regression.lsp

   1 ;;; -*- mode: lisp -*-
   2 ;;;
   3 ;;; Copyright (c) 2005--2007, by A.J. Rossini <blindglobe@gmail.com>
   4 ;;; See COPYRIGHT file for any additional restrictions (BSD license).
   5 ;;; Since 1991, ANSI was finally finished.  Modified to match ANSI
   6 ;;; Common Lisp.
   7
   8 ;;;;
   9 ;;;; regression.lsp XLISP-STAT regression model proto and methods
  10 ;;;; XLISP-STAT 2.1 Copyright (c) 1990, by Luke Tierney
  11 ;;;; Additions to Xlisp 2.1, Copyright (c) 1989 by David Michael Betz
  12 ;;;; You may give out copies of this software; for conditions see the file
  13 ;;;; COPYING included with this distribution.
  14 ;;;;
  15 ;;;;
  16 ;;;; Incorporates modifications suggested by Sandy Weisberg.
  17 ;;;;
  18
  19 (in-package :cl-user)
  20
  21 (defpackage :lisp-stat-regression-linear
  22   (:use :common-lisp
  23         :lisp-stat-object-system
  24         :lisp-stat-basics
  25         :lisp-stat-compound-data
  26         :lisp-stat-math
  27         :lisp-stat-matrix
  28         :lisp-stat-linalg
  29         :lisp-stat-descriptive-statistics)
  30   (:shadowing-import-from :lisp-stat-object-system
  31                           slot-value call-method call-next-method)
  32   (:shadowing-import-from :lisp-stat-math
  33       expt + - * / ** mod rem abs 1+ 1- log exp sqrt sin cos tan
  34       asin acos atan sinh cosh tanh asinh acosh atanh float random
  35       truncate floor ceiling round minusp zerop plusp evenp oddp
  36       < <= = /= >= > ;; complex
  37       conjugate realpart imagpart phase
  38       min max logand logior logxor lognot ffloor fceiling
  39       ftruncate fround signum cis)
  40   (:export regression-model regression-model-proto x y intercept sweep-matrix
  41            basis weights included total-sum-of-squares residual-sum-of-squares
  42            predictor-names response-name case-labels))
  43
  44 (in-package :lisp-stat-regression-linear)
  45
  46 ;;; Regresion Model Prototype
  47
  48 (defvar regression-model-proto)
  49 (defproto regression-model-proto
  50           '(x y intercept sweep-matrix basis weights
  51               included
  52               total-sum-of-squares
  53               residual-sum-of-squares
  54               predictor-names
  55               response-name
  56               case-labels)
  57           ()
  58           *object*
  59           "Normal Linear Regression Model")
  60
  61 (defun regression-model (x y &key
  62                            (intercept T)
  63                            (print T)
  64                            (weights nil)
  65                            (included (repeat t (length y)))
  66                            predictor-names
  67                            response-name
  68                            case-labels)
  69 "Args: (x y &key (intercept T) (print T) (weights nil)
  70           included predictor-names response-name case-labels)
  71 X           - list of independent variables or X matrix
  72 Y           - dependent variable.
  73 INTERCEPT   - T to include (default), NIL for no intercept
  74 PRINT       - if not NIL print summary information
  75 WEIGHTS     - if supplied should be the same length as Y; error variances are
  76               assumed to be inversely proportional to WEIGHTS
  77 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
  78             - sequences of strings or symbols.
  79 INCLUDED    - if supplied should be the same length as Y, with elements nil
  80               to skip a in computing estimates (but not in residual analysis).
  81 Returns a regression model object. To examine the model further assign the
  82 result to a variable and send it messages.
  83 Example (data are in file absorbtion.lsp in the sample data directory/folder):
  84   (def m (regression-model (list iron aluminum) absorbtion))
  85   (send m :help) (send m :plot-residuals)"
  86   (let ((x (cond
  87             ((matrixp x) x)
  88             ((vectorp x) (list x))
  89             ((and (consp x) (numberp (car x))) (list x))
  90             (t x)))
  91         (m (send regression-model-proto :new)))
  92     (send m :x (if (matrixp x) x (apply #'bind-columns x)))
  93     (send m :y y)
  94     (send m :intercept intercept)
  95     (send m :weights weights)
  96     (send m :included included)
  97     (send m :predictor-names predictor-names)
  98     (send m :response-name response-name)
  99     (send m :case-labels case-labels)
 100     (if print (send m :display))
 101     m))
 102
 103 (defmeth regression-model-proto :isnew ()
 104   (send self :needs-computing t))
 105
 106 (defmeth regression-model-proto :save ()
 107 "Message args: ()
 108 Returns an expression that will reconstruct the regression model."
 109   `(regression-model ',(send self :x)
 110                      ',(send self :y)
 111                      :intercept ',(send self :intercept)
 112                      :weights ',(send self :weights)
 113                      :included ',(send self :included)
 114                      :predictor-names ',(send self :predictor-names)
 115                      :response-name ',(send self :response-name)
 116                      :case-labels ',(send self :case-labels)))
 117
 118 ;;; Computing and Display Methods
 119
 120 (defmeth regression-model-proto :compute ()
 121 "Message args: ()
 122 Recomputes the estimates. For internal use by other messages"
 123   (let* ((included (if-else (send self :included) 1 0))
 124          (x (send self :x))
 125          (y (send self :y))
 126          (intercept (send self :intercept))
 127          (weights (send self :weights))
 128          (w (if weights (* included weights) included))
 129          (m (make-sweep-matrix x y w))
 130          (n (array-dimension x 1))
 131          (p (- (array-dimension m 0) 1))
 132          (tss (aref m p p))
 133          (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation (column-list x)))))
 134          ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
 135          (sweep-result
 136           (if intercept
 137               (sweep-operator m (iseq 1 n) tol)
 138               (sweep-operator m (iseq 0 n) (cons 0.0 tol)))))
 139     (setf (slot-value 'sweep-matrix) (first sweep-result))
 140     (setf (slot-value 'total-sum-of-squares) tss)
 141     (setf (slot-value 'residual-sum-of-squares)
 142           (aref (first sweep-result) p p))
 143     (setf (slot-value 'basis)
 144           (let ((b (remove 0 (second sweep-result))))
 145             (if b (- (reduce #'- (reverse b)) 1)
 146                 (error "no columns could be swept"))))))
 147
 148 (defmeth regression-model-proto :needs-computing (&optional set)
 149    ;;(declare (ignore self))
 150    (if set (setf (slot-value 'sweep-matrix) nil))
 151    (null (slot-value 'sweep-matrix)))
 152
 153 (defmeth regression-model-proto :display ()
 154 "Message args: ()
 155 Prints the least squares regression summary. Variables not used in the fit
 156 are marked as aliased."
 157   (let ((coefs (coerce (send self :coef-estimates) 'list))
 158         (se-s (send self :coef-standard-errors))
 159         (x (send self :x))
 160         (p-names (send self :predictor-names)))
 161     (if (send self :weights)
 162         (format t "~%Weighted Least Squares Estimates:~2%")
 163         (format t "~%Least Squares Estimates:~2%"))
 164     (when (send self :intercept)
 165           (format t "Constant               ~10f   ~A~%"
 166                   (car coefs) (list (car se-s)))
 167           (setf coefs (cdr coefs))
 168           (setf se-s (cdr se-s)))
 169     (dotimes (i (array-dimension x 1))
 170              (cond
 171                ((member i (send self :basis))
 172                 (format t "~22a ~10f   ~A~%"
 173                         (select p-names i) (car coefs) (list (car se-s)))
 174                 (setf coefs (cdr coefs) se-s (cdr se-s)))
 175                (t (format t "~22a    aliased~%" (select p-names i)))))
 176     (format t "~%")
 177     (format t "R Squared:             ~10f~%" (send self :r-squared))
 178     (format t "Sigma hat:             ~10f~%" (send self :sigma-hat))
 179     (format t "Number of cases:       ~10d~%" (send self :num-cases))
 180     (if (/= (send self :num-cases) (send self :num-included))
 181         (format t "Number of cases used:  ~10d~%" (send self :num-included)))
 182     (format t "Degrees of freedom:    ~10d~%" (send self :df))
 183     (format t "~%")))
 184
 185 ;;; Slot accessors and mutators
 186
 187 (defmeth regression-model-proto :x (&optional new-x)
 188 "Message args: (&optional new-x)
 189 With no argument returns the x matrix as supplied to m. With an argument
 190 NEW-X sets the x matrix to NEW-X and recomputes the estimates."
 191   (when (and new-x (matrixp new-x))
 192         (setf (slot-value 'x) new-x)
 193         (send self :needs-computing t))
 194   (slot-value 'x))
 195
 196 (defmeth regression-model-proto :y (&optional new-y)
 197 "Message args: (&optional new-y)
 198 With no argument returns the y sequence as supplied to m. With an argument
 199 NEW-Y sets the y sequence to NEW-Y and recomputes the estimates."
 200   (when (and new-y (or (matrixp new-y) (sequencep new-y)))
 201         (setf (slot-value 'y) new-y)
 202         (send self :needs-computing t))
 203   (slot-value 'y))
 204
 205 (defmeth regression-model-proto :intercept (&optional (val nil set))
 206 "Message args: (&optional new-intercept)
 207 With no argument returns T if the model includes an intercept term, nil if
 208 not. With an argument NEW-INTERCEPT the model is changed to include or
 209 exclude an intercept, according to the value of NEW-INTERCEPT."
 210   (when set
 211         (setf (slot-value 'intercept) val)
 212         (send self :needs-computing t))
 213   (slot-value 'intercept))
 214
 215 (defmeth regression-model-proto :weights (&optional (new-w nil set))
 216 "Message args: (&optional new-w)
 217 With no argument returns the weight sequence as supplied to m; NIL means
 218 an unweighted model. NEW-W sets the weights sequence to NEW-W and
 219 recomputes the estimates."
 220   (when set
 221         (setf (slot-value 'weights) new-w)
 222         (send self :needs-computing t))
 223   (slot-value 'weights))
 224
 225 (defmeth regression-model-proto :total-sum-of-squares ()
 226 "Message args: ()
 227 Returns the total sum of squares around the mean."
 228   (if (send self :needs-computing) (send self :compute))
 229   (slot-value 'total-sum-of-squares))
 230
 231 (defmeth regression-model-proto :residual-sum-of-squares ()
 232 "Message args: ()
 233 Returns the residual sum of squares for the model."
 234   (if (send self :needs-computing) (send self :compute))
 235   (slot-value 'residual-sum-of-squares))
 236
 237 (defmeth regression-model-proto :basis ()
 238 "Message args: ()
 239 Returns the indices of the variables used in fitting the model."
 240   (if (send self :needs-computing) (send self :compute))
 241   (slot-value 'basis))
 242
 243 (defmeth regression-model-proto :sweep-matrix ()
 244 "Message args: ()
 245 Returns the swept sweep matrix. For internal use"
 246   (if (send self :needs-computing) (send self :compute))
 247   (slot-value 'sweep-matrix))
 248
 249 (defmeth regression-model-proto :included (&optional new-included)
 250 "Message args: (&optional new-included)
 251 With no argument,  NIL means a case is not used in calculating estimates, and non-nil means it is used.  NEW-INCLUDED is a sequence of length of y of nil and t to select cases.  Estimates are recomputed."
 252   (when (and new-included
 253              (= (length new-included) (send self :num-cases)))
 254         (setf (slot-value 'included) (copy-seq new-included))
 255         (send self :needs-computing t))
 256   (if (slot-value 'included)
 257       (slot-value 'included)
 258       (repeat t (send self :num-cases))))
 259
 260 (defmeth regression-model-proto :predictor-names (&optional (names nil set))
 261 "Message args: (&optional (names nil set))
 262 With no argument returns the predictor names. NAMES sets the names."
 263   (if set (setf (slot-value 'predictor-names) (mapcar #'string names)))
 264   (let ((p (array-dimension (send self :x) 1))
 265         (p-names (slot-value 'predictor-names)))
 266     (if (not (and p-names (= (length p-names) p)))
 267         (setf (slot-value 'predictor-names)
 268               (mapcar #'(lambda (a) (format nil "Variable ~a" a))
 269                       (iseq 0 (- p 1))))))
 270   (slot-value 'predictor-names))
 271
 272 (defmeth regression-model-proto :response-name (&optional (name "Y" set))
 273    "Message args: (&optional name)
 274 With no argument returns the response name. NAME sets the name."
 275    ;;(declare (ignore self))
 276    (if set (setf (slot-value 'response-name) (if name (string name) "Y")))
 277    (slot-value 'response-name))
 278
 279 (defmeth regression-model-proto :case-labels (&optional (labels nil set))
 280 "Message args: (&optional labels)
 281 With no argument returns the case-labels. LABELS sets the labels."
 282   (if set (setf (slot-value 'case-labels)
 283                 (if labels
 284                     (mapcar #'string labels)
 285                     (mapcar #'(lambda (x) (format nil "~d" x))
 286                             (iseq 0 (- (send self :num-cases) 1))))))
 287   (slot-value 'case-labels))
 288
 289 ;;;
 290 ;;; Other Methods
 291 ;;; None of these methods access any slots directly.
 292 ;;;
 293
 294 (defmeth regression-model-proto :num-cases ()
 295 "Message args: ()
 296 Returns the number of cases in the model."
 297   (length (send self :y)))
 298
 299 (defmeth regression-model-proto :num-included ()
 300 "Message args: ()
 301 Returns the number of cases used in the computations."
 302   (sum (if-else (send self :included) 1 0)))
 303
 304 (defmeth regression-model-proto :num-coefs ()
 305 "Message args: ()
 306 Returns the number of coefficients in the fit model (including the
 307 intercept if the model includes one)."
 308   (if (send self :intercept)
 309       (+ 1 (length (send self :basis)))
 310       (length (send self :basis))))
 311
 312 (defmeth regression-model-proto :df ()
 313 "Message args: ()
 314 Returns the number of degrees of freedom in the model."
 315   (- (send self :num-included) (send self :num-coefs)))
 316
 317 (defmeth regression-model-proto :x-matrix ()
 318 "Message args: ()
 319 Returns the X matrix for the model, including a column of 1's, if
 320 appropriate. Columns of X matrix correspond to entries in basis."
 321   (let ((m (select (send self :x)
 322                    (iseq 0 (- (send self :num-cases) 1))
 323                    (send self :basis))))
 324     (if (send self :intercept)
 325         (bind-columns (repeat 1 (send self :num-cases)) m)
 326         m)))
 327
 328 (defmeth regression-model-proto :leverages ()
 329 "Message args: ()
 330 Returns the diagonal elements of the hat matrix."
 331   (let* ((weights (send self :weights))
 332          (x (send self :x-matrix))
 333          (raw-levs
 334           (matmult (* (matmult x (send self :xtxinv)) x)
 335                    (repeat 1 (send self :num-coefs)))))
 336     (if weights (* weights raw-levs) raw-levs)))
 337
 338 (defmeth regression-model-proto :fit-values ()
 339 "Message args: ()
 340 Returns the fitted values for the model."
 341   (matmult (send self :x-matrix) (send self :coef-estimates)))
 342
 343 (defmeth regression-model-proto :raw-residuals ()
 344 "Message args: ()
 345 Returns the raw residuals for a model."
 346   (- (send self :y) (send self :fit-values)))
 347
 348 (defmeth regression-model-proto :residuals ()
 349 "Message args: ()
 350 Returns the raw residuals for a model without weights. If the model
 351 includes weights the raw residuals times the square roots of the weights
 352 are returned."
 353   (let ((raw-residuals (send self :raw-residuals))
 354         (weights (send self :weights)))
 355     (if weights (* (sqrt weights) raw-residuals) raw-residuals)))
 356
 357 (defmeth regression-model-proto :sum-of-squares ()
 358 "Message args: ()
 359 Returns the error sum of squares for the model."
 360   (send self :residual-sum-of-squares))
 361
 362 (defmeth regression-model-proto :sigma-hat ()
 363 "Message args: ()
 364 Returns the estimated standard deviation of the deviations about the
 365 regression line."
 366   (let ((ss (send self :sum-of-squares))
 367         (df (send self :df)))
 368     (if (/= df 0) (sqrt (/ ss df)))))
 369
 370 ;; for models without an intercept the 'usual' formula for R^2 can give
 371 ;; negative results; hence the max.
 372 (defmeth regression-model-proto :r-squared ()
 373 "Message args: ()
 374 Returns the sample squared multiple correlation coefficient, R squared, for
 375 the regression."
 376   (max (- 1 (/ (send self :sum-of-squares) (send self :total-sum-of-squares)))
 377        0))
 378
 379 (defmeth regression-model-proto :coef-estimates ()
 380 "Message args: ()
 381 Returns the OLS (ordinary least squares) estimates of the regression
 382 coefficients. Entries beyond the intercept correspond to entries in basis."
 383   (let ((n (array-dimension (send self :x) 1))
 384         (indices (if (send self :intercept)
 385                      (cons 0 (+ 1 (send self :basis)))
 386                      (+ 1 (send self :basis))))
 387         (m (send self :sweep-matrix)))
 388     (coerce (compound-data-seq (select m (+ 1 n) indices)) 'list)))
 389
 390 (defmeth regression-model-proto :xtxinv ()
 391 "Message args: ()
 392 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
 393   (let ((indices (if (send self :intercept)
 394                      (cons 0 (1+ (send self :basis)))
 395                      (1+ (send self :basis)))))
 396     (select (send self :sweep-matrix) indices indices)))
 397
 398 (defmeth regression-model-proto :coef-standard-errors ()
 399 "Message args: ()
 400 Returns estimated standard errors of coefficients. Entries beyond the
 401 intercept correspond to entries in basis."
 402   (let ((s (send self :sigma-hat)))
 403     (if s (* (send self :sigma-hat) (sqrt (diagonal (send self :xtxinv)))))))
 404
 405 (defmeth regression-model-proto :studentized-residuals ()
 406 "Message args:  ()
 407 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
 408   (let ((res (send self :residuals))
 409         (lev (send self :leverages))
 410         (sig (send self :sigma-hat))
 411         (inc (send self :included)))
 412     (if-else inc
 413              (/ res (* sig (sqrt (pmax .00001 (- 1 lev)))))
 414              (/ res (* sig (sqrt (+ 1 lev)))))))
 415
 416 (defmeth regression-model-proto :externally-studentized-residuals ()
 417 "Message args:  ()
 418 Computes the externally studentized residuals."
 419   (let* ((res (send self :studentized-residuals))
 420          (df (send self :df)))
 421     (if-else (send self :included)
 422              (* res (sqrt (/ (- df 1) (- df (^ res 2)))))
 423              res)))
 424
 425 (defmeth regression-model-proto :cooks-distances ()
 426 "Message args: ()
 427 Computes Cook's distances."
 428   (let ((lev (send self :leverages))
 429         (res (/ (^ (send self :studentized-residuals) 2)
 430                 (send self :num-coefs))))
 431     (if-else (send self :included) (* res (/ lev (- 1 lev) )) (* res lev))))
 432
 433
 434 (defun plot-points (x y &rest args)
 435   "FIXME!!"
 436   (declare (ignore x y args))
 437   (error "Graphics not implemented yet."))
 438
 439 ;; Can not plot points yet!!
 440 (defmeth regression-model-proto :plot-residuals (&optional x-values)
 441 "Message args: (&optional x-values)
 442 Opens a window with a plot of the residuals. If X-VALUES are not supplied
 443 the fitted values are used. The plot can be linked to other plots with the
 444 link-views function. Returns a plot object."
 445   (plot-points (if x-values x-values (send self :fit-values))
 446                (send self :residuals)
 447                :title "Residual Plot"
 448                :point-labels (send self :case-labels)))
 449
 450 (defmeth regression-model-proto :plot-bayes-residuals
 451   (&optional x-values)
 452   "Message args: (&optional x-values)
 453 Opens a window with a plot of the standardized residuals and two standard
 454 error bars for the posterior distribution of the actual deviations from the
 455 line. See Chaloner and Brant. If X-VALUES are not supplied  the fitted values
 456 are used. The plot can be linked to other plots with the link-views function.
 457 Returns a plot object."
 458   (let* ((r (/ (send self :residuals) (send self :sigma-hat)))
 459          (d (* 2 (sqrt (send self :leverages))))
 460          (low (- r d))
 461          (high (+ r d))
 462          (x-values (if x-values x-values (send self :fit-values)))
 463          (p (plot-points x-values r
 464                          :title "Bayes Residual Plot"
 465                          :point-labels (send self :case-labels))))
 466     (map 'list #'(lambda (a b c d) (send p :plotline a b c d nil))
 467          x-values low x-values high)
 468     (send p :adjust-to-data)
 469     p))