moment.rs

// Some formulae:
//     mean_x = sum(weight[i] * x[i]) / sum(weight)
//     dp_xy = weighted sum of deviation products of variables x, y, written in
//             the paper as simply XY.
//     dp_xy = sum(weight[i] * (x[i] - mean_x) * (y[i] - mean_y))
//
//     cov(x, y) = dp_xy / sum(weight)
//     var(x) = cov(x, x)
//
// Algorithms from:
// Numerically stable parallel computation of (co-)variance.
// Schubert, E. & Gertz, M. (2018).
//
// Key equations from the paper:
// (17) for mean update, (23) for dp update (and also Table 1).
//
//
// For higher moments we refer to:
// Numerically Stable, Scalable Formulas for Parallel and Online Computation of
// Higher-Order Multivariate Central Moments with Arbitrary Weights.
// Pébay, P. & Terriberry, T. B. & Kolla, H. & Bennett J. (2016)
//
// Key equations from paper:
// (3.26) mean update, (3.27) moment update.
//
// Here we use mk to mean the weighted kth central moment:
//    mk = sum(weight[i] * (x[i] - mean_x)**k)
// Note that we'll use the terms m2 = dp = dp_xx if unambiguous.

#![allow(clippy::collapsible_else_if)]

use arrow::array::{Array, PrimitiveArray};
use arrow::types::NativeType;
use num_traits::AsPrimitive;
use polars_utils::algebraic_ops::*;

const CHUNK_SIZE: usize = 128;

#[derive(Default, Clone)]
pub struct VarState {
    weight: f64,
    mean: f64,
    dp: f64,
}

#[derive(Default, Clone)]
pub struct CovState {
    weight: f64,
    mean_x: f64,
    mean_y: f64,
    dp_xy: f64,
}

#[derive(Default, Clone)]
pub struct PearsonState {
    weight: f64,
    mean_x: f64,
    mean_y: f64,
    dp_xx: f64,
    dp_xy: f64,
    dp_yy: f64,
}

impl VarState {
    fn new(x: &[f64]) -> Self {
        if x.is_empty() {
            return Self::default();
        }

        let weight = x.len() as f64;
        let mean = alg_sum_f64(x.iter().copied()) / weight;
        Self {
            weight,
            mean,
            dp: alg_sum_f64(x.iter().map(|&xi| (xi - mean) * (xi - mean))),
        }
    }

    fn clear_zero_weight_nan(&mut self) {
        // Clear NaNs due to division by zero.
        if self.weight == 0.0 {
            self.mean = 0.0;
            self.dp = 0.0;
        }
    }

    pub fn insert_one(&mut self, x: f64) {
        // Just a specialized version of
        // self.combine(&Self { weight: 1.0, mean: x, dp: 0.0 })
        let new_weight = self.weight + 1.0;
        let delta_mean = x - self.mean;
        let new_mean = self.mean + delta_mean / new_weight;
        self.dp += (x - new_mean) * delta_mean;
        self.weight = new_weight;
        self.mean = new_mean;
        self.clear_zero_weight_nan();
    }

    pub fn remove_one(&mut self, x: f64) {
        // Just a specialized version of
        // self.combine(&Self { weight: -1.0, mean: x, dp: 0.0 })
        let new_weight = self.weight - 1.0;
        let delta_mean = x - self.mean;
        let new_mean = self.mean - delta_mean / new_weight;
        self.dp -= (x - new_mean) * delta_mean;
        self.weight = new_weight;
        self.mean = new_mean;
        self.clear_zero_weight_nan();
    }

    pub fn combine(&mut self, other: &Self) {
        if other.weight == 0.0 {
            return;
        }

        let new_weight = self.weight + other.weight;
        let other_weight_frac = other.weight / new_weight;
        let delta_mean = other.mean - self.mean;
        let new_mean = self.mean + delta_mean * other_weight_frac;
        self.dp += other.dp + other.weight * (other.mean - new_mean) * delta_mean;
        self.weight = new_weight;
        self.mean = new_mean;
        self.clear_zero_weight_nan();
    }

    pub fn finalize(&self, ddof: u8) -> Option<f64> {
        if self.weight <= ddof as f64 {
            None
        } else {
            let var = self.dp / (self.weight - ddof as f64);
            Some(if var < 0.0 {
                // Variance can't be negative, except through numerical instability.
                // We don't use f64::max here so we propagate nans.
                0.0
            } else {
                var
            })
        }
    }
}

impl CovState {
    fn new(x: &[f64], y: &[f64]) -> Self {
        assert!(x.len() == y.len());
        if x.is_empty() {
            return Self::default();
        }

        let weight = x.len() as f64;
        let inv_weight = 1.0 / weight;
        let mean_x = alg_sum_f64(x.iter().copied()) * inv_weight;
        let mean_y = alg_sum_f64(y.iter().copied()) * inv_weight;
        Self {
            weight,
            mean_x,
            mean_y,
            dp_xy: alg_sum_f64(
                x.iter()
                    .zip(y)
                    .map(|(&xi, &yi)| (xi - mean_x) * (yi - mean_y)),
            ),
        }
    }

    pub fn combine(&mut self, other: &Self) {
        if other.weight == 0.0 {
            return;
        } else if self.weight == 0.0 {
            *self = other.clone();
            return;
        }

        let new_weight = self.weight + other.weight;
        let other_weight_frac = other.weight / new_weight;
        let delta_mean_x = other.mean_x - self.mean_x;
        let delta_mean_y = other.mean_y - self.mean_y;
        let new_mean_x = self.mean_x + delta_mean_x * other_weight_frac;
        let new_mean_y = self.mean_y + delta_mean_y * other_weight_frac;
        self.dp_xy += other.dp_xy + other.weight * (other.mean_x - new_mean_x) * delta_mean_y;
        self.weight = new_weight;
        self.mean_x = new_mean_x;
        self.mean_y = new_mean_y;
    }

    pub fn finalize(&self, ddof: u8) -> Option<f64> {
        if self.weight <= ddof as f64 {
            None
        } else {
            Some(self.dp_xy / (self.weight - ddof as f64))
        }
    }
}

impl PearsonState {
    fn new(x: &[f64], y: &[f64]) -> Self {
        assert!(x.len() == y.len());
        if x.is_empty() {
            return Self::default();
        }

        let weight = x.len() as f64;
        let inv_weight = 1.0 / weight;
        let mean_x = alg_sum_f64(x.iter().copied()) * inv_weight;
        let mean_y = alg_sum_f64(y.iter().copied()) * inv_weight;
        let mut dp_xx = 0.0;
        let mut dp_xy = 0.0;
        let mut dp_yy = 0.0;
        for (xi, yi) in x.iter().zip(y.iter()) {
            dp_xx = alg_add_f64(dp_xx, (xi - mean_x) * (xi - mean_x));
            dp_xy = alg_add_f64(dp_xy, (xi - mean_x) * (yi - mean_y));
            dp_yy = alg_add_f64(dp_yy, (yi - mean_y) * (yi - mean_y));
        }
        Self {
            weight,
            mean_x,
            mean_y,
            dp_xx,
            dp_xy,
            dp_yy,
        }
    }

    pub fn combine(&mut self, other: &Self) {
        if other.weight == 0.0 {
            return;
        } else if self.weight == 0.0 {
            *self = other.clone();
            return;
        }

        let new_weight = self.weight + other.weight;
        let other_weight_frac = other.weight / new_weight;
        let delta_mean_x = other.mean_x - self.mean_x;
        let delta_mean_y = other.mean_y - self.mean_y;
        let new_mean_x = self.mean_x + delta_mean_x * other_weight_frac;
        let new_mean_y = self.mean_y + delta_mean_y * other_weight_frac;
        self.dp_xx += other.dp_xx + other.weight * (other.mean_x - new_mean_x) * delta_mean_x;
        self.dp_xy += other.dp_xy + other.weight * (other.mean_x - new_mean_x) * delta_mean_y;
        self.dp_yy += other.dp_yy + other.weight * (other.mean_y - new_mean_y) * delta_mean_y;
        self.weight = new_weight;
        self.mean_x = new_mean_x;
        self.mean_y = new_mean_y;
    }

    pub fn finalize(&self) -> f64 {
        let denom_sq = self.dp_xx * self.dp_yy;
        if denom_sq > 0.0 {
            self.dp_xy / denom_sq.sqrt()
        } else {
            f64::NAN
        }
    }
}

#[derive(Default, Clone)]
pub struct SkewState {
    weight: f64,
    mean: f64,
    m2: f64,
    m3: f64,
}

impl SkewState {
    fn new(x: &[f64]) -> Self {
        if x.is_empty() {
            return Self::default();
        }

        let weight = x.len() as f64;
        let mean = alg_sum_f64(x.iter().copied()) / weight;
        let mut m2 = 0.0;
        let mut m3 = 0.0;
        for xi in x.iter() {
            let d = xi - mean;
            let d2 = d * d;
            let d3 = d * d2;
            m2 = alg_add_f64(m2, d2);
            m3 = alg_add_f64(m3, d3);
        }
        Self {
            weight,
            mean,
            m2,
            m3,
        }
    }

    fn clear_zero_weight_nan(&mut self) {
        // Clear NaNs due to division by zero.
        if self.weight == 0.0 {
            self.mean = 0.0;
            self.m2 = 0.0;
            self.m3 = 0.0;
        }
    }

    pub fn insert_one(&mut self, x: f64) {
        // Specialization of self.combine(&SkewState { weight: 1.0, mean: x, m2: 0.0, m3: 0.0 });
        let new_weight = self.weight + 1.0;
        let delta_mean = x - self.mean;
        let delta_mean_weight = delta_mean / new_weight;
        let new_mean = self.mean + delta_mean_weight;

        let weight_diff = self.weight - 1.0;
        let m2_update = (x - new_mean) * delta_mean;
        let new_m2 = self.m2 + m2_update;
        let new_m3 = self.m3 + delta_mean_weight * (m2_update * weight_diff - 3.0 * self.m2);

        self.weight = new_weight;
        self.mean = new_mean;
        self.m2 = new_m2;
        self.m3 = new_m3;
        self.clear_zero_weight_nan();
    }

    pub fn remove_one(&mut self, x: f64) {
        // Specialization of self.combine(&SkewState { weight: -1.0, mean: x, m2: 0.0, m3: 0.0 });
        let new_weight = self.weight - 1.0;
        let delta_mean = x - self.mean;
        let delta_mean_weight = delta_mean / new_weight;
        let new_mean = self.mean - delta_mean_weight;

        let weight_diff = self.weight + 1.0;
        let m2_update = (new_mean - x) * delta_mean;
        let new_m2 = self.m2 + m2_update;
        let new_m3 = self.m3 + delta_mean_weight * (m2_update * weight_diff + 3.0 * self.m2);

        self.weight = new_weight;
        self.mean = new_mean;
        self.m2 = new_m2;
        self.m3 = new_m3;
        self.clear_zero_weight_nan();
    }

    pub fn combine(&mut self, other: &Self) {
        if other.weight == 0.0 {
            return;
        } else if self.weight == 0.0 {
            *self = other.clone();
            return;
        }

        let new_weight = self.weight + other.weight;
        let delta_mean = other.mean - self.mean;
        let delta_mean_weight = delta_mean / new_weight;
        let new_mean = self.mean + other.weight * delta_mean_weight;

        let weight_diff = self.weight - other.weight;
        let self_weight_other_m2 = self.weight * other.m2;
        let other_weight_self_m2 = other.weight * self.m2;
        let m2_update = other.weight * (other.mean - new_mean) * delta_mean;
        let new_m2 = self.m2 + other.m2 + m2_update;
        let new_m3 = self.m3
            + other.m3
            + delta_mean_weight
                * (m2_update * weight_diff + 3.0 * (self_weight_other_m2 - other_weight_self_m2));

        self.weight = new_weight;
        self.mean = new_mean;
        self.m2 = new_m2;
        self.m3 = new_m3;
        self.clear_zero_weight_nan();
    }

    pub fn finalize(&self, bias: bool) -> Option<f64> {
        let m2 = self.m2 / self.weight;
        let m3 = self.m3 / self.weight;
        let is_zero = m2 <= (f64::EPSILON * self.mean).powi(2);
        let biased_est = if is_zero { f64::NAN } else { m3 / m2.powf(1.5) };
        if bias {
            if self.weight == 0.0 {
                None
            } else {
                Some(biased_est)
            }
        } else {
            if self.weight <= 2.0 {
                None
            } else {
                let correction = (self.weight * (self.weight - 1.0)).sqrt() / (self.weight - 2.0);
                Some(correction * biased_est)
            }
        }
    }
}

#[derive(Default, Clone)]
pub struct KurtosisState {
    weight: f64,
    mean: f64,
    m2: f64,
    m3: f64,
    m4: f64,
}

impl KurtosisState {
    fn new(x: &[f64]) -> Self {
        if x.is_empty() {
            return Self::default();
        }

        let weight = x.len() as f64;
        let mean = alg_sum_f64(x.iter().copied()) / weight;
        let mut m2 = 0.0;
        let mut m3 = 0.0;
        let mut m4 = 0.0;
        for xi in x.iter() {
            let d = xi - mean;
            let d2 = d * d;
            let d3 = d * d2;
            let d4 = d2 * d2;
            m2 = alg_add_f64(m2, d2);
            m3 = alg_add_f64(m3, d3);
            m4 = alg_add_f64(m4, d4);
        }
        Self {
            weight,
            mean,
            m2,
            m3,
            m4,
        }
    }

    fn clear_zero_weight_nan(&mut self) {
        // Clear NaNs due to division by zero.
        if self.weight == 0.0 {
            self.mean = 0.0;
            self.m2 = 0.0;
            self.m3 = 0.0;
            self.m4 = 0.0;
        }
    }

    pub fn insert_one(&mut self, x: f64) {
        // Specialization of self.combine(&KurtosisState { weight: 1.0, mean: x, m2: 0.0, m3: 0.0, m4: 0.0 });
        let new_weight = self.weight + 1.0;
        let delta_mean = x - self.mean;
        let delta_mean_weight = delta_mean / new_weight;
        let new_mean = self.mean + delta_mean_weight;

        let weight_diff = self.weight - 1.0;
        let m2_update = (x - new_mean) * delta_mean;
        let new_m2 = self.m2 + m2_update;
        let new_m3 = self.m3 + delta_mean_weight * (m2_update * weight_diff - 3.0 * self.m2);
        let new_m4 = self.m4
            + delta_mean_weight
                * (delta_mean_weight
                    * (m2_update * (self.weight * weight_diff + 1.0) + 6.0 * self.m2)
                    - 4.0 * self.m3);

        self.weight = new_weight;
        self.mean = new_mean;
        self.m2 = new_m2;
        self.m3 = new_m3;
        self.m4 = new_m4;
        self.clear_zero_weight_nan();
    }

    pub fn remove_one(&mut self, x: f64) {
        // Specialization of self.combine(&KurtosisState { weight: -1.0, mean: x, m2: 0.0, m3: 0.0, m4: 0.0 });
        let new_weight = self.weight - 1.0;
        let delta_mean = x - self.mean;
        let delta_mean_weight = delta_mean / new_weight;
        let new_mean = self.mean - delta_mean_weight;

        let weight_diff = self.weight + 1.0;
        let m2_update = (new_mean - x) * delta_mean;
        let new_m2 = self.m2 + m2_update;
        let new_m3 = self.m3 + delta_mean_weight * (m2_update * weight_diff + 3.0 * self.m2);
        let new_m4 = self.m4
            + delta_mean_weight
                * (delta_mean_weight
                    * (m2_update * (self.weight * weight_diff + 1.0) + 6.0 * self.m2)
                    + 4.0 * self.m3);

        self.weight = new_weight;
        self.mean = new_mean;
        self.m2 = new_m2;
        self.m3 = new_m3;
        self.m4 = new_m4;
        self.clear_zero_weight_nan();
    }

    pub fn combine(&mut self, other: &Self) {
        if other.weight == 0.0 {
            return;
        } else if self.weight == 0.0 {
            *self = other.clone();
            return;
        }

        let new_weight = self.weight + other.weight;
        let delta_mean = other.mean - self.mean;
        let delta_mean_weight = delta_mean / new_weight;
        let new_mean = self.mean + other.weight * delta_mean_weight;

        let weight_diff = self.weight - other.weight;
        let self_weight_other_m2 = self.weight * other.m2;
        let other_weight_self_m2 = other.weight * self.m2;
        let m2_update = other.weight * (other.mean - new_mean) * delta_mean;
        let new_m2 = self.m2 + other.m2 + m2_update;
        let new_m3 = self.m3
            + other.m3
            + delta_mean_weight
                * (m2_update * weight_diff + 3.0 * (self_weight_other_m2 - other_weight_self_m2));
        let new_m4 = self.m4
            + other.m4
            + delta_mean_weight
                * (delta_mean_weight
                    * (m2_update * (self.weight * weight_diff + other.weight * other.weight)
                        + 6.0
                            * (self.weight * self_weight_other_m2
                                + other.weight * other_weight_self_m2))
                    + 4.0 * (self.weight * other.m3 - other.weight * self.m3));

        self.weight = new_weight;
        self.mean = new_mean;
        self.m2 = new_m2;
        self.m3 = new_m3;
        self.m4 = new_m4;
        self.clear_zero_weight_nan();
    }

    pub fn finalize(&self, fisher: bool, bias: bool) -> Option<f64> {
        let m4 = self.m4 / self.weight;
        let m2 = self.m2 / self.weight;
        let is_zero = m2 <= (f64::EPSILON * self.mean).powi(2);
        let biased_est = if is_zero { f64::NAN } else { m4 / (m2 * m2) };
        let out = if bias {
            if self.weight == 0.0 {
                return None;
            }

            biased_est
        } else {
            if self.weight <= 3.0 {
                return None;
            }

            let n = self.weight;
            let nm1_nm2 = (n - 1.0) / (n - 2.0);
            let np1_nm3 = (n + 1.0) / (n - 3.0);
            let nm1_nm3 = (n - 1.0) / (n - 3.0);
            nm1_nm2 * (np1_nm3 * biased_est - 3.0 * nm1_nm3) + 3.0
        };

        if fisher { Some(out - 3.0) } else { Some(out) }
    }
}

fn chunk_as_float<T, I, F>(it: I, mut f: F)
where
    T: NativeType + AsPrimitive<f64>,
    I: IntoIterator<Item = T>,
    F: FnMut(&[f64]),
{
    let mut chunk = [0.0; CHUNK_SIZE];
    let mut i = 0;
    for val in it {
        if i >= CHUNK_SIZE {
            f(&chunk);
            i = 0;
        }
        chunk[i] = val.as_();
        i += 1;
    }
    if i > 0 {
        f(&chunk[..i]);
    }
}

fn chunk_as_float_binary<T, U, I, F>(it: I, mut f: F)
where
    T: NativeType + AsPrimitive<f64>,
    U: NativeType + AsPrimitive<f64>,
    I: IntoIterator<Item = (T, U)>,
    F: FnMut(&[f64], &[f64]),
{
    let mut left_chunk = [0.0; CHUNK_SIZE];
    let mut right_chunk = [0.0; CHUNK_SIZE];
    let mut i = 0;
    for (l, r) in it {
        if i >= CHUNK_SIZE {
            f(&left_chunk, &right_chunk);
            i = 0;
        }
        left_chunk[i] = l.as_();
        right_chunk[i] = r.as_();
        i += 1;
    }
    if i > 0 {
        f(&left_chunk[..i], &right_chunk[..i]);
    }
}

pub fn var<T>(arr: &PrimitiveArray<T>) -> VarState
where
    T: NativeType + AsPrimitive<f64>,
{
    let mut out = VarState::default();
    if arr.has_nulls() {
        chunk_as_float(arr.non_null_values_iter(), |chunk| {
            out.combine(&VarState::new(chunk))
        });
    } else {
        chunk_as_float(arr.values().iter().copied(), |chunk| {
            out.combine(&VarState::new(chunk))
        });
    }
    out
}

pub fn cov<T, U>(x: &PrimitiveArray<T>, y: &PrimitiveArray<U>) -> CovState
where
    T: NativeType + AsPrimitive<f64>,
    U: NativeType + AsPrimitive<f64>,
{
    assert!(x.len() == y.len());
    let mut out = CovState::default();
    if x.has_nulls() || y.has_nulls() {
        chunk_as_float_binary(
            x.iter()
                .zip(y.iter())
                .filter_map(|(l, r)| l.copied().zip(r.copied())),
            |l, r| out.combine(&CovState::new(l, r)),
        );
    } else {
        chunk_as_float_binary(
            x.values().iter().copied().zip(y.values().iter().copied()),
            |l, r| out.combine(&CovState::new(l, r)),
        );
    }
    out
}

pub fn pearson_corr<T, U>(x: &PrimitiveArray<T>, y: &PrimitiveArray<U>) -> PearsonState
where
    T: NativeType + AsPrimitive<f64>,
    U: NativeType + AsPrimitive<f64>,
{
    assert!(x.len() == y.len());
    let mut out = PearsonState::default();
    if x.has_nulls() || y.has_nulls() {
        chunk_as_float_binary(
            x.iter()
                .zip(y.iter())
                .filter_map(|(l, r)| l.copied().zip(r.copied())),
            |l, r| out.combine(&PearsonState::new(l, r)),
        );
    } else {
        chunk_as_float_binary(
            x.values().iter().copied().zip(y.values().iter().copied()),
            |l, r| out.combine(&PearsonState::new(l, r)),
        );
    }
    out
}

pub fn skew<T>(arr: &PrimitiveArray<T>) -> SkewState
where
    T: NativeType + AsPrimitive<f64>,
{
    let mut out = SkewState::default();
    if arr.has_nulls() {
        chunk_as_float(arr.non_null_values_iter(), |chunk| {
            out.combine(&SkewState::new(chunk))
        });
    } else {
        chunk_as_float(arr.values().iter().copied(), |chunk| {
            out.combine(&SkewState::new(chunk))
        });
    }
    out
}

pub fn kurtosis<T>(arr: &PrimitiveArray<T>) -> KurtosisState
where
    T: NativeType + AsPrimitive<f64>,
{
    let mut out = KurtosisState::default();
    if arr.has_nulls() {
        chunk_as_float(arr.non_null_values_iter(), |chunk| {
            out.combine(&KurtosisState::new(chunk))
        });
    } else {
        chunk_as_float(arr.values().iter().copied(), |chunk| {
            out.combine(&KurtosisState::new(chunk))
        });
    }
    out
}