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