8 Statistics

This chapter describes the statistical functions provided by the Science Collection. The basic statistical functions include functions to compute the mean, variance, and standard deviation, More advanced functions allow you to calculate absolute deviation, skewness, and kurtosis, as well as the median and arbitrary percentiles. The algorithms use recurrance relations to compute average quantities in a stable way, without large intermediate values that might overflow.

The functions described in this chapter are defined in the "statistics.rkt" file in the Science Collection and are made available using the form:

(require (planet williams/science/statistics))

8.1 Running Statistics

A running statistics object accumulates a minimal set of statistics (n, min, max, mean, variance, and standard deviation) for a set of data values. A running statistics object does not require that a sequence (e.g., list or vector) of the data value be maintained.

8.2 Running Statistics Example

This example generated 100 random numbers between 0.0 and 10.0 and maintains running statistics on the values.

#lang racket

(require (planet williams/science/statistics)

(planet williams/science/random-distributions))

(define (test)

(let ((stat (make-statistics)))

(for ((i (in-range 100)))

(statistics-tally! stat (random-flat 0.0 10.0)))

(printf "Running Statistics Example~n")

(printf "                 n = ~a~n" (statistics-n stat))

(printf "               min = ~a~n" (statistics-min stat))

(printf "               max = ~a~n" (statistics-max stat))

(printf "              mean = ~a~n" (statistics-mean stat))

(printf "          variance = ~a~n" (statistics-variance stat))

(printf "standard deviation = ~a~n" (statistics-standard-deviation stat))))

(test)

Produces the following results.

Running Statistics Example

n = 100

min = 0.11100957474903939

max = 9.938914540059452

mean = 5.466640451797567

variance = 8.677003172428925

standard deviation = 2.945675333846031

8.3 Mean, Standard Deviation, and Variance

8.4 Absolute Deviation

8.5 Higher Moments (Skewness and Kurtosis)

8.6 Autocorrelation

8.7 Covariance

8.8 Correlation

8.9 Weighted Samples

8.10 Maximum and Minimum

8.11 Median and Quantiles

Thw median and quantile functions described in this section operate on sorted data. The contracts for these functions enforce this. Also, for convenience we use quantiles measured on a scale of 0 to 1 instead of percentiles, which use a scale of 0 to 100).

The quantile is found by interpolation using the formula:

quantile = 1 - delta(x[i]) + delta(x(i + 1))

where i is floor((n - 1) × f) and delta is (n - 1) × f - 1.

8.12 Statistics Example

This example generates two vectors from a unit Gaussian distribution and a vector of conse squared weighting data. All of the vectors are of length 1,000. Thes data are used to test all of the statistics functions.

#lang racket

(require (planet williams/science/random-distributions/gaussian)

(planet williams/science/statistics)

(planet williams/science/math))

(define (naive-sort! data)

(let loop ()

(let ((n (vector-length data))

(sorted? #t))

(do ((i 1 (+ i 1)))

((= i n) data)

(when (< (vector-ref data i)

(vector-ref data (- i 1)))

(let ((t (vector-ref data i)))

(vector-set! data i (vector-ref data (- i 1)))

(vector-set! data (- i 1) t)

(set! sorted? #f))))

(unless sorted?

(loop)))))

(let ((data1 (make-vector 1000))

(data2 (make-vector 1000))

(w     (make-vector 1000)))

(for ((i (in-range 1000)))

(vector-set! data1 i (random-unit-gaussian))

(vector-set! data2 i (random-unit-gaussian))

(vector-set! w i

(expt (cos (- (* 2.0 pi (/ i 1000.0)) pi)) 2)))

(printf "Statistics Example~n")

(printf "                                mean = ~a~n"

(mean data1))

(printf "                            variance = ~a~n"

(variance data1))

(printf "                  standard deviation = ~a~n"

(standard-deviation data1))

(printf "                   variance from 0.0 = ~a~n"

(variance-with-fixed-mean data1 0.0))

(printf "         standard deviation from 0.0 = ~a~n"

(standard-deviation-with-fixed-mean data1 0.0))

(printf "                  absolute deviation = ~a~n"

(absolute-deviation data1))

(printf "         absolute deviation from 0.0 = ~a~n"

(absolute-deviation data1 0.0))

(printf "                                skew = ~a~n"

(skew data1))

(printf "                            kurtosis = ~a~n"

(kurtosis data1))

(printf "               lag-1 autocorrelation = ~a~n"

(lag-1-autocorrelation data1))

(printf "                          covariance = ~a~n"

(covariance data1 data2))

(printf "                       weighted mean = ~a~n"

(weighted-mean w data1))

(printf "                   weighted variance = ~a~n"

(weighted-variance w data1))

(printf "         weighted standard deviation = ~a~n"

(weighted-standard-deviation w data1))

(printf "          weighted variance from 0.0 = ~a~n"

(weighted-variance-with-fixed-mean w data1 0.0))

(printf "weighted standard deviation from 0.0 = ~a~n"

(weighted-standard-deviation-with-fixed-mean w data1 0.0))

(printf "         weighted absolute deviation = ~a~n"

(weighted-absolute-deviation w data1))

(printf "weighted absolute deviation from 0.0 = ~a~n"

(weighted-absolute-deviation w data1 0.0))

(printf "                       weighted skew = ~a~n"

(weighted-skew w data1))

(printf "                   weighted kurtosis = ~a~n"

(weighted-kurtosis w data1))

(printf "                             maximum = ~a~n"

(maximum data1))

(printf "                             minimum = ~a~n"

(minimum data1))

(printf "              index of maximum value = ~a~n"

(maximum-index data1))

(printf "              index of minimum value = ~a~n"

(minimum-index data1))

(naive-sort! data1)

(printf "                              median = ~a~n"

(median-from-sorted-data data1))

(printf "                        10% quantile = ~a~n"

(quantile-from-sorted-data data1 0.1))

(printf "                        20% quantile = ~a~n"

(quantile-from-sorted-data data1 0.2))

(printf "                        30% quantile = ~a~n"

(quantile-from-sorted-data data1 0.3))

(printf "                        40% quantile = ~a~n"

(quantile-from-sorted-data data1 0.4))

(printf "                        50% quantile = ~a~n"

(quantile-from-sorted-data data1 0.5))

(printf "                        60% quantile = ~a~n"

(quantile-from-sorted-data data1 0.6))

(printf "                        70% quantile = ~a~n"

(quantile-from-sorted-data data1 0.7))

(printf "                        80% quantile = ~a~n"

(quantile-from-sorted-data data1 0.8))

(printf "                        90% quantile = ~a~n"

(quantile-from-sorted-data data1 0.9)))

Produces the following output:

Statistics Example

mean = 0.03457693091555611

variance = 1.0285343857083435

standard deviation = 1.0141668431320083

variance from 0.0 = 1.028701415474174

standard deviation from 0.0 = 1.014249188056946

absolute deviation = 0.7987180852601665

absolute deviation from 0.0 = 0.7987898146946209

skew = 0.04340293467117837

kurtosis = 0.17722452271702993

lag-1 autocorrelation = 0.0029930889831972143

covariance = 0.005782911085590894

weighted mean = 0.05096139259270008

weighted variance = 1.0500293763787367

weighted standard deviation = 1.0247094107007786

weighted variance from 0.0 = 1.0510513958491579

weighted standard deviation from 0.0 = 1.0252079768755011

weighted absolute deviation = 0.8054378524718832

weighted absolute deviation from 0.0 = 0.8052440544958938

weighted skew = 0.046448729539282155

weighted kurtosis = 0.3050060704791675

maximum = 3.731148814104969

minimum = -3.327265864298485

index of maximum value = 502

index of minimum value = 476

median = 0.019281803306206644

10% quantile = -1.243869878615807

20% quantile = -0.7816243947573505

30% quantile = -0.4708703241429585

40% quantile = -0.2299309332835332

50% quantile = 0.019281803306206644

60% quantile = 0.30022966479982344

70% quantile = 0.5317978807508836

80% quantile = 0.832291888537874

90% quantile = 1.3061151234700463