + Div<Output = F>>(
points: &ArrayView2<F>,
values: &ArrayView1<F>,
cov_fn: CovarianceFunction,
length_scale: F,
max_neighbors: usize,
) -> InterpolateResult<FastKriging<F>> {
let mut builder = FastKriging::builder()
.points(points.to_owned())
.values(values.to_owned())
.covariance_function(cov_fn)
.approximation_method(FastKrigingMethod::Local)
.max_neighbors(max_neighbors);
// Set the length scale
builder = builder.length_scale(length_scale);
builder.build()
}
/// Creates a new FastKriging model with fixed rank approximation
///
/// The fixed rank approximation provides significant speedup for large
/// datasets by using a low-rank representation of the covariance matrix.
///
/// # Arguments
///
/// * `points` - Coordinates of sample points (n_points × n_dimensions)
/// * `values` - Values at sample points (n_points)
/// * `cov_fn` - Covariance function to use
/// * `length_scale` - Isotropic length scale parameter
/// * `rank` - Rank of the low-rank approximation (smaller = faster, less accurate)
///
/// # Returns
///
/// A FastKriging interpolator with fixed rank approximation method
///
/// # Example
///
/// ```rust,no_run
/// use ndarray::{Array1, Array2};
/// use scirs2_interpolate::advanced::fast_kriging::{
/// make_fixed_rank_kriging, CovarianceFunction
/// };
///
/// // Create sample data
/// let points = Array2::<f64>::zeros((5000, 2));
/// let values = Array1::<f64>::zeros(5000);
///
/// // Create a fixed rank kriging model
/// let kriging = make_fixed_rank_kriging(
/// &points.view(),
/// &values.view(),
/// CovarianceFunction::Matern52,
/// 1.0, // length_scale
/// 100 // rank
/// ).unwrap();
///
/// // Make a prediction
/// let query_point = Array2::<f64>::zeros((1, 2));
/// let pred = kriging.predict(&query_point.view()).unwrap();
/// ```
pub fn make_fixed_rank_kriging<F: Float + FromPrimitive + Debug + Add<Output = F> + Sub<Output = F> + Mul<Output = F> + Div<Output = F>>(
points: &ArrayView2<F>,
values: &ArrayView1<F>,
cov_fn: CovarianceFunction,
length_scale: F,
rank: usize,
) -> InterpolateResult<FastKriging<F>> {
let mut builder = FastKriging::builder()
.points(points.to_owned())
.values(values.to_owned())
.covariance_function(cov_fn)
.approximation_method(FastKrigingMethod::FixedRank(rank));
// Set the length scale
builder = builder.length_scale(length_scale);
builder.build()
}
/// Creates a new FastKriging model with HODLR (Hierarchical Off-Diagonal Low-Rank) approximation
///
/// The HODLR approximation uses a hierarchical approach to partition the dataset
/// and approximate far-field interactions with low-rank representations.
///
/// # Arguments
///
/// * `points` - Coordinates of sample points (n_points × n_dimensions)
/// * `values` - Values at sample points (n_points)
/// * `cov_fn` - Covariance function to use
/// * `length_scale` - Isotropic length scale parameter
/// * `leaf_size` - Maximum size of leaf nodes in the hierarchical decomposition
///
/// # Returns
///
/// A FastKriging interpolator with HODLR approximation method
///
/// # Example
///
/// ```rust,no_run
/// use ndarray::{Array1, Array2};
/// use scirs2_interpolate::advanced::fast_kriging::{
/// make_hodlr_kriging, CovarianceFunction
/// };
///
/// // Create sample data
/// let points = Array2::<f64>::zeros((5000, 2));
/// let values = Array1::<f64>::zeros(5000);
///
/// // Create a HODLR kriging model
/// let kriging = make_hodlr_kriging(
/// &points.view(),
/// &values.view(),
/// CovarianceFunction::Matern52,
/// 1.0, // length_scale
/// 64 // leaf_size
/// ).unwrap();
///
/// // Make a prediction
/// let query_point = Array2::<f64>::zeros((1, 2));
/// let pred = kriging.predict(&query_point.view()).unwrap();
/// ```
pub fn make_hodlr_kriging<F: Float + FromPrimitive + Debug + Add<Output = F> + Sub<Output = F> + Mul<Output = F> + Div<Output = F>>(
points: &ArrayView2<F>,
values: &ArrayView1<F>,
cov_fn: CovarianceFunction,
length_scale: F,
leaf_size: usize,
) -> InterpolateResult<FastKriging<F>> {
let mut builder = FastKriging::builder()
.points(points.to_owned())
.values(values.to_owned())
.covariance_function(cov_fn)
.approximation_method(FastKrigingMethod::HODLR(leaf_size));
// Set the length scale
builder = builder.length_scale(length_scale);
builder.build()
}
/// Creates a new FastKriging model with covariance tapering
///
/// The tapering approach introduces sparsity in the covariance matrix
/// by setting small values to zero, enabling efficient sparse matrix operations.
///
/// # Arguments
///
/// * `points` - Coordinates of sample points (n_points × n_dimensions)
/// * `values` - Values at sample points (n_points)
/// * `cov_fn` - Covariance function to use
/// * `length_scale` - Isotropic length scale parameter
/// * `taper_range` - Distance beyond which covariances are set to zero
///
/// # Returns
///
/// A FastKriging interpolator with tapering approximation method
///
/// # Example
///
/// ```rust,no_run
/// use ndarray::{Array1, Array2};
/// use scirs2_interpolate::advanced::fast_kriging::{
/// make_tapered_kriging, CovarianceFunction
/// };
///
/// // Create sample data
/// let points = Array2::<f64>::zeros((2000, 2));
/// let values = Array1::<f64>::zeros(2000);
///
/// // Create a tapered kriging model
/// let kriging = make_tapered_kriging(
/// &points.view(),
/// &values.view(),
/// CovarianceFunction::Matern52,
/// 1.0, // length_scale
/// 2.5 // taper_range
/// ).unwrap();
///
/// // Make a prediction
/// let query_point = Array2::<f64>::zeros((1, 2));
/// let pred = kriging.predict(&query_point.view()).unwrap();
/// ```
pub fn make_tapered_kriging<F: Float + FromPrimitive + Debug + Add<Output = F> + Sub<Output = F> + Mul<Output = F> + Div<Output = F>>(
points: &ArrayView2<F>,
values: &ArrayView1<F>,
cov_fn: CovarianceFunction,
length_scale: F,
taper_range: f64,
) -> InterpolateResult<FastKriging<F>> {
let mut builder = FastKriging::builder()
.points(points.to_owned())
.values(values.to_owned())
.covariance_function(cov_fn)
.approximation_method(FastKrigingMethod::Tapering(taper_range));
// Set the length scale
builder = builder.length_scale(length_scale);
builder.build()
}
#[cfg(test)]
mod tests {
use super::*;
use approx::assert_abs_diff_eq;
use ndarray::array;
#[test]
fn test_fast_kriging_builder() {
// Create 2D points
let points = Array2::from_shape_vec(
(5, 2),
vec![0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.5, 0.5],
)
.unwrap();
// Create values at those points
let values = array![0.0, 1.0, 1.0, 2.0, 1.0];
// Create a fast kriging interpolator with local approximation
let kriging = FastKriging::builder()
.points(points.clone())
.values(values.clone())
.covariance_function(CovarianceFunction::SquaredExponential)
.approximation_method(FastKrigingMethod::Local)
.max_neighbors(3)
.build()
.unwrap();
// Test prediction at a single point
let test_point = Array2::from_shape_vec((1, 2), vec![0.5, 0.0]).unwrap();
let result = kriging.predict(&test_point.view()).unwrap();
// Prediction should be approximately 0.5 (halfway between 0 and 1)
// Using a larger epsilon due to the approximation
assert!((result.value[0] - 0.5).abs() < 0.3);
}
}