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