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
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.
15 ;;;; Incorporates modifications suggested by Sandy Weisberg.
17 (in-package :lisp-stat-regression-linear
)
19 ;;; Regresion Model Prototype
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.
31 ;; Note that the above is a stream-of-conscience response to the
32 ;; challenge of reproducibility in the setting of prototype "on-line"
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
41 residual-sum-of-squares
48 "Normal Linear Regression Model")
51 (defun regression-model (x y
&key
55 (included (repeat t
(length y
)))
59 (doc "Undocumented Regression Model Instance")
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
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)"
82 ((typep x
'vector
) (list x
))
84 (numberp (car x
))) (list x
))
86 (m (send regression-model-proto
:new
)))
89 (send m
:x
(if (matrixp x
) x
(apply #'bind-columns x
)))
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
)
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
))
106 (defmeth regression-model-proto
:isnew
()
107 (send self
:needs-computing t
))
109 (defmeth regression-model-proto
:save
()
111 Returns an expression that will reconstruct the regression model."
112 `(regression-model ',(send self
:x
)
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
)))
121 ;;; Computing and Display Methods
123 (defmeth regression-model-proto
:compute
()
125 Recomputes the estimates. For internal use by other messages"
126 (let* ((included (if-else (send self
:included
) 1 0))
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))
136 (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation
(column-list x
)))))
137 ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
140 (sweep-operator m
(iseq 1 n
) tol
)
141 (sweep-operator m
(iseq 0 n
) (cons 0.0 tol
)))))
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"))))))
155 (defmeth regression-model-proto
:needs-computing
(&optional set
)
156 "Message args: ( &optional set )
158 If value given, sets the flag for whether (re)computation is needed to
159 update the model fits."
161 (if set
(setf (slot-value 'sweep-matrix
) nil
))
162 (null (slot-value 'sweep-matrix
)))
164 (defmeth regression-model-proto
:display
()
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
))
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))
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
)))))
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
))
197 ;;; Slot accessors and mutators
199 (defmeth regression-model-proto
:doc
(&optional new-doc append
)
200 "Message args: (&optional new-doc)
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."
207 (when (and new-doc
(stringp new-doc
))
208 (setf (slot-value 'doc
)
217 (defmeth regression-model-proto
:x
(&optional new-x
)
218 "Message args: (&optional new-x)
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
223 (when (and new-x
(matrixp new-x
))
224 (setf (slot-value 'x
) new-x
)
225 (send self
:needs-computing t
))
228 (defmeth regression-model-proto
:y
(&optional new-y
)
229 "Message args: (&optional new-y)
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
236 (typep new-y
'sequence
)))
237 (setf (slot-value 'y
) new-y
)
238 (send self
:needs-computing t
))
241 (defmeth regression-model-proto
:intercept
(&optional
(val nil set
))
242 "Message args: (&optional new-intercept)
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
249 (setf (slot-value 'intercept
) val
)
250 (send self
:needs-computing t
))
251 (slot-value 'intercept
))
253 (defmeth regression-model-proto
:weights
(&optional
(new-w nil set
))
254 "Message args: (&optional new-w)
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."
260 (setf (slot-value 'weights
) new-w
)
261 (send self
:needs-computing t
))
262 (slot-value 'weights
))
264 (defmeth regression-model-proto
:total-sum-of-squares
()
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
))
271 (defmeth regression-model-proto
:residual-sum-of-squares
()
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
))
278 (defmeth regression-model-proto
:basis
()
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
)
287 (list (slot-value 'basis
))))
290 (defmeth regression-model-proto
:sweep-matrix
()
293 Returns the swept sweep matrix. For internal use"
294 (if (send self
:needs-computing
)
295 (send self
:compute
))
296 (slot-value 'sweep-matrix
))
298 (defmeth regression-model-proto
:included
(&optional new-included
)
299 "Message args: (&optional new-included)
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
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
))))
313 (defmeth regression-model-proto
:predictor-names
(&optional
(names nil set
))
314 "Message args: (&optional (names nil set))
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
))
324 (slot-value 'predictor-names
))
326 (defmeth regression-model-proto
:response-name
(&optional
(name "Y" set
))
327 "Message args: (&optional name)
329 With no argument returns the response name. NAME sets the name."
331 (if set
(setf (slot-value 'response-name
) (if name
(string name
) "Y")))
332 (slot-value 'response-name
))
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
)
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
))
346 ;;; None of these methods access any slots directly.
349 (defmeth regression-model-proto
:num-cases
()
351 Returns the number of cases in the model."
352 (length (send self
:y
)))
354 (defmeth regression-model-proto
:num-included
()
356 Returns the number of cases used in the computations."
357 (sum (if-else (send self
:included
) 1 0)))
359 (defmeth regression-model-proto
:num-coefs
()
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
))))
367 (defmeth regression-model-proto
:df
()
369 Returns the number of degrees of freedom in the model."
370 (- (send self
:num-included
) (send self
:num-coefs
)))
372 (defmeth regression-model-proto
:x-matrix
()
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
)
383 (defmeth regression-model-proto
:leverages
()
385 Returns the diagonal elements of the hat matrix."
386 (let* ((weights (send self
:weights
))
387 (x (send self
:x-matrix
))
389 (matmult (* (matmult x
(send self
:xtxinv
)) x
)
390 (repeat 1 (send self
:num-coefs
)))))
391 (if weights
(* weights raw-levs
) raw-levs
)))
393 (defmeth regression-model-proto
:fit-values
()
395 Returns the fitted values for the model."
396 (matmult (send self
:x-matrix
) (send self
:coef-estimates
)))
398 (defmeth regression-model-proto
:raw-residuals
()
400 Returns the raw residuals for a model."
401 (- (send self
:y
) (send self
:fit-values
)))
403 (defmeth regression-model-proto
:residuals
()
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
408 (let ((raw-residuals (send self
:raw-residuals
))
409 (weights (send self
:weights
)))
410 (if weights
(* (sqrt weights
) raw-residuals
) raw-residuals
)))
412 (defmeth regression-model-proto
:sum-of-squares
()
414 Returns the error sum of squares for the model."
415 (send self
:residual-sum-of-squares
))
417 (defmeth regression-model-proto
:sigma-hat
()
419 Returns the estimated standard deviation of the deviations about the
421 (let ((ss (send self
:sum-of-squares
))
422 (df (send self
:df
)))
423 (if (/= df
0) (sqrt (/ ss df
)))))
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
()
429 Returns the sample squared multiple correlation coefficient, R squared, for
431 (max (- 1 (/ (send self
:sum-of-squares
) (send self
:total-sum-of-squares
)))
434 (defmeth regression-model-proto
:coef-estimates
()
437 Returns the OLS (ordinary least squares) estimates of the regression
438 coefficients. Entries beyond the intercept correspond to entries in
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
450 (defmeth regression-model-proto
:xtxinv
()
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
)))
458 (defmeth regression-model-proto
:coef-standard-errors
()
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
)))))))
465 (defmeth regression-model-proto
:studentized-residuals
()
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
)))
473 (/ res
(* sig
(sqrt (pmax .00001 (- 1 lev
)))))
474 (/ res
(* sig
(sqrt (+ 1 lev
)))))))
476 (defmeth regression-model-proto
:externally-studentized-residuals
()
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)))))
485 (defmeth regression-model-proto
:cooks-distances
()
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
))))
494 (defun plot-points (x y
&rest args
)
496 (declare (ignore x y args
))
497 (error "Graphics not implemented yet."))
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
)))
510 (defmeth regression-model-proto
:plot-bayes-residuals
512 "Message args: (&optional x-values)
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."
520 (let* ((r (/ (send self
:residuals
)
521 (send self
:sigma-hat
)))
522 (d (* 2 (sqrt (send self
:leverages
))))
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
)