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Remove inferior estimate of error of weighted average

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It was a biased estimator, while the alternative one isn't.
This commit is contained in:
Vinzent Steinberg 2017-05-22 13:24:57 +02:00
parent efaca98d0c
commit 77fa8b4ed2
5 changed files with 126 additions and 378 deletions
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6
src/lib.rs
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@ -2,8 +2,8 @@
//! sequence of numbers, and for their standard errors. The typical workflow
//! looks like this:
//!
//! 1. Initialize your estimator of choice (`Average`, `WeightedAverage` or
//! `WeightedAverage2`) with `new()`.
//! 1. Initialize your estimator of choice (`Average` or `WeightedAverage`) with
//! `new()`.
//! 2. Add some subset (called "samples") of the sequence of numbers (called
//! "population") for which you want to estimate the average, using `add()`
//! or `collect()`.
@ -32,8 +32,6 @@ extern crate conv;
#[macro_use] mod macros;
mod average;
mod weighted_average;
mod weighted_average2;
pub use average::Average;
pub use weighted_average::WeightedAverage;
pub use weighted_average2::WeightedAverage as WeightedAverage2;

162
src/weighted_average.rs
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@ -1,7 +1,9 @@
use core;
/// Estimate the weighted arithmetic mean and the weighted variance of a
/// sequence of numbers ("population").
use conv::ApproxFrom;
/// Estimate the weighted and unweighted arithmetic mean and the unweighted
/// variance of a sequence of numbers ("population").
///
/// This can be used to estimate the standard error of the weighted mean.
///
@ -13,41 +15,58 @@ use core;
///
/// let a: WeightedAverage = (1..6).zip(1..6)
/// .map(|(x, w)| (f64::from(x), f64::from(w))).collect();
/// println!("The weighted average is {} ± {}.", a.mean(), a.error());
/// println!("The weighted average is {} ± {}.", a.weighted_mean(), a.error());
/// ```
#[derive(Debug, Clone)]
pub struct WeightedAverage {
/// Sum of the weights.
weight_sum: f64,
/// Average value.
avg: f64,
/// Intermediate sum of squares for calculating the variance.
/// Sum of the squares of the weights.
weight_sum_sq: f64,
/// Weighted average value.
weighted_avg: f64,
/// Number of samples.
n: u64,
/// Unweighted average value.
unweighted_avg: f64,
/// Intermediate sum of squares for calculating the *unweighted* variance.
v: f64,
}
impl WeightedAverage {
/// Create a new weighted average estimator.
/// Create a new weighted and unweighted average estimator.
pub fn new() -> WeightedAverage {
WeightedAverage { weight_sum: 0., avg: 0., v: 0. }
WeightedAverage {
weight_sum: 0., weight_sum_sq: 0., weighted_avg: 0.,
n: 0, unweighted_avg: 0., v: 0.,
}
}
/// Add a weighted element sampled from the population.
pub fn add(&mut self, sample: f64, weight: f64) {
// This algorithm was suggested by West in 1979.
// The algorithm for the unweighted average was suggested by Welford in 1962.
// The algorithm for the weighted average was suggested by West in 1979.
//
// See
// https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
// and
// http://people.ds.cam.ac.uk/fanf2/hermes/doc/antiforgery/stats.pdf.
self.weight_sum += weight;
let prev_avg = self.avg;
self.avg = prev_avg + (weight / self.weight_sum) * (sample - prev_avg);
self.v += weight * (sample - prev_avg) * (sample - self.avg);
self.weight_sum_sq += weight*weight;
let prev_avg = self.weighted_avg;
self.weighted_avg = prev_avg + (weight / self.weight_sum) * (sample - prev_avg);
self.n += 1;
let delta = sample - self.unweighted_avg;
self.unweighted_avg += delta / f64::approx_from(self.n).unwrap();
self.v += delta * (sample - self.unweighted_avg);
}
/// Determine whether the samples are empty.
/// Determine whether the sample is empty.
pub fn is_empty(&self) -> bool {
self.weight_sum == 0. && self.v == 0. && self.avg == 0.
self.n == 0
}
/// Return the sum of the weights.
@ -55,57 +74,70 @@ impl WeightedAverage {
self.weight_sum
}
/// Estimate the weighted mean of the population.
pub fn mean(&self) -> f64 {
self.avg
/// Return the sum of the squared weights.
pub fn sum_weights_sq(&self) -> f64 {
self.weight_sum_sq
}
/// Calculate the weighted population variance of the sample.
///
/// This is a biased estimator of the weighted variance of the population.
pub fn population_variance(&self) -> f64 {
/// Estimate the weighted mean of the sequence.
pub fn weighted_mean(&self) -> f64 {
self.weighted_avg
}
/// Estimate the unweighted mean of the sequence.
pub fn unweighted_mean(&self) -> f64 {
self.unweighted_avg
}
/// Return sample size.
pub fn len(&self) -> u64 {
self.n
}
/// Calculate the effective sample size.
pub fn effective_len(&self) -> f64 {
if self.is_empty() {
0.
} else {
self.v / self.weight_sum
return 0.
}
self.weight_sum * self.weight_sum / self.weight_sum_sq
}
/// Calculate the weighted sample variance.
/// Calculate the *unweighted* population variance of the sample.
///
/// This is an unbiased estimator of the weighted variance of the population.
/// This is a biased estimator of the variance of the population.
pub fn population_variance(&self) -> f64 {
if self.n < 2 {
return 0.;
}
self.v / f64::approx_from(self.n).unwrap()
}
/// Calculate the *unweighted* sample variance.
///
/// Note that this will return 0 if the sum of the weights is <= 1.
/// This is an unbiased estimator of the variance of the population.
pub fn sample_variance(&self) -> f64 {
if self.weight_sum <= 1. {
0.
} else {
self.v / (self.weight_sum - 1.0)
if self.n < 2 {
return 0.;
}
self.v / f64::approx_from(self.n - 1).unwrap()
}
/// Estimate the standard error of the weighted mean of the population.
/// Estimate the standard error of the *weighted* mean of the sequence.
///
/// Note that this will return 0 if the sum of the weights is 0.
/// For this estimator, the sum of weights should be larger than 1.
/// Returns 0 if the sum of weights is 0.
///
/// This biased estimator uses the weighted variance and the sum of weights.
/// It considers the weights as (noninteger) counts of how often the sample
/// has been observed, applying the standard formulas to calculate mean,
/// variance and sample size across all "repeats".
/// This unbiased estimator assumes that the samples were independently
/// drawn from the same population with constant variance.
pub fn error(&self) -> f64 {
// This uses the same estimate as SPSS.
// This uses the same estimate as WinCross, which should provide better
// results than the ones used by SPSS or Mentor.
//
// See http://www.analyticalgroup.com/download/WEIGHTED_MEAN.pdf.
// See http://www.analyticalgroup.com/download/WEIGHTED_VARIANCE.pdf.
if self.weight_sum == 0. {
return 0.;
}
let variance = if self.weight_sum <= 1. {
self.population_variance()
} else {
self.sample_variance()
};
(variance / self.weight_sum).sqrt()
let inv_effective_len = self.weight_sum_sq / (self.weight_sum * self.weight_sum);
(self.sample_variance() * inv_effective_len).sqrt()
}
/// Merge another sample into this one.
@ -124,20 +156,32 @@ impl WeightedAverage {
/// let mut avg_left: WeightedAverage = left.iter().map(|&x| x).collect();
/// let avg_right: WeightedAverage = right.iter().map(|&x| x).collect();
/// avg_left.merge(&avg_right);
/// assert!((avg_total.mean() - avg_left.mean()).abs() < 1e-15);
/// assert!((avg_total.weighted_mean() - avg_left.weighted_mean()).abs() < 1e-15);
/// assert!((avg_total.error() - avg_left.error()).abs() < 1e-15);
/// ```
pub fn merge(&mut self, other: &WeightedAverage) {
// This is similar to the algorithm proposed by Chan et al. in 1979.
//
// See https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance.
let delta = other.avg - self.avg;
let total_weight_sum = self.weight_sum + other.weight_sum;
self.avg = (self.weight_sum * self.avg + other.weight_sum * other.avg)
/ (self.weight_sum + other.weight_sum);
self.v += other.v + delta*delta * self.weight_sum * other.weight_sum
/ total_weight_sum;
self.weight_sum = total_weight_sum;
{
let total_weight_sum = self.weight_sum + other.weight_sum;
self.weighted_avg = (self.weight_sum * self.weighted_avg
+ other.weight_sum * other.weighted_avg)
/ (self.weight_sum + other.weight_sum);
self.weight_sum = total_weight_sum;
self.weight_sum_sq += other.weight_sum_sq;
}
{
let delta = other.unweighted_avg - self.unweighted_avg;
let len_self = f64::approx_from(self.n).unwrap();
let len_other = f64::approx_from(other.n).unwrap();
let len_total = len_self + len_other;
self.n += other.n;
self.unweighted_avg = (len_self * self.unweighted_avg
+ len_other * other.unweighted_avg)
/ len_total;
self.v += other.v + delta*delta * len_self * len_other / len_total;
}
}
}
@ -172,8 +216,11 @@ mod tests {
let mut avg_left: WeightedAverage = left.iter().map(|x| (*x, 1.)).collect();
let avg_right: WeightedAverage = right.iter().map(|x| (*x, 1.)).collect();
avg_left.merge(&avg_right);
assert_eq!(avg_total.n, avg_left.n);
assert_eq!(avg_total.weight_sum, avg_left.weight_sum);
assert_eq!(avg_total.avg, avg_left.avg);
assert_eq!(avg_total.weight_sum_sq, avg_left.weight_sum_sq);
assert_eq!(avg_total.weighted_avg, avg_left.weighted_avg);
assert_eq!(avg_total.unweighted_avg, avg_left.unweighted_avg);
assert_eq!(avg_total.v, avg_left.v);
}
}
@ -189,8 +236,11 @@ mod tests {
let mut avg_left: WeightedAverage = left.iter().map(|&(x, w)| (x, w)).collect();
let avg_right: WeightedAverage = right.iter().map(|&(x, w)| (x, w)).collect();
avg_left.merge(&avg_right);
assert_eq!(avg_total.n, avg_left.n);
assert_almost_eq!(avg_total.weight_sum, avg_left.weight_sum, 1e-15);
assert_almost_eq!(avg_total.avg, avg_left.avg, 1e-15);
assert_eq!(avg_total.weight_sum_sq, avg_left.weight_sum_sq);
assert_almost_eq!(avg_total.weighted_avg, avg_left.weighted_avg, 1e-15);
assert_almost_eq!(avg_total.unweighted_avg, avg_left.unweighted_avg, 1e-15);
assert_almost_eq!(avg_total.v, avg_left.v, 1e-14);
}
}

246
src/weighted_average2.rs
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@ -1,246 +0,0 @@
use core;
use conv::ApproxFrom;
/// Estimate the weighted and unweighted arithmetic mean and the unweighted
/// variance of a sequence of numbers ("population").
///
/// This can be used to estimate the standard error of the weighted mean.
///
///
/// ## Example
///
/// ```
/// use average::WeightedAverage2 as WeightedAverage;
///
/// let a: WeightedAverage = (1..6).zip(1..6)
/// .map(|(x, w)| (f64::from(x), f64::from(w))).collect();
/// println!("The weighted average is {} ± {}.", a.weighted_mean(), a.error());
/// ```
#[derive(Debug, Clone)]
pub struct WeightedAverage {
/// Sum of the weights.
weight_sum: f64,
/// Sum of the squares of the weights.
weight_sum_sq: f64,
/// Weighted average value.
weighted_avg: f64,
/// Number of samples.
n: u64,
/// Unweighted average value.
unweighted_avg: f64,
/// Intermediate sum of squares for calculating the *unweighted* variance.
v: f64,
}
impl WeightedAverage {
/// Create a new weighted and unweighted average estimator.
pub fn new() -> WeightedAverage {
WeightedAverage {
weight_sum: 0., weight_sum_sq: 0., weighted_avg: 0.,
n: 0, unweighted_avg: 0., v: 0.,
}
}
/// Add a weighted element sampled from the population.
pub fn add(&mut self, sample: f64, weight: f64) {
// The algorithm for the unweighted average was suggested by Welford in 1962.
// The algorithm for the weighted average was suggested by West in 1979.
//
// See
// https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
// and
// http://people.ds.cam.ac.uk/fanf2/hermes/doc/antiforgery/stats.pdf.
self.weight_sum += weight;
self.weight_sum_sq += weight*weight;
let prev_avg = self.weighted_avg;
self.weighted_avg = prev_avg + (weight / self.weight_sum) * (sample - prev_avg);
self.n += 1;
let delta = sample - self.unweighted_avg;
self.unweighted_avg += delta / f64::approx_from(self.n).unwrap();
self.v += delta * (sample - self.unweighted_avg);
}
/// Determine whether the sample is empty.
pub fn is_empty(&self) -> bool {
self.n == 0
}
/// Return the sum of the weights.
pub fn sum_weights(&self) -> f64 {
self.weight_sum
}
/// Return the sum of the squared weights.
pub fn sum_weights_sq(&self) -> f64 {
self.weight_sum_sq
}
/// Estimate the weighted mean of the sequence.
pub fn weighted_mean(&self) -> f64 {
self.weighted_avg
}
/// Estimate the unweighted mean of the sequence.
pub fn unweighted_mean(&self) -> f64 {
self.unweighted_avg
}
/// Return sample size.
pub fn len(&self) -> u64 {
self.n
}
/// Calculate the effective sample size.
pub fn effective_len(&self) -> f64 {
if self.is_empty() {
return 0.
}
self.weight_sum * self.weight_sum / self.weight_sum_sq
}
/// Calculate the *unweighted* population variance of the sample.
///
/// This is a biased estimator of the variance of the population.
pub fn population_variance(&self) -> f64 {
if self.n < 2 {
return 0.;
}
self.v / f64::approx_from(self.n).unwrap()
}
/// Calculate the *unweighted* sample variance.
///
/// This is an unbiased estimator of the variance of the population.
pub fn sample_variance(&self) -> f64 {
if self.n < 2 {
return 0.;
}
self.v / f64::approx_from(self.n - 1).unwrap()
}
/// Estimate the standard error of the *weighted* mean of the sequence.
///
/// Returns 0 if the sum of weights is 0.
///
/// This unbiased estimator assumes that the samples were independently
/// drawn from the same population with constant variance.
pub fn error(&self) -> f64 {
// This uses the same estimate as WinCross.
//
// See http://www.analyticalgroup.com/download/WEIGHTED_MEAN.pdf.
if self.weight_sum_sq == 0. || self.weight_sum == 0. {
return 0.;
}
let effective_base = self.weight_sum * self.weight_sum / self.weight_sum_sq;
(self.sample_variance() / effective_base).sqrt()
}
/// Merge another sample into this one.
///
///
/// ## Example
///
/// ```
/// use average::WeightedAverage2 as WeightedAverage;
///
/// let weighted_sequence: &[(f64, f64)] = &[
/// (1., 0.1), (2., 0.2), (3., 0.3), (4., 0.4), (5., 0.5),
/// (6., 0.6), (7., 0.7), (8., 0.8), (9., 0.9)];
/// let (left, right) = weighted_sequence.split_at(3);
/// let avg_total: WeightedAverage = weighted_sequence.iter().map(|&x| x).collect();
/// let mut avg_left: WeightedAverage = left.iter().map(|&x| x).collect();
/// let avg_right: WeightedAverage = right.iter().map(|&x| x).collect();
/// avg_left.merge(&avg_right);
/// assert!((avg_total.weighted_mean() - avg_left.weighted_mean()).abs() < 1e-15);
/// assert!((avg_total.error() - avg_left.error()).abs() < 1e-15);
/// ```
pub fn merge(&mut self, other: &WeightedAverage) {
// This is similar to the algorithm proposed by Chan et al. in 1979.
//
// See https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance.
{
let total_weight_sum = self.weight_sum + other.weight_sum;
self.weighted_avg = (self.weight_sum * self.weighted_avg
+ other.weight_sum * other.weighted_avg)
/ (self.weight_sum + other.weight_sum);
self.weight_sum = total_weight_sum;
self.weight_sum_sq += other.weight_sum_sq;
}
{
let delta = other.unweighted_avg - self.unweighted_avg;
let len_self = f64::approx_from(self.n).unwrap();
let len_other = f64::approx_from(other.n).unwrap();
let len_total = len_self + len_other;
self.n += other.n;
self.unweighted_avg = (len_self * self.unweighted_avg
+ len_other * other.unweighted_avg)
/ len_total;
self.v += other.v + delta*delta * len_self * len_other / len_total;
}
}
}
impl core::default::Default for WeightedAverage {
fn default() -> WeightedAverage {
WeightedAverage::new()
}
}
impl core::iter::FromIterator<(f64, f64)> for WeightedAverage {
fn from_iter<T>(iter: T) -> WeightedAverage
where T: IntoIterator<Item=(f64, f64)>
{
let mut a = WeightedAverage::new();
for (i, w) in iter {
a.add(i, w);
}
a
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn merge_unweighted() {
let sequence: &[f64] = &[1., 2., 3., 4., 5., 6., 7., 8., 9.];
for mid in 0..sequence.len() {
let (left, right) = sequence.split_at(mid);
let avg_total: WeightedAverage = sequence.iter().map(|x| (*x, 1.)).collect();
let mut avg_left: WeightedAverage = left.iter().map(|x| (*x, 1.)).collect();
let avg_right: WeightedAverage = right.iter().map(|x| (*x, 1.)).collect();
avg_left.merge(&avg_right);
assert_eq!(avg_total.n, avg_left.n);
assert_eq!(avg_total.weight_sum, avg_left.weight_sum);
assert_eq!(avg_total.weight_sum_sq, avg_left.weight_sum_sq);
assert_eq!(avg_total.weighted_avg, avg_left.weighted_avg);
assert_eq!(avg_total.unweighted_avg, avg_left.unweighted_avg);
assert_eq!(avg_total.v, avg_left.v);
}
}
#[test]
fn merge_weighted() {
let sequence: &[(f64, f64)] = &[
(1., 0.1), (2., 0.2), (3., 0.3), (4., 0.4), (5., 0.5),
(6., 0.6), (7., 0.7), (8., 0.8), (9., 0.)];
for mid in 0..sequence.len() {
let (left, right) = sequence.split_at(mid);
let avg_total: WeightedAverage = sequence.iter().map(|&(x, w)| (x, w)).collect();
let mut avg_left: WeightedAverage = left.iter().map(|&(x, w)| (x, w)).collect();
let avg_right: WeightedAverage = right.iter().map(|&(x, w)| (x, w)).collect();
avg_left.merge(&avg_right);
assert_eq!(avg_total.n, avg_left.n);
assert_almost_eq!(avg_total.weight_sum, avg_left.weight_sum, 1e-15);
assert_eq!(avg_total.weight_sum_sq, avg_left.weight_sum_sq);
assert_almost_eq!(avg_total.weighted_avg, avg_left.weighted_avg, 1e-15);
assert_almost_eq!(avg_total.unweighted_avg, avg_left.unweighted_avg, 1e-15);
assert_almost_eq!(avg_total.v, avg_left.v, 1e-14);
}
}
}

24
tests/weighted_average.rs
View File

@ -9,15 +9,23 @@ use average::WeightedAverage;
#[test]
fn trivial() {
let mut a = WeightedAverage::new();
assert_eq!(a.len(), 0);
assert_eq!(a.sum_weights(), 0.);
assert_eq!(a.sum_weights_sq(), 0.);
a.add(1.0, 1.0);
assert_eq!(a.mean(), 1.0);
assert_eq!(a.len(), 1);
assert_eq!(a.weighted_mean(), 1.0);
assert_eq!(a.unweighted_mean(), 1.0);
assert_eq!(a.sum_weights(), 1.0);
assert_eq!(a.sum_weights_sq(), 1.0);
assert_eq!(a.population_variance(), 0.0);
assert_eq!(a.error(), 0.0);
a.add(1.0, 1.0);
assert_eq!(a.mean(), 1.0);
assert_eq!(a.len(), 2);
assert_eq!(a.weighted_mean(), 1.0);
assert_eq!(a.unweighted_mean(), 1.0);
assert_eq!(a.sum_weights(), 2.0);
assert_eq!(a.sum_weights_sq(), 2.0);
assert_eq!(a.population_variance(), 0.0);
assert_eq!(a.error(), 0.0);
}
@ -25,7 +33,9 @@ fn trivial() {
#[test]
fn simple() {
let a: WeightedAverage = (1..6).map(|x| (f64::from(x), 1.0)).collect();
assert_eq!(a.mean(), 3.0);
assert_eq!(a.len(), 5);
assert_eq!(a.weighted_mean(), 3.0);
assert_eq!(a.unweighted_mean(), 3.0);
assert_eq!(a.sum_weights(), 5.0);
assert_eq!(a.sample_variance(), 2.5);
assert_almost_eq!(a.error(), f64::sqrt(0.5), 1e-16);
@ -38,10 +48,12 @@ fn reference() {
let weights = &[1.23, 2.12, 1.23, 0.32, 1.53, 0.59, 0.94, 0.94, 0.84, 0.73];
let a: WeightedAverage = values.iter().zip(weights.iter())
.map(|(x, w)| (*x, *w)).collect();
assert_almost_eq!(a.mean(), 3.53486, 1e-5);
assert_almost_eq!(a.sample_variance(), 1.8210, 1e-4);
assert_almost_eq!(a.weighted_mean(), 3.53486, 1e-5);
assert_almost_eq!(a.sample_variance(), 1.7889, 1e-4);
assert_eq!(a.sum_weights(), 10.47);
assert_almost_eq!(a.error(), f64::sqrt(0.1739), 1e-4);
assert_eq!(a.len(), 10);
assert_almost_eq!(a.effective_len(), 8.2315, 1e-4);
assert_almost_eq!(a.error(), f64::sqrt(0.2173), 1e-4);
}
#[test]

66
tests/weighted_average2.rs
View File

@ -1,66 +0,0 @@
#[macro_use] extern crate average;
extern crate core;
use core::iter::Iterator;
use average::WeightedAverage2 as WeightedAverage;
#[test]
fn trivial() {
let mut a = WeightedAverage::new();
assert_eq!(a.len(), 0);
assert_eq!(a.sum_weights(), 0.);
assert_eq!(a.sum_weights_sq(), 0.);
a.add(1.0, 1.0);
assert_eq!(a.len(), 1);
assert_eq!(a.weighted_mean(), 1.0);
assert_eq!(a.unweighted_mean(), 1.0);
assert_eq!(a.sum_weights(), 1.0);
assert_eq!(a.sum_weights_sq(), 1.0);
assert_eq!(a.population_variance(), 0.0);
assert_eq!(a.error(), 0.0);
a.add(1.0, 1.0);
assert_eq!(a.len(), 2);
assert_eq!(a.weighted_mean(), 1.0);
assert_eq!(a.unweighted_mean(), 1.0);
assert_eq!(a.sum_weights(), 2.0);
assert_eq!(a.sum_weights_sq(), 2.0);
assert_eq!(a.population_variance(), 0.0);
assert_eq!(a.error(), 0.0);
}
#[test]
fn simple() {
let a: WeightedAverage = (1..6).map(|x| (f64::from(x), 1.0)).collect();
assert_eq!(a.len(), 5);
assert_eq!(a.weighted_mean(), 3.0);
assert_eq!(a.unweighted_mean(), 3.0);
assert_eq!(a.sum_weights(), 5.0);
assert_eq!(a.sample_variance(), 2.5);
assert_almost_eq!(a.error(), f64::sqrt(0.5), 1e-16);
}
#[test]
fn reference() {
// Example from http://www.analyticalgroup.com/download/WEIGHTED_MEAN.pdf.
let values = &[5., 5., 4., 4., 3., 4., 3., 2., 2., 1.];
let weights = &[1.23, 2.12, 1.23, 0.32, 1.53, 0.59, 0.94, 0.94, 0.84, 0.73];
let a: WeightedAverage = values.iter().zip(weights.iter())
.map(|(x, w)| (*x, *w)).collect();
assert_almost_eq!(a.weighted_mean(), 3.53486, 1e-5);
assert_almost_eq!(a.sample_variance(), 1.7889, 1e-4);
assert_eq!(a.sum_weights(), 10.47);
assert_eq!(a.len(), 10);
assert_almost_eq!(a.effective_len(), 8.2315, 1e-4);
assert_almost_eq!(a.error(), f64::sqrt(0.2173), 1e-4);
}
#[test]
fn error_corner_case() {
let values = &[1., 2.];
let weights = &[0.5, 0.5];
let a: WeightedAverage = values.iter().zip(weights.iter())
.map(|(x, w)| (*x, *w)).collect();
assert_eq!(a.error(), 0.5);
}