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