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
  49 ;; The general strategy behind the fitting of models using prototypes
  50 ;; is that we need to think about want the actual fits are, and then
  51 ;; the fits can be used to recompute as components are changes.  One
  52 ;; catch here is that we'd like some notion of trace-ability, in
  53 ;; particular, there is not necessarily a fixed way to take care of the
  54 ;; audit trail.  save-nd-die might be a means of recording the final
  55 ;; approach, but we are challenged by the problem of using advice and
  56 ;; other such features to capture stages and steps that are considered
  57 ;; along the goals of estimating a model.
  58
  59 (defvar regression-model-proto nil
  60   "Prototype for all regression model instances.")
  61 (defproto regression-model-proto
  62     '(x y intercept sweep-matrix basis weights
  63       included
  64       total-sum-of-squares
  65       residual-sum-of-squares
  66       predictor-names
  67       response-name
  68       case-labels
  69       doc)
  70   ()
  71   *object*
  72   "Normal Linear Regression Model")
  73
  74 (defun regression-model (x y &key
  75                          (intercept T)
  76                          (print T)
  77                          (weights nil)
  78                          (included (repeat t (length y)))
  79                          predictor-names
  80                          response-name
  81                          case-labels
  82                          (doc "Undocumented Regression Model Instance")
  83                          (debug T))
  84   "Args: (x y &key (intercept T) (print T) (weights nil)
  85           included predictor-names response-name case-labels)
  86 X           - list of independent variables or X matrix
  87 Y           - dependent variable.
  88 INTERCEPT   - T to include (default), NIL for no intercept
  89 PRINT       - if not NIL print summary information
  90 WEIGHTS     - if supplied should be the same length as Y; error
  91               variances are
  92               assumed to be inversely proportional to WEIGHTS
  93 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
  94             - sequences of strings or symbols.
  95 INCLUDED    - if supplied should be the same length as Y, with
  96               elements nil to skip a in computing estimates (but not
  97               in residual analysis).
  98 Returns a regression model object. To examine the model further assign the
  99 result to a variable and send it messages.
 100 Example (data are in file absorbtion.lsp in the sample data directory):
 101   (def m (regression-model (list iron aluminum) absorbtion))
 102   (send m :help) (send m :plot-residuals)"
 103   (let ((x (cond
 104             ((matrixp x) x)
 105             ((typep x 'vector) (list x))
 106             ((and (consp x)
 107                   (numberp (car x))) (list x))
 108             (t x)))
 109         (m (send regression-model-proto :new)))
 110     (format t "~%")
 111     (send m :doc doc)
 112     (send m :x (if (matrixp x) x (apply #'bind-columns x)))
 113     (send m :y y)
 114     (send m :intercept intercept)
 115     (send m :weights weights)
 116     (send m :included included)
 117     (send m :predictor-names predictor-names)
 118     (send m :response-name response-name)
 119     (send m :case-labels case-labels)
 120     (if debug
 121         (progn
 122           (format t "~%")
 123           (format t "~S~%" (send m :doc))
 124           (format t "X: ~S~%" (send m :x))
 125           (format t "Y: ~S~%" (send m :y))))
 126     (if print (send m :display))
 127     m))
 128
 129 (defmeth regression-model-proto :isnew ()
 130   (send self :needs-computing t))
 131
 132 (defmeth regression-model-proto :save ()
 133 "Message args: ()
 134 Returns an expression that will reconstruct the regression model."
 135   `(regression-model ',(send self :x)
 136                      ',(send self :y)
 137                      :intercept ',(send self :intercept)
 138                      :weights ',(send self :weights)
 139                      :included ',(send self :included)
 140                      :predictor-names ',(send self :predictor-names)
 141                      :response-name ',(send self :response-name)
 142                      :case-labels ',(send self :case-labels)))
 143
 144 ;;; Computing and Display Methods
 145
 146 (defmeth regression-model-proto :compute ()
 147 "Message args: ()
 148 Recomputes the estimates. For internal use by other messages"
 149   (let* ((included (if-else (send self :included) 1 0))
 150          (x (send self :x))
 151          (y (send self :y))
 152          (intercept (send self :intercept))
 153          (weights (send self :weights))
 154          (w (if weights (* included weights) included))
 155          (m (make-sweep-matrix x y w)) ;;; ERROR HERE
 156          (n (array-dimension x 1))
 157          (p (- (array-dimension m 0) 1))
 158          (tss (aref m p p))
 159          (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation (column-list x)))))
 160          ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
 161          (sweep-result
 162           (if intercept
 163               (sweep-operator m (iseq 1 n) tol)
 164               (sweep-operator m (iseq 0 n) (cons 0.0 tol)))))
 165     (format t
 166            "~%REMOVEME: regr-mdl-prto :compute =~A~%~A~%~A~%~A~%~A~%"
 167            sweep-result x y m tss)
 168     (setf (slot-value 'sweep-matrix) (first sweep-result))
 169     (setf (slot-value 'total-sum-of-squares) tss)
 170     (setf (slot-value 'residual-sum-of-squares)
 171           (aref (first sweep-result) p p))
 172     (setf (slot-value 'basis)
 173           (let ((b (remove 0 (second sweep-result))))
 174             (if b (- (reduce #'- (reverse b)) 1)
 175                 (error "no columns could be swept"))))))
 176
 177 (defmeth regression-model-proto :needs-computing (&optional set)
 178 "Message args: ( &optional set )
 179
 180 If value given, sets the flag for whether (re)computation is needed to
 181 update the model fits."
 182    (send self :nop)
 183    (if set (setf (slot-value 'sweep-matrix) nil))
 184    (null (slot-value 'sweep-matrix)))
 185
 186 (defmeth regression-model-proto :display ()
 187 "Message args: ()
 188
 189 Prints the least squares regression summary. Variables not used in the fit
 190 are marked as aliased."
 191   (let ((coefs (coerce (send self :coef-estimates) 'list))
 192         (se-s (send self :coef-standard-errors))
 193         (x (send self :x))
 194         (p-names (send self :predictor-names)))
 195     (if (send self :weights)
 196         (format t "~%Weighted Least Squares Estimates:~2%")
 197         (format t "~%Least Squares Estimates:~2%"))
 198     (when (send self :intercept)
 199           (format t "Constant               ~10f   ~A~%"
 200                   (car coefs) (list (car se-s)))
 201           (setf coefs (cdr coefs))
 202           (setf se-s (cdr se-s)))
 203     (dotimes (i (array-dimension x 1))
 204              (cond
 205                ((member i (send self :basis))
 206                 (format t "~22a ~10f   ~A~%"
 207                         (select p-names i) (car coefs) (list (car se-s)))
 208                 (setf coefs (cdr coefs) se-s (cdr se-s)))
 209                (t (format t "~22a    aliased~%" (select p-names i)))))
 210     (format t "~%")
 211     (format t "R Squared:             ~10f~%" (send self :r-squared))
 212     (format t "Sigma hat:             ~10f~%" (send self :sigma-hat))
 213     (format t "Number of cases:       ~10d~%" (send self :num-cases))
 214     (if (/= (send self :num-cases) (send self :num-included))
 215         (format t "Number of cases used:  ~10d~%" (send self :num-included)))
 216     (format t "Degrees of freedom:    ~10d~%" (send self :df))
 217     (format t "~%")))
 218
 219 ;;; Slot accessors and mutators
 220
 221 (defmeth regression-model-proto :doc (&optional new-doc append)
 222 "Message args: (&optional new-doc)
 223
 224 Returns the DOC-STRING as supplied to m.
 225 Additionally, with an argument NEW-DOC, sets the DOC-STRING to
 226 NEW-DOC.  In this setting, when APPEND is T, append NEW-DOC to DOC
 227 rather than doing replacement."
 228   (send self :nop)
 229   (when (and new-doc (stringp new-doc))
 230         (setf (slot-value 'doc)
 231               (if append
 232                   (concatenate 'string
 233                                (slot-value 'doc)
 234                                new-doc)
 235                   new-doc)))
 236   (slot-value 'doc))
 237
 238
 239 (defmeth regression-model-proto :x (&optional new-x)
 240 "Message args: (&optional new-x)
 241
 242 With no argument returns the x matrix as supplied to m. With an
 243 argument, NEW-X sets the x matrix to NEW-X and recomputes the
 244 estimates."
 245   (when (and new-x (matrixp new-x))
 246         (setf (slot-value 'x) new-x)
 247         (send self :needs-computing t))
 248   (slot-value 'x))
 249
 250 (defmeth regression-model-proto :y (&optional new-y)
 251 "Message args: (&optional new-y)
 252
 253 With no argument returns the y sequence as supplied to m. With an
 254 argument, NEW-Y sets the y sequence to NEW-Y and recomputes the
 255 estimates."
 256   (when (and new-y
 257              (or (matrixp new-y)
 258                  (typep  new-y 'sequence)))
 259     (let ((mat-y (coerce-seq-to-1d-col-matrix new-y)))
 260       (setf (slot-value 'y) new-y)
 261       (send self :needs-computing t)))
 262   (slot-value 'y))
 263
 264 (defmeth regression-model-proto :intercept (&optional (val nil set))
 265 "Message args: (&optional new-intercept)
 266
 267 With no argument returns T if the model includes an intercept term,
 268 nil if not. With an argument NEW-INTERCEPT the model is changed to
 269 include or exclude an intercept, according to the value of
 270 NEW-INTERCEPT."
 271   (when set
 272         (setf (slot-value 'intercept) val)
 273         (send self :needs-computing t))
 274   (slot-value 'intercept))
 275
 276 (defmeth regression-model-proto :weights (&optional (new-w nil set))
 277 "Message args: (&optional new-w)
 278
 279 With no argument returns the weight sequence as supplied to m; NIL
 280 means an unweighted model. NEW-W sets the weights sequence to NEW-W
 281 and recomputes the estimates."
 282   (when set
 283         (setf (slot-value 'weights) new-w)
 284         (send self :needs-computing t))
 285   (slot-value 'weights))
 286
 287 (defmeth regression-model-proto :total-sum-of-squares ()
 288 "Message args: ()
 289
 290 Returns the total sum of squares around the mean."
 291   (if (send self :needs-computing) (send self :compute))
 292   (slot-value 'total-sum-of-squares))
 293
 294 (defmeth regression-model-proto :residual-sum-of-squares ()
 295 "Message args: ()
 296
 297 Returns the residual sum of squares for the model."
 298   (if (send self :needs-computing) (send self :compute))
 299   (slot-value 'residual-sum-of-squares))
 300
 301 (defmeth regression-model-proto :basis ()
 302 "Message args: ()
 303
 304 Returns the indices of the variables used in fitting the model, in a
 305 sequence."
 306   (if (send self :needs-computing)
 307       (send self :compute))
 308   (if (typep (slot-value 'basis) 'sequence)
 309       (slot-value 'basis)
 310       (list (slot-value 'basis))))
 311
 312
 313 (defmeth regression-model-proto :sweep-matrix ()
 314 "Message args: ()
 315
 316 Returns the swept sweep matrix. For internal use"
 317   (if (send self :needs-computing)
 318       (send self :compute))
 319   (slot-value 'sweep-matrix))
 320
 321 (defmeth regression-model-proto :included (&optional new-included)
 322 "Message args: (&optional new-included)
 323
 324 With no argument, NIL means a case is not used in calculating
 325 estimates, and non-nil means it is used.  NEW-INCLUDED is a sequence
 326 of length of y of nil and t to select cases.  Estimates are
 327 recomputed."
 328   (when (and new-included
 329              (= (length new-included) (send self :num-cases)))
 330         (setf (slot-value 'included) (copy-seq new-included))
 331         (send self :needs-computing t))
 332   (if (slot-value 'included)
 333       (slot-value 'included)
 334       (repeat t (send self :num-cases))))
 335
 336 (defmeth regression-model-proto :predictor-names (&optional (names nil set))
 337 "Message args: (&optional (names nil set))
 338
 339 With no argument returns the predictor names. NAMES sets the names."
 340   (if set (setf (slot-value 'predictor-names) (mapcar #'string names)))
 341   (let ((p (array-dimension (send self :x) 1))
 342         (p-names (slot-value 'predictor-names)))
 343     (if (not (and p-names (= (length p-names) p)))
 344         (setf (slot-value 'predictor-names)
 345               (mapcar #'(lambda (a) (format nil "Variable ~a" a))
 346                       (iseq 0 (- p 1))))))
 347   (slot-value 'predictor-names))
 348
 349 (defmeth regression-model-proto :response-name (&optional (name "Y" set))
 350    "Message args: (&optional name)
 351
 352 With no argument returns the response name. NAME sets the name."
 353    (send self :nop)
 354    (if set (setf (slot-value 'response-name) (if name (string name) "Y")))
 355    (slot-value 'response-name))
 356
 357 (defmeth regression-model-proto :case-labels (&optional (labels nil set))
 358 "Message args: (&optional labels)
 359 With no argument returns the case-labels. LABELS sets the labels."
 360   (if set (setf (slot-value 'case-labels)
 361                 (if labels
 362                     (mapcar #'string labels)
 363                     (mapcar #'(lambda (x) (format nil "~d" x))
 364                             (iseq 0 (- (send self :num-cases) 1))))))
 365   (slot-value 'case-labels))
 366
 367 ;;;
 368 ;;; Other Methods
 369 ;;; None of these methods access any slots directly.
 370 ;;;
 371
 372 (defmeth regression-model-proto :num-cases ()
 373 "Message args: ()
 374 Returns the number of cases in the model."
 375   (length (send self :y)))
 376
 377 (defmeth regression-model-proto :num-included ()
 378 "Message args: ()
 379 Returns the number of cases used in the computations."
 380   (sum (if-else (send self :included) 1 0)))
 381
 382 (defmeth regression-model-proto :num-coefs ()
 383 "Message args: ()
 384 Returns the number of coefficients in the fit model (including the
 385 intercept if the model includes one)."
 386   (if (send self :intercept)
 387       (+ 1 (length (send self :basis)))
 388       (length (send self :basis))))
 389
 390 (defmeth regression-model-proto :df ()
 391 "Message args: ()
 392 Returns the number of degrees of freedom in the model."
 393   (- (send self :num-included) (send self :num-coefs)))
 394
 395 (defmeth regression-model-proto :x-matrix ()
 396 "Message args: ()
 397 Returns the X matrix for the model, including a column of 1's, if
 398 appropriate. Columns of X matrix correspond to entries in basis."
 399   (let ((m (select (send self :x)
 400                    (iseq 0 (- (send self :num-cases) 1))
 401                    (send self :basis))))
 402     (if (send self :intercept)
 403         (bind-columns (repeat 1 (send self :num-cases)) m)
 404         m)))
 405
 406 (defmeth regression-model-proto :leverages ()
 407 "Message args: ()
 408 Returns the diagonal elements of the hat matrix."
 409   (let* ((weights (send self :weights))
 410          (x (send self :x-matrix))
 411          (raw-levs
 412           (matmult (* (matmult x (send self :xtxinv)) x)
 413                    (repeat 1 (send self :num-coefs)))))
 414     (if weights (* weights raw-levs) raw-levs)))
 415
 416 (defmeth regression-model-proto :fit-values ()
 417 "Message args: ()
 418 Returns the fitted values for the model."
 419   (matmult (send self :x-matrix) (send self :coef-estimates)))
 420
 421 (defmeth regression-model-proto :raw-residuals ()
 422 "Message args: ()
 423 Returns the raw residuals for a model."
 424   (- (send self :y) (send self :fit-values)))
 425
 426 (defmeth regression-model-proto :residuals ()
 427 "Message args: ()
 428 Returns the raw residuals for a model without weights. If the model
 429 includes weights the raw residuals times the square roots of the weights
 430 are returned."
 431   (let ((raw-residuals (send self :raw-residuals))
 432         (weights (send self :weights)))
 433     (if weights (* (sqrt weights) raw-residuals) raw-residuals)))
 434
 435 (defmeth regression-model-proto :sum-of-squares ()
 436 "Message args: ()
 437 Returns the error sum of squares for the model."
 438   (send self :residual-sum-of-squares))
 439
 440 (defmeth regression-model-proto :sigma-hat ()
 441 "Message args: ()
 442 Returns the estimated standard deviation of the deviations about the
 443 regression line."
 444   (let ((ss (send self :sum-of-squares))
 445         (df (send self :df)))
 446     (if (/= df 0) (sqrt (/ ss df)))))
 447
 448 ;; for models without an intercept the 'usual' formula for R^2 can give
 449 ;; negative results; hence the max.
 450 (defmeth regression-model-proto :r-squared ()
 451 "Message args: ()
 452 Returns the sample squared multiple correlation coefficient, R squared, for
 453 the regression."
 454   (max (- 1 (/ (send self :sum-of-squares) (send self :total-sum-of-squares)))
 455        0))
 456
 457 (defmeth regression-model-proto :coef-estimates ()
 458 "Message args: ()
 459
 460 Returns the OLS (ordinary least squares) estimates of the regression
 461 coefficients. Entries beyond the intercept correspond to entries in
 462 basis."
 463   (let ((n (array-dimension (send self :x) 1))
 464         (indices (flatten-list
 465                   (if (send self :intercept)
 466                      (list 0 (+ 1 (send self :basis)))  ;; was cons -- why?
 467                      (list (+ 1 (send self :basis))))))
 468         (m (send self :sweep-matrix)))
 469     (format t "~%REMOVEME2: Coef-ests: ~A ~% ~A ~% ~A ~% ~A"
 470             m n indices (send self :basis))
 471     (coerce (compound-data-seq (select m (+ 1 n) indices)) 'list)))
 472
 473 (defmeth regression-model-proto :xtxinv ()
 474 "Message args: ()
 475 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
 476   (let ((indices (if (send self :intercept)
 477                      (cons 0 (1+ (send self :basis)))
 478                      (1+ (send self :basis)))))
 479     (select (send self :sweep-matrix) indices indices)))
 480
 481 (defmeth regression-model-proto :coef-standard-errors ()
 482 "Message args: ()
 483 Returns estimated standard errors of coefficients. Entries beyond the
 484 intercept correspond to entries in basis."
 485   (let ((s (send self :sigma-hat)))
 486     (if s (* (send self :sigma-hat) (sqrt (diagonal (send self :xtxinv)))))))
 487
 488 (defmeth regression-model-proto :studentized-residuals ()
 489 "Message args:  ()
 490 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
 491   (let ((res (send self :residuals))
 492         (lev (send self :leverages))
 493         (sig (send self :sigma-hat))
 494         (inc (send self :included)))
 495     (if-else inc
 496              (/ res (* sig (sqrt (pmax .00001 (- 1 lev)))))
 497              (/ res (* sig (sqrt (+ 1 lev)))))))
 498
 499 (defmeth regression-model-proto :externally-studentized-residuals ()
 500 "Message args:  ()
 501 Computes the externally studentized residuals."
 502   (let* ((res (send self :studentized-residuals))
 503          (df (send self :df)))
 504     (if-else (send self :included)
 505              (* res (sqrt (/ (- df 1) (- df (^ res 2)))))
 506              res)))
 507
 508 (defmeth regression-model-proto :cooks-distances ()
 509 "Message args: ()
 510 Computes Cook's distances."
 511   (let ((lev (send self :leverages))
 512         (res (/ (^ (send self :studentized-residuals) 2)
 513                 (send self :num-coefs))))
 514     (if-else (send self :included) (* res (/ lev (- 1 lev) )) (* res lev))))
 515
 516
 517 (defun plot-points (x y &rest args)
 518   "FIXME!!"
 519   (declare (ignore x y args))
 520   (error "Graphics not implemented yet."))
 521
 522 ;; Can not plot points yet!!
 523 (defmeth regression-model-proto :plot-residuals (&optional x-values)
 524 "Message args: (&optional x-values)
 525 Opens a window with a plot of the residuals. If X-VALUES are not supplied
 526 the fitted values are used. The plot can be linked to other plots with the
 527 link-views function. Returns a plot object."
 528   (plot-points (if x-values x-values (send self :fit-values))
 529                (send self :residuals)
 530                :title "Residual Plot"
 531                :point-labels (send self :case-labels)))
 532
 533 (defmeth regression-model-proto :plot-bayes-residuals
 534   (&optional x-values)
 535   "Message args: (&optional x-values)
 536
 537 Opens a window with a plot of the standardized residuals and two
 538 standard error bars for the posterior distribution of the actual
 539 deviations from the line. See Chaloner and Brant. If X-VALUES are not
 540 supplied the fitted values are used. The plot can be linked to other
 541 plots with the link-views function.  Returns a plot object."
 542
 543   (let* ((r (/ (send self :residuals)
 544                (send self :sigma-hat)))
 545          (d (* 2 (sqrt (send self :leverages))))
 546          (low (- r d))
 547          (high (+ r d))
 548          (x-values (if x-values x-values (send self :fit-values)))
 549          (p (plot-points x-values r
 550                          :title "Bayes Residual Plot"
 551                          :point-labels (send self :case-labels))))
 552     (map 'list #'(lambda (a b c d) (send p :plotline a b c d nil))
 553          x-values low x-values high)
 554     (send p :adjust-to-data)
 555     p))