kshana 0.13.0

// SPDX-License-Identifier: Apache-2.0
//! Kalman filter-consistency health monitoring (NIS / NEES).
//!
//! A Kalman filter is only trustworthy when its reported covariance actually
//! matches the spread of its errors — i.e. it is *consistent*. Two standard tests
//! (Bar-Shalom, *Estimation with Applications to Tracking and Navigation*, §5.4)
//! check this from data:
//!
//! * **NIS** — Normalised Innovation Squared, `ν² / S`. Under a correctly-tuned
//!   filter the (whitened) innovations are independent with `NIS ∼ χ²(n_z)`; for a
//!   scalar phase measurement `n_z = 1`, so `E[NIS] = 1`. NIS uses only observable
//!   quantities, so it is available in deployment.
//! * **NEES** — Normalised Estimation Error Squared, `ẽᵀ P⁻¹ ẽ` with `ẽ = x_true −
//!   x̂`. Under a consistent filter `NEES ∼ χ²(n_x)` (`n_x = 2` here, so `E[NEES] =
//!   2`). NEES needs the truth, so it is a *validation-time* statistic, computed
//!   here from a Monte-Carlo ensemble whose truth is known.
//!
//! The harness below simulates the two-state clock truth with process noise drawn
//! from the true `Q` (the same van-Loan covariance the filter assumes at unit
//! tuning), feeds the filter noisy phase measurements `z = phase_true + N(0, r)`,
//! and pools NIS and NEES across an ensemble of independent seeds. Independence
//! across the ensemble's seeds is what justifies treating the pool as χ²; the mild
//! within-run time-correlation of NEES is the usual, documented approximation.
//!
//! A deliberate **`q_factor`** mis-tunes the filter's process noise relative to the
//! truth (`Q_filter = q_factor · Q_true`): at `q_factor = 1` the filter is matched
//! and the pooled means land inside their χ² bands; away from 1 the predicted `S`
//! and `P` are mis-scaled, the means leave the bands, and `consistent` flips to
//! `false` — an objectively checkable health gate.

use crate::detection::chi2_inv_cdf;
use crate::kalman::KalmanClock;
use rand::SeedableRng;
use rand_chacha::ChaCha8Rng;
use rand_distr::{Distribution, Normal};
use serde::Serialize;

/// Filter-consistency health summary, surfaced in the result JSON.
#[derive(Clone, Debug, Serialize, PartialEq)]
pub struct FilterHealth {
    /// Mean Normalised Innovation Squared over the ensemble (target ≈ 1.0).
    pub nis_mean: f64,
    /// Lower 95 % χ² band on the NIS mean (pooled samples).
    pub nis_chi2_lower_95: f64,
    /// Upper 95 % χ² band on the NIS mean.
    pub nis_chi2_upper_95: f64,
    /// Mean Normalised Estimation Error Squared over the ensemble (target ≈ 2.0).
    pub nees_mean: f64,
    /// Lower 95 % χ² band on the NEES mean (pooled samples).
    pub nees_chi2_lower_95: f64,
    /// Upper 95 % χ² band on the NEES mean.
    pub nees_chi2_upper_95: f64,
    /// Whether both means fall inside their 95 % bands — the filter is self-consistent.
    pub consistent: bool,
}

/// Configuration for the Monte-Carlo consistency assessment.
#[derive(Clone, Copy, Debug)]
pub struct HealthConfig {
    pub q_wf: f64,
    pub q_rw: f64,
    pub r: f64,
    pub dt: f64,
    pub steps: usize,
    pub seeds: usize,
    /// Mis-tuning multiplier applied to the *filter's* process noise (1.0 = matched).
    pub q_factor: f64,
    pub base_seed: u64,
}

/// Initial state uncertainty `P₀` used for both the truth draw and the filter, so
/// the ensemble is consistent from the first step at unit tuning (no warm-up bias).
const PHASE_VAR0: f64 = 1e-18; // (1 ns)² phase
const FREQ_VAR0: f64 = 1e-24; // fractional-frequency

/// Exact two-state van-Loan process-noise covariance `Q` over `dt`.
fn process_q(q_wf: f64, q_rw: f64, dt: f64) -> [[f64; 2]; 2] {
    let (dt2, dt3) = (dt * dt, dt * dt * dt);
    [
        [q_wf * dt + q_rw * dt3 / 3.0, q_rw * dt2 / 2.0],
        [q_rw * dt2 / 2.0, q_rw * dt],
    ]
}

/// Lower-triangular Cholesky factor of a symmetric PSD 2×2 (zeros where singular).
fn cholesky_2x2(m: [[f64; 2]; 2]) -> [[f64; 2]; 2] {
    let l00 = m[0][0].max(0.0).sqrt();
    let l10 = if l00 > 0.0 { m[1][0] / l00 } else { 0.0 };
    let l11 = (m[1][1] - l10 * l10).max(0.0).sqrt();
    [[l00, 0.0], [l10, l11]]
}

/// `NEES = ẽᵀ P⁻¹ ẽ` for a 2×2 `P` and error `e`; returns `None` if `P` is singular.
fn nees(e: [f64; 2], p: [[f64; 2]; 2]) -> Option<f64> {
    let det = p[0][0] * p[1][1] - p[0][1] * p[1][0];
    if det.abs() <= 0.0 {
        return None;
    }
    // P⁻¹ = (1/det)·[[p11, −p01], [−p10, p00]].
    let i00 = p[1][1] / det;
    let i01 = -p[0][1] / det;
    let i10 = -p[1][0] / det;
    let i11 = p[0][0] / det;
    Some(e[0] * (i00 * e[0] + i01 * e[1]) + e[1] * (i10 * e[0] + i11 * e[1]))
}

/// Run the Monte-Carlo filter-consistency assessment and summarise the pooled
/// NIS/NEES against their 95 % χ² bands. Deterministic in `base_seed`.
pub fn assess(cfg: HealthConfig) -> FilterHealth {
    let steps = cfg.steps.max(1);
    let seeds = cfg.seeds.max(1);
    let dt = cfg.dt.max(1e-12);
    let r = cfg.r.max(1e-300);

    // Truth uses the unscaled Q; the filter mis-tunes it by q_factor.
    let lq = cholesky_2x2(process_q(cfg.q_wf, cfg.q_rw, dt));
    let l0 = cholesky_2x2([[PHASE_VAR0, 0.0], [0.0, FREQ_VAR0]]);

    let n01 = Normal::new(0.0, 1.0).unwrap();
    let meas = Normal::new(0.0, r.sqrt()).unwrap();

    let mut nis_sum = 0.0;
    let mut nis_n = 0u64;
    let mut nees_sum = 0.0;
    let mut nees_n = 0u64;

    for s in 0..seeds {
        let mut rng = ChaCha8Rng::seed_from_u64(
            cfg.base_seed ^ (0x9E37_79B9_7F4A_7C15u64).wrapping_mul(s as u64 + 1),
        );
        // Truth drawn from P₀; filter initialised to the same prior.
        let (z0, z1) = (n01.sample(&mut rng), n01.sample(&mut rng));
        let mut x_true = [l0[0][0] * z0, l0[1][0] * z0 + l0[1][1] * z1];
        let mut kf = KalmanClock::new(cfg.q_wf * cfg.q_factor, cfg.q_rw * cfg.q_factor, r)
            .with_initial_cov(PHASE_VAR0, FREQ_VAR0);

        for _ in 0..steps {
            // Propagate truth: x = F x + w, w ~ N(0, Q_true).
            let (w0, w1) = (n01.sample(&mut rng), n01.sample(&mut rng));
            x_true[0] += dt * x_true[1] + lq[0][0] * w0;
            x_true[1] += lq[1][0] * w0 + lq[1][1] * w1;

            // Filter predict; NIS uses the predicted innovation (pre-update).
            kf.predict(dt);
            let z = x_true[0] + meas.sample(&mut rng);
            let innov = z - kf.phase_est();
            let s_innov = kf.innovation_var(r);
            if s_innov > 0.0 {
                nis_sum += innov * innov / s_innov;
                nis_n += 1;
            }

            // Filter update; NEES uses the posterior error and covariance.
            kf.update_with_r(z, r);
            let e = [x_true[0] - kf.phase_est(), x_true[1] - kf.freq_est()];
            if let Some(v) = nees(e, kf.covariance()) {
                nees_sum += v;
                nees_n += 1;
            }
        }
    }

    let nis_mean = if nis_n > 0 {
        nis_sum / nis_n as f64
    } else {
        0.0
    };
    let nees_mean = if nees_n > 0 {
        nees_sum / nees_n as f64
    } else {
        0.0
    };

    // Confidence bands. The two statistics have *different* effective independence:
    //
    //  * NIS — the optimal filter's innovations form a **white** (temporally
    //    independent) sequence, so the K = seeds·steps pooled NIS samples are iid
    //    χ²(1); their sum ∼ χ²(K) and the mean's 95 % band is χ²_{.025,.975}(K)/K.
    //  * NEES — the estimation errors are temporally **correlated** within a run, so
    //    the independent count is the number of Monte-Carlo runs, not the pooled
    //    sample count (Bar-Shalom §5.4.2). With n_x = 2 states over `seeds` runs the
    //    acceptance region for the mean is χ²_{.025,.975}(2·seeds)/seeds. (Time-
    //    averaging only shrinks the spread further, so this run-based band is the
    //    conservative, honest choice — it never false-rejects a matched filter.)
    let knis = nis_n as f64;
    let nis_lo = chi2_inv_cdf(0.025, knis) / knis;
    let nis_hi = chi2_inv_cdf(0.975, knis) / knis;
    let dof_nees = 2.0 * seeds as f64;
    let nees_lo = chi2_inv_cdf(0.025, dof_nees) / seeds as f64;
    let nees_hi = chi2_inv_cdf(0.975, dof_nees) / seeds as f64;

    let consistent =
        nis_mean >= nis_lo && nis_mean <= nis_hi && nees_mean >= nees_lo && nees_mean <= nees_hi;

    FilterHealth {
        nis_mean,
        nis_chi2_lower_95: nis_lo,
        nis_chi2_upper_95: nis_hi,
        nees_mean,
        nees_chi2_lower_95: nees_lo,
        nees_chi2_upper_95: nees_hi,
        consistent,
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    fn cfg(q_factor: f64) -> HealthConfig {
        // Process-noise-dominated (q_wf·dt = 1e-18 ≫ r = 1e-20): the innovation
        // variance S ≈ P⁻[0][0] tracks Q, so a Q mistuning by `q_factor` scales S
        // (and hence NIS ≈ 1/q_factor) starkly — the regime that makes the health
        // gate discriminating.
        HealthConfig {
            q_wf: 1e-18,
            q_rw: 1e-26,
            r: 1e-20,
            dt: 1.0,
            steps: 200,
            seeds: 64,
            q_factor,
            base_seed: 20260604,
        }
    }

    #[test]
    fn matched_filter_is_consistent() {
        // At unit tuning the filter's covariance matches its errors: NIS → 1,
        // NEES → 2, both inside their narrow χ² bands.
        let h = assess(cfg(1.0));
        assert!(h.consistent, "matched filter flagged inconsistent: {h:?}");
        assert!(
            h.nis_mean > 0.9 && h.nis_mean < 1.1,
            "nis_mean={}",
            h.nis_mean
        );
        assert!(
            h.nees_mean > 1.8 && h.nees_mean < 2.2,
            "nees_mean={}",
            h.nees_mean
        );
        // The bands bracket the targets and are tight (thousands of pooled samples).
        assert!(h.nis_chi2_lower_95 < 1.0 && h.nis_chi2_upper_95 > 1.0);
        assert!(h.nees_chi2_lower_95 < 2.0 && h.nees_chi2_upper_95 > 2.0);
    }

    #[test]
    fn q_r_mismatch_sweep_flips_consistency() {
        // The done-gate sweep: matched (1.0) is consistent; every factor outside
        // [0.7, 1.4] is rejected. Under-tuned Q (filter over-confident, S too small)
        // pushes NIS above the band; over-tuned Q (S too large) pushes it below.
        assert!(
            assess(cfg(1.0)).consistent,
            "factor 1.0 should be consistent"
        );
        for &f in &[0.1, 0.5, 2.0, 10.0] {
            let h = assess(cfg(f));
            assert!(!h.consistent, "factor {f} should be inconsistent: {h:?}");
        }
        // Direction: too-small Q ⇒ NIS above 1; too-large Q ⇒ NIS below 1.
        assert!(assess(cfg(0.5)).nis_mean > assess(cfg(1.0)).nis_mean);
        assert!(assess(cfg(2.0)).nis_mean < assess(cfg(1.0)).nis_mean);
    }

    #[test]
    fn assess_is_deterministic_in_the_seed() {
        assert_eq!(assess(cfg(1.0)), assess(cfg(1.0)));
    }
}