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 :lisp-stat)
  20
  21 (export '(regression-model regression-model-proto x y intercept sweep-matrix
  22           basis weights included total-sum-of-squares residual-sum-of-squares
  23           predictor-names response-name case-labels))
  24
  25 ;;;;
  26 ;;;;
  27 ;;;; Regresion Model Prototype
  28 ;;;;
  29 ;;;;
  30
  31 (defproto regression-model-proto
  32           '(x y intercept sweep-matrix basis weights
  33               included
  34               total-sum-of-squares
  35               residual-sum-of-squares
  36               predictor-names
  37               response-name
  38               case-labels)
  39           ()
  40           *object*
  41           "Normal Linear Regression Model")
  42
  43 ;; The doc for this function string is at the limit of XLISP's string
  44 ;; constant size - making it longer may cause problems
  45 (defun regression-model (x y &key
  46                            (intercept T)
  47                            (print T)
  48                            weights
  49                            (included (repeat t (length y)))
  50                            predictor-names
  51                            response-name
  52                            case-labels)
  53 "Args: (x y &key (intercept T) (print T) weights
  54           included predictor-names response-name case-labels)
  55 X           - list of independent variables or X matrix
  56 Y           - dependent variable.
  57 INTERCEPT   - T to include (default), NIL for no intercept
  58 PRINT       - if not NIL print summary information
  59 WEIGHTS     - if supplied should be the same length as Y; error variances are
  60                assumed to be inversely proportional to WEIGHTS
  61 PREDICTOR-NAMES
  62 RESPONSE-NAME
  63 CASE-LABELS - sequences of strings or symbols.
  64 INCLUDED    - if supplied should be the same length as Y, with elements nil
  65               to skip a in computing estimates (but not in residual analysis).
  66 Returns a regression model object. To examine the model further assign the
  67 result to a variable and send it messages.
  68 Example (data are in file absorbtion.lsp in the sample data directory/folder):
  69   (def m (regression-model (list iron aluminum) absorbtion))
  70   (send m :help)
  71   (send m :plot-residuals)"
  72   (let ((x (cond
  73              ((matrixp x) x)
  74              ((vectorp x) (list x))
  75              ((and (consp x) (numberp (car x))) (list x))
  76              (t x)))
  77         (m (send regression-model-proto :new)))
  78     (send m :x (if (matrixp x) x (apply #'bind-columns x)))
  79     (send m :y y)
  80     (send m :intercept intercept)
  81     (send m :weights weights)
  82     (send m :included included)
  83     (send m :predictor-names predictor-names)
  84     (send m :response-name response-name)
  85     (send m :case-labels case-labels)
  86     (if print (send m :display))
  87     m))
  88
  89 (defmeth regression-model-proto :isnew () (send self :needs-computing t))
  90
  91 (defmeth regression-model-proto :save ()
  92 "Message args: ()
  93 Returns an expression that will reconstruct the regression model."
  94   `(regression-model ',(send self :x)
  95                      ',(send self :y)
  96                      :intercept ',(send self :intercept)
  97                      :weights ',(send self :weights)
  98                      :included ',(send self :included)
  99                      :predictor-names ',(send self :predictor-names)
 100                      :response-name ',(send self :response-name)
 101                      :case-labels ',(send self :case-labels)))
 102
 103 ;;;
 104 ;;; Computing and Display Methods
 105 ;;;
 106
 107 (defmeth regression-model-proto :compute ()
 108 "Message args: ()
 109 Recomputes the estimates. For internal use by other messages"
 110   (let* ((included (if-else (send self :included) 1 0))
 111          (x (send self :x))
 112          (y (send self :y))
 113          (intercept (send self :intercept))
 114          (weights (send self :weights))
 115          (w (if weights (* included weights) included))
 116          (m (make-sweep-matrix x y w))
 117          (n (array-dimension x 1))
 118          (p (- (array-dimension m 0) 1))
 119          (tss (aref m p p))
 120          (tol (* .0001 (mapcar #'standard-deviation (column-list x))))
 121          (sweep-result
 122           (if intercept
 123               (sweep-operator m (iseq 1 n) tol)
 124               (sweep-operator m (iseq 0 n) (cons 0.0 tol)))))
 125     (setf (slot-value 'sweep-matrix) (first sweep-result))
 126     (setf (slot-value 'total-sum-of-squares) tss)
 127     (setf (slot-value 'residual-sum-of-squares)
 128           (aref (first sweep-result) p p))
 129     (setf (slot-value 'basis)
 130           (let ((b (remove 0 (second sweep-result))))
 131             (if b
 132                 (- (reverse b) 1)
 133                 (error "no columns could be swept"))))))
 134
 135 (defmeth regression-model-proto :needs-computing (&optional set)
 136   (if set (setf (slot-value 'sweep-matrix) nil))
 137   (null (slot-value 'sweep-matrix)))
 138
 139 (defmeth regression-model-proto :display ()
 140 "Message args: ()
 141 Prints the least squares regression summary. Variables not used in the fit
 142 are marked as aliased."
 143   (let ((coefs (coerce (send self :coef-estimates) 'list))
 144         (se-s (send self :coef-standard-errors))
 145         (x (send self :x))
 146         (p-names (send self :predictor-names)))
 147     (if (send self :weights)
 148         (format t "~%Weighted Least Squares Estimates:~2%")
 149         (format t "~%Least Squares Estimates:~2%"))
 150     (when (send self :intercept)
 151           (format t "Constant               ~10f   ~A~%"
 152                   (car coefs) (list (car se-s)))
 153           (setf coefs (cdr coefs))
 154           (setf se-s (cdr se-s)))
 155     (dotimes (i (array-dimension x 1))
 156              (cond
 157                ((member i (send self :basis))
 158                 (format t "~22a ~10f   ~A~%"
 159                         (select p-names i) (car coefs) (list (car se-s)))
 160                 (setf coefs (cdr coefs) se-s (cdr se-s)))
 161                (t (format t "~22a    aliased~%" (select p-names i)))))
 162     (format t "~%")
 163     (format t "R Squared:             ~10f~%" (send self :r-squared))
 164     (format t "Sigma hat:             ~10f~%" (send self :sigma-hat))
 165     (format t "Number of cases:       ~10d~%" (send self :num-cases))
 166     (if (/= (send self :num-cases) (send self :num-included))
 167         (format t "Number of cases used:  ~10d~%" (send self :num-included)))
 168     (format t "Degrees of freedom:    ~10d~%" (send self :df))
 169     (format t "~%")))
 170
 171 ;;;
 172 ;;; Slot accessors and mutators
 173 ;;;
 174
 175 (defmeth regression-model-proto :x (&optional new-x)
 176 "Message args: (&optional new-x)
 177 With no argument returns the x matrix as supplied to m. With an argument
 178 NEW-X sets the x matrix to NEW-X and recomputes the estimates."
 179   (when (and new-x (matrixp new-x))
 180         (setf (slot-value 'x) new-x)
 181         (send self :needs-computing t))
 182   (slot-value 'x))
 183
 184 (defmeth regression-model-proto :y (&optional new-y)
 185 "Message args: (&optional new-y)
 186 With no argument returns the y sequence as supplied to m. With an argument
 187 NEW-Y sets the y sequence to NEW-Y and recomputes the estimates."
 188   (when (and new-y (or (matrixp new-y) (sequencep new-y)))
 189         (setf (slot-value 'y) new-y)
 190         (send self :needs-computing t))
 191   (slot-value 'y))
 192
 193 (defmeth regression-model-proto :intercept (&optional (val nil set))
 194 "Message args: (&optional new-intercept)
 195 With no argument returns T if the model includes an intercept term, nil if
 196 not. With an argument NEW-INTERCEPT the model is changed to include or
 197 exclude an intercept, according to the value of NEW-INTERCEPT."
 198   (when set
 199         (setf (slot-value 'intercept) val)
 200         (send self :needs-computing t))
 201   (slot-value 'intercept))
 202
 203 (defmeth regression-model-proto :weights (&optional (new-w nil set))
 204 "Message args: (&optional new-w)
 205 With no argument returns the weight sequence as supplied to m; NIL means
 206 an unweighted model. NEW-W sets the weights sequence to NEW-W and
 207 recomputes the estimates."
 208   (when set
 209         (setf (slot-value 'weights) new-w)
 210         (send self :needs-computing t))
 211   (slot-value 'weights))
 212
 213 (defmeth regression-model-proto :total-sum-of-squares ()
 214 "Message args: ()
 215 Returns the total sum of squares around the mean."
 216   (if (send self :needs-computing) (send self :compute))
 217   (slot-value 'total-sum-of-squares))
 218
 219 (defmeth regression-model-proto :residual-sum-of-squares ()
 220 "Message args: ()
 221 Returns the residual sum of squares for the model."
 222   (if (send self :needs-computing) (send self :compute))
 223   (slot-value 'residual-sum-of-squares))
 224
 225 (defmeth regression-model-proto :basis ()
 226 "Message args: ()
 227 Returns the indices of the variables used in fitting the model."
 228   (if (send self :needs-computing) (send self :compute))
 229   (slot-value 'basis))
 230
 231 (defmeth regression-model-proto :sweep-matrix ()
 232 "Message args: ()
 233 Returns the swept sweep matrix. For internal use"
 234   (if (send self :needs-computing) (send self :compute))
 235   (slot-value 'sweep-matrix))
 236
 237 (defmeth regression-model-proto :included (&optional new-included)
 238 "Message args: (&optional new-included)
 239 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."
 240   (when (and new-included
 241              (= (length new-included) (send self :num-cases)))
 242         (setf (slot-value 'included) (copy-seq new-included))
 243         (send self :needs-computing t))
 244   (if (slot-value 'included)
 245       (slot-value 'included)
 246       (repeat t (send self :num-cases))))
 247
 248 (defmeth regression-model-proto :predictor-names (&optional (names nil set))
 249 "Message args: (&optional (names nil set))
 250 With no argument returns the predictor names. NAMES sets the names."
 251   (if set (setf (slot-value 'predictor-names) (mapcar #'string names)))
 252   (let ((p (array-dimension (send self :x) 1))
 253         (p-names (slot-value 'predictor-names)))
 254     (if (not (and p-names (= (length p-names) p)))
 255         (setf (slot-value 'predictor-names)
 256               (mapcar #'(lambda (a) (format nil "Variable ~a" a))
 257                       (iseq 0 (- p 1))))))
 258   (slot-value 'predictor-names))
 259
 260 (defmeth regression-model-proto :response-name (&optional (name "Y" set))
 261 "Message args: (&optional name)
 262 With no argument returns the response name. NAME sets the name."
 263   (if set (setf (slot-value 'response-name) (if name (string name) "Y")))
 264   (slot-value 'response-name))
 265
 266 (defmeth regression-model-proto :case-labels (&optional (labels nil set))
 267 "Message args: (&optional labels)
 268 With no argument returns the case-labels. LABELS sets the labels."
 269   (if set (setf (slot-value 'case-labels)
 270                 (if labels
 271                     (mapcar #'string labels)
 272                     (mapcar #'(lambda (x) (format nil "~d" x))
 273                             (iseq 0 (- (send self :num-cases) 1))))))
 274   (slot-value 'case-labels))
 275
 276 ;;;
 277 ;;; Other Methods
 278 ;;; None of these methods access any slots directly.
 279 ;;;
 280
 281 (defmeth regression-model-proto :num-cases ()
 282 "Message args: ()
 283 Returns the number of cases in the model."
 284   (length (send self :y)))
 285
 286 (defmeth regression-model-proto :num-included ()
 287 "Message args: ()
 288 Returns the number of cases used in the computations."
 289   (sum (if-else (send self :included) 1 0)))
 290
 291 (defmeth regression-model-proto :num-coefs ()
 292 "Message args: ()
 293 Returns the number of coefficients in the fit model (including the
 294 intercept if the model includes one)."
 295   (if (send self :intercept)
 296       (+ 1 (length (send self :basis)))
 297       (length (send self :basis))))
 298
 299 (defmeth regression-model-proto :df ()
 300 "Message args: ()
 301 Returns the number of degrees of freedom in the model."
 302   (- (send self :num-included) (send self :num-coefs)))
 303
 304 (defmeth regression-model-proto :x-matrix ()
 305 "Message args: ()
 306 Returns the X matrix for the model, including a column of 1's, if
 307 appropriate. Columns of X matrix correspond to entries in basis."
 308   (let ((m (select (send self :x)
 309                    (iseq 0 (- (send self :num-cases) 1))
 310                    (send self :basis))))
 311     (if (send self :intercept)
 312         (bind-columns (repeat 1 (send self :num-cases)) m)
 313         m)))
 314
 315 (defmeth regression-model-proto :leverages ()
 316 "Message args: ()
 317 Returns the diagonal elements of the hat matrix."
 318   (let* ((weights (send self :weights))
 319          (x (send self :x-matrix))
 320          (raw-levs
 321           (matmult (* (matmult x (send self :xtxinv)) x)
 322                    (repeat 1 (send self :num-coefs)))))
 323     (if weights (* weights raw-levs) raw-levs)))
 324
 325 (defmeth regression-model-proto :fit-values ()
 326 "Message args: ()
 327 Returns the fitted values for the model."
 328   (matmult (send self :x-matrix) (send self :coef-estimates)))
 329
 330 (defmeth regression-model-proto :raw-residuals ()
 331 "Message args: ()
 332 Returns the raw residuals for a model."
 333   (- (send self :y) (send self :fit-values)))
 334
 335 (defmeth regression-model-proto :residuals ()
 336 "Message args: ()
 337 Returns the raw residuals for a model without weights. If the model
 338 includes weights the raw residuals times the square roots of the weights
 339 are returned."
 340   (let ((raw-residuals (send self :raw-residuals))
 341         (weights (send self :weights)))
 342     (if weights (* (sqrt weights) raw-residuals) raw-residuals)))
 343
 344 (defmeth regression-model-proto :sum-of-squares ()
 345 "Message args: ()
 346 Returns the error sum of squares for the model."
 347   (send self :residual-sum-of-squares))
 348
 349 (defmeth regression-model-proto :sigma-hat ()
 350 "Message args: ()
 351 Returns the estimated standard deviation of the deviations about the
 352 regression line."
 353   (let ((ss (send self :sum-of-squares))
 354         (df (send self :df)))
 355     (if (/= df 0) (sqrt (/ ss df)))))
 356
 357 ;; for models without an intercept the 'usual' formula for R^2 can give
 358 ;; negative results; hence the max.
 359 (defmeth regression-model-proto :r-squared ()
 360 "Message args: ()
 361 Returns the sample squared multiple correlation coefficient, R squared, for
 362 the regression."
 363   (max (- 1 (/ (send self :sum-of-squares) (send self :total-sum-of-squares)))
 364        0))
 365
 366 (defmeth regression-model-proto :coef-estimates ()
 367 "Message args: ()
 368 Returns the OLS (ordinary least squares) estimates of the regression
 369 coefficients. Entries beyond the intercept correspond to entries in basis."
 370   (let ((n (array-dimension (send self :x) 1))
 371         (indices (if (send self :intercept)
 372                      (cons 0 (+ 1 (send self :basis)))
 373                      (+ 1 (send self :basis))))
 374         (m (send self :sweep-matrix)))
 375     (coerce (compound-data-seq (select m (+ 1 n) indices)) 'list)))
 376
 377 (defmeth regression-model-proto :xtxinv ()
 378 "Message args: ()
 379 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
 380   (let ((indices (if (send self :intercept)
 381                      (cons 0 (1+ (send self :basis)))
 382                      (1+ (send self :basis)))))
 383     (select (send self :sweep-matrix) indices indices)))
 384
 385 (defmeth regression-model-proto :coef-standard-errors ()
 386 "Message args: ()
 387 Returns estimated standard errors of coefficients. Entries beyond the
 388 intercept correspond to entries in basis."
 389   (let ((s (send self :sigma-hat)))
 390     (if s (* (send self :sigma-hat) (sqrt (diagonal (send self :xtxinv)))))))
 391
 392 (defmeth regression-model-proto :studentized-residuals ()
 393 "Message args:  ()
 394 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
 395   (let ((res (send self :residuals))
 396         (lev (send self :leverages))
 397         (sig (send self :sigma-hat))
 398         (inc (send self :included)))
 399     (if-else inc
 400              (/ res (* sig (sqrt (pmax .00001 (- 1 lev)))))
 401              (/ res (* sig (sqrt (+ 1 lev)))))))
 402
 403 (defmeth regression-model-proto :externally-studentized-residuals ()
 404 "Message args:  ()
 405 Computes the externally studentized residuals."
 406   (let* ((res (send self :studentized-residuals))
 407          (df (send self :df)))
 408     (if-else (send self :included)
 409              (* res (sqrt (/ (- df 1) (- df (^ res 2)))))
 410              res)))
 411
 412 (defmeth regression-model-proto :cooks-distances ()
 413 "Message args: ()
 414 Computes Cook's distances."
 415   (let ((lev (send self :leverages))
 416         (res (/ (^ (send self :studentized-residuals) 2)
 417                 (send self :num-coefs))))
 418     (if-else (send self :included) (* res (/ lev (- 1 lev) )) (* res lev))))
 419
 420 (defmeth regression-model-proto :plot-residuals (&optional x-values)
 421 "Message args: (&optional x-values)
 422 Opens a window with a plot of the residuals. If X-VALUES are not supplied
 423 the fitted values are used. The plot can be linked to other plots with the
 424 link-views function. Returns a plot object."
 425   (plot-points (if x-values x-values (send self :fit-values))
 426                (send self :residuals)
 427                :title "Residual Plot"
 428                :point-labels (send self :case-labels)))
 429
 430 (defmeth regression-model-proto :plot-bayes-residuals
 431   (&optional x-values)
 432 "Message args: (&optional x-values)
 433 Opens a window with a plot of the standardized residuals and two standard
 434 error bars for the posterior distribution of the actual deviations from the
 435 line. See Chaloner and Brant. If X-VALUES are not supplied  the fitted values
 436 are used. The plot can be linked to other plots with the link-views function.
 437 Returns a plot object."
 438   (let* ((r (/ (send self :residuals) (send self :sigma-hat)))
 439          (d (* 2 (sqrt (send self :leverages))))
 440          (low (- r d))
 441          (high (+ r d))
 442          (x-values (if x-values x-values (send self :fit-values)))
 443          (p (plot-points x-values r :title "Bayes Residual Plot"
 444                          :point-labels (send self :case-labels))))
 445     (map 'list #'(lambda (a b c d) (send p :plotline a b c d nil))
 446                x-values low x-values high)
 447     (send p :adjust-to-data)
 448     p))