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
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.
16 ;;;; Incorporates modifications suggested by Sandy Weisberg.
21 (defpackage :lisp-stat-regression-linear
23 :lisp-stat-object-system
25 :lisp-stat-compound-data
29 :lisp-stat-descriptive-statistics
)
30 (:shadowing-import-from
:lisp-stat-object-system
31 slot-value call-method call-next-method
)
32 (:shadowing-import-from
:lisp-stat-math
33 expt
+ -
* / ** mod rem abs
1+ 1- log exp sqrt sin cos tan
34 asin acos atan sinh cosh tanh asinh acosh atanh float random
35 truncate floor ceiling round minusp zerop plusp evenp oddp
36 < <= = /= >= > ;; complex
37 conjugate realpart imagpart phase
38 min max logand logior logxor lognot ffloor fceiling
39 ftruncate fround signum cis
)
40 (:export regression-model regression-model-proto x y intercept sweep-matrix
41 basis weights included total-sum-of-squares residual-sum-of-squares
42 predictor-names response-name case-labels
))
44 (in-package :lisp-stat-regression-linear
)
46 ;;; Regresion Model Prototype
48 (defvar regression-model-proto nil
49 "Prototype for all regression model instances.")
50 (defproto regression-model-proto
51 '(x y intercept sweep-matrix basis weights
54 residual-sum-of-squares
61 "Normal Linear Regression Model")
63 (defun regression-model (x y
&key
67 (included (repeat t
(length y
)))
71 (doc "Undocumented Regression Model Instance")
73 "Args: (x y &key (intercept T) (print T) (weights nil)
74 included predictor-names response-name case-labels)
75 X - list of independent variables or X matrix
76 Y - dependent variable.
77 INTERCEPT - T to include (default), NIL for no intercept
78 PRINT - if not NIL print summary information
79 WEIGHTS - if supplied should be the same length as Y; error
81 assumed to be inversely proportional to WEIGHTS
82 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
83 - sequences of strings or symbols.
84 INCLUDED - if supplied should be the same length as Y, with
85 elements nil to skip a in computing estimates (but not
86 in residual analysis).
87 Returns a regression model object. To examine the model further assign the
88 result to a variable and send it messages.
89 Example (data are in file absorbtion.lsp in the sample data directory):
90 (def m (regression-model (list iron aluminum) absorbtion))
91 (send m :help) (send m :plot-residuals)"
94 ((typep x
'vector
) (list x
))
96 (numberp (car x
))) (list x
))
98 (m (send regression-model-proto
:new
)))
101 (send m
:x
(if (matrixp x
) x
(apply #'bind-columns x
)))
103 (send m
:intercept intercept
)
104 (send m
:weights weights
)
105 (send m
:included included
)
106 (send m
:predictor-names predictor-names
)
107 (send m
:response-name response-name
)
108 (send m
:case-labels case-labels
)
112 (format t
"~S~%" (send m
:doc
))
113 (format t
"X: ~S~%" (send m
:x
))
114 (format t
"Y: ~S~%" (send m
:y
))))
115 (if print
(send m
:display
))
118 (defmeth regression-model-proto
:isnew
()
119 (send self
:needs-computing t
))
121 (defmeth regression-model-proto
:save
()
123 Returns an expression that will reconstruct the regression model."
124 `(regression-model ',(send self
:x
)
126 :intercept
',(send self
:intercept
)
127 :weights
',(send self
:weights
)
128 :included
',(send self
:included
)
129 :predictor-names
',(send self
:predictor-names
)
130 :response-name
',(send self
:response-name
)
131 :case-labels
',(send self
:case-labels
)))
133 ;;; Computing and Display Methods
135 (defmeth regression-model-proto
:compute
()
137 Recomputes the estimates. For internal use by other messages"
138 (let* ((included (if-else (send self
:included
) 1 0))
141 (intercept (send self
:intercept
))
142 (weights (send self
:weights
))
143 (w (if weights
(* included weights
) included
))
144 (m (make-sweep-matrix x y w
)) ;;; ERROR HERE
145 (n (array-dimension x
1))
146 (p (- (array-dimension m
0) 1))
148 (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation
(column-list x
)))))
149 ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
152 (sweep-operator m
(iseq 1 n
) tol
)
153 (sweep-operator m
(iseq 0 n
) (cons 0.0 tol
)))))
154 (setf (slot-value 'sweep-matrix
) (first sweep-result
))
155 (setf (slot-value 'total-sum-of-squares
) tss
)
156 (setf (slot-value 'residual-sum-of-squares
)
157 (aref (first sweep-result
) p p
))
158 (setf (slot-value 'basis
)
159 (let ((b (remove 0 (second sweep-result
))))
160 (if b
(- (reduce #'-
(reverse b
)) 1)
161 (error "no columns could be swept"))))))
163 (defmeth regression-model-proto
:needs-computing
(&optional set
)
164 ;;(declare (ignore self))
165 (if set
(setf (slot-value 'sweep-matrix
) nil
))
166 (null (slot-value 'sweep-matrix
)))
168 (defmeth regression-model-proto
:display
()
170 Prints the least squares regression summary. Variables not used in the fit
171 are marked as aliased."
172 (let ((coefs (coerce (send self
:coef-estimates
) 'list
))
173 (se-s (send self
:coef-standard-errors
))
175 (p-names (send self
:predictor-names
)))
176 (if (send self
:weights
)
177 (format t
"~%Weighted Least Squares Estimates:~2%")
178 (format t
"~%Least Squares Estimates:~2%"))
179 (when (send self
:intercept
)
180 (format t
"Constant ~10f ~A~%"
181 (car coefs
) (list (car se-s
)))
182 (setf coefs
(cdr coefs
))
183 (setf se-s
(cdr se-s
)))
184 (dotimes (i (array-dimension x
1))
186 ((member i
(send self
:basis
))
187 (format t
"~22a ~10f ~A~%"
188 (select p-names i
) (car coefs
) (list (car se-s
)))
189 (setf coefs
(cdr coefs
) se-s
(cdr se-s
)))
190 (t (format t
"~22a aliased~%" (select p-names i
)))))
192 (format t
"R Squared: ~10f~%" (send self
:r-squared
))
193 (format t
"Sigma hat: ~10f~%" (send self
:sigma-hat
))
194 (format t
"Number of cases: ~10d~%" (send self
:num-cases
))
195 (if (/= (send self
:num-cases
) (send self
:num-included
))
196 (format t
"Number of cases used: ~10d~%" (send self
:num-included
)))
197 (format t
"Degrees of freedom: ~10d~%" (send self
:df
))
200 ;;; Slot accessors and mutators
202 (defmeth regression-model-proto
:doc
(&optional new-doc
)
203 "Message args: (&optional new-doc)
204 With no argument returns the DOC-STRING as supplied to m. With an argument
205 NEW-DOC sets the DOC-STRING to NEW-DOC."
206 (when (and new-doc
(stringp new-doc
))
207 (setf (slot-value 'doc
) new-doc
))
211 (defmeth regression-model-proto
:x
(&optional new-x
)
212 "Message args: (&optional new-x)
213 With no argument returns the x matrix as supplied to m. With an argument
214 NEW-X sets the x matrix to NEW-X and recomputes the estimates."
215 (when (and new-x
(matrixp new-x
))
216 (setf (slot-value 'x
) new-x
)
217 (send self
:needs-computing t
))
220 (defmeth regression-model-proto
:y
(&optional new-y
)
221 "Message args: (&optional new-y)
222 With no argument returns the y sequence as supplied to m. With an argument
223 NEW-Y sets the y sequence to NEW-Y and recomputes the estimates."
225 (or (matrixp new-y
) (typep new-y
'sequence
)))
226 (setf (slot-value 'y
) new-y
)
227 (send self
:needs-computing t
))
230 (defmeth regression-model-proto
:intercept
(&optional
(val nil set
))
231 "Message args: (&optional new-intercept)
232 With no argument returns T if the model includes an intercept term, nil if
233 not. With an argument NEW-INTERCEPT the model is changed to include or
234 exclude an intercept, according to the value of NEW-INTERCEPT."
236 (setf (slot-value 'intercept
) val
)
237 (send self
:needs-computing t
))
238 (slot-value 'intercept
))
240 (defmeth regression-model-proto
:weights
(&optional
(new-w nil set
))
241 "Message args: (&optional new-w)
242 With no argument returns the weight sequence as supplied to m; NIL means
243 an unweighted model. NEW-W sets the weights sequence to NEW-W and
244 recomputes the estimates."
246 (setf (slot-value 'weights
) new-w
)
247 (send self
:needs-computing t
))
248 (slot-value 'weights
))
250 (defmeth regression-model-proto
:total-sum-of-squares
()
252 Returns the total sum of squares around the mean."
253 (if (send self
:needs-computing
) (send self
:compute
))
254 (slot-value 'total-sum-of-squares
))
256 (defmeth regression-model-proto
:residual-sum-of-squares
()
258 Returns the residual sum of squares for the model."
259 (if (send self
:needs-computing
) (send self
:compute
))
260 (slot-value 'residual-sum-of-squares
))
262 (defmeth regression-model-proto
:basis
()
264 Returns the indices of the variables used in fitting the model."
265 (if (send self
:needs-computing
) (send self
:compute
))
268 (defmeth regression-model-proto
:sweep-matrix
()
270 Returns the swept sweep matrix. For internal use"
271 (if (send self
:needs-computing
) (send self
:compute
))
272 (slot-value 'sweep-matrix
))
274 (defmeth regression-model-proto
:included
(&optional new-included
)
275 "Message args: (&optional new-included)
276 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."
277 (when (and new-included
278 (= (length new-included
) (send self
:num-cases
)))
279 (setf (slot-value 'included
) (copy-seq new-included
))
280 (send self
:needs-computing t
))
281 (if (slot-value 'included
)
282 (slot-value 'included
)
283 (repeat t
(send self
:num-cases
))))
285 (defmeth regression-model-proto
:predictor-names
(&optional
(names nil set
))
286 "Message args: (&optional (names nil set))
287 With no argument returns the predictor names. NAMES sets the names."
288 (if set
(setf (slot-value 'predictor-names
) (mapcar #'string names
)))
289 (let ((p (array-dimension (send self
:x
) 1))
290 (p-names (slot-value 'predictor-names
)))
291 (if (not (and p-names
(= (length p-names
) p
)))
292 (setf (slot-value 'predictor-names
)
293 (mapcar #'(lambda (a) (format nil
"Variable ~a" a
))
295 (slot-value 'predictor-names
))
297 (defmeth regression-model-proto
:response-name
(&optional
(name "Y" set
))
298 "Message args: (&optional name)
299 With no argument returns the response name. NAME sets the name."
300 ;;(declare (ignore self))
301 (if set
(setf (slot-value 'response-name
) (if name
(string name
) "Y")))
302 (slot-value 'response-name
))
304 (defmeth regression-model-proto
:case-labels
(&optional
(labels nil set
))
305 "Message args: (&optional labels)
306 With no argument returns the case-labels. LABELS sets the labels."
307 (if set
(setf (slot-value 'case-labels
)
309 (mapcar #'string labels
)
310 (mapcar #'(lambda (x) (format nil
"~d" x
))
311 (iseq 0 (- (send self
:num-cases
) 1))))))
312 (slot-value 'case-labels
))
316 ;;; None of these methods access any slots directly.
319 (defmeth regression-model-proto
:num-cases
()
321 Returns the number of cases in the model."
322 (length (send self
:y
)))
324 (defmeth regression-model-proto
:num-included
()
326 Returns the number of cases used in the computations."
327 (sum (if-else (send self
:included
) 1 0)))
329 (defmeth regression-model-proto
:num-coefs
()
331 Returns the number of coefficients in the fit model (including the
332 intercept if the model includes one)."
333 (if (send self
:intercept
)
334 (+ 1 (length (send self
:basis
)))
335 (length (send self
:basis
))))
337 (defmeth regression-model-proto
:df
()
339 Returns the number of degrees of freedom in the model."
340 (- (send self
:num-included
) (send self
:num-coefs
)))
342 (defmeth regression-model-proto
:x-matrix
()
344 Returns the X matrix for the model, including a column of 1's, if
345 appropriate. Columns of X matrix correspond to entries in basis."
346 (let ((m (select (send self
:x
)
347 (iseq 0 (- (send self
:num-cases
) 1))
348 (send self
:basis
))))
349 (if (send self
:intercept
)
350 (bind-columns (repeat 1 (send self
:num-cases
)) m
)
353 (defmeth regression-model-proto
:leverages
()
355 Returns the diagonal elements of the hat matrix."
356 (let* ((weights (send self
:weights
))
357 (x (send self
:x-matrix
))
359 (matmult (* (matmult x
(send self
:xtxinv
)) x
)
360 (repeat 1 (send self
:num-coefs
)))))
361 (if weights
(* weights raw-levs
) raw-levs
)))
363 (defmeth regression-model-proto
:fit-values
()
365 Returns the fitted values for the model."
366 (matmult (send self
:x-matrix
) (send self
:coef-estimates
)))
368 (defmeth regression-model-proto
:raw-residuals
()
370 Returns the raw residuals for a model."
371 (- (send self
:y
) (send self
:fit-values
)))
373 (defmeth regression-model-proto
:residuals
()
375 Returns the raw residuals for a model without weights. If the model
376 includes weights the raw residuals times the square roots of the weights
378 (let ((raw-residuals (send self
:raw-residuals
))
379 (weights (send self
:weights
)))
380 (if weights
(* (sqrt weights
) raw-residuals
) raw-residuals
)))
382 (defmeth regression-model-proto
:sum-of-squares
()
384 Returns the error sum of squares for the model."
385 (send self
:residual-sum-of-squares
))
387 (defmeth regression-model-proto
:sigma-hat
()
389 Returns the estimated standard deviation of the deviations about the
391 (let ((ss (send self
:sum-of-squares
))
392 (df (send self
:df
)))
393 (if (/= df
0) (sqrt (/ ss df
)))))
395 ;; for models without an intercept the 'usual' formula for R^2 can give
396 ;; negative results; hence the max.
397 (defmeth regression-model-proto
:r-squared
()
399 Returns the sample squared multiple correlation coefficient, R squared, for
401 (max (- 1 (/ (send self
:sum-of-squares
) (send self
:total-sum-of-squares
)))
404 (defmeth regression-model-proto
:coef-estimates
()
406 Returns the OLS (ordinary least squares) estimates of the regression
407 coefficients. Entries beyond the intercept correspond to entries in basis."
408 (let ((n (array-dimension (send self
:x
) 1))
409 (indices (if (send self
:intercept
)
410 (cons 0 (+ 1 (send self
:basis
)))
411 (+ 1 (send self
:basis
))))
412 (m (send self
:sweep-matrix
)))
413 (coerce (compound-data-seq (select m
(+ 1 n
) indices
)) 'list
)))
415 (defmeth regression-model-proto
:xtxinv
()
417 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
418 (let ((indices (if (send self
:intercept
)
419 (cons 0 (1+ (send self
:basis
)))
420 (1+ (send self
:basis
)))))
421 (select (send self
:sweep-matrix
) indices indices
)))
423 (defmeth regression-model-proto
:coef-standard-errors
()
425 Returns estimated standard errors of coefficients. Entries beyond the
426 intercept correspond to entries in basis."
427 (let ((s (send self
:sigma-hat
)))
428 (if s
(* (send self
:sigma-hat
) (sqrt (diagonal (send self
:xtxinv
)))))))
430 (defmeth regression-model-proto
:studentized-residuals
()
432 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
433 (let ((res (send self
:residuals
))
434 (lev (send self
:leverages
))
435 (sig (send self
:sigma-hat
))
436 (inc (send self
:included
)))
438 (/ res
(* sig
(sqrt (pmax .00001 (- 1 lev
)))))
439 (/ res
(* sig
(sqrt (+ 1 lev
)))))))
441 (defmeth regression-model-proto
:externally-studentized-residuals
()
443 Computes the externally studentized residuals."
444 (let* ((res (send self
:studentized-residuals
))
445 (df (send self
:df
)))
446 (if-else (send self
:included
)
447 (* res
(sqrt (/ (- df
1) (- df
(^ res
2)))))
450 (defmeth regression-model-proto
:cooks-distances
()
452 Computes Cook's distances."
453 (let ((lev (send self
:leverages
))
454 (res (/ (^
(send self
:studentized-residuals
) 2)
455 (send self
:num-coefs
))))
456 (if-else (send self
:included
) (* res
(/ lev
(- 1 lev
) )) (* res lev
))))
459 (defun plot-points (x y
&rest args
)
461 (declare (ignore x y args
))
462 (error "Graphics not implemented yet."))
464 ;; Can not plot points yet!!
465 (defmeth regression-model-proto
:plot-residuals
(&optional x-values
)
466 "Message args: (&optional x-values)
467 Opens a window with a plot of the residuals. If X-VALUES are not supplied
468 the fitted values are used. The plot can be linked to other plots with the
469 link-views function. Returns a plot object."
470 (plot-points (if x-values x-values
(send self
:fit-values
))
471 (send self
:residuals
)
472 :title
"Residual Plot"
473 :point-labels
(send self
:case-labels
)))
475 (defmeth regression-model-proto
:plot-bayes-residuals
477 "Message args: (&optional x-values)
479 Opens a window with a plot of the standardized residuals and two
480 standard error bars for the posterior distribution of the actual
481 deviations from the line. See Chaloner and Brant. If X-VALUES are not
482 supplied the fitted values are used. The plot can be linked to other
483 plots with the link-views function. Returns a plot object."
485 (let* ((r (/ (send self
:residuals
)
486 (send self
:sigma-hat
)))
487 (d (* 2 (sqrt (send self
:leverages
))))
490 (x-values (if x-values x-values
(send self
:fit-values
)))
491 (p (plot-points x-values r
492 :title
"Bayes Residual Plot"
493 :point-labels
(send self
:case-labels
))))
494 (map 'list
#'(lambda (a b c d
) (send p
:plotline a b c d nil
))
495 x-values low x-values high
)
496 (send p
:adjust-to-data
)