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 (in-package :cl-user)
  14
  15 (defpackage :lisp-stat-descriptive-statistics
  16  (:use :common-lisp
  17        :lisp-stat-data)
  18  (:export ;; descriptive stats
  19           standard-deviation quantile median interquartile-range
  20           fivnum
  21
  22           sample
  23
  24           ;; the following are more matrix-centric
  25           covariance-matrix matrix print-matrix solve
  26           backsolve eigenvalues eigenvectors accumulate cumsum combine
  27           lowess))
  28
  29 (in-package :lisp-stat-descriptive-statistics)
  30
  31 ;;;;
  32 ;;;; Basic Summary Statistics
  33 ;;;;
  34
  35 (defun standard-deviation (x)
  36 "Args: (x)
  37 Returns the standard deviation of the elements x. Vector reducing."
  38   (let ((n (count-elements x))
  39         (r (- x (mean x))))
  40     (sqrt (* (mean (* r r)) (/ n (- n 1))))))
  41
  42 (defun quantile (x p)
  43 "Args: (x p)
  44 Returns the P-th quantile(s) of sequence X. P can be a number or a sequence."
  45   (let* ((x (sort-data x))
  46          (n (length x))
  47          (np (* p (- n 1)))
  48          (low (floor np))
  49          (high (ceiling np)))
  50     (/ (+ (select x low) (select x high)) 2)))
  51
  52 (defun median (x)
  53 "Args: (x)
  54 Returns the median of the elements of X."
  55   (quantile x 0.5))
  56
  57 (defun interquartile-range (x)
  58 "Args: (number-data)
  59 Returns the interquartile range of the elements of X."
  60   (apply #'- (quantile x '(0.75 0.25))))
  61
  62 (defun fivnum (x)
  63 "Args: (number-data)
  64 Returns the five number summary (min, 1st quartile, medinan, 3rd quartile,
  65 max) of the elements X."
  66   (quantile x '(0 .25 .5 .75 1)))
  67
  68 (defun covariance-matrix (&rest args)
  69 "Args: (&rest args)
  70 Returns the sample covariance matrix of the data columns in ARGS. ARGS may
  71 consist of lists, vectors or matrices."
  72   (let ((columns (apply #'append
  73                         (mapcar #'(lambda (x)
  74                                     (if (matrixp x) (column-list x) (list x)))
  75                                 args))))
  76     (/ (cross-product (apply #'bind-columns
  77                              (- columns (mapcar #'mean columns))))
  78        (- (length (car columns)) 1))))
  79
  80 ;;;; Sampling / Resampling
  81
  82 (defun sample (x ssize &optional replace)
  83 "Args: (x n &optional (replace nil))
  84 Returns a list of a random sample of size N from sequence X drawn with or
  85 without replacement."
  86   (check-sequence x)
  87   (let ((n (length x))
  88         (x (if (consp x) (coerce x 'vector) (copy-vector x)))
  89         (result nil))
  90     (if (< 0 n)
  91         (dotimes (i ssize result)
  92                  (let ((j (if replace (random n) (+ i (random (- n i))))))
  93                    (setf result (cons (aref x j) result))
  94                    (unless replace     ;; swap elements i and j
  95                            (let ((temp (aref x i)))
  96                              (setf (aref x i) (aref x j))
  97                              (setf (aref x j) temp))))))))
  98
  99
 100
 101 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 102 ;;;;
 103 ;;;;                         Sorting Functions
 104 ;;;;
 105 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
 106
 107 (defun sort-data (x)
 108 "Args: (sequence)
 109 Returns a sequence with the numbers or strings in the sequence X in order."
 110   (flet ((less (x y) (if (numberp x) (< x y) (string-lessp x y))))
 111     (stable-sort (copy-seq (compound-data-seq x)) #'less)))
 112
 113 (defun order (x)
 114 "Args (x)
 115 Returns a sequence of the indices of elements in the sequence of numbers
 116 or strings X in order."
 117   (let* ((seq (compound-data-seq x))
 118          (type (if (consp seq) 'list 'vector))
 119          (i -1))
 120     (flet ((entry (x) (setf i (+ i 1)) (list x i))
 121            (less (a b)
 122                  (let ((x (first a))
 123                        (y (first b)))
 124                    (if (numberp x) (< x y) (string-lessp x y)))))
 125       (let ((sorted-seq (stable-sort (map type #'entry seq) #'less)))
 126         (map type #'second sorted-seq)))))
 127
 128 ;; this isn't destructive -- do we document destructive only, or any
 129 ;; variant?
 130 (defun rank (x)
 131 "Args (x)
 132 Returns a sequence with the elements of the list or array of numbers or
 133 strings X replaced by their ranks."
 134   (let ((ranked-seq (order (order x))))
 135     (make-compound-data (compound-data-shape x) ranked-seq)))
 136
 137 (defun mean (x)
 138 "Args: (x)
 139 Returns the mean of the elements x. Vector reducing."
 140   (let ((mean 0.0)
 141         (count 0.0))
 142     (labels ((add-to-mean (x)
 143               (let ((count+1 (+ count 1.0)))
 144                 (setf mean (+ (* (/ count count+1) mean) (* (/ count+1) x)))
 145                 (setf count count+1)))
 146              (find-mean (x)
 147                (if (numberp x)
 148                  (add-to-mean x)
 149                  (let ((seq (compound-data-seq x)))
 150                    (if (consp seq)
 151                      (dolist (x seq)
 152                        (if (numberp x) (add-to-mean x) (find-mean x)))
 153                      (let ((n (length seq)))
 154                        (dotimes (i n)
 155                          (declare (fixnum i))
 156                          (let ((x (aref seq i)))
 157                            (if (numberp x)
 158                                (add-to-mean x)
 159                                (find-mean x))))))))))
 160       (find-mean x)
 161       mean)))
 162
 163 ;;;;
 164 ;;;; Linear Algebra Functions
 165 ;;;;
 166
 167 (defun matrix (dim data)
 168 "Args: (dim data)
 169 returns a matrix of dimensions DIM initialized using sequence DATA
 170 in row major order."
 171   (let ((dim (coerce dim 'list))
 172         (data (coerce data 'list)))
 173     (make-array dim :initial-contents (split-list data (nth 1 dim)))))
 174
 175 (defun print-matrix (a &optional (stream *standard-output*))
 176 "Args: (matrix &optional stream)
 177 Prints MATRIX to STREAM in a nice form that is still machine readable"
 178   (unless (matrixp a) (error "not a matrix - ~a" a))
 179   (let ((size (min 15 (max (map-elements #'flatsize a)))))
 180     (format stream "#2a(~%")
 181     (dolist (x (row-list a))
 182             (format stream "    (")
 183             (let ((n (length x)))
 184               (dotimes (i n)
 185                        (let ((y (aref x i)))
 186                          (cond
 187                            ((integerp y) (format stream "~vd" size y))
 188                            ((floatp y) (format stream "~vg" size y))
 189                            (t (format stream "~va" size y))))
 190                        (if (< i (- n 1)) (format stream " "))))
 191             (format stream ")~%"))
 192     (format stream "   )~%")
 193     nil))
 194
 195 (defun solve (a b)
 196 "Args: (a b)
 197 Solves A x = B using LU decomposition and backsolving. B can be a sequence
 198 or a matrix."
 199   (let ((lu (lu-decomp a)))
 200     (if (matrixp b)
 201         (apply #'bind-columns
 202                (mapcar #'(lambda (x) (lu-solve lu x)) (column-list b)))
 203         (lu-solve lu b))))
 204
 205 (defun backsolve (a b)
 206 "Args: (a b)
 207 Solves A x = B by backsolving, assuming A is upper triangular. B must be a
 208 sequence. For use with qr-decomp."
 209   (let* ((n (length b))
 210          (sol (make-array n)))
 211     (dotimes (i n)
 212              (let* ((k (- n i 1))
 213                     (val (elt b k)))
 214                (dotimes (j i)
 215                         (let ((l (- n j 1)))
 216                           (setq val (- val (* (aref sol l) (aref a k l))))))
 217                (setf (aref sol k) (/ val (aref a k k)))))
 218     (if (listp b) (coerce sol 'list) sol)))
 219
 220 (defun eigenvalues (a)
 221 "Args: (a)
 222 Returns list of eigenvalues of square, symmetric matrix A"
 223   (first (eigen a)))
 224
 225 (defun eigenvectors (a)
 226 "Args: (a)
 227 Returns list of eigenvectors of square, symmetric matrix A"
 228   (second (eigen a)))
 229
 230 (defun accumulate (f s)
 231 "Args: (f s)
 232 Accumulates elements of sequence S using binary function F.
 233 (accumulate #'+ x) returns the cumulative sum of x."
 234   (let* ((result (list (elt s 0)))
 235          (tail result))
 236     (flet ((acc (dummy x)
 237                 (rplacd tail (list (funcall f (first tail) x)))
 238                 (setf tail (cdr tail))))
 239       (reduce #'acc s))
 240     (if (vectorp s) (coerce result 'vector) result)))
 241
 242 (defun cumsum (x)
 243   "Args: (x)
 244 Returns the cumulative sum of X."
 245   (accumulate #'+ x))
 246
 247 (defun combine (&rest args)
 248   "Args (&rest args)
 249 Returns sequence of elements of all arguments."
 250   (copy-seq (element-seq args)))
 251
 252 (defun lowess (x y &key (f .25) (steps 2) (delta -1) sorted)
 253 "Args: (x y &key (f .25) (steps 2) delta sorted)
 254 Returns (list X YS) with YS the LOWESS fit. F is the fraction of data used for
 255 each point, STEPS is the number of robust iterations. Fits for points within
 256 DELTA of each other are interpolated linearly. If the X values setting SORTED
 257 to T speeds up the computation."
 258   (let ((x (if sorted x (sort-data x)))
 259         (y (if sorted y (select y (order x))))
 260         (delta (if (> delta 0.0) delta (/ (- (max x) (min x)) 50))))
 261     (list x)));; (|base-lowess| x y f steps delta))))