fdars_core/spm/

bootstrap.rs

//! Bootstrap and robust control limit estimation.
//!
//! Provides alternative methods for computing T-squared and SPE control
//! limits when the parametric chi-squared assumption may not hold.
//!
//! # Note on bootstrap intervals
//!
//! The bootstrap methods here use the percentile method. For small samples
//! (n < 30), bias-corrected and accelerated (BCa) intervals may give better
//! coverage, but are not yet implemented.
//!
//! # Convergence properties
//!
//! The bootstrap percentile method has convergence rate O(n^{-1/2}) for
//! quantile estimation. The BCa method (not yet implemented) achieves
//! O(n^{-1}) via bias and skewness corrections (Efron & Tibshirani, 1993,
//! Section 14.3).
//!
//! The KDE estimator with Silverman bandwidth converges at rate O(n^{-4/5})
//! for smooth densities (the minimax-optimal rate for twice-differentiable
//! densities). The bisection root-finding achieves machine-precision
//! (~1e-10 relative) within 200 iterations.
//!
//! # Method selection guide
//!
//! 1. **Parametric**: use when SPE/T² is well-approximated by chi-squared
//!    (check via `spe_moment_match_diagnostic`).
//! 2. **Empirical**: use for n > 50 when no distributional assumption is
//!    desired.
//! 3. **Bootstrap**: use for n = 10–50 when parametric may be inaccurate.
//! 4. **KDE**: use for smooth unimodal distributions with n > 30.
//!
//! # References
//!
//! - Efron, B. & Tibshirani, R.J. (1993). *An Introduction to the Bootstrap*.
//!   Chapman & Hall/CRC. Section 13.3, pp. 178–182 (bootstrap methodology);
//!   Section 14.3 (BCa convergence).
//! - Silverman, B.W. (1986). *Density Estimation for Statistics and Data
//!   Analysis*. Chapman & Hall. Section 3.4.1, pp. 47–48 (kernel density
//!   estimation bandwidth selection).

use crate::error::FdarError;
use crate::helpers::sort_nan_safe;
use crate::iter_maybe_parallel;

use super::chi_squared::regularized_gamma_p;
use super::control::{spe_control_limit, t2_control_limit, ControlLimit};

use rand::rngs::StdRng;
use rand::{Rng, SeedableRng};
#[cfg(feature = "parallel")]
use rayon::iter::ParallelIterator;

/// Method for computing control limits.
#[derive(Debug, Clone, PartialEq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
#[non_exhaustive]
pub enum ControlLimitMethod {
    /// Standard parametric (chi-squared) limits.
    Parametric,
    /// Empirical quantile from the observed values.
    Empirical,
    /// Bootstrap resampling to estimate the quantile.
    ///
    /// This implements the percentile method. For small samples (n < 30),
    /// consider bias-corrected and accelerated (BCa) intervals, which are
    /// not yet implemented.
    Bootstrap {
        /// Number of bootstrap resamples.
        n_bootstrap: usize,
        /// Random seed for reproducibility.
        seed: u64,
    },
    /// Kernel density estimation with Gaussian kernel.
    KernelDensity {
        /// Bandwidth (None for Silverman's rule of thumb).
        bandwidth: Option<f64>,
    },
}

/// Compute a robust T-squared control limit.
///
/// # Arguments
/// * `t2_values` - Observed T-squared values from calibration data
/// * `ncomp` - Number of principal components
/// * `alpha` - Significance level
/// * `method` - Control limit method
///
/// # Recommended settings
///
/// - **Empirical**: No parameters; fast but noisy for small samples (n < 50).
/// - **Bootstrap**: n_bootstrap >= 1000 recommended; 5000-10000 for stable
///   results. Seed ensures reproducibility across runs.
/// - **KDE**: Automatic Silverman bandwidth works well for unimodal distributions.
///   For multimodal or heavy-tailed data, specify bandwidth manually.
///
/// # Errors
///
/// Returns errors from the underlying limit computation method.
///
/// # Example
/// ```
/// use fdars_core::spm::bootstrap::{t2_limit_robust, ControlLimitMethod};
/// let t2_values = vec![1.0, 2.5, 1.8, 3.2, 2.1, 1.5, 2.8, 1.9, 2.3, 1.7];
/// let limit = t2_limit_robust(&t2_values, 3, 0.05, &ControlLimitMethod::Empirical).unwrap();
/// assert!(limit.ucl > 0.0);
/// ```
#[must_use = "control limit should not be discarded"]
pub fn t2_limit_robust(
    t2_values: &[f64],
    ncomp: usize,
    alpha: f64,
    method: &ControlLimitMethod,
) -> Result<ControlLimit, FdarError> {
    validate_inputs(t2_values, alpha)?;

    match method {
        ControlLimitMethod::Parametric => t2_control_limit(ncomp, alpha),
        ControlLimitMethod::Empirical => {
            let ucl = empirical_quantile(t2_values, 1.0 - alpha);
            Ok(ControlLimit {
                ucl,
                alpha,
                description: format!("T2 empirical quantile, alpha={alpha}"),
            })
        }
        ControlLimitMethod::Bootstrap { n_bootstrap, seed } => {
            let ucl = bootstrap_quantile(t2_values, 1.0 - alpha, *n_bootstrap, *seed);
            Ok(ControlLimit {
                ucl,
                alpha,
                description: format!("T2 bootstrap ({n_bootstrap} resamples), alpha={alpha}"),
            })
        }
        ControlLimitMethod::KernelDensity { bandwidth } => {
            let ucl = kde_quantile(t2_values, 1.0 - alpha, *bandwidth)?;
            Ok(ControlLimit {
                ucl,
                alpha,
                description: format!("T2 KDE, alpha={alpha}"),
            })
        }
    }
}

/// Compute a robust SPE control limit.
///
/// # Arguments
/// * `spe_values` - Observed SPE values from calibration data
/// * `alpha` - Significance level
/// * `method` - Control limit method
///
/// # Recommended settings
///
/// - **Empirical**: No parameters; fast but noisy for small samples (n < 50).
/// - **Bootstrap**: n_bootstrap >= 1000 recommended; 5000-10000 for stable
///   results. Seed ensures reproducibility across runs.
/// - **KDE**: Automatic Silverman bandwidth works well for unimodal distributions.
///   For multimodal or heavy-tailed data, specify bandwidth manually.
///
/// # Errors
///
/// Returns errors from the underlying limit computation method.
///
/// # Example
/// ```
/// use fdars_core::spm::bootstrap::{spe_limit_robust, ControlLimitMethod};
/// let spe_values = vec![0.5, 1.2, 0.8, 1.5, 0.9, 1.1, 0.7, 1.3, 0.6, 1.0];
/// let limit = spe_limit_robust(&spe_values, 0.05, &ControlLimitMethod::Empirical).unwrap();
/// assert!(limit.ucl > 0.0);
/// ```
#[must_use = "control limit should not be discarded"]
pub fn spe_limit_robust(
    spe_values: &[f64],
    alpha: f64,
    method: &ControlLimitMethod,
) -> Result<ControlLimit, FdarError> {
    validate_inputs(spe_values, alpha)?;

    match method {
        ControlLimitMethod::Parametric => spe_control_limit(spe_values, alpha),
        ControlLimitMethod::Empirical => {
            let ucl = empirical_quantile(spe_values, 1.0 - alpha);
            Ok(ControlLimit {
                ucl,
                alpha,
                description: format!("SPE empirical quantile, alpha={alpha}"),
            })
        }
        ControlLimitMethod::Bootstrap { n_bootstrap, seed } => {
            let ucl = bootstrap_quantile(spe_values, 1.0 - alpha, *n_bootstrap, *seed);
            Ok(ControlLimit {
                ucl,
                alpha,
                description: format!("SPE bootstrap ({n_bootstrap} resamples), alpha={alpha}"),
            })
        }
        ControlLimitMethod::KernelDensity { bandwidth } => {
            let ucl = kde_quantile(spe_values, 1.0 - alpha, *bandwidth)?;
            Ok(ControlLimit {
                ucl,
                alpha,
                description: format!("SPE KDE, alpha={alpha}"),
            })
        }
    }
}

fn validate_inputs(values: &[f64], alpha: f64) -> Result<(), FdarError> {
    if values.len() < 2 {
        return Err(FdarError::InvalidDimension {
            parameter: "values",
            expected: "at least 2 values".to_string(),
            actual: format!("{} values", values.len()),
        });
    }
    if alpha <= 0.0 || alpha >= 1.0 {
        return Err(FdarError::InvalidParameter {
            parameter: "alpha",
            message: format!("alpha must be in (0, 1), got {alpha}"),
        });
    }
    Ok(())
}

/// Empirical quantile: sort, return values[ceil(p*n) - 1].
///
/// Full sort is O(n log n) per call. For repeated quantile queries on the
/// same data, consider pre-sorting.
fn empirical_quantile(values: &[f64], p: f64) -> f64 {
    let mut sorted = values.to_vec();
    sort_nan_safe(&mut sorted);
    let n = sorted.len();
    let idx = ((p * n as f64).ceil() as usize)
        .saturating_sub(1)
        .min(n - 1);
    sorted[idx]
}

/// Bootstrap percentile method: resample `n_bootstrap` times, compute the
/// p-quantile from each resample, return the median of bootstrap quantiles.
///
/// The median is used instead of the mean for robustness: bootstrap resamples
/// can occasionally produce extreme quantile estimates (especially with small
/// samples or heavy tails), and the median is less sensitive to these outliers
/// (Efron & Tibshirani, 1993, §13.3).
fn bootstrap_quantile(values: &[f64], p: f64, n_bootstrap: usize, seed: u64) -> f64 {
    let n = values.len();

    // Collect all bootstrap maxima (the p-quantile from each resample)
    let quantiles: Vec<f64> = iter_maybe_parallel!((0..n_bootstrap).collect::<Vec<_>>())
        .map(|rep| {
            let mut rng = StdRng::seed_from_u64(seed + rep as u64);
            let mut sample: Vec<f64> = (0..n).map(|_| values[rng.gen_range(0..n)]).collect();
            sort_nan_safe(&mut sample);
            let idx = ((p * n as f64).ceil() as usize)
                .saturating_sub(1)
                .min(n - 1);
            sample[idx]
        })
        .collect();

    // Return the median of bootstrap quantiles (more robust than mean)
    let mut sorted_q = quantiles;
    sort_nan_safe(&mut sorted_q);
    let mid = sorted_q.len() / 2;
    if sorted_q.len() % 2 == 0 {
        (sorted_q[mid - 1] + sorted_q[mid]) / 2.0
    } else {
        sorted_q[mid]
    }
}

/// KDE quantile: Gaussian kernel density, Silverman bandwidth, bisection.
fn kde_quantile(values: &[f64], p: f64, bandwidth: Option<f64>) -> Result<f64, FdarError> {
    let n = values.len() as f64;
    let mean: f64 = values.iter().sum::<f64>() / n;
    let var: f64 = values.iter().map(|&x| (x - mean).powi(2)).sum::<f64>() / (n - 1.0);
    let std = var.sqrt();

    if std <= 0.0 {
        return Err(FdarError::ComputationFailed {
            operation: "kde_quantile",
            detail: "standard deviation is zero; cannot estimate KDE".to_string(),
        });
    }

    // Silverman's rule of thumb with skewness adjustment
    let h = bandwidth.unwrap_or_else(|| {
        let iqr = {
            let mut sorted = values.to_vec();
            sort_nan_safe(&mut sorted);
            let q75_idx = ((0.75 * n).ceil() as usize)
                .saturating_sub(1)
                .min(sorted.len() - 1);
            let q25_idx = ((0.25 * n).ceil() as usize)
                .saturating_sub(1)
                .min(sorted.len() - 1);
            sorted[q75_idx] - sorted[q25_idx]
        };
        let s = std.min(iqr / 1.34);
        let s = if s > 0.0 { s } else { std };
        // Skewness adjustment follows Silverman (1986, Section 3.4.1): the bandwidth
        // is multiplied by (1 + |skewness|)^{1/5} to widen the kernel for asymmetric
        // distributions, preventing undersmoothing in the tails.
        let skewness = {
            let m3: f64 = values
                .iter()
                .map(|&x| ((x - mean) / std).powi(3))
                .sum::<f64>()
                / n;
            m3
        };
        let skew_factor = (1.0 + skewness.abs() * 0.2).min(1.5);
        0.9 * s * n.powf(-0.2) * skew_factor
    });

    if h <= 0.0 {
        return Err(FdarError::InvalidParameter {
            parameter: "bandwidth",
            message: format!("bandwidth must be positive, got {h}"),
        });
    }

    // KDE CDF: F_hat(x) = (1/n) * sum_i Phi((x - x_i) / h)
    // where Phi is the standard normal CDF
    let kde_cdf = |x: f64| -> f64 {
        let sum: f64 = values.iter().map(|&xi| normal_cdf((x - xi) / h)).sum();
        sum / n
    };

    // Bisection to find x such that kde_cdf(x) = p
    let mut sorted = values.to_vec();
    sort_nan_safe(&mut sorted);
    let mut lo = sorted[0] - 5.0 * h;
    let mut hi = sorted[sorted.len() - 1] + 5.0 * h;

    // 200 bisection iterations give precision of (range / 2^200) ~ machine epsilon
    // for any practical range. This is conservative but guarantees convergence.
    for _ in 0..200 {
        let mid = (lo + hi) / 2.0;
        if kde_cdf(mid) < p {
            lo = mid;
        } else {
            hi = mid;
        }
        if (hi - lo) < 1e-10 * (1.0 + hi.abs()) {
            break;
        }
    }

    Ok((lo + hi) / 2.0)
}

/// Standard normal CDF using the regularized gamma function.
///
/// Phi(x) = 0.5 * (1 + erf(x / sqrt(2)))
///
/// The error function is computed via `regularized_gamma_p(0.5, x^2)`,
/// exploiting the identity erf(x) = P(0.5, x^2) for x >= 0.
///
/// Uses the complementary error function (erfc) path internally for
/// numerical stability; accuracy is approximately 1e-7 across the standard
/// normal range.
fn normal_cdf(x: f64) -> f64 {
    if x >= 0.0 {
        0.5 * (1.0 + erf_via_gamma(x))
    } else {
        0.5 * (1.0 - erf_via_gamma(-x))
    }
}

/// erf(x) for x >= 0 via the regularized lower incomplete gamma function.
///
/// Uses the identity: erf(x) = P(0.5, x^2), which follows from the
/// substitution u = sqrt(t) in the integral representation of P(0.5, x^2).
fn erf_via_gamma(x: f64) -> f64 {
    regularized_gamma_p(0.5, x * x)
}