#lang scribble/doc @(require scribble/manual scribble/struct scribblings/icons (for-label scheme/base "../science-with-graphics.ss")) @title[#:tag "statistics"]{Statistics} @local-table-of-contents[] This chapter describes the statistical functions provided by the PLT Scheme 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 @filepath{statistics.ss} file in the science collection and are made available using the form: @defmodule[(planet williams/science/statistics)] @;---------- @; Mean, Standard Deviation, and Variance @;---------- @section{Mean, Standard Deviation, and Variance} @defproc[(mean (data (vectorof real?))) real?]{ Returns the arithmetic mean of @scheme{data}.} @defproc*[(((variance (data (vectorof real?)) (mu real?)) (>=/c 0.0)) ((variance (data (vectorof real?))) (>=/c 0.0)))]{ Returns the sample variance of @scheme[data]. If @scheme[mu] is not provided, it is calculated by a call to @scheme[(mean data)].} @defproc*[(((standard-deviation (data (vectorof real?)) (mu real?)) (>=/c 0.0)) ((standard-deviation (data (vectorof real?))) (>=/c 0.0)))]{ Returns the standard deviation of @scheme[data]. The @tech{standard deviation} is defined as the square root of the variance. The result is the square root of the corresponding @scheme[variance] function.} @defproc[(variance-with-fixed-mean (data (vectorof real?)) (mu real?)) (>=/c 0.0)]{ Returns an unbiased estimate of the variance of @scheme[data] when the population mean @scheme[mu] of the underlying distribution is known @italic{a priori}.} @defproc[(standard-deviation-with-fixed-mean (data (vectorof real?)) (mu real?)) (>=/c 0.0)]{ Returns the standard deviation of @scheme[data] with a fixed population mean @scheme[mu]. The result is the square root of the @scheme[variance-with-fixed-mean] function.} @;---------- @; Absolute Deviation @;---------- @section{Absolute Deviation} @defproc*[(((absolute-deviation (data (vectorof real?)) (mu real?)) (>=/c 0.0)) ((absolute-deviation (data (vectorof real?))) (>=/c 0.0)))]{ Returns the absolute devistion of @scheme[data] relative to the given value of the mean @scheme[mu]. If @scheme[mu] is not provided, it is calculated by a call to @scheme[(mean data)]. This function is also useful if you want to calculate the absolute deviation to any value other than the mean, such as zero or the median.} @;---------- @; Higher Moments (Skewness and Kurtosis) @;---------- @section{Higher Moments (Skewness and Kurtosis)} @defproc*[(((skew (data (vectorof real?)) (mu real?) (sigma (>=/c 0.0))) real?) ((skew (data (vectorof real?))) real?))]{ Returns the skewness of @scheme[data] using the given values of the mean @scheme[mu] and standard deviation @scheme[sigma]. The @tech{skewness} measures the symmetry of the tails of a distribution. If @scheme[mu] and @scheme[sigma] are not provided, they are calculated by calls to @scheme[(mean data)] and @scheme[(standard-deviation data mu)].} @defproc*[(((kurtosis (data (vectorof real?)) (mu real?) (sigma (>=/c 0.0))) real?) ((kurtosis (data (vectorof real?))) real?))]{ Returns the kurtosis of @scheme[data] using the given values of the mean @scheme[mu] and standard deviation @scheme[sigma]. The @tech{kurtosis} measures how sharply peaked a distribution is relative to its width. If @scheme[mu] and @scheme[sigma] are not provided, they are calculated by calls to @scheme[(mean data)] and @scheme[(standard-deviation data mu)].} @;---------- @; Autocorrelation @;---------- @section{Autocorrelation} @defproc*[(((lag-1-autocorrelation (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0)))) (mu real?) (sigma (>=/c 0.0))) real?) ((lag-1-autocorrelation (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) real?))]{ Returns the lag-1 autocorrelation of @scheme[data] using the given value of the mean @scheme[mu]. If @scheme[mu] is not provided, it is calculated by a call to @scheme[(mean data)].} @;---------- @; Covariance @;---------- @section{Covariance} @defproc*[(((covariance (data1 (vectorof real?)) (data2 (and/c (vectorof real?) (lambda (x) (= (vector-length data1) (vector-length data2))))) (mu1 real?) (mu2 real?)) real?) ((covariance (data1 (vectorof real?)) (data2 (and/c (vectorof real?) (lambda (x) (= (vector-length data1) (vector-length data2)))))) real?))]{ Returns the covariance of @scheme[data1] and @scheme[data2], which must both be the same length, using the given values of @scheme[mu1] and @scheme[mu2]. If the values of @scheme[mu1] and @scheme[mu2] are not given, they are calculated using calls to @scheme[(mean data1)] and @scheme[(mean data2)], respectively.} @defproc[(covariance-with-fixed-means (data1 (vectorof real?)) (data2 (and/c (vectorof real?) (lambda (x) (= (vector-length data1) (vector-length data2))))) (mu1 real?) (mu2 real?)) real?]{ Returns the covariance of @scheme[data1] and @scheme[data2], which must both be the same length, when the population means @scheme[mu1] and @scheme[mu2] of the underlying distributions are known @italic{a priori}.} @;---------- @; Wighted Samples @;---------- @section{Weighted Samples} @defproc[(weighted-mean (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data)))))) real?]{ Returns the weighted mean of @scheme[data] using weights @scheme[w].} @defproc*[(((weighted-variance (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?)) (>=/c 0.0)) ((weighted-variance (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data)))))) (>=/c 0.0)))]{ Returns the weighted variance of @scheme[data] using weights @scheme[w] and the given weighted mean @scheme[wmu]. If @scheme[wmu] is not provided, it is calculated by a call ro @scheme[(weighted-mean w data)].} @defproc*[(((weighted-standard-deviation (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?)) (>=/c 0.0)) ((weighted-standard-deviation (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data)))))) (>=/c 0.0)))]{ Returns the weighted standard deviation of @scheme[data] using weights @scheme[w]. The @tech{standard deviation} is defined as the square root of the variance. The result is the square root of the corresponding @scheme[weighted-variance] function.} @defproc[(weighted-variance-with-fixed-mean (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?)) (>=/c 0.0)]{ Returns an unbiased estimate of the weighted variance of @scheme[data] using weights @scheme[w] when the weighted population mean @scheme[wmu] of the underlying population is known @italic{a priori}.} @defproc[(weighted-standard-deviation-with-fixed-mean (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?)) (>=/c 0.0)]{ Returns the weighted standard deviation of @scheme[data] using weights @scheme[w] with a fixed population mean @scheme[mu]. The result is the square root of the @scheme[weighted-variance-with-fixed-mean] function.} @defproc*[(((weighted-absolute-deviation (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?)) (>=/c 0.0)) ((weighted-absolute-deviation (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data)))))) (>=/c 0.0)))]{ Returns the weighted absolute devistion of @scheme[data] using weights @scheme[w] relative to the given value of the weighted mean @scheme[wmu]. If @scheme[wmu] is not provided, it is calculated by a call to @scheme[(weighted-mean w data)]. This function is also useful if you want to calculate the weighted absolute deviation to any value other than the mean, such as zero or the weighted median.} @defproc*[(((weighted-skew (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?) (wsigma (>=/c 0.0))) (>=/c 0.0)) ((weighted-skew (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data)))))) (>=/c 0.0)))]{ Returns the weighted skewness of @scheme[data] using weights @scheme[w] using the given values of the weighted mean @scheme[wmu] and weighted standard deviation @scheme[wsigma]. The @tech{skewness} measures the symmetry of the tails of a distribution. If @scheme[wmu] and @scheme[wsigma] are not provided, they are calculated by calls to @scheme[(weighted-mean w data)] and @scheme[(weighted-standard-deviation w data wmu)].} @defproc*[(((weighted-kurtosis (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data))))) (wmu real?) (wsigma (>=/c 0.0))) (>=/c 0.0)) ((weighted-kurtosis (w (vectorof real?)) (data (and/c (vectorof real?) (lambda (x) (= (vector-length w) (vector-length data)))))) (>=/c 0.0)))]{ Returns the weighted kurtosis of @scheme[data] using weights @scheme[w] using the given values of the weighted mean @scheme[wmu] and weighted standard deviation @scheme[wsigma]. The @tech{kurtosis} measures how sharply peaked a distribution is relative to its width. If @scheme[wmu] and @scheme[wsigma] are not provided, they are calculated by calls to @scheme[(weighted-mean w data)] and @scheme[(weighted-standard-deviation w data wmu)].} @;---------- @; Maximum and Minimum @;---------- @section{Maximum and Minimum} @defproc[(maximum (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) real?]{ Returns the maximum value in @scheme[data].} @defproc[(minimum (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) real?]{ Returns the minimum value in @scheme[data].} @defproc[(minimum-maximum (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) (values real? real?)]{ Returns the minimum and maximum values on @scheme[data] as multiple values.} @defproc[(maximum-index (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) natural-number/c]{ Returns the index of the maximum value in @scheme[data]. When there are several equal maximum elements, the index of the first one is chosen.} @defproc[(minimum-index (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) natural-number/c]{ Returns the index of the minimum value in @scheme[data]. When there are several equal minimum elements, the index of the first one is chosen.} @defproc[(minimum-maximum-index (data (and/c (vectorof real?) (lambda (x) (> (vector-length data) 0))))) (values natural-number/c natural-number/c)]{ Returns the indices of the minimum and maximum values in @scheme[data] as multiple values. When there are several equal minimum or maximum elements, the index of the first ones are chosen.} @;---------- @; Median and Percentiles @;---------- @section{Median and Percentiles} Thw median and percentile 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 percentiled, which ise a scale of 0 to 100). @defproc[(median-from-sorted-data (sorted-data (and/c (vectorof real?) (lambda (x) (> (vector-length sorted-data) 0)) sorted?))) real?]{ Returns the median value of @scheme[sorted-data]. When the dataset has an odd number of elements, the median is the value of element @math{(n - 1)/2}. When the dataset has an even number of elements, the median is the mean of the two nearest middle values, elements @math{(n - 1)/2} and @math{n/2}.} @defproc[(percentile-from-sorted-data (sorted-data (and/c (vectorof real?) (lambda (x) (> (vector-length sorted-data) 0)) sorted?)) (f (real-in 0.0 1.0))) real?]{ Returns a quantile value of @scheme[sorted-data]. The quantile is determined by the value @scheme[f], a fraction between 0 and 1. For example to compute the 75@superscript{th} percentile, @scheme[f] should have the value 0.75. The quantile is found by interpolation using the formula: @math{quantile = 1 - delta(x[i]) + delta(x(i + 1))} where @math{i} is @math{floor((n - 1) × f)} and @math{delta} is @math{(n - 1) × f - 1}.} @;---------- @; Statistics Example @;---------- @section{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. @schememod[ scheme (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))) (do ((i 0 (+ i 1))) ((= i 1000) (void)) ;; Random data from unit gaussian (vector-set! data1 i (random-unit-gaussian)) (vector-set! data2 i (random-unit-gaussian)) ;; Cos^2 weighting (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 .1)) (printf " 20% quantile = ~a~n" (quantile-from-sorted-data data1 .2)) (printf " 30% quantile = ~a~n" (quantile-from-sorted-data data1 .3)) (printf " 40% quantile = ~a~n" (quantile-from-sorted-data data1 .4)) (printf " 50% quantile = ~a~n" (quantile-from-sorted-data data1 .5)) (printf " 60% quantile = ~a~n" (quantile-from-sorted-data data1 .6)) (printf " 70% quantile = ~a~n" (quantile-from-sorted-data data1 .7)) (printf " 80% quantile = ~a~n" (quantile-from-sorted-data data1 .8)) (printf " 90% quantile = ~a~n" (quantile-from-sorted-data data1 .9)))] Produces the following output: @verbatim{ Statistics Example mean = 0.03457693091555611 variance = 1.0285343857083422 standard deviation = 1.0141668431320077 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 = 0.043402934671178436 kurtosis = 0.17722452271704014 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}