feat(kde): kernel functions with statistical properties and LOESS support

[thisisamirv]

Add kernel functions with statistical properties and LOESS support

This PR prepares the foundation for KDE and LOESS implementations.

Summary

Major enhancement to the kernel module adding:

1. Kernel density estimation (KDE) constants to consts.rs
2. Complete rewrite of kernel.rs with AMISE efficiency metrics, dual evaluation modes, and LOESS integration
3. Compile-time validation of statistical properties

Changes to consts.rs

• Kernel variance (μ₂): ∫ u² K(u) du for 9 kernels
• Kernel roughness (R(K)): ∫ K(u)² du for 9 kernels
• Module docs explaining kernel properties and bandwidth conversion
• References to Silverman (1986) Table 3.1 and Wand & Jones (1995)
• Added missing SQRT_PI constant

Changes to kernel.rs

1. Statistical Correctness

• Dual evaluation modes:
  • evaluate(): Normalized for KDE (integrates to 1)
  • evaluate_weight(): Unnormalized for LOESS (local regression weights)
• AMISE properties: Efficiency, variance, roughness, bandwidth factors from Silverman (1986)
• Compile-time validation: Ensures efficiency calculations match theoretical values

2. New Kernels & Renaming

• Renamed Quartic → Bisquare (it's a more standard name)
• Added Cosine kernel (high efficiency ≈ 0.9995)
• Added Logistic kernel (heavy-tailed unbounded)
• Added Sigmoid kernel (hyperbolic secant)
• Fixed Sigmoid formula: was exp(πx), now correctly exp(x)

3. Enhanced API

• KernelType enum: Runtime kernel selection for LOESS and other applications
• CustomKernel: User-defined kernels with metadata
• Batch operations: evaluate_batch(), compute_distance_weights()
• Utilities: robust_reweights(), normalize_weights()
• Recommendations: recommended_for_kde(), recommended_for_loess(), most_efficient()

4. Boundary Behavior Fix

• Changed boundary handling from |x| <= 1 to |x| >= 1 for consistency
• Support now returns open interval (-1, 1) as mathematically correct
• All bounded kernels return 0 at exactly x = ±1

5. Documentation & Testing

• Comprehensive table: Formula, support, efficiency, R(K), μ₂(K) for all kernels
• Compile-time tests: Verify efficiency matches Silverman (1986) exact values
• Many runtime tests: Integration, symmetry, monotonicity, normalization, edge cases

Breaking Changes

1. Quartic → Bisquare: Renamed for statistical consistency
2. Boundary behavior: |x| = 1 now returns 0 (was non-zero)
3. Sigmoid formula: Fixed to match hyperbolic secant definition
4. Tricube normalization: Now (70/81)(1-|x|³)³ (was unnormalized)

Fixed

• Fix unused import warnings for distribution (removed unused import crate::distribution::internal::testing_boiler).
• Fix Geometric::inverse_cdf platform-dependent behavior on Windows:

Problem

The test_inverse_cdf test was failing on Windows with:

• Expected: 1
• Got: 2 for inverse_cdf(0.0)

Root Cause

Floating-point precision differences across platforms caused inverse_cdf(0.0) to compute inconsistent results when using the formula ceil(log(1-p) / log(1-self.p)).

Solution

Added an explicit implementation of inverse_cdf for the Geometric distribution that handles edge cases consistently:

• Returns min() (1) when input probability p <= 0.0
• Returns 1 when distribution parameter self.p == 1.0
• Returns max() (u64::MAX) when input probability p >= 1.0
• Uses the mathematical formula for other cases

This ensures consistent behavior across all platforms (macOS, Linux, Windows).

[codecov]

Codecov Report

❌ Patch coverage is 95.62044% with 54 lines in your changes missing coverage. Please review.
✅ Project coverage is 94.73%. Comparing base (5da3470) to head (8325b1a).
⚠️ Report is 6 commits behind head on master.

Files with missing lines	Patch %	Lines
src/function/kernel.rs	95.65%	53 Missing ⚠️
src/distribution/geometric.rs	92.30%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #360      +/-   ##
==========================================
- Coverage   94.99%   94.73%   -0.27%     
==========================================
  Files          61       59       -2     
  Lines       13615    14004     +389     
==========================================
+ Hits        12934    13266     +332     
- Misses        681      738      +57     

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[YeungOnion]

Hey, would you be willing to break this one out across a few PRs?

One partition could be,

• fixes and changes to kernel functions, additions for consts and compile time testing
• kerneltype enum and custom kernel (happy to discuss this, but we've not implemented a lot in the way of dynamic dispatch)
• fixing geometric inverse cdf is worked on in feat: add specialised inverse cdf implementation for geometric distribution #343 (I should move that forward and make additional tests into a separate issue)
• fix for macro imports and use - I don't know why this lint appears in this context.

[thisisamirv]

Hi @YeungOnion , sorry for going radio silent on this PR earlier. I got excited about exploring optimizations and ended up building a much more feature-complete and performant standalone LOWESS/LOESS implementation instead. I’ve since added parallelism, streaming/online modes, confidence intervals, better robustness, cross-validation, no-std support, and validated it against R’s implementations, which are highly optimized and are built on fortran/BLAS (https://github.com/thisisamirv/lowess.git and https://github.com/thisisamirv/loess-rs.git). For the lowess crate the only dependencies are num traits and wide, and for the loess crate it is num traits, wide, and nalgebra.
Would you be interested in integrating this improved version into statsrs (perhaps as a new module or optional feature)? Happy to open a fresh PR tailored to your style/guide. No worries if it doesn’t fit the scope!