1 // Copyright 2012 The Rust Project Developers. See the COPYRIGHT
2 // file at the top-level directory of this distribution and at
3 // http://rust-lang.org/COPYRIGHT.
5 // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
6 // http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
7 // <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
8 // option. This file may not be copied, modified, or distributed
9 // except according to those terms.
11 #![allow(missing_docs)]
13 use std::collections::hash_map;
14 use std::collections::hash_map::{Occupied, Vacant};
19 use std::num::{Float, FloatMath};
21 fn local_cmp<T:Float>(x: T, y: T) -> Ordering {
22 // arbitrarily decide that NaNs are larger than everything.
25 } else if x.is_nan() {
36 fn local_sort<T: Float>(v: &mut [T]) {
37 v.sort_by(|x: &T, y: &T| local_cmp(*x, *y));
40 /// Trait that provides simple descriptive statistics on a univariate set of numeric samples.
41 pub trait Stats <T: FloatMath + FromPrimitive> for Sized? {
43 /// Sum of the samples.
45 /// Note: this method sacrifices performance at the altar of accuracy
46 /// Depends on IEEE-754 arithmetic guarantees. See proof of correctness at:
47 /// ["Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates"]
48 /// (http://www.cs.cmu.edu/~quake-papers/robust-arithmetic.ps)
49 /// *Discrete & Computational Geometry 18*, 3 (Oct 1997), 305-363, Shewchuk J.R.
52 /// Minimum value of the samples.
55 /// Maximum value of the samples.
58 /// Arithmetic mean (average) of the samples: sum divided by sample-count.
60 /// See: https://en.wikipedia.org/wiki/Arithmetic_mean
63 /// Median of the samples: value separating the lower half of the samples from the higher half.
64 /// Equal to `self.percentile(50.0)`.
66 /// See: https://en.wikipedia.org/wiki/Median
67 fn median(&self) -> T;
69 /// Variance of the samples: bias-corrected mean of the squares of the differences of each
70 /// sample from the sample mean. Note that this calculates the _sample variance_ rather than the
71 /// population variance, which is assumed to be unknown. It therefore corrects the `(n-1)/n`
72 /// bias that would appear if we calculated a population variance, by dividing by `(n-1)` rather
75 /// See: https://en.wikipedia.org/wiki/Variance
78 /// Standard deviation: the square root of the sample variance.
80 /// Note: this is not a robust statistic for non-normal distributions. Prefer the
81 /// `median_abs_dev` for unknown distributions.
83 /// See: https://en.wikipedia.org/wiki/Standard_deviation
84 fn std_dev(&self) -> T;
86 /// Standard deviation as a percent of the mean value. See `std_dev` and `mean`.
88 /// Note: this is not a robust statistic for non-normal distributions. Prefer the
89 /// `median_abs_dev_pct` for unknown distributions.
90 fn std_dev_pct(&self) -> T;
92 /// Scaled median of the absolute deviations of each sample from the sample median. This is a
93 /// robust (distribution-agnostic) estimator of sample variability. Use this in preference to
94 /// `std_dev` if you cannot assume your sample is normally distributed. Note that this is scaled
95 /// by the constant `1.4826` to allow its use as a consistent estimator for the standard
98 /// See: http://en.wikipedia.org/wiki/Median_absolute_deviation
99 fn median_abs_dev(&self) -> T;
101 /// Median absolute deviation as a percent of the median. See `median_abs_dev` and `median`.
102 fn median_abs_dev_pct(&self) -> T;
104 /// Percentile: the value below which `pct` percent of the values in `self` fall. For example,
105 /// percentile(95.0) will return the value `v` such that 95% of the samples `s` in `self`
106 /// satisfy `s <= v`.
108 /// Calculated by linear interpolation between closest ranks.
110 /// See: http://en.wikipedia.org/wiki/Percentile
111 fn percentile(&self, pct: T) -> T;
113 /// Quartiles of the sample: three values that divide the sample into four equal groups, each
114 /// with 1/4 of the data. The middle value is the median. See `median` and `percentile`. This
115 /// function may calculate the 3 quartiles more efficiently than 3 calls to `percentile`, but
116 /// is otherwise equivalent.
118 /// See also: https://en.wikipedia.org/wiki/Quartile
119 fn quartiles(&self) -> (T,T,T);
121 /// Inter-quartile range: the difference between the 25th percentile (1st quartile) and the 75th
122 /// percentile (3rd quartile). See `quartiles`.
124 /// See also: https://en.wikipedia.org/wiki/Interquartile_range
128 /// Extracted collection of all the summary statistics of a sample set.
129 #[deriving(Clone, PartialEq)]
130 #[allow(missing_docs)]
131 pub struct Summary<T> {
140 pub median_abs_dev: T,
141 pub median_abs_dev_pct: T,
142 pub quartiles: (T,T,T),
146 impl<T: FloatMath + FromPrimitive> Summary<T> {
147 /// Construct a new summary of a sample set.
148 pub fn new(samples: &[T]) -> Summary<T> {
153 mean: samples.mean(),
154 median: samples.median(),
156 std_dev: samples.std_dev(),
157 std_dev_pct: samples.std_dev_pct(),
158 median_abs_dev: samples.median_abs_dev(),
159 median_abs_dev_pct: samples.median_abs_dev_pct(),
160 quartiles: samples.quartiles(),
166 impl<T: FloatMath + FromPrimitive> Stats<T> for [T] {
167 // FIXME #11059 handle NaN, inf and overflow
169 let mut partials = vec![];
171 for &mut x in self.iter() {
173 // This inner loop applies `hi`/`lo` summation to each
174 // partial so that the list of partial sums remains exact.
175 for i in range(0, partials.len()) {
176 let mut y: T = partials[i];
177 if x.abs() < y.abs() {
178 mem::swap(&mut x, &mut y);
180 // Rounded `x+y` is stored in `hi` with round-off stored in
181 // `lo`. Together `hi+lo` are exactly equal to `x+y`.
183 let lo = y - (hi - x);
184 if lo != Float::zero() {
190 if j >= partials.len() {
194 partials.truncate(j+1);
197 let zero: T = Float::zero();
198 partials.iter().fold(zero, |p, q| p + *q)
202 assert!(self.len() != 0);
203 self.iter().fold(self[0], |p, q| p.min(*q))
207 assert!(self.len() != 0);
208 self.iter().fold(self[0], |p, q| p.max(*q))
211 fn mean(&self) -> T {
212 assert!(self.len() != 0);
213 self.sum() / FromPrimitive::from_uint(self.len()).unwrap()
216 fn median(&self) -> T {
217 self.percentile(FromPrimitive::from_uint(50).unwrap())
224 let mean = self.mean();
225 let mut v: T = Float::zero();
226 for s in self.iter() {
230 // NB: this is _supposed to be_ len-1, not len. If you
231 // change it back to len, you will be calculating a
232 // population variance, not a sample variance.
233 let denom = FromPrimitive::from_uint(self.len()-1).unwrap();
238 fn std_dev(&self) -> T {
242 fn std_dev_pct(&self) -> T {
243 let hundred = FromPrimitive::from_uint(100).unwrap();
244 (self.std_dev() / self.mean()) * hundred
247 fn median_abs_dev(&self) -> T {
248 let med = self.median();
249 let abs_devs: Vec<T> = self.iter().map(|&v| (med - v).abs()).collect();
250 // This constant is derived by smarter statistics brains than me, but it is
251 // consistent with how R and other packages treat the MAD.
252 let number = FromPrimitive::from_f64(1.4826).unwrap();
253 abs_devs.median() * number
256 fn median_abs_dev_pct(&self) -> T {
257 let hundred = FromPrimitive::from_uint(100).unwrap();
258 (self.median_abs_dev() / self.median()) * hundred
261 fn percentile(&self, pct: T) -> T {
262 let mut tmp = self.to_vec();
263 local_sort(tmp.as_mut_slice());
264 percentile_of_sorted(tmp.as_slice(), pct)
267 fn quartiles(&self) -> (T,T,T) {
268 let mut tmp = self.to_vec();
269 local_sort(tmp.as_mut_slice());
270 let first = FromPrimitive::from_uint(25).unwrap();
271 let a = percentile_of_sorted(tmp.as_slice(), first);
272 let secound = FromPrimitive::from_uint(50).unwrap();
273 let b = percentile_of_sorted(tmp.as_slice(), secound);
274 let third = FromPrimitive::from_uint(75).unwrap();
275 let c = percentile_of_sorted(tmp.as_slice(), third);
280 let (a,_,c) = self.quartiles();
286 // Helper function: extract a value representing the `pct` percentile of a sorted sample-set, using
287 // linear interpolation. If samples are not sorted, return nonsensical value.
288 fn percentile_of_sorted<T: Float + FromPrimitive>(sorted_samples: &[T],
290 assert!(sorted_samples.len() != 0);
291 if sorted_samples.len() == 1 {
292 return sorted_samples[0];
294 let zero: T = Float::zero();
295 assert!(zero <= pct);
296 let hundred = FromPrimitive::from_uint(100).unwrap();
297 assert!(pct <= hundred);
299 return sorted_samples[sorted_samples.len() - 1];
301 let length = FromPrimitive::from_uint(sorted_samples.len() - 1).unwrap();
302 let rank = (pct / hundred) * length;
303 let lrank = rank.floor();
304 let d = rank - lrank;
305 let n = lrank.to_uint().unwrap();
306 let lo = sorted_samples[n];
307 let hi = sorted_samples[n+1];
312 /// Winsorize a set of samples, replacing values above the `100-pct` percentile and below the `pct`
313 /// percentile with those percentiles themselves. This is a way of minimizing the effect of
314 /// outliers, at the cost of biasing the sample. It differs from trimming in that it does not
315 /// change the number of samples, just changes the values of those that are outliers.
317 /// See: http://en.wikipedia.org/wiki/Winsorising
318 pub fn winsorize<T: Float + FromPrimitive>(samples: &mut [T], pct: T) {
319 let mut tmp = samples.to_vec();
320 local_sort(tmp.as_mut_slice());
321 let lo = percentile_of_sorted(tmp.as_slice(), pct);
322 let hundred: T = FromPrimitive::from_uint(100).unwrap();
323 let hi = percentile_of_sorted(tmp.as_slice(), hundred-pct);
324 for samp in samples.iter_mut() {
327 } else if *samp < lo {
333 /// Render writes the min, max and quartiles of the provided `Summary` to the provided `Writer`.
334 pub fn write_5_number_summary<W: Writer, T: Float + Show>(w: &mut W,
335 s: &Summary<T>) -> io::IoResult<()> {
336 let (q1,q2,q3) = s.quartiles;
337 write!(w, "(min={}, q1={}, med={}, q3={}, max={})",
345 /// Render a boxplot to the provided writer. The boxplot shows the min, max and quartiles of the
346 /// provided `Summary` (thus includes the mean) and is scaled to display within the range of the
347 /// nearest multiple-of-a-power-of-ten above and below the min and max of possible values, and
348 /// target `width_hint` characters of display (though it will be wider if necessary).
350 /// As an example, the summary with 5-number-summary `(min=15, q1=17, med=20, q3=24, max=31)` might
354 /// 10 | [--****#******----------] | 40
356 pub fn write_boxplot<W: Writer, T: Float + Show + FromPrimitive>(
360 -> io::IoResult<()> {
362 let (q1,q2,q3) = s.quartiles;
364 // the .abs() handles the case where numbers are negative
365 let ten: T = FromPrimitive::from_uint(10).unwrap();
366 let lomag = ten.powf(s.min.abs().log10().floor());
367 let himag = ten.powf(s.max.abs().log10().floor());
369 // need to consider when the limit is zero
370 let zero: T = Float::zero();
371 let lo = if lomag == Float::zero() {
374 (s.min / lomag).floor() * lomag
377 let hi = if himag == Float::zero() {
380 (s.max / himag).ceil() * himag
385 let lostr = lo.to_string();
386 let histr = hi.to_string();
388 let overhead_width = lostr.len() + histr.len() + 4;
389 let range_width = width_hint - overhead_width;
390 let range_float = FromPrimitive::from_uint(range_width).unwrap();
391 let char_step = range / range_float;
393 try!(write!(w, "{} |", lostr));
398 while c < range_width && v < s.min {
399 try!(write!(w, " "));
403 try!(write!(w, "["));
405 while c < range_width && v < q1 {
406 try!(write!(w, "-"));
410 while c < range_width && v < q2 {
411 try!(write!(w, "*"));
415 try!(write!(w, "#"));
417 while c < range_width && v < q3 {
418 try!(write!(w, "*"));
422 while c < range_width && v < s.max {
423 try!(write!(w, "-"));
427 try!(write!(w, "]"));
428 while c < range_width {
429 try!(write!(w, " "));
434 try!(write!(w, "| {}", histr));
438 /// Returns a HashMap with the number of occurrences of every element in the
439 /// sequence that the iterator exposes.
440 pub fn freq_count<T: Iterator<U>, U: Eq+Hash>(mut iter: T) -> hash_map::HashMap<U, uint> {
441 let mut map: hash_map::HashMap<U,uint> = hash_map::HashMap::new();
443 match map.entry(elem) {
444 Occupied(mut entry) => { *entry.get_mut() += 1; },
445 Vacant(entry) => { entry.set(1); },
451 // Test vectors generated from R, using the script src/etc/stat-test-vectors.r.
457 use stats::write_5_number_summary;
458 use stats::write_boxplot;
462 macro_rules! assert_approx_eq(
463 ($a:expr, $b:expr) => ({
465 let (a, b) = (&$a, &$b);
466 assert!((*a - *b).abs() < 1.0e-6,
467 "{} is not approximately equal to {}", *a, *b);
471 fn check(samples: &[f64], summ: &Summary<f64>) {
473 let summ2 = Summary::new(samples);
475 let mut w = io::stdout();
477 (write!(w, "\n")).unwrap();
478 write_5_number_summary(w, &summ2).unwrap();
479 (write!(w, "\n")).unwrap();
480 write_boxplot(w, &summ2, 50).unwrap();
481 (write!(w, "\n")).unwrap();
483 assert_eq!(summ.sum, summ2.sum);
484 assert_eq!(summ.min, summ2.min);
485 assert_eq!(summ.max, summ2.max);
486 assert_eq!(summ.mean, summ2.mean);
487 assert_eq!(summ.median, summ2.median);
489 // We needed a few more digits to get exact equality on these
490 // but they're within float epsilon, which is 1.0e-6.
491 assert_approx_eq!(summ.var, summ2.var);
492 assert_approx_eq!(summ.std_dev, summ2.std_dev);
493 assert_approx_eq!(summ.std_dev_pct, summ2.std_dev_pct);
494 assert_approx_eq!(summ.median_abs_dev, summ2.median_abs_dev);
495 assert_approx_eq!(summ.median_abs_dev_pct, summ2.median_abs_dev_pct);
497 assert_eq!(summ.quartiles, summ2.quartiles);
498 assert_eq!(summ.iqr, summ2.iqr);
502 fn test_min_max_nan() {
503 let xs = &[1.0, 2.0, f64::NAN, 3.0, 4.0];
504 let summary = Summary::new(xs);
505 assert_eq!(summary.min, 1.0);
506 assert_eq!(summary.max, 4.0);
515 let summ = &Summary {
516 sum: 1882.0000000000,
519 mean: 941.0000000000,
520 median: 941.0000000000,
522 std_dev: 24.0416305603,
523 std_dev_pct: 2.5549022912,
524 median_abs_dev: 25.2042000000,
525 median_abs_dev_pct: 2.6784484591,
526 quartiles: (932.5000000000,941.0000000000,949.5000000000),
532 fn test_norm10narrow() {
545 let summ = &Summary {
546 sum: 9996.0000000000,
548 max: 1217.0000000000,
549 mean: 999.6000000000,
550 median: 970.5000000000,
551 var: 16050.7111111111,
552 std_dev: 126.6914010938,
553 std_dev_pct: 12.6742097933,
554 median_abs_dev: 102.2994000000,
555 median_abs_dev_pct: 10.5408964451,
556 quartiles: (956.7500000000,970.5000000000,1078.7500000000),
562 fn test_norm10medium() {
575 let summ = &Summary {
576 sum: 8653.0000000000,
578 max: 1084.0000000000,
579 mean: 865.3000000000,
580 median: 911.5000000000,
581 var: 48628.4555555556,
582 std_dev: 220.5186059170,
583 std_dev_pct: 25.4846418487,
584 median_abs_dev: 195.7032000000,
585 median_abs_dev_pct: 21.4704552935,
586 quartiles: (771.0000000000,911.5000000000,1017.2500000000),
592 fn test_norm10wide() {
605 let summ = &Summary {
606 sum: 9349.0000000000,
608 max: 1591.0000000000,
609 mean: 934.9000000000,
610 median: 913.5000000000,
611 var: 239208.9888888889,
612 std_dev: 489.0899599142,
613 std_dev_pct: 52.3146817750,
614 median_abs_dev: 611.5725000000,
615 median_abs_dev_pct: 66.9482758621,
616 quartiles: (567.2500000000,913.5000000000,1331.2500000000),
622 fn test_norm25verynarrow() {
650 let summ = &Summary {
651 sum: 24926.0000000000,
653 max: 1040.0000000000,
654 mean: 997.0400000000,
655 median: 998.0000000000,
657 std_dev: 19.8294393937,
658 std_dev_pct: 1.9888308788,
659 median_abs_dev: 22.2390000000,
660 median_abs_dev_pct: 2.2283567134,
661 quartiles: (983.0000000000,998.0000000000,1013.0000000000),
680 let summ = &Summary {
685 median: 11.5000000000,
687 std_dev: 16.9643416875,
688 std_dev_pct: 101.5828843560,
689 median_abs_dev: 13.3434000000,
690 median_abs_dev_pct: 116.0295652174,
691 quartiles: (4.2500000000,11.5000000000,22.5000000000),
710 let summ = &Summary {
715 median: 24.5000000000,
717 std_dev: 19.5848580967,
718 std_dev_pct: 74.4671410520,
719 median_abs_dev: 22.9803000000,
720 median_abs_dev_pct: 93.7971428571,
721 quartiles: (9.5000000000,24.5000000000,36.5000000000),
740 let summ = &Summary {
745 median: 22.0000000000,
747 std_dev: 21.4050876611,
748 std_dev_pct: 88.4507754589,
749 median_abs_dev: 21.4977000000,
750 median_abs_dev_pct: 97.7168181818,
751 quartiles: (7.7500000000,22.0000000000,35.0000000000),
785 let summ = &Summary {
790 median: 19.0000000000,
792 std_dev: 24.5161851301,
793 std_dev_pct: 103.3565983562,
794 median_abs_dev: 19.2738000000,
795 median_abs_dev_pct: 101.4410526316,
796 quartiles: (6.0000000000,19.0000000000,31.0000000000),
830 let summ = &Summary {
835 median: 20.0000000000,
837 std_dev: 4.5650848842,
838 std_dev_pct: 22.2037202539,
839 median_abs_dev: 5.9304000000,
840 median_abs_dev_pct: 29.6520000000,
841 quartiles: (17.0000000000,20.0000000000,24.0000000000),
847 fn test_pois25lambda30() {
875 let summ = &Summary {
880 median: 32.0000000000,
882 std_dev: 5.1568724372,
883 std_dev_pct: 16.3814245145,
884 median_abs_dev: 5.9304000000,
885 median_abs_dev_pct: 18.5325000000,
886 quartiles: (28.0000000000,32.0000000000,34.0000000000),
892 fn test_pois25lambda40() {
920 let summ = &Summary {
921 sum: 1019.0000000000,
925 median: 42.0000000000,
927 std_dev: 5.8685603004,
928 std_dev_pct: 14.3978417577,
929 median_abs_dev: 5.9304000000,
930 median_abs_dev_pct: 14.1200000000,
931 quartiles: (37.0000000000,42.0000000000,45.0000000000),
937 fn test_pois25lambda50() {
965 let summ = &Summary {
966 sum: 1235.0000000000,
970 median: 50.0000000000,
972 std_dev: 5.6273143387,
973 std_dev_pct: 11.3913245723,
974 median_abs_dev: 4.4478000000,
975 median_abs_dev_pct: 8.8956000000,
976 quartiles: (44.0000000000,50.0000000000,52.0000000000),
1010 let summ = &Summary {
1011 sum: 1242.0000000000,
1014 mean: 49.6800000000,
1015 median: 45.0000000000,
1016 var: 1015.6433333333,
1017 std_dev: 31.8691595957,
1018 std_dev_pct: 64.1488719719,
1019 median_abs_dev: 45.9606000000,
1020 median_abs_dev_pct: 102.1346666667,
1021 quartiles: (29.0000000000,45.0000000000,79.0000000000),
1028 fn test_boxplot_nonpositive() {
1029 fn t(s: &Summary<f64>, expected: String) {
1030 let mut m = Vec::new();
1031 write_boxplot(&mut m, s, 30).unwrap();
1032 let out = String::from_utf8(m).unwrap();
1033 assert_eq!(out, expected);
1036 t(&Summary::new(&[-2.0f64, -1.0f64]),
1037 "-2 |[------******#*****---]| -1".to_string());
1038 t(&Summary::new(&[0.0f64, 2.0f64]),
1039 "0 |[-------*****#*******---]| 2".to_string());
1040 t(&Summary::new(&[-2.0f64, 0.0f64]),
1041 "-2 |[------******#******---]| 0".to_string());
1045 fn test_sum_f64s() {
1046 assert_eq!([0.5f64, 3.2321f64, 1.5678f64].sum(), 5.2999);
1049 fn test_sum_f64_between_ints_that_sum_to_0() {
1050 assert_eq!([1e30f64, 1.2f64, -1e30f64].sum(), 1.2);
1060 pub fn sum_three_items(b: &mut Bencher) {
1062 [1e20f64, 1.5f64, -1e20f64].sum();
1066 pub fn sum_many_f64(b: &mut Bencher) {
1067 let nums = [-1e30f64, 1e60, 1e30, 1.0, -1e60];
1068 let v = Vec::from_fn(500, |i| nums[i%5]);