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 conjugate realpart imagpart phase
37 min max logand logior logxor lognot ffloor fceiling
38 ftruncate fround signum cis
)
39 (:export regression-model regression-model-proto x y intercept sweep-matrix
40 basis weights included total-sum-of-squares residual-sum-of-squares
41 predictor-names response-name case-labels
))
43 (in-package :lisp-stat-regression-linear
)
45 ;;; Regresion Model Prototype
47 (defproto regression-model-proto
48 '(x y intercept sweep-matrix basis weights
51 residual-sum-of-squares
57 "Normal Linear Regression Model")
59 (defun regression-model (x y
&key
63 (included (repeat t
(length y
)))
67 "Args: (x y &key (intercept T) (print T) weights
68 included predictor-names response-name case-labels)
69 X - list of independent variables or X matrix
70 Y - dependent variable.
71 INTERCEPT - T to include (default), NIL for no intercept
72 PRINT - if not NIL print summary information
73 WEIGHTS - if supplied should be the same length as Y; error variances are
74 assumed to be inversely proportional to WEIGHTS
75 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
76 - sequences of strings or symbols.
77 INCLUDED - if supplied should be the same length as Y, with elements nil
78 to skip a in computing estimates (but not in residual analysis).
79 Returns a regression model object. To examine the model further assign the
80 result to a variable and send it messages.
81 Example (data are in file absorbtion.lsp in the sample data directory/folder):
82 (def m (regression-model (list iron aluminum) absorbtion))
83 (send m :help) (send m :plot-residuals)"
86 ((vectorp x
) (list x
))
87 ((and (consp x
) (numberp (car x
))) (list x
))
89 (m (send regression-model-proto
:new
)))
90 (send m
:x
(if (matrixp x
) x
(apply #'bind-columns x
)))
92 (send m
:intercept intercept
)
93 (send m
:weights weights
)
94 (send m
:included included
)
95 (send m
:predictor-names predictor-names
)
96 (send m
:response-name response-name
)
97 (send m
:case-labels case-labels
)
98 (if print
(send m
:display
))
101 (defmeth regression-model-proto
:isnew
()
102 (send self
:needs-computing t
))
104 (defmeth regression-model-proto
:save
()
106 Returns an expression that will reconstruct the regression model."
107 `(regression-model ',(send self
:x
)
109 :intercept
',(send self
:intercept
)
110 :weights
',(send self
:weights
)
111 :included
',(send self
:included
)
112 :predictor-names
',(send self
:predictor-names
)
113 :response-name
',(send self
:response-name
)
114 :case-labels
',(send self
:case-labels
)))
116 ;;; Computing and Display Methods
118 (defmeth regression-model-proto
:compute
()
120 Recomputes the estimates. For internal use by other messages"
121 (let* ((included (if-else (send self
:included
) 1 0))
124 (intercept (send self
:intercept
))
125 (weights (send self
:weights
))
126 (w (if weights
(* included weights
) included
))
127 (m (make-sweep-matrix x y w
))
128 (n (array-dimension x
1))
129 (p (- (array-dimension m
0) 1))
131 (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation
(column-list x
)))))
132 ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
135 (sweep-operator m
(iseq 1 n
) tol
)
136 (sweep-operator m
(iseq 0 n
) (cons 0.0 tol
)))))
137 (setf (slot-value 'sweep-matrix
) (first sweep-result
))
138 (setf (slot-value 'total-sum-of-squares
) tss
)
139 (setf (slot-value 'residual-sum-of-squares
)
140 (aref (first sweep-result
) p p
))
141 (setf (slot-value 'basis
)
142 (let ((b (remove 0 (second sweep-result
))))
143 (if b
(- (reduce #'-
(reverse b
)) 1)
144 (error "no columns could be swept"))))))
146 (defmeth regression-model-proto
:needs-computing
(&optional set
)
147 (if set
(setf (slot-value 'sweep-matrix
) nil
))
148 (null (slot-value 'sweep-matrix
)))
150 (defmeth regression-model-proto
:display
()
152 Prints the least squares regression summary. Variables not used in the fit
153 are marked as aliased."
154 (let ((coefs (coerce (send self
:coef-estimates
) 'list
))
155 (se-s (send self
:coef-standard-errors
))
157 (p-names (send self
:predictor-names
)))
158 (if (send self
:weights
)
159 (format t
"~%Weighted Least Squares Estimates:~2%")
160 (format t
"~%Least Squares Estimates:~2%"))
161 (when (send self
:intercept
)
162 (format t
"Constant ~10f ~A~%"
163 (car coefs
) (list (car se-s
)))
164 (setf coefs
(cdr coefs
))
165 (setf se-s
(cdr se-s
)))
166 (dotimes (i (array-dimension x
1))
168 ((member i
(send self
:basis
))
169 (format t
"~22a ~10f ~A~%"
170 (select p-names i
) (car coefs
) (list (car se-s
)))
171 (setf coefs
(cdr coefs
) se-s
(cdr se-s
)))
172 (t (format t
"~22a aliased~%" (select p-names i
)))))
174 (format t
"R Squared: ~10f~%" (send self
:r-squared
))
175 (format t
"Sigma hat: ~10f~%" (send self
:sigma-hat
))
176 (format t
"Number of cases: ~10d~%" (send self
:num-cases
))
177 (if (/= (send self
:num-cases
) (send self
:num-included
))
178 (format t
"Number of cases used: ~10d~%" (send self
:num-included
)))
179 (format t
"Degrees of freedom: ~10d~%" (send self
:df
))
182 ;;; Slot accessors and mutators
184 (defmeth regression-model-proto
:x
(&optional new-x
)
185 "Message args: (&optional new-x)
186 With no argument returns the x matrix as supplied to m. With an argument
187 NEW-X sets the x matrix to NEW-X and recomputes the estimates."
188 (when (and new-x
(matrixp new-x
))
189 (setf (slot-value 'x
) new-x
)
190 (send self
:needs-computing t
))
193 (defmeth regression-model-proto
:y
(&optional new-y
)
194 "Message args: (&optional new-y)
195 With no argument returns the y sequence as supplied to m. With an argument
196 NEW-Y sets the y sequence to NEW-Y and recomputes the estimates."
197 (when (and new-y
(or (matrixp new-y
) (sequencep new-y
)))
198 (setf (slot-value 'y
) new-y
)
199 (send self
:needs-computing t
))
202 (defmeth regression-model-proto
:intercept
(&optional
(val nil set
))
203 "Message args: (&optional new-intercept)
204 With no argument returns T if the model includes an intercept term, nil if
205 not. With an argument NEW-INTERCEPT the model is changed to include or
206 exclude an intercept, according to the value of NEW-INTERCEPT."
208 (setf (slot-value 'intercept
) val
)
209 (send self
:needs-computing t
))
210 (slot-value 'intercept
))
212 (defmeth regression-model-proto
:weights
(&optional
(new-w nil set
))
213 "Message args: (&optional new-w)
214 With no argument returns the weight sequence as supplied to m; NIL means
215 an unweighted model. NEW-W sets the weights sequence to NEW-W and
216 recomputes the estimates."
218 (setf (slot-value 'weights
) new-w
)
219 (send self
:needs-computing t
))
220 (slot-value 'weights
))
222 (defmeth regression-model-proto
:total-sum-of-squares
()
224 Returns the total sum of squares around the mean."
225 (if (send self
:needs-computing
) (send self
:compute
))
226 (slot-value 'total-sum-of-squares
))
228 (defmeth regression-model-proto
:residual-sum-of-squares
()
230 Returns the residual sum of squares for the model."
231 (if (send self
:needs-computing
) (send self
:compute
))
232 (slot-value 'residual-sum-of-squares
))
234 (defmeth regression-model-proto
:basis
()
236 Returns the indices of the variables used in fitting the model."
237 (if (send self
:needs-computing
) (send self
:compute
))
240 (defmeth regression-model-proto
:sweep-matrix
()
242 Returns the swept sweep matrix. For internal use"
243 (if (send self
:needs-computing
) (send self
:compute
))
244 (slot-value 'sweep-matrix
))
246 (defmeth regression-model-proto
:included
(&optional new-included
)
247 "Message args: (&optional new-included)
248 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."
249 (when (and new-included
250 (= (length new-included
) (send self
:num-cases
)))
251 (setf (slot-value 'included
) (copy-seq new-included
))
252 (send self
:needs-computing t
))
253 (if (slot-value 'included
)
254 (slot-value 'included
)
255 (repeat t
(send self
:num-cases
))))
257 (defmeth regression-model-proto
:predictor-names
(&optional
(names nil set
))
258 "Message args: (&optional (names nil set))
259 With no argument returns the predictor names. NAMES sets the names."
260 (if set
(setf (slot-value 'predictor-names
) (mapcar #'string names
)))
261 (let ((p (array-dimension (send self
:x
) 1))
262 (p-names (slot-value 'predictor-names
)))
263 (if (not (and p-names
(= (length p-names
) p
)))
264 (setf (slot-value 'predictor-names
)
265 (mapcar #'(lambda (a) (format nil
"Variable ~a" a
))
267 (slot-value 'predictor-names
))
269 (defmeth regression-model-proto
:response-name
(&optional
(name "Y" set
))
270 "Message args: (&optional name)
271 With no argument returns the response name. NAME sets the name."
272 (if set
(setf (slot-value 'response-name
) (if name
(string name
) "Y")))
273 (slot-value 'response-name
))
275 (defmeth regression-model-proto
:case-labels
(&optional
(labels nil set
))
276 "Message args: (&optional labels)
277 With no argument returns the case-labels. LABELS sets the labels."
278 (if set
(setf (slot-value 'case-labels
)
280 (mapcar #'string labels
)
281 (mapcar #'(lambda (x) (format nil
"~d" x
))
282 (iseq 0 (- (send self
:num-cases
) 1))))))
283 (slot-value 'case-labels
))
287 ;;; None of these methods access any slots directly.
290 (defmeth regression-model-proto
:num-cases
()
292 Returns the number of cases in the model."
293 (length (send self
:y
)))
295 (defmeth regression-model-proto
:num-included
()
297 Returns the number of cases used in the computations."
298 (sum (if-else (send self
:included
) 1 0)))
300 (defmeth regression-model-proto
:num-coefs
()
302 Returns the number of coefficients in the fit model (including the
303 intercept if the model includes one)."
304 (if (send self
:intercept
)
305 (+ 1 (length (send self
:basis
)))
306 (length (send self
:basis
))))
308 (defmeth regression-model-proto
:df
()
310 Returns the number of degrees of freedom in the model."
311 (- (send self
:num-included
) (send self
:num-coefs
)))
313 (defmeth regression-model-proto
:x-matrix
()
315 Returns the X matrix for the model, including a column of 1's, if
316 appropriate. Columns of X matrix correspond to entries in basis."
317 (let ((m (select (send self
:x
)
318 (iseq 0 (- (send self
:num-cases
) 1))
319 (send self
:basis
))))
320 (if (send self
:intercept
)
321 (bind-columns (repeat 1 (send self
:num-cases
)) m
)
324 (defmeth regression-model-proto
:leverages
()
326 Returns the diagonal elements of the hat matrix."
327 (let* ((weights (send self
:weights
))
328 (x (send self
:x-matrix
))
330 (matmult (* (matmult x
(send self
:xtxinv
)) x
)
331 (repeat 1 (send self
:num-coefs
)))))
332 (if weights
(* weights raw-levs
) raw-levs
)))
334 (defmeth regression-model-proto
:fit-values
()
336 Returns the fitted values for the model."
337 (matmult (send self
:x-matrix
) (send self
:coef-estimates
)))
339 (defmeth regression-model-proto
:raw-residuals
()
341 Returns the raw residuals for a model."
342 (- (send self
:y
) (send self
:fit-values
)))
344 (defmeth regression-model-proto
:residuals
()
346 Returns the raw residuals for a model without weights. If the model
347 includes weights the raw residuals times the square roots of the weights
349 (let ((raw-residuals (send self
:raw-residuals
))
350 (weights (send self
:weights
)))
351 (if weights
(* (sqrt weights
) raw-residuals
) raw-residuals
)))
353 (defmeth regression-model-proto
:sum-of-squares
()
355 Returns the error sum of squares for the model."
356 (send self
:residual-sum-of-squares
))
358 (defmeth regression-model-proto
:sigma-hat
()
360 Returns the estimated standard deviation of the deviations about the
362 (let ((ss (send self
:sum-of-squares
))
363 (df (send self
:df
)))
364 (if (/= df
0) (sqrt (/ ss df
)))))
366 ;; for models without an intercept the 'usual' formula for R^2 can give
367 ;; negative results; hence the max.
368 (defmeth regression-model-proto
:r-squared
()
370 Returns the sample squared multiple correlation coefficient, R squared, for
372 (max (- 1 (/ (send self
:sum-of-squares
) (send self
:total-sum-of-squares
)))
375 (defmeth regression-model-proto
:coef-estimates
()
377 Returns the OLS (ordinary least squares) estimates of the regression
378 coefficients. Entries beyond the intercept correspond to entries in basis."
379 (let ((n (array-dimension (send self
:x
) 1))
380 (indices (if (send self
:intercept
)
381 (cons 0 (+ 1 (send self
:basis
)))
382 (+ 1 (send self
:basis
))))
383 (m (send self
:sweep-matrix
)))
384 (coerce (compound-data-seq (select m
(+ 1 n
) indices
)) 'list
)))
386 (defmeth regression-model-proto
:xtxinv
()
388 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
389 (let ((indices (if (send self
:intercept
)
390 (cons 0 (1+ (send self
:basis
)))
391 (1+ (send self
:basis
)))))
392 (select (send self
:sweep-matrix
) indices indices
)))
394 (defmeth regression-model-proto
:coef-standard-errors
()
396 Returns estimated standard errors of coefficients. Entries beyond the
397 intercept correspond to entries in basis."
398 (let ((s (send self
:sigma-hat
)))
399 (if s
(* (send self
:sigma-hat
) (sqrt (diagonal (send self
:xtxinv
)))))))
401 (defmeth regression-model-proto
:studentized-residuals
()
403 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
404 (let ((res (send self
:residuals
))
405 (lev (send self
:leverages
))
406 (sig (send self
:sigma-hat
))
407 (inc (send self
:included
)))
409 (/ res
(* sig
(sqrt (pmax .00001 (- 1 lev
)))))
410 (/ res
(* sig
(sqrt (+ 1 lev
)))))))
412 (defmeth regression-model-proto
:externally-studentized-residuals
()
414 Computes the externally studentized residuals."
415 (let* ((res (send self
:studentized-residuals
))
416 (df (send self
:df
)))
417 (if-else (send self
:included
)
418 (* res
(sqrt (/ (- df
1) (- df
(^ res
2)))))
421 (defmeth regression-model-proto
:cooks-distances
()
423 Computes Cook's distances."
424 (let ((lev (send self
:leverages
))
425 (res (/ (^
(send self
:studentized-residuals
) 2)
426 (send self
:num-coefs
))))
427 (if-else (send self
:included
) (* res
(/ lev
(- 1 lev
) )) (* res lev
))))
430 (defun plot-points (x y
&rest args
)
432 (error "Graphics not implemented yet."))
434 ;; Can not plot points yet!!
435 (defmeth regression-model-proto
:plot-residuals
(&optional x-values
)
436 "Message args: (&optional x-values)
437 Opens a window with a plot of the residuals. If X-VALUES are not supplied
438 the fitted values are used. The plot can be linked to other plots with the
439 link-views function. Returns a plot object."
440 (plot-points (if x-values x-values
(send self
:fit-values
))
441 (send self
:residuals
)
442 :title
"Residual Plot"
443 :point-labels
(send self
:case-labels
)))
445 (defmeth regression-model-proto
:plot-bayes-residuals
447 "Message args: (&optional x-values)
448 Opens a window with a plot of the standardized residuals and two standard
449 error bars for the posterior distribution of the actual deviations from the
450 line. See Chaloner and Brant. If X-VALUES are not supplied the fitted values
451 are used. The plot can be linked to other plots with the link-views function.
452 Returns a plot object."
453 (let* ((r (/ (send self
:residuals
) (send self
:sigma-hat
)))
454 (d (* 2 (sqrt (send self
:leverages
))))
457 (x-values (if x-values x-values
(send self
:fit-values
)))
458 (p (plot-points x-values r
459 :title
"Bayes Residual Plot"
460 :point-labels
(send self
:case-labels
))))
462 ;; the lambda needs to be something that fits into list
464 ;; #'(lambda (a b c d) (send p :plotline a b c d nil))
465 ;; x-values low x-values high)
466 (send p
:adjust-to-data
)