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use crate::math::MulAdd;
macro_rules! impl_welford {
($name:ident, $ty:ty) => {
/// Welford — Online mean, variance, and standard deviation.
///
/// Numerically stable single-pass computation using Welford's algorithm.
/// No catastrophic cancellation. Supports merging partial results via
/// Chan's algorithm for parallel aggregation.
///
/// # Use Cases
/// - Running statistics on latency, throughput, PnL
/// - Z-score computation (combine with EMA for baseline)
/// - Input to adaptive thresholds
#[derive(Debug, Clone)]
pub struct $name {
count: u64,
mean: $ty,
m2: $ty,
}
impl $name {
/// Creates a new empty accumulator.
#[inline]
#[must_use]
pub const fn new() -> Self {
Self {
count: 0,
mean: 0.0 as $ty,
m2: 0.0 as $ty,
}
}
/// Creates an accumulator pre-loaded from known statistics.
///
/// `m2` is the sum of squared deviations from the mean
/// (`variance * (count - 1)` for sample variance).
#[inline]
#[must_use]
pub const fn from_parts(count: u64, mean: $ty, m2: $ty) -> Self {
Self { count, mean, m2 }
}
/// Feeds a sample.
#[inline]
pub fn update(&mut self, sample: $ty) {
self.count += 1;
let delta = sample - self.mean;
self.mean += delta / self.count as $ty;
let delta2 = sample - self.mean;
self.m2 += delta * delta2;
}
/// Number of samples processed.
#[inline]
#[must_use]
pub fn count(&self) -> u64 {
self.count
}
/// Running mean, or `None` if empty.
#[inline]
#[must_use]
pub fn mean(&self) -> Option<$ty> {
if self.count == 0 {
Option::None
} else {
Option::Some(self.mean)
}
}
/// Sample variance (N-1 denominator), or `None` if < 2 samples.
#[inline]
#[must_use]
pub fn variance(&self) -> Option<$ty> {
if self.count < 2 {
Option::None
} else {
Option::Some(self.m2 / (self.count - 1) as $ty)
}
}
/// Population variance (N denominator), or `None` if empty.
#[inline]
#[must_use]
pub fn population_variance(&self) -> Option<$ty> {
if self.count == 0 {
Option::None
} else {
Option::Some(self.m2 / self.count as $ty)
}
}
/// Sample standard deviation, or `None` if < 2 samples.
#[inline]
#[must_use]
#[cfg(any(feature = "std", feature = "libm"))]
pub fn std_dev(&self) -> Option<$ty> {
self.variance().map(|v| {
#[allow(clippy::cast_possible_truncation)]
{
crate::math::sqrt(v as f64) as $ty
}
})
}
/// Merges another accumulator into this one (Chan's algorithm).
///
/// After merging, `self` contains the statistics of the combined
/// dataset. The other accumulator is unchanged.
#[inline]
pub fn merge(&mut self, other: &Self) {
if other.count == 0 {
return;
}
if self.count == 0 {
self.count = other.count;
self.mean = other.mean;
self.m2 = other.m2;
return;
}
let combined_count = self.count + other.count;
let delta = other.mean - self.mean;
let weight = other.count as $ty / combined_count as $ty;
let new_mean = delta.fma(weight, self.mean);
let cross = self.count as $ty * other.count as $ty / combined_count as $ty;
let new_m2 = delta.fma(delta * cross, self.m2 + other.m2);
self.count = combined_count;
self.mean = new_mean;
self.m2 = new_m2;
}
/// Resets to empty state.
#[inline]
pub fn reset(&mut self) {
self.count = 0;
self.mean = 0.0 as $ty;
self.m2 = 0.0 as $ty;
}
}
impl Default for $name {
#[inline]
fn default() -> Self {
Self::new()
}
}
};
}
impl_welford!(WelfordF64, f64);
impl_welford!(WelfordF32, f32);
#[cfg(test)]
mod tests {
use super::*;
// =========================================================================
// Basic correctness
// =========================================================================
#[test]
fn empty_returns_none() {
let w = WelfordF64::new();
assert_eq!(w.count(), 0);
assert!(w.mean().is_none());
assert!(w.variance().is_none());
assert!(w.population_variance().is_none());
assert!(w.std_dev().is_none());
}
#[test]
fn single_sample() {
let mut w = WelfordF64::new();
w.update(42.0);
assert_eq!(w.count(), 1);
assert_eq!(w.mean(), Some(42.0));
// Variance needs at least 2 samples
assert!(w.variance().is_none());
// Population variance with 1 sample is 0
assert_eq!(w.population_variance(), Some(0.0));
}
#[test]
fn known_values() {
let mut w = WelfordF64::new();
// Dataset: [2, 4, 4, 4, 5, 5, 7, 9]
for &x in &[2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0] {
w.update(x);
}
assert_eq!(w.count(), 8);
// Mean = 40/8 = 5.0
let mean = w.mean().unwrap();
assert!((mean - 5.0).abs() < 1e-10, "mean should be 5.0, got {mean}");
// Population variance = 4.0
let pop_var = w.population_variance().unwrap();
assert!(
(pop_var - 4.0).abs() < 1e-10,
"pop variance should be 4.0, got {pop_var}"
);
// Sample variance = 32/7 ≈ 4.571428
let var = w.variance().unwrap();
assert!(
(var - 32.0 / 7.0).abs() < 1e-10,
"sample variance should be 32/7, got {var}"
);
// Std dev = sqrt(32/7) ≈ 2.138
let sd = w.std_dev().unwrap();
assert!(
(sd - (32.0_f64 / 7.0).sqrt()).abs() < 1e-6,
"std dev got {sd}"
);
}
#[test]
fn two_samples() {
let mut w = WelfordF64::new();
w.update(10.0);
w.update(20.0);
assert_eq!(w.count(), 2);
assert!((w.mean().unwrap() - 15.0).abs() < 1e-10);
// Sample variance of [10, 20] = 50.0
assert!((w.variance().unwrap() - 50.0).abs() < 1e-10);
// Pop variance = 25.0
assert!((w.population_variance().unwrap() - 25.0).abs() < 1e-10);
}
// =========================================================================
// Numerical stability
// =========================================================================
#[test]
fn numerical_stability_large_offset() {
// Classic failure case for naive variance: large mean, small deltas
let mut w = WelfordF64::new();
let base = 1e8;
for i in 0..1000 {
w.update((i as f64).fma(0.001, base));
}
let var = w.variance().unwrap();
// Variance of 0, 0.001, 0.002, ..., 0.999 ≈ 0.08341...
// (doesn't matter exactly — just check it's positive and reasonable)
assert!(var > 0.0, "variance should be positive, got {var}");
assert!(var < 1.0, "variance should be small, got {var}");
}
// =========================================================================
// Merge (Chan's algorithm)
// =========================================================================
#[test]
fn merge_empty_into_empty() {
let mut a = WelfordF64::new();
let b = WelfordF64::new();
a.merge(&b);
assert_eq!(a.count(), 0);
assert!(a.mean().is_none());
}
#[test]
fn merge_into_empty() {
let mut a = WelfordF64::new();
let mut b = WelfordF64::new();
b.update(10.0);
b.update(20.0);
a.merge(&b);
assert_eq!(a.count(), 2);
assert!((a.mean().unwrap() - 15.0).abs() < 1e-10);
}
#[test]
fn merge_empty_into_existing() {
let mut a = WelfordF64::new();
a.update(10.0);
a.update(20.0);
let b = WelfordF64::new();
a.merge(&b);
assert_eq!(a.count(), 2);
assert!((a.mean().unwrap() - 15.0).abs() < 1e-10);
}
#[test]
fn merge_matches_single_accumulator() {
let data = [2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0];
// Single accumulator
let mut single = WelfordF64::new();
for &x in &data {
single.update(x);
}
// Split into two halves and merge
let mut first = WelfordF64::new();
let mut second = WelfordF64::new();
for &x in &data[..4] {
first.update(x);
}
for &x in &data[4..] {
second.update(x);
}
first.merge(&second);
assert_eq!(first.count(), single.count());
assert!((first.mean().unwrap() - single.mean().unwrap()).abs() < 1e-10);
assert!((first.variance().unwrap() - single.variance().unwrap()).abs() < 1e-10);
}
#[test]
fn merge_uneven_split() {
let mut single = WelfordF64::new();
let mut a = WelfordF64::new();
let mut b = WelfordF64::new();
for i in 0..100 {
let x = i as f64;
single.update(x);
if i < 7 {
a.update(x);
} else {
b.update(x);
}
}
a.merge(&b);
assert_eq!(a.count(), 100);
assert!((a.mean().unwrap() - single.mean().unwrap()).abs() < 1e-10);
assert!((a.variance().unwrap() - single.variance().unwrap()).abs() < 1e-6);
}
// =========================================================================
// Reset
// =========================================================================
#[test]
fn reset_clears_state() {
let mut w = WelfordF64::new();
for i in 0..100 {
w.update(i as f64);
}
w.reset();
assert_eq!(w.count(), 0);
assert!(w.mean().is_none());
assert!(w.variance().is_none());
}
// =========================================================================
// Default
// =========================================================================
#[test]
fn default_is_empty() {
let w = WelfordF64::default();
assert_eq!(w.count(), 0);
}
// =========================================================================
// f32 variant
// =========================================================================
// =========================================================================
// from_parts
// =========================================================================
#[test]
fn from_parts_round_trip() {
let mut w = WelfordF64::new();
for &x in &[2.0, 4.0, 4.0, 4.0, 5.0, 5.0, 7.0, 9.0] {
w.update(x);
}
let count = w.count();
let mean = w.mean().unwrap();
// m2 = variance * (count - 1)
let m2 = w.variance().unwrap() * (count - 1) as f64;
let w2 = WelfordF64::from_parts(count, mean, m2);
assert_eq!(w2.count(), count);
assert!((w2.mean().unwrap() - mean).abs() < 1e-10);
assert!((w2.variance().unwrap() - w.variance().unwrap()).abs() < 1e-10);
}
#[test]
fn f32_basic() {
let mut w = WelfordF32::new();
w.update(10.0);
w.update(20.0);
assert_eq!(w.count(), 2);
assert!((w.mean().unwrap() - 15.0).abs() < 1e-5);
}
#[test]
fn f32_default() {
let w = WelfordF32::default();
assert_eq!(w.count(), 0);
}
}