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
28 (:shadowing-import-from
:lisp-stat-object-system
29 slot-value call-method call-next-method
)
31 (:export regression-model regression-model-proto x y intercept sweep-matrix
32 basis weights included total-sum-of-squares residual-sum-of-squares
33 predictor-names response-name case-labels
))
35 (in-package :lisp-stat-regression-linear
)
37 ;;; Regresion Model Prototype
39 (defproto regression-model-proto
40 '(x y intercept sweep-matrix basis weights
43 residual-sum-of-squares
49 "Normal Linear Regression Model")
51 (defun regression-model (x y
&key
55 (included (repeat t
(length y
)))
59 "Args: (x y &key (intercept T) (print T) weights
60 included predictor-names response-name case-labels)
61 X - list of independent variables or X matrix
62 Y - dependent variable.
63 INTERCEPT - T to include (default), NIL for no intercept
64 PRINT - if not NIL print summary information
65 WEIGHTS - if supplied should be the same length as Y; error variances are
66 assumed to be inversely proportional to WEIGHTS
67 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
68 - sequences of strings or symbols.
69 INCLUDED - if supplied should be the same length as Y, with elements nil
70 to skip a in computing estimates (but not in residual analysis).
71 Returns a regression model object. To examine the model further assign the
72 result to a variable and send it messages.
73 Example (data are in file absorbtion.lsp in the sample data directory/folder):
74 (def m (regression-model (list iron aluminum) absorbtion))
75 (send m :help) (send m :plot-residuals)"
78 ((vectorp x
) (list x
))
79 ((and (consp x
) (numberp (car x
))) (list x
))
81 (m (send regression-model-proto
:new
)))
82 (send m
:x
(if (matrixp x
) x
(apply #'bind-columns x
)))
84 (send m
:intercept intercept
)
85 (send m
:weights weights
)
86 (send m
:included included
)
87 (send m
:predictor-names predictor-names
)
88 (send m
:response-name response-name
)
89 (send m
:case-labels case-labels
)
90 (if print
(send m
:display
))
93 (defmeth regression-model-proto
:isnew
()
94 (send self
:needs-computing t
))
96 (defmeth regression-model-proto
:save
()
98 Returns an expression that will reconstruct the regression model."
99 `(regression-model ',(send self
:x
)
101 :intercept
',(send self
:intercept
)
102 :weights
',(send self
:weights
)
103 :included
',(send self
:included
)
104 :predictor-names
',(send self
:predictor-names
)
105 :response-name
',(send self
:response-name
)
106 :case-labels
',(send self
:case-labels
)))
108 ;;; Computing and Display Methods
110 (defmeth regression-model-proto
:compute
()
112 Recomputes the estimates. For internal use by other messages"
113 (let* ((included (if-else (send self
:included
) 1 0))
116 (intercept (send self
:intercept
))
117 (weights (send self
:weights
))
118 (w (if weights
(* included weights
) included
))
119 (m (make-sweep-matrix x y w
))
120 (n (array-dimension x
1))
121 (p (- (array-dimension m
0) 1))
123 (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation
(column-list x
)))))
124 ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
127 (sweep-operator m
(iseq 1 n
) tol
)
128 (sweep-operator m
(iseq 0 n
) (cons 0.0 tol
)))))
129 (setf (slot-value 'sweep-matrix
) (first sweep-result
))
130 (setf (slot-value 'total-sum-of-squares
) tss
)
131 (setf (slot-value 'residual-sum-of-squares
)
132 (aref (first sweep-result
) p p
))
133 (setf (slot-value 'basis
)
134 (let ((b (remove 0 (second sweep-result
))))
135 (if b
(- (reduce #'-
(reverse b
)) 1)
136 (error "no columns could be swept"))))))
138 (defmeth regression-model-proto
:needs-computing
(&optional set
)
139 (if set
(setf (slot-value 'sweep-matrix
) nil
))
140 (null (slot-value 'sweep-matrix
)))
142 (defmeth regression-model-proto
:display
()
144 Prints the least squares regression summary. Variables not used in the fit
145 are marked as aliased."
146 (let ((coefs (coerce (send self
:coef-estimates
) 'list
))
147 (se-s (send self
:coef-standard-errors
))
149 (p-names (send self
:predictor-names
)))
150 (if (send self
:weights
)
151 (format t
"~%Weighted Least Squares Estimates:~2%")
152 (format t
"~%Least Squares Estimates:~2%"))
153 (when (send self
:intercept
)
154 (format t
"Constant ~10f ~A~%"
155 (car coefs
) (list (car se-s
)))
156 (setf coefs
(cdr coefs
))
157 (setf se-s
(cdr se-s
)))
158 (dotimes (i (array-dimension x
1))
160 ((member i
(send self
:basis
))
161 (format t
"~22a ~10f ~A~%"
162 (select p-names i
) (car coefs
) (list (car se-s
)))
163 (setf coefs
(cdr coefs
) se-s
(cdr se-s
)))
164 (t (format t
"~22a aliased~%" (select p-names i
)))))
166 (format t
"R Squared: ~10f~%" (send self
:r-squared
))
167 (format t
"Sigma hat: ~10f~%" (send self
:sigma-hat
))
168 (format t
"Number of cases: ~10d~%" (send self
:num-cases
))
169 (if (/= (send self
:num-cases
) (send self
:num-included
))
170 (format t
"Number of cases used: ~10d~%" (send self
:num-included
)))
171 (format t
"Degrees of freedom: ~10d~%" (send self
:df
))
174 ;;; Slot accessors and mutators
176 (defmeth regression-model-proto
:x
(&optional new-x
)
177 "Message args: (&optional new-x)
178 With no argument returns the x matrix as supplied to m. With an argument
179 NEW-X sets the x matrix to NEW-X and recomputes the estimates."
180 (when (and new-x
(matrixp new-x
))
181 (setf (slot-value 'x
) new-x
)
182 (send self
:needs-computing t
))
185 (defmeth regression-model-proto
:y
(&optional new-y
)
186 "Message args: (&optional new-y)
187 With no argument returns the y sequence as supplied to m. With an argument
188 NEW-Y sets the y sequence to NEW-Y and recomputes the estimates."
189 (when (and new-y
(or (matrixp new-y
) (sequencep new-y
)))
190 (setf (slot-value 'y
) new-y
)
191 (send self
:needs-computing t
))
194 (defmeth regression-model-proto
:intercept
(&optional
(val nil set
))
195 "Message args: (&optional new-intercept)
196 With no argument returns T if the model includes an intercept term, nil if
197 not. With an argument NEW-INTERCEPT the model is changed to include or
198 exclude an intercept, according to the value of NEW-INTERCEPT."
200 (setf (slot-value 'intercept
) val
)
201 (send self
:needs-computing t
))
202 (slot-value 'intercept
))
204 (defmeth regression-model-proto
:weights
(&optional
(new-w nil set
))
205 "Message args: (&optional new-w)
206 With no argument returns the weight sequence as supplied to m; NIL means
207 an unweighted model. NEW-W sets the weights sequence to NEW-W and
208 recomputes the estimates."
210 (setf (slot-value 'weights
) new-w
)
211 (send self
:needs-computing t
))
212 (slot-value 'weights
))
214 (defmeth regression-model-proto
:total-sum-of-squares
()
216 Returns the total sum of squares around the mean."
217 (if (send self
:needs-computing
) (send self
:compute
))
218 (slot-value 'total-sum-of-squares
))
220 (defmeth regression-model-proto
:residual-sum-of-squares
()
222 Returns the residual sum of squares for the model."
223 (if (send self
:needs-computing
) (send self
:compute
))
224 (slot-value 'residual-sum-of-squares
))
226 (defmeth regression-model-proto
:basis
()
228 Returns the indices of the variables used in fitting the model."
229 (if (send self
:needs-computing
) (send self
:compute
))
232 (defmeth regression-model-proto
:sweep-matrix
()
234 Returns the swept sweep matrix. For internal use"
235 (if (send self
:needs-computing
) (send self
:compute
))
236 (slot-value 'sweep-matrix
))
238 (defmeth regression-model-proto
:included
(&optional new-included
)
239 "Message args: (&optional new-included)
240 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."
241 (when (and new-included
242 (= (length new-included
) (send self
:num-cases
)))
243 (setf (slot-value 'included
) (copy-seq new-included
))
244 (send self
:needs-computing t
))
245 (if (slot-value 'included
)
246 (slot-value 'included
)
247 (repeat t
(send self
:num-cases
))))
249 (defmeth regression-model-proto
:predictor-names
(&optional
(names nil set
))
250 "Message args: (&optional (names nil set))
251 With no argument returns the predictor names. NAMES sets the names."
252 (if set
(setf (slot-value 'predictor-names
) (mapcar #'string names
)))
253 (let ((p (array-dimension (send self
:x
) 1))
254 (p-names (slot-value 'predictor-names
)))
255 (if (not (and p-names
(= (length p-names
) p
)))
256 (setf (slot-value 'predictor-names
)
257 (mapcar #'(lambda (a) (format nil
"Variable ~a" a
))
259 (slot-value 'predictor-names
))
261 (defmeth regression-model-proto
:response-name
(&optional
(name "Y" set
))
262 "Message args: (&optional name)
263 With no argument returns the response name. NAME sets the name."
264 (if set
(setf (slot-value 'response-name
) (if name
(string name
) "Y")))
265 (slot-value 'response-name
))
267 (defmeth regression-model-proto
:case-labels
(&optional
(labels nil set
))
268 "Message args: (&optional labels)
269 With no argument returns the case-labels. LABELS sets the labels."
270 (if set
(setf (slot-value 'case-labels
)
272 (mapcar #'string labels
)
273 (mapcar #'(lambda (x) (format nil
"~d" x
))
274 (iseq 0 (- (send self
:num-cases
) 1))))))
275 (slot-value 'case-labels
))
279 ;;; None of these methods access any slots directly.
282 (defmeth regression-model-proto
:num-cases
()
284 Returns the number of cases in the model."
285 (length (send self
:y
)))
287 (defmeth regression-model-proto
:num-included
()
289 Returns the number of cases used in the computations."
290 (sum (if-else (send self
:included
) 1 0)))
292 (defmeth regression-model-proto
:num-coefs
()
294 Returns the number of coefficients in the fit model (including the
295 intercept if the model includes one)."
296 (if (send self
:intercept
)
297 (+ 1 (length (send self
:basis
)))
298 (length (send self
:basis
))))
300 (defmeth regression-model-proto
:df
()
302 Returns the number of degrees of freedom in the model."
303 (- (send self
:num-included
) (send self
:num-coefs
)))
305 (defmeth regression-model-proto
:x-matrix
()
307 Returns the X matrix for the model, including a column of 1's, if
308 appropriate. Columns of X matrix correspond to entries in basis."
309 (let ((m (select (send self
:x
)
310 (iseq 0 (- (send self
:num-cases
) 1))
311 (send self
:basis
))))
312 (if (send self
:intercept
)
313 (bind-columns (repeat 1 (send self
:num-cases
)) m
)
316 (defmeth regression-model-proto
:leverages
()
318 Returns the diagonal elements of the hat matrix."
319 (let* ((weights (send self
:weights
))
320 (x (send self
:x-matrix
))
322 (matmult (* (matmult x
(send self
:xtxinv
)) x
)
323 (repeat 1 (send self
:num-coefs
)))))
324 (if weights
(* weights raw-levs
) raw-levs
)))
326 (defmeth regression-model-proto
:fit-values
()
328 Returns the fitted values for the model."
329 (matmult (send self
:x-matrix
) (send self
:coef-estimates
)))
331 (defmeth regression-model-proto
:raw-residuals
()
333 Returns the raw residuals for a model."
334 (- (send self
:y
) (send self
:fit-values
)))
336 (defmeth regression-model-proto
:residuals
()
338 Returns the raw residuals for a model without weights. If the model
339 includes weights the raw residuals times the square roots of the weights
341 (let ((raw-residuals (send self
:raw-residuals
))
342 (weights (send self
:weights
)))
343 (if weights
(* (sqrt weights
) raw-residuals
) raw-residuals
)))
345 (defmeth regression-model-proto
:sum-of-squares
()
347 Returns the error sum of squares for the model."
348 (send self
:residual-sum-of-squares
))
350 (defmeth regression-model-proto
:sigma-hat
()
352 Returns the estimated standard deviation of the deviations about the
354 (let ((ss (send self
:sum-of-squares
))
355 (df (send self
:df
)))
356 (if (/= df
0) (sqrt (/ ss df
)))))
358 ;; for models without an intercept the 'usual' formula for R^2 can give
359 ;; negative results; hence the max.
360 (defmeth regression-model-proto
:r-squared
()
362 Returns the sample squared multiple correlation coefficient, R squared, for
364 (max (- 1 (/ (send self
:sum-of-squares
) (send self
:total-sum-of-squares
)))
367 (defmeth regression-model-proto
:coef-estimates
()
369 Returns the OLS (ordinary least squares) estimates of the regression
370 coefficients. Entries beyond the intercept correspond to entries in basis."
371 (let ((n (array-dimension (send self
:x
) 1))
372 (indices (if (send self
:intercept
)
373 (cons 0 (+ 1 (send self
:basis
)))
374 (+ 1 (send self
:basis
))))
375 (m (send self
:sweep-matrix
)))
376 (coerce (compound-data-seq (select m
(+ 1 n
) indices
)) 'list
)))
378 (defmeth regression-model-proto
:xtxinv
()
380 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
381 (let ((indices (if (send self
:intercept
)
382 (cons 0 (1+ (send self
:basis
)))
383 (1+ (send self
:basis
)))))
384 (select (send self
:sweep-matrix
) indices indices
)))
386 (defmeth regression-model-proto
:coef-standard-errors
()
388 Returns estimated standard errors of coefficients. Entries beyond the
389 intercept correspond to entries in basis."
390 (let ((s (send self
:sigma-hat
)))
391 (if s
(* (send self
:sigma-hat
) (sqrt (diagonal (send self
:xtxinv
)))))))
393 (defmeth regression-model-proto
:studentized-residuals
()
395 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
396 (let ((res (send self
:residuals
))
397 (lev (send self
:leverages
))
398 (sig (send self
:sigma-hat
))
399 (inc (send self
:included
)))
401 (/ res
(* sig
(sqrt (pmax .00001 (- 1 lev
)))))
402 (/ res
(* sig
(sqrt (+ 1 lev
)))))))
404 (defmeth regression-model-proto
:externally-studentized-residuals
()
406 Computes the externally studentized residuals."
407 (let* ((res (send self
:studentized-residuals
))
408 (df (send self
:df
)))
409 (if-else (send self
:included
)
410 (* res
(sqrt (/ (- df
1) (- df
(^ res
2)))))
413 (defmeth regression-model-proto
:cooks-distances
()
415 Computes Cook's distances."
416 (let ((lev (send self
:leverages
))
417 (res (/ (^
(send self
:studentized-residuals
) 2)
418 (send self
:num-coefs
))))
419 (if-else (send self
:included
) (* res
(/ lev
(- 1 lev
) )) (* res lev
))))
421 (defmeth regression-model-proto
:plot-residuals
(&optional x-values
)
422 "Message args: (&optional x-values)
423 Opens a window with a plot of the residuals. If X-VALUES are not supplied
424 the fitted values are used. The plot can be linked to other plots with the
425 link-views function. Returns a plot object."
426 (plot-points (if x-values x-values
(send self
:fit-values
))
427 (send self
:residuals
)
428 :title
"Residual Plot"
429 :point-labels
(send self
:case-labels
)))
431 (defmeth regression-model-proto
:plot-bayes-residuals
433 "Message args: (&optional x-values)
434 Opens a window with a plot of the standardized residuals and two standard
435 error bars for the posterior distribution of the actual deviations from the
436 line. See Chaloner and Brant. If X-VALUES are not supplied the fitted values
437 are used. The plot can be linked to other plots with the link-views function.
438 Returns a plot object."
439 (let* ((r (/ (send self
:residuals
) (send self
:sigma-hat
)))
440 (d (* 2 (sqrt (send self
:leverages
))))
443 (x-values (if x-values x-values
(send self
:fit-values
)))
444 (p (plot-points x-values r
445 :title
"Bayes Residual Plot"
446 :point-labels
(send self
:case-labels
))))
448 ;; the lambda needs to be something that fits into list
450 ;; #'(lambda (a b c d) (send p :plotline a b c d nil))
451 ;; x-values low x-values high)
452 (send p
:adjust-to-data
)