5.3.3

### 8Statistics Functions

 (require math/statistics)

This module exports functions that compute statistics, meaning summary values for collections of samples, and functions for managing sequences of weighted or unweighted samples.

Most of the functions that compute statistics accept a sequence of nonnegative reals that correspond one-to-one with sample values. These are used as weights; equivalently counts, pseudocounts or unnormalized probabilities. While this makes it easy to work with weighted samples, it introduces some subtleties in bias correction. In particular, central moments must be computed without bias correction by default. See Expected Values for a discussion.

#### 8.1Expected Values

Functions documented in this section that compute higher central moments, such as variance, stddev and skewness, can optionally apply bias correction to their estimates. For example, when variance is given the argument #:bias #t, it multiplies the result by (/ n (- n 1)), where n is the number of samples.

The meaning of “bias correction” becomes less clear with weighted samples, however. Often, the weights represent counts, so when moment-estimating functions receive #:bias #t, they interpret it as “use the sum of ws for n.” In the following example, the sample 4 is first counted twice and then given weight 2; therefore n = 5 in both cases:
 > (variance '(1 2 3 4 4) #:bias #t) - : Real [generalized from Nonnegative-Real] 17/10 > (variance '(1 2 3 4) '(1 1 1 2) #:bias #t) - : Real [generalized from Nonnegative-Real] 17/10

However, sample weights often do not represent counts. For these cases, the #:bias keyword can be followed by a real-valued pseudocount, which is used for n:
 > (variance '(1 2 3 4) '(1/2 1/2 1/2 1) #:bias 5) - : Real [generalized from Nonnegative-Real] 17/10

Because the magnitude of the bias correction for weighted samples cannot be known without user guidance, in all cases, the bias argument defaults to #f.

 procedure(mean xs [ws]) → Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f
When ws is #f (the default), returns the sample mean of the values in xs. Otherwise, returns the weighted sample mean of the values in xs with corresponding weights ws.

Examples:

> (mean '(1 2 3 4 5))

- : Real

3

> (mean '(1 2 3 4 5) '(1 1 1 1 10.0))

- : Real

4.285714285714286

> (define d (normal-dist))
> (mean (sample d 10000))

- : Real

-0.002790029375320962

> (define arr (array-strict (build-array #(5 1000) (λ (_) (sample d)))))
> (array-map mean (array->list-array arr 1))

- : (Array Real)

 (array #[0.07004206683471068 -0.022323034835281134 0.0709180096914594 -0.008581740552876894 -0.022665553867497364])

 procedure(variance xs [ws #:bias bias]) → Nonnegative-Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(stddev xs [ws #:bias bias]) → Nonnegative-Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(skewness xs [ws #:bias bias]) → Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(kurtosis xs [ws #:bias bias]) → Nonnegative-Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
If ws is #f, these compute the sample variance, standard deviation, skewness and excess kurtosis the samples in xs. If ws is not #f, they compute weighted variations of the same.

Examples:

 > (stddev '(1 2 3 4 5)) - : Real [generalized from Nonnegative-Real] 1.4142135623730951 > (stddev '(1 2 3 4 5) '(1 1 1 1 10)) - : Real [generalized from Nonnegative-Real] 1.2777531299998799
See Expected Values for the meaning of the bias keyword argument.

 procedure(variance/mean mean xs [ws #:bias bias]) → Nonnegative-Real mean : Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(stddev/mean mean xs [ws #:bias bias]) → Nonnegative-Real mean : Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(skewness/mean mean xs [ws #:bias bias]) → Real mean : Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(kurtosis/mean mean xs [ws #:bias bias]) → Nonnegative-Real mean : Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
Like variance, stddev, skewness and kurtosis, but computed using known mean mean.

#### 8.2Running Expected Values

The statistics object allows computing the sample minimum, maximum, count, mean, variance, skewness, and excess kurtosis of a sequence of samples in O(1) space.

To use it, start with empty-statistics, then use update-statistics to obtain a new statistics object with updated values. Use statistics-min, statistics-mean, and similar functions to get the current estimates.

Example:

 > (let* ([s  empty-statistics] [s  (update-statistics s 1)] [s  (update-statistics s 2)] [s  (update-statistics s 3)] [s  (update-statistics s 4 2)]) (values (statistics-mean s) (statistics-stddev s #:bias #t)))

- : (Values Flonum Flonum) [generalized from (Values Flonum Nonnegative-Flonum)]

 2.8 1.30384

 struct(struct statistics (min max count)) min : Flonum max : Flonum count : Nonnegative-Flonum
Represents running statistics.

The min and max fields are the minimum and maximum value observed so far, and the count field is the total weight of the samples (which is the number of samples if all samples are unweighted). The remaining, hidden fields are used to compute moments, and their number and meaning may change in future releases.

 value
The empty statistics object.

Examples:

 > (statistics-min empty-statistics) - : Flonum +inf.0 > (statistics-max empty-statistics) - : Flonum -inf.0 > (statistics-range empty-statistics) - : Flonum [generalized from Nonnegative-Flonum] +nan.0 > (statistics-count empty-statistics) - : Flonum [generalized from Nonnegative-Flonum] 0.0 > (statistics-mean empty-statistics) - : Flonum +nan.0 > (statistics-variance empty-statistics) - : Flonum [generalized from Nonnegative-Flonum] +nan.0 > (statistics-skewness empty-statistics) - : Flonum +nan.0 > (statistics-kurtosis empty-statistics) - : Flonum [generalized from Nonnegative-Flonum] +nan.0

 procedure(update-statistics s x [w]) → statistics s : statistics x : Real w : Real = 1.0
Returns a new statistics object that includes x in the computed statistics. If w is given, x is weighted by w in the moment computations.

 procedure(update-statistics* s xs [ws]) → statistics s : statistics xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f
Like update-statistics, but includes all of xs, possibly weighted by corresponding elements in ws, in the returned statistics object.

Examples:

> (define s (update-statistics* empty-statistics '(1 2 3 4) '(1 1 1 2)))
> (statistics-mean s)

- : Flonum

2.8

> (statistics-stddev s #:bias #t)

- : Flonum [generalized from Nonnegative-Flonum]

1.3038404810405297

This function uses O(1) space regardless of the length of xs.

 procedure s : statistics
 procedure s : statistics
 procedure(statistics-variance s [#:bias bias]) → Nonnegative-Flonum s : statistics bias : (U #t #f Real) = #f
 procedure(statistics-stddev s [#:bias bias]) → Nonnegative-Flonum s : statistics bias : (U #t #f Real) = #f
 procedure(statistics-skewness s [#:bias bias]) → Flonum s : statistics bias : (U #t #f Real) = #f
 procedure(statistics-kurtosis s [#:bias bias]) → Nonnegative-Flonum s : statistics bias : (U #t #f Real) = #f
Compute the range, mean, variance, standard deviation, skewness, and excess kurtosis of the observations summarized in s.

See Expected Values for the meaning of the bias keyword argument.

#### 8.3Correlation

 procedure(covariance xs ys [ws #:bias bias]) → Real xs : (Sequenceof Real) ys : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(correlation xs ys [ws #:bias bias]) → Real xs : (Sequenceof Real) ys : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
Compute the sample covariance and correlation of xs and ys, optionally weighted by ws.

Examples:

> (define xs (sample (normal-dist) 10000))
> (define ys (map (λ: ([x : Real]) (sample (normal-dist x))) xs))
> (correlation xs ys)

- : Real

0.7104292619038987

Removing the correlation using importance weights:
 > (define ws (map (λ: ([x : Real] [y : Real]) (/ (pdf (normal-dist) y) (pdf (normal-dist x) y))) xs ys))
> (correlation xs ys (ann ws (Sequenceof Real)))

- : Real

0.01856334379507927

See Expected Values for the meaning of the bias keyword argument.

 procedure(covariance/means μx μy xs ys [ws #:bias bias]) → Real μx : Real μy : Real xs : (Sequenceof Real) ys : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
 procedure(correlation/means μx μy xs ys [ws #:bias bias]) → Real μx : Real μy : Real xs : (Sequenceof Real) ys : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f bias : (U #t #f Real) = #f
Like covariance and correlation, but computed using known means μx and μy.

#### 8.4Counting and Binning

 procedure xs : (Sequenceof A) (samples->hash xs ws) → (HashTable A Nonnegative-Real) xs : (Sequenceof A) ws : (U #f (Sequenceof Real))
Returns a hash table mapping each unique element in xs (under equal?) to its count, or, if ws is not #f, to its total weight.

Examples:

 > (samples->hash '(1 2 3 4 4)) - : (HashTable Integer Positive-Integer) '#hash((4 . 2) (3 . 1) (2 . 1) (1 . 1)) > (samples->hash '(1 1 2 3 4) '(1/2 1/2 1 1 2)) - : (HashTable Integer Nonnegative-Real) '#hash((4 . 2) (3 . 1) (2 . 1) (1 . 1))

 procedure(count-samples xs) → (Values (Listof A) (Listof Positive-Integer)) xs : (Sequenceof A) (count-samples xs ws) → (Values (Listof A) (Listof Nonnegative-Real)) xs : (Sequenceof A) ws : (U #f (Sequenceof Real))
Like samples->hash, but returns two lists. The elements in the returned (Listof A) are in order of first appearance in xs.

Examples:

> (count-samples '(1 2 3 4 4))

- : (Values (Listof Positive-Byte) (Listof Positive-Integer))

 '(1 2 3 4) '(1 1 1 2)
> (count-samples '(1 1 2 3 4) '(1/2 1/2 1 1 2))

- : (Values (Listof Positive-Byte) (Listof Nonnegative-Real))

 '(1 2 3 4) '(1 1 1 2)

 struct(struct sample-bin (min max values weights)) min : B max : B values : (Listof A) weights : (U #f (Listof Nonnegative-Real))
Represents a bin, or a group of samples within an interval in a total order. The values and bounds have a different type to allow bin-samples/key to group elements based on a function of their values.

 procedure(bin-samples bounds lte? xs ws) → (Listof (sample-bin A A)) bounds : (Sequenceof A) lte? : (A A -> Any) xs : (Sequenceof A) ws : (U #f (Sequenceof Real))
Similar to (sort xs lte?), but additionally groups samples into bins. The bins’ bounds are sorted before binning xs.

If n = (length bounds), then bin-samples returns at least (- n 1) bins, one for each pair of adjacent (sorted) bounds. If some values in xs are less than the smallest bound, they are grouped into a single bin in front. If some are greater than the largest bound, they are grouped into a single bin at the end.

Examples:

> (bin-samples '() <= '(0 1 2 3 4 5 6))

- : (Listof (sample-bin Byte Byte))

(list (sample-bin 0 6 '(0 1 2 3 4 5 6) #f))

> (bin-samples '(3) <= '(0 1 2 3 4 5 6))

- : (Listof (sample-bin Byte Byte))

(list (sample-bin 0 3 '(0 1 2 3) #f) (sample-bin 3 6 '(4 5 6) #f))

> (bin-samples '(2 4) <= '(0 1 2 3 4 5 6))

- : (Listof (sample-bin Byte Byte))

 (list (sample-bin 0 2 '(0 1 2) #f) (sample-bin 2 4 '(3 4) #f) (sample-bin 4 6 '(5 6) #f))
 > (bin-samples '(2 4) <= '(0 1 2 3 4 5 6) '(10 20 30 40 50 60 70))

- : (Listof (sample-bin Byte Byte))

 (list (sample-bin 0 2 '(0 1 2) '(10 20 30)) (sample-bin 2 4 '(3 4) '(40 50)) (sample-bin 4 6 '(5 6) '(60 70)))

If lte? is a less-than-or-equal relation, the bins represent half-open intervals (min, max] (except possibly the first, which may be closed). If lte? is a less-than relation, the bins represent half-open intervals [min, max) (except possibly the last, which may be closed). In either case, the sorts applied to bounds and xs are stable.

Because intervals used in probability measurements are normally open on the left, prefer to use less-than-or-equal relations for lte?.

If ws is #f, bin-samples returns bins with #f weights.

 procedure(bin-samples/key bounds lte? key xs ws) → (Listof (sample-bin A B)) bounds : (Sequenceof B) lte? : (B B -> Any) key : (A -> B) xs : (Sequenceof A) ws : (U #f (Sequenceof Real))
Similar to (sort xs lte? #:key key #:cache-keys? #t), but additionally groups samples into bins.

Example:

 > (bin-samples/key '(2 4) <= (inst car Real String) '((1 . "1") (2 . "2") (3 . "3") (4 . "4") (5 . "5")))

- : (Listof (sample-bin (Pairof Positive-Byte String) Real))

 (list (sample-bin 1 2 '((1 . "1") (2 . "2")) #f) (sample-bin 2 4 '((3 . "3") (4 . "4")) #f) (sample-bin 4 5 '((5 . "5")) #f))
See bin-samples for the simpler, one-type variant.

 procedure(sample-bin-compact bin) → (sample-bin A B) bin : (sample-bin A B)
Compacts bin by applying count-samples to its values and weights.

Example:

 > (sample-bin-compact (sample-bin 1 4 '(1 2 3 4 4) #f)) - : (sample-bin Positive-Byte Positive-Byte) (sample-bin 1 4 '(1 2 3 4) '(1 1 1 2))

 procedure bin : (sample-bin A B)
If (sample-bin-weights bin) is #f, returns the number of samples in bin; otherwise, returns the sum of their weights.

Examples:

 > (sample-bin-total (sample-bin 1 4 '(1 2 3 4 4) #f)) - : Real [generalized from Nonnegative-Real] 5 > (sample-bin-total (sample-bin-compact (sample-bin 1 4 '(1 2 3 4 4) #f))) - : Real [generalized from Nonnegative-Real] 5

#### 8.5Order Statistics

 procedure(sort-samples lt? xs) → (Listof A) lt? : (A A -> Any) xs : (Sequenceof A) (sort-samples lt? xs ws) → (Values (Listof A) (Listof Nonnegative-Real)) lt? : (A A -> Any) xs : (Sequenceof A) ws : (U #f (Sequenceof Real))
Sorts possibly weighted samples according to lt?, which is assumed to define a total order over the elements in xs.

Examples:

> (sort-samples < '(5 2 3 1))

- : (Listof Positive-Byte)

'(1 2 3 5)

> (sort-samples < '(5 2 3 1) '(50 20 30 10))

- : (Values (Listof Positive-Byte) (Listof Nonnegative-Real))

 '(1 2 3 5) '(10 20 30 50)
> (sort-samples < '(5 2 3 1) #f)

- : (Values (Listof Positive-Byte) (Listof Nonnegative-Real))

 '(1 2 3 5) '(1 1 1 1)
Because sort-samples is defined in terms of sort, the sort is only guaranteed to be stable if lt? is strictly a less-than relation.

 procedure(median lt? xs [ws]) → A lt? : (A A -> Any) xs : (Sequenceof A) ws : (U #f (Sequenceof Real)) = #f
Equivalent to (quantile 1/2 lt? xs ws).

 procedure(quantile p lt? xs [ws]) → A p : Real lt? : (A A -> Any) xs : (Sequenceof A) ws : (U #f (Sequenceof Real)) = #f
Computes the inverse of the empirical cdf represented by the samples xs, which are optionally weighted by ws.

Examples:

 > (quantile 0 < '(1 3 5)) - : Integer [generalized from Positive-Byte] 1 > (quantile 0.5 < '(1 2 3 4)) - : Integer [generalized from Positive-Byte] 2 > (quantile 0.5 < '(1 2 3 4) '(0.25 0.2 0.2 0.35)) - : Integer [generalized from Positive-Byte] 3

If p = 0, quantile returns the smallest element of xs under the ordering relation lt?. If p = 1, it returns the largest element.

For weighted samples, quantile sorts xs and ws together (using sort-samples), then finds the least x for which the proportion of its cumulative weight is greater than or equal to p.

For unweighted samples, quantile uses the quickselect algorithm to find the element that would be at index (ceiling (- (* p n) 1)) if xs were sorted, where n is the length of xs.

 procedure(absdev xs [ws]) → Nonnegative-Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f
Computes the average absolute difference between the elements in xs and (median < xs ws). If ws is not #f, it is a weighted average.

 procedure(absdev/median median xs [ws]) → Nonnegative-Real median : Real xs : (Sequenceof Real) ws : (U #f (Sequenceof Real)) = #f
Like (absdev xs ws), but computed using known median median.