src/stat-models/regression.lsp

   1 ;;; -*- mode: lisp -*-
   2 ;;;
   3 ;;; Copyright (c) 2005--2008, 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 ;;;; Originally from:
   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 ;;;; Incorporates modifications suggested by Sandy Weisberg.
  16
  17 (in-package :lisp-stat-regression-linear)
  18
  19 ;;; Regresion Model Prototype
  20
  21 ;; The general strategy behind the fitting of models using prototypes
  22 ;; is that we need to think about want the actual fits are, and then
  23 ;; the fits can be used to recompute as components are changes.  One
  24 ;; catch here is that we'd like some notion of trace-ability, in
  25 ;; particular, there is not necessarily a fixed way to take care of the
  26 ;; audit trail.  save-and-die might be a means of recording the final
  27 ;; approach, but we are challenged by the problem of using advice and
  28 ;; other such features to capture stages and steps that are considered
  29 ;; along the goals of estimating a model.
  30
  31 ;; Note that the above is a stream-of-conscience response to the
  32 ;; challenge of reproducibility in the setting of prototype "on-line"
  33 ;; computation.
  34
  35 (defvar regression-model-proto nil
  36   "Prototype for all regression model instances.")
  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       doc)
  46   ()
  47   *object*
  48   "Normal Linear Regression Model")
  49
  50
  51 (defun regression-model (x y &key
  52                          (intercept T)
  53                          (print T)
  54                          (weights nil)
  55                          (included (repeat t (length y)))
  56                          predictor-names
  57                          response-name
  58                          case-labels
  59                          (doc "Undocumented Regression Model Instance")
  60                          (debug T))
  61   "Args: (x y &key (intercept T) (print T) (weights nil)
  62           included predictor-names response-name case-labels)
  63 X           - list of independent variables or X matrix
  64 Y           - dependent variable.
  65 INTERCEPT   - T to include (default), NIL for no intercept
  66 PRINT       - if not NIL print summary information
  67 WEIGHTS     - if supplied should be the same length as Y; error
  68               variances are
  69               assumed to be inversely proportional to WEIGHTS
  70 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
  71             - sequences of strings or symbols.
  72 INCLUDED    - if supplied should be the same length as Y, with
  73               elements nil to skip a in computing estimates (but not
  74               in residual analysis).
  75 Returns a regression model object. To examine the model further assign the
  76 result to a variable and send it messages.
  77 Example (data are in file absorbtion.lsp in the sample data directory):
  78   (def m (regression-model (list iron aluminum) absorbtion))
  79   (send m :help) (send m :plot-residuals)"
  80   (let ((x (cond
  81             ((matrixp x) x)
  82             ((typep x 'vector) (list x))
  83             ((and (consp x)
  84                   (numberp (car x))) (list x))
  85             (t x)))
  86         (m (send regression-model-proto :new)))
  87     (format t "~%")
  88     (send m :doc doc)
  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 debug
  98         (progn
  99           (format t "~%")
 100           (format t "~S~%" (send m :doc))
 101           (format t "X: ~S~%" (send m :x))
 102           (format t "Y: ~S~%" (send m :y))))
 103     (if print (send m :display))
 104     m))
 105
 106 (defmeth regression-model-proto :isnew ()
 107   (send self :needs-computing t))
 108
 109 (defmeth regression-model-proto :save ()
 110 "Message args: ()
 111 Returns an expression that will reconstruct the regression model."
 112   `(regression-model ',(send self :x)
 113                      ',(send self :y)
 114                      :intercept ',(send self :intercept)
 115                      :weights ',(send self :weights)
 116                      :included ',(send self :included)
 117                      :predictor-names ',(send self :predictor-names)
 118                      :response-name ',(send self :response-name)
 119                      :case-labels ',(send self :case-labels)))
 120
 121 ;;; Computing and Display Methods
 122
 123 (defmeth regression-model-proto :compute ()
 124 "Message args: ()
 125 Recomputes the estimates. For internal use by other messages"
 126   (let* ((included (if-else (send self :included) 1 0))
 127          (x (send self :x))
 128          (y (send self :y))
 129          (intercept (send self :intercept))
 130          (weights (send self :weights))
 131          (w (if weights (* included weights) included))
 132          (m (make-sweep-matrix x y w)) ;;; ERROR HERE
 133          (n (array-dimension x 1))
 134          (p (- (array-dimension m 0) 1))
 135          (tss (aref m p p))
 136          (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation (column-list x)))))
 137          ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
 138          (sweep-result
 139           (if intercept
 140               (sweep-operator m (iseq 1 n) tol)
 141               (sweep-operator m (iseq 0 n) (cons 0.0 tol)))))
 142     (format t
 143             "~%REMOVEME: regr-mdl-prto :compute~%Sweep= ~A~%x= ~A~%y= ~A~%m= ~A~%tss= ~A~%"
 144            sweep-result x y m tss)
 145     (setf (slot-value 'sweep-matrix) (first sweep-result))
 146     (setf (slot-value 'total-sum-of-squares) tss)
 147     (setf (slot-value 'residual-sum-of-squares)
 148           (aref (first sweep-result) p p))
 149     ;; SOMETHING WRONG HERE! FIX-ME
 150     (setf (slot-value 'basis)
 151           (let ((b (remove 0 (second sweep-result))))
 152             (if b (- (reduce #'- (reverse b)) 1)
 153                 (error "no columns could be swept"))))))
 154
 155 (defmeth regression-model-proto :needs-computing (&optional set)
 156 "Message args: ( &optional set )
 157
 158 If value given, sets the flag for whether (re)computation is needed to
 159 update the model fits."
 160    (send self :nop)
 161    (if set (setf (slot-value 'sweep-matrix) nil))
 162    (null (slot-value 'sweep-matrix)))
 163
 164 (defmeth regression-model-proto :display ()
 165 "Message args: ()
 166
 167 Prints the least squares regression summary. Variables not used in the fit
 168 are marked as aliased."
 169   (let ((coefs (coerce (send self :coef-estimates) 'list))
 170         (se-s (send self :coef-standard-errors))
 171         (x (send self :x))
 172         (p-names (send self :predictor-names)))
 173     (if (send self :weights)
 174         (format t "~%Weighted Least Squares Estimates:~2%")
 175         (format t "~%Least Squares Estimates:~2%"))
 176     (when (send self :intercept)
 177           (format t "Constant               ~10f   ~A~%"
 178                   (car coefs) (list (car se-s)))
 179           (setf coefs (cdr coefs))
 180           (setf se-s (cdr se-s)))
 181     (dotimes (i (array-dimension x 1))
 182              (cond
 183                ((member i (send self :basis))
 184                 (format t "~22a ~10f   ~A~%"
 185                         (select p-names i) (car coefs) (list (car se-s)))
 186                 (setf coefs (cdr coefs) se-s (cdr se-s)))
 187                (t (format t "~22a    aliased~%" (select p-names i)))))
 188     (format t "~%")
 189     (format t "R Squared:             ~10f~%" (send self :r-squared))
 190     (format t "Sigma hat:             ~10f~%" (send self :sigma-hat))
 191     (format t "Number of cases:       ~10d~%" (send self :num-cases))
 192     (if (/= (send self :num-cases) (send self :num-included))
 193         (format t "Number of cases used:  ~10d~%" (send self :num-included)))
 194     (format t "Degrees of freedom:    ~10d~%" (send self :df))
 195     (format t "~%")))
 196
 197 ;;; Slot accessors and mutators
 198
 199 (defmeth regression-model-proto :doc (&optional new-doc append)
 200 "Message args: (&optional new-doc)
 201
 202 Returns the DOC-STRING as supplied to m.
 203 Additionally, with an argument NEW-DOC, sets the DOC-STRING to
 204 NEW-DOC.  In this setting, when APPEND is T, append NEW-DOC to DOC
 205 rather than doing replacement."
 206   (send self :nop)
 207   (when (and new-doc (stringp new-doc))
 208         (setf (slot-value 'doc)
 209               (if append
 210                   (concatenate 'string
 211                                (slot-value 'doc)
 212                                new-doc)
 213                   new-doc)))
 214   (slot-value 'doc))
 215
 216
 217 (defmeth regression-model-proto :x (&optional new-x)
 218 "Message args: (&optional new-x)
 219
 220 With no argument returns the x matrix as supplied to m. With an
 221 argument, NEW-X sets the x matrix to NEW-X and recomputes the
 222 estimates."
 223   (when (and new-x (matrixp new-x))
 224         (setf (slot-value 'x) new-x)
 225         (send self :needs-computing t))
 226   (slot-value 'x))
 227
 228 (defmeth regression-model-proto :y (&optional new-y)
 229 "Message args: (&optional new-y)
 230
 231 With no argument returns the y sequence as supplied to m. With an
 232 argument, NEW-Y sets the y sequence to NEW-Y and recomputes the
 233 estimates."
 234   (when (and new-y
 235              (or (matrixp new-y)
 236                  (typep  new-y 'sequence)))
 237     (setf (slot-value 'y) new-y)
 238     (send self :needs-computing t))
 239   (slot-value 'y))
 240
 241 (defmeth regression-model-proto :intercept (&optional (val nil set))
 242 "Message args: (&optional new-intercept)
 243
 244 With no argument returns T if the model includes an intercept term,
 245 nil if not. With an argument NEW-INTERCEPT the model is changed to
 246 include or exclude an intercept, according to the value of
 247 NEW-INTERCEPT."
 248   (when set
 249         (setf (slot-value 'intercept) val)
 250         (send self :needs-computing t))
 251   (slot-value 'intercept))
 252
 253 (defmeth regression-model-proto :weights (&optional (new-w nil set))
 254 "Message args: (&optional new-w)
 255
 256 With no argument returns the weight sequence as supplied to m; NIL
 257 means an unweighted model. NEW-W sets the weights sequence to NEW-W
 258 and recomputes the estimates."
 259   (when set
 260         (setf (slot-value 'weights) new-w)
 261         (send self :needs-computing t))
 262   (slot-value 'weights))
 263
 264 (defmeth regression-model-proto :total-sum-of-squares ()
 265 "Message args: ()
 266
 267 Returns the total sum of squares around the mean."
 268   (if (send self :needs-computing) (send self :compute))
 269   (slot-value 'total-sum-of-squares))
 270
 271 (defmeth regression-model-proto :residual-sum-of-squares ()
 272 "Message args: ()
 273
 274 Returns the residual sum of squares for the model."
 275   (if (send self :needs-computing) (send self :compute))
 276   (slot-value 'residual-sum-of-squares))
 277
 278 (defmeth regression-model-proto :basis ()
 279 "Message args: ()
 280
 281 Returns the indices of the variables used in fitting the model, in a
 282 sequence.  Recompute before this, if needed."
 283   (if (send self :needs-computing)
 284       (send self :compute))
 285   (if (typep (slot-value 'basis) 'sequence)
 286       (slot-value 'basis)
 287       (list (slot-value 'basis))))
 288
 289
 290 (defmeth regression-model-proto :sweep-matrix ()
 291 "Message args: ()
 292
 293 Returns the swept sweep matrix. For internal use"
 294   (if (send self :needs-computing)
 295       (send self :compute))
 296   (slot-value 'sweep-matrix))
 297
 298 (defmeth regression-model-proto :included (&optional new-included)
 299 "Message args: (&optional new-included)
 300
 301 With no argument, NIL means a case is not used in calculating
 302 estimates, and non-nil means it is used.  NEW-INCLUDED is a sequence
 303 of length of y of nil and t to select cases.  Estimates are
 304 recomputed."
 305   (when (and new-included
 306              (= (length new-included) (send self :num-cases)))
 307         (setf (slot-value 'included) (copy-seq new-included))
 308         (send self :needs-computing t))
 309   (if (slot-value 'included)
 310       (slot-value 'included)
 311       (repeat t (send self :num-cases))))
 312
 313 (defmeth regression-model-proto :predictor-names (&optional (names nil set))
 314 "Message args: (&optional (names nil set))
 315
 316 With no argument returns the predictor names. NAMES sets the names."
 317   (if set (setf (slot-value 'predictor-names) (mapcar #'string names)))
 318   (let ((p (array-dimension (send self :x) 1))
 319         (p-names (slot-value 'predictor-names)))
 320     (if (not (and p-names (= (length p-names) p)))
 321         (setf (slot-value 'predictor-names)
 322               (mapcar #'(lambda (a) (format nil "Variable ~a" a))
 323                       (iseq 0 (- p 1))))))
 324   (slot-value 'predictor-names))
 325
 326 (defmeth regression-model-proto :response-name (&optional (name "Y" set))
 327    "Message args: (&optional name)
 328
 329 With no argument returns the response name. NAME sets the name."
 330    (send self :nop)
 331    (if set (setf (slot-value 'response-name) (if name (string name) "Y")))
 332    (slot-value 'response-name))
 333
 334 (defmeth regression-model-proto :case-labels (&optional (labels nil set))
 335 "Message args: (&optional labels)
 336 With no argument returns the case-labels. LABELS sets the labels."
 337   (if set (setf (slot-value 'case-labels)
 338                 (if labels
 339                     (mapcar #'string labels)
 340                     (mapcar #'(lambda (x) (format nil "~d" x))
 341                             (iseq 0 (- (send self :num-cases) 1))))))
 342   (slot-value 'case-labels))
 343
 344 ;;;
 345 ;;; Other Methods
 346 ;;; None of these methods access any slots directly.
 347 ;;;
 348
 349 (defmeth regression-model-proto :num-cases ()
 350 "Message args: ()
 351 Returns the number of cases in the model."
 352   (length (send self :y)))
 353
 354 (defmeth regression-model-proto :num-included ()
 355 "Message args: ()
 356 Returns the number of cases used in the computations."
 357   (sum (if-else (send self :included) 1 0)))
 358
 359 (defmeth regression-model-proto :num-coefs ()
 360 "Message args: ()
 361 Returns the number of coefficients in the fit model (including the
 362 intercept if the model includes one)."
 363   (if (send self :intercept)
 364       (+ 1 (length (send self :basis)))
 365       (length (send self :basis))))
 366
 367 (defmeth regression-model-proto :df ()
 368 "Message args: ()
 369 Returns the number of degrees of freedom in the model."
 370   (- (send self :num-included) (send self :num-coefs)))
 371
 372 (defmeth regression-model-proto :x-matrix ()
 373 "Message args: ()
 374 Returns the X matrix for the model, including a column of 1's, if
 375 appropriate. Columns of X matrix correspond to entries in basis."
 376   (let ((m (select (send self :x)
 377                    (iseq 0 (- (send self :num-cases) 1))
 378                    (send self :basis))))
 379     (if (send self :intercept)
 380         (bind-columns (repeat 1 (send self :num-cases)) m)
 381         m)))
 382
 383 (defmeth regression-model-proto :leverages ()
 384 "Message args: ()
 385 Returns the diagonal elements of the hat matrix."
 386   (let* ((weights (send self :weights))
 387          (x (send self :x-matrix))
 388          (raw-levs
 389           (matmult (* (matmult x (send self :xtxinv)) x)
 390                    (repeat 1 (send self :num-coefs)))))
 391     (if weights (* weights raw-levs) raw-levs)))
 392
 393 (defmeth regression-model-proto :fit-values ()
 394 "Message args: ()
 395 Returns the fitted values for the model."
 396   (matmult (send self :x-matrix) (send self :coef-estimates)))
 397
 398 (defmeth regression-model-proto :raw-residuals ()
 399 "Message args: ()
 400 Returns the raw residuals for a model."
 401   (- (send self :y) (send self :fit-values)))
 402
 403 (defmeth regression-model-proto :residuals ()
 404 "Message args: ()
 405 Returns the raw residuals for a model without weights. If the model
 406 includes weights the raw residuals times the square roots of the weights
 407 are returned."
 408   (let ((raw-residuals (send self :raw-residuals))
 409         (weights (send self :weights)))
 410     (if weights (* (sqrt weights) raw-residuals) raw-residuals)))
 411
 412 (defmeth regression-model-proto :sum-of-squares ()
 413 "Message args: ()
 414 Returns the error sum of squares for the model."
 415   (send self :residual-sum-of-squares))
 416
 417 (defmeth regression-model-proto :sigma-hat ()
 418 "Message args: ()
 419 Returns the estimated standard deviation of the deviations about the
 420 regression line."
 421   (let ((ss (send self :sum-of-squares))
 422         (df (send self :df)))
 423     (if (/= df 0) (sqrt (/ ss df)))))
 424
 425 ;; for models without an intercept the 'usual' formula for R^2 can give
 426 ;; negative results; hence the max.
 427 (defmeth regression-model-proto :r-squared ()
 428 "Message args: ()
 429 Returns the sample squared multiple correlation coefficient, R squared, for
 430 the regression."
 431   (max (- 1 (/ (send self :sum-of-squares) (send self :total-sum-of-squares)))
 432        0))
 433
 434 (defmeth regression-model-proto :coef-estimates ()
 435 "Message args: ()
 436
 437 Returns the OLS (ordinary least squares) estimates of the regression
 438 coefficients. Entries beyond the intercept correspond to entries in
 439 basis."
 440   (let ((n (array-dimension (send self :x) 1))
 441         (indices (flatten-list
 442                   (if (send self :intercept)
 443                      (cons 0 (+ 1 (send self :basis)))
 444                      (list (+ 1 (send self :basis))))))
 445         (m (send self :sweep-matrix)))
 446     (format t "~%REMOVEME2: Coef-ests: ~% Sweep Matrix: ~A ~% array dim 1: ~A ~% Swept indices:  ~A ~% basis: ~A"
 447             m n indices (send self :basis))
 448     (coerce (compound-data-seq (select m (+ 1 n) indices)) 'list))) ;; ERROR
 449
 450 (defmeth regression-model-proto :xtxinv ()
 451 "Message args: ()
 452 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
 453   (let ((indices (if (send self :intercept)
 454                      (cons 0 (1+ (send self :basis)))
 455                      (1+ (send self :basis)))))
 456     (select (send self :sweep-matrix) indices indices)))
 457
 458 (defmeth regression-model-proto :coef-standard-errors ()
 459 "Message args: ()
 460 Returns estimated standard errors of coefficients. Entries beyond the
 461 intercept correspond to entries in basis."
 462   (let ((s (send self :sigma-hat)))
 463     (if s (* (send self :sigma-hat) (sqrt (diagonal (send self :xtxinv)))))))
 464
 465 (defmeth regression-model-proto :studentized-residuals ()
 466 "Message args:  ()
 467 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
 468   (let ((res (send self :residuals))
 469         (lev (send self :leverages))
 470         (sig (send self :sigma-hat))
 471         (inc (send self :included)))
 472     (if-else inc
 473              (/ res (* sig (sqrt (pmax .00001 (- 1 lev)))))
 474              (/ res (* sig (sqrt (+ 1 lev)))))))
 475
 476 (defmeth regression-model-proto :externally-studentized-residuals ()
 477 "Message args:  ()
 478 Computes the externally studentized residuals."
 479   (let* ((res (send self :studentized-residuals))
 480          (df (send self :df)))
 481     (if-else (send self :included)
 482              (* res (sqrt (/ (- df 1) (- df (^ res 2)))))
 483              res)))
 484
 485 (defmeth regression-model-proto :cooks-distances ()
 486 "Message args: ()
 487 Computes Cook's distances."
 488   (let ((lev (send self :leverages))
 489         (res (/ (^ (send self :studentized-residuals) 2)
 490                 (send self :num-coefs))))
 491     (if-else (send self :included) (* res (/ lev (- 1 lev) )) (* res lev))))
 492
 493
 494 (defun plot-points (x y &rest args)
 495   "need to fix."
 496   (declare (ignore x y args))
 497   (error "Graphics not implemented yet."))
 498
 499 ;; Can not plot points yet!!
 500 (defmeth regression-model-proto :plot-residuals (&optional x-values)
 501 "Message args: (&optional x-values)
 502 Opens a window with a plot of the residuals. If X-VALUES are not supplied
 503 the fitted values are used. The plot can be linked to other plots with the
 504 link-views function. Returns a plot object."
 505   (plot-points (if x-values x-values (send self :fit-values))
 506                (send self :residuals)
 507                :title "Residual Plot"
 508                :point-labels (send self :case-labels)))
 509
 510 (defmeth regression-model-proto :plot-bayes-residuals
 511   (&optional x-values)
 512   "Message args: (&optional x-values)
 513
 514 Opens a window with a plot of the standardized residuals and two
 515 standard error bars for the posterior distribution of the actual
 516 deviations from the line. See Chaloner and Brant. If X-VALUES are not
 517 supplied the fitted values are used. The plot can be linked to other
 518 plots with the link-views function.  Returns a plot object."
 519
 520   (let* ((r (/ (send self :residuals)
 521                (send self :sigma-hat)))
 522          (d (* 2 (sqrt (send self :leverages))))
 523          (low (- r d))
 524          (high (+ r d))
 525          (x-values (if x-values x-values (send self :fit-values)))
 526          (p (plot-points x-values r
 527                          :title "Bayes Residual Plot"
 528                          :point-labels (send self :case-labels))))
 529     (map 'list #'(lambda (a b c d) (send p :plotline a b c d nil))
 530          x-values low x-values high)
 531     (send p :adjust-to-data)
 532     p))