statistics.lsp

   1 ;;; -*- mode: lisp -*-
   2 ;;; Copyright (c) 2005--2007, by A.J. Rossini <blindglobe@gmail.com>
   3 ;;; See COPYRIGHT file for any additional restrictions (BSD license).
   4 ;;; Since 1991, ANSI was finally finished.  Edited for ANSI Common Lisp.
   5
   6 ;;; XLisp-ism's removed to focus on Common Lisp.  Original source from:
   7 ;;;; statistics.lsp XLISP-STAT statistics functions
   8 ;;;; XLISP-STAT 2.1 Copyright (c) 1990, by Luke Tierney
   9 ;;;; Additions to Xlisp 2.1, Copyright (c) 1989 by David Michael Betz
  10 ;;;; You may give out copies of this software; for conditions see the file
  11 ;;;; COPYING included with this distribution.
  12
  13 (defpackage :lisp-stat-descriptive-statistics
  14  (:use :common-lisp
  15        :lisp-stat-data)
  16  (:export ;; descriptive stats
  17           standard-deviation quantile median interquartile-range
  18           fivnum
  19
  20
  21           ;; the following are more matrix-centric
  22           covariance-matrix matrix print-matrix solve
  23           backsolve eigenvalues eigenvectors accumulate cumsum combine
  24           lowess))
  25
  26 (:in-package :lisp-stat-descriptive-statistics)
  27
  28 ;;;;
  29 ;;;; Basic Summary Statistics
  30 ;;;;
  31
  32 (defun standard-deviation (x)
  33 "Args: (x)
  34 Returns the standard deviation of the elements x. Vector reducing."
  35   (let ((n (count-elements x))
  36         (r (- x (mean x))))
  37     (sqrt (* (mean (* r r)) (/ n (- n 1))))))
  38
  39 (defun quantile (x p)
  40 "Args: (x p)
  41 Returns the P-th quantile(s) of sequence X. P can be a number or a sequence."
  42   (let* ((x (sort-data x))
  43          (n (length x))
  44          (np (* p (- n 1)))
  45          (low (floor np))
  46          (high (ceiling np)))
  47     (/ (+ (select x low) (select x high)) 2)))
  48
  49 (defun median (x)
  50 "Args: (x)
  51 Returns the median of the elements of X."
  52   (quantile x 0.5))
  53
  54 (defun interquartile-range (x)
  55 "Args: (number-data)
  56 Returns the interquartile range of the elements of X."
  57   (apply #'- (quantile x '(0.75 0.25))))
  58
  59 (defun fivnum (x)
  60 "Args: (number-data)
  61 Returns the five number summary (min, 1st quartile, medinan, 3rd quartile,
  62 max) of the elements X."
  63   (quantile x '(0 .25 .5 .75 1)))
  64
  65 (defun covariance-matrix (&rest args)
  66 "Args: (&rest args)
  67 Returns the sample covariance matrix of the data columns in ARGS. ARGS may
  68 consist of lists, vectors or matrices."
  69   (let ((columns (apply #'append
  70                         (mapcar #'(lambda (x)
  71                                     (if (matrixp x) (column-list x) (list x)))
  72                                 args))))
  73     (/ (cross-product (apply #'bind-columns
  74                              (- columns (mapcar #'mean columns))))
  75        (- (length (car columns)) 1))))
  76
  77
  78 ;;;;
  79 ;;;; Linear Algebra Functions
  80 ;;;;
  81
  82 (defun matrix (dim data)
  83 "Args: (dim data)
  84 returns a matrix of dimensions DIM initialized using sequence DATA
  85 in row major order."
  86   (let ((dim (coerce dim 'list))
  87         (data (coerce data 'list)))
  88     (make-array dim :initial-contents (split-list data (nth 1 dim)))))
  89
  90 (defun print-matrix (a &optional (stream *standard-output*))
  91 "Args: (matrix &optional stream)
  92 Prints MATRIX to STREAM in a nice form that is still machine readable"
  93   (unless (matrixp a) (error "not a matrix - ~a" a))
  94   (let ((size (min 15 (max (map-elements #'flatsize a)))))
  95     (format stream "#2a(~%")
  96     (dolist (x (row-list a))
  97             (format stream "    (")
  98             (let ((n (length x)))
  99               (dotimes (i n)
 100                        (let ((y (aref x i)))
 101                          (cond
 102                            ((integerp y) (format stream "~vd" size y))
 103                            ((floatp y) (format stream "~vg" size y))
 104                            (t (format stream "~va" size y))))
 105                        (if (< i (- n 1)) (format stream " "))))
 106             (format stream ")~%"))
 107     (format stream "   )~%")
 108     nil))
 109
 110 (defun solve (a b)
 111 "Args: (a b)
 112 Solves A x = B using LU decomposition and backsolving. B can be a sequence
 113 or a matrix."
 114   (let ((lu (lu-decomp a)))
 115     (if (matrixp b)
 116         (apply #'bind-columns
 117                (mapcar #'(lambda (x) (lu-solve lu x)) (column-list b)))
 118         (lu-solve lu b))))
 119
 120 (defun backsolve (a b)
 121 "Args: (a b)
 122 Solves A x = B by backsolving, assuming A is upper triangular. B must be a
 123 sequence. For use with qr-decomp."
 124   (let* ((n (length b))
 125          (sol (make-array n)))
 126     (dotimes (i n)
 127              (let* ((k (- n i 1))
 128                     (val (elt b k)))
 129                (dotimes (j i)
 130                         (let ((l (- n j 1)))
 131                           (setq val (- val (* (aref sol l) (aref a k l))))))
 132                (setf (aref sol k) (/ val (aref a k k)))))
 133     (if (listp b) (coerce sol 'list) sol)))
 134
 135 (defun eigenvalues (a)
 136 "Args: (a)
 137 Returns list of eigenvalues of square, symmetric matrix A"
 138   (first (eigen a)))
 139
 140 (defun eigenvectors (a)
 141 "Args: (a)
 142 Returns list of eigenvectors of square, symmetric matrix A"
 143   (second (eigen a)))
 144
 145 (defun accumulate (f s)
 146 "Args: (f s)
 147 Accumulates elements of sequence S using binary function F.
 148 (accumulate #'+ x) returns the cumulative sum of x."
 149   (let* ((result (list (elt s 0)))
 150          (tail result))
 151     (flet ((acc (dummy x)
 152                 (rplacd tail (list (funcall f (first tail) x)))
 153                 (setf tail (cdr tail))))
 154       (reduce #'acc s))
 155     (if (vectorp s) (coerce result 'vector) result)))
 156
 157 (defun cumsum (x)
 158   "Args: (x)
 159 Returns the cumulative sum of X."
 160   (accumulate #'+ x))
 161
 162 (defun combine (&rest args)
 163   "Args (&rest args)
 164 Returns sequence of elements of all arguments."
 165   (copy-seq (element-seq args)))
 166
 167 (defun lowess (x y &key (f .25) (steps 2) (delta -1) sorted)
 168 "Args: (x y &key (f .25) (steps 2) delta sorted)
 169 Returns (list X YS) with YS the LOWESS fit. F is the fraction of data used for
 170 each point, STEPS is the number of robust iterations. Fits for points within
 171 DELTA of each other are interpolated linearly. If the X values setting SORTED
 172 to T speeds up the computation."
 173   (let ((x (if sorted x (sort-data x)))
 174         (y (if sorted y (select y (order x))))
 175         (delta (if (> delta 0.0) delta (/ (- (max x) (min x)) 50))))
 176     (list x)));; (|base-lowess| x y f steps delta))))