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