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 (:shadowing-import-from
:lisp-stat-object-system
30 slot-value call-method call-next-method
)
31 (:shadowing-import-from
:lisp-stat-math
32 expt
+ -
* / ** mod rem abs
1+ 1- log exp sqrt sin cos tan
33 asin acos atan sinh cosh tanh asinh acosh atanh float random
34 truncate floor ceiling round minusp zerop plusp evenp oddp
35 < <= = /= >= > complex conjugate realpart imagpart phase
36 min max logand logior logxor lognot ffloor fceiling
37 ftruncate fround signum cis
)
38 (:export regression-model regression-model-proto x y intercept sweep-matrix
39 basis weights included total-sum-of-squares residual-sum-of-squares
40 predictor-names response-name case-labels
))
42 (in-package :lisp-stat-regression-linear
)
44 ;;; Regresion Model Prototype
46 (defproto regression-model-proto
47 '(x y intercept sweep-matrix basis weights
50 residual-sum-of-squares
56 "Normal Linear Regression Model")
58 (defun regression-model (x y
&key
62 (included (repeat t
(length y
)))
66 "Args: (x y &key (intercept T) (print T) weights
67 included predictor-names response-name case-labels)
68 X - list of independent variables or X matrix
69 Y - dependent variable.
70 INTERCEPT - T to include (default), NIL for no intercept
71 PRINT - if not NIL print summary information
72 WEIGHTS - if supplied should be the same length as Y; error variances are
73 assumed to be inversely proportional to WEIGHTS
74 PREDICTOR-NAMES, RESPONSE-NAME, CASE-LABELS
75 - sequences of strings or symbols.
76 INCLUDED - if supplied should be the same length as Y, with elements nil
77 to skip a in computing estimates (but not in residual analysis).
78 Returns a regression model object. To examine the model further assign the
79 result to a variable and send it messages.
80 Example (data are in file absorbtion.lsp in the sample data directory/folder):
81 (def m (regression-model (list iron aluminum) absorbtion))
82 (send m :help) (send m :plot-residuals)"
85 ((vectorp x
) (list x
))
86 ((and (consp x
) (numberp (car x
))) (list x
))
88 (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
)
97 (if print
(send m
:display
))
100 (defmeth regression-model-proto
:isnew
()
101 (send self
:needs-computing t
))
103 (defmeth regression-model-proto
:save
()
105 Returns an expression that will reconstruct the regression model."
106 `(regression-model ',(send self
:x
)
108 :intercept
',(send self
:intercept
)
109 :weights
',(send self
:weights
)
110 :included
',(send self
:included
)
111 :predictor-names
',(send self
:predictor-names
)
112 :response-name
',(send self
:response-name
)
113 :case-labels
',(send self
:case-labels
)))
115 ;;; Computing and Display Methods
117 (defmeth regression-model-proto
:compute
()
119 Recomputes the estimates. For internal use by other messages"
120 (let* ((included (if-else (send self
:included
) 1 0))
123 (intercept (send self
:intercept
))
124 (weights (send self
:weights
))
125 (w (if weights
(* included weights
) included
))
126 (m (make-sweep-matrix x y w
))
127 (n (array-dimension x
1))
128 (p (- (array-dimension m
0) 1))
130 (tol (* 0.001 (reduce #'* (mapcar #'standard-deviation
(column-list x
)))))
131 ;; (tol (* 0.001 (apply #'* (mapcar #'standard-deviation (column-list x)))))
134 (sweep-operator m
(iseq 1 n
) tol
)
135 (sweep-operator m
(iseq 0 n
) (cons 0.0 tol
)))))
136 (setf (slot-value 'sweep-matrix
) (first sweep-result
))
137 (setf (slot-value 'total-sum-of-squares
) tss
)
138 (setf (slot-value 'residual-sum-of-squares
)
139 (aref (first sweep-result
) p p
))
140 (setf (slot-value 'basis
)
141 (let ((b (remove 0 (second sweep-result
))))
142 (if b
(- (reduce #'-
(reverse b
)) 1)
143 (error "no columns could be swept"))))))
145 (defmeth regression-model-proto
:needs-computing
(&optional set
)
146 (if set
(setf (slot-value 'sweep-matrix
) nil
))
147 (null (slot-value 'sweep-matrix
)))
149 (defmeth regression-model-proto
:display
()
151 Prints the least squares regression summary. Variables not used in the fit
152 are marked as aliased."
153 (let ((coefs (coerce (send self
:coef-estimates
) 'list
))
154 (se-s (send self
:coef-standard-errors
))
156 (p-names (send self
:predictor-names
)))
157 (if (send self
:weights
)
158 (format t
"~%Weighted Least Squares Estimates:~2%")
159 (format t
"~%Least Squares Estimates:~2%"))
160 (when (send self
:intercept
)
161 (format t
"Constant ~10f ~A~%"
162 (car coefs
) (list (car se-s
)))
163 (setf coefs
(cdr coefs
))
164 (setf se-s
(cdr se-s
)))
165 (dotimes (i (array-dimension x
1))
167 ((member i
(send self
:basis
))
168 (format t
"~22a ~10f ~A~%"
169 (select p-names i
) (car coefs
) (list (car se-s
)))
170 (setf coefs
(cdr coefs
) se-s
(cdr se-s
)))
171 (t (format t
"~22a aliased~%" (select p-names i
)))))
173 (format t
"R Squared: ~10f~%" (send self
:r-squared
))
174 (format t
"Sigma hat: ~10f~%" (send self
:sigma-hat
))
175 (format t
"Number of cases: ~10d~%" (send self
:num-cases
))
176 (if (/= (send self
:num-cases
) (send self
:num-included
))
177 (format t
"Number of cases used: ~10d~%" (send self
:num-included
)))
178 (format t
"Degrees of freedom: ~10d~%" (send self
:df
))
181 ;;; Slot accessors and mutators
183 (defmeth regression-model-proto
:x
(&optional new-x
)
184 "Message args: (&optional new-x)
185 With no argument returns the x matrix as supplied to m. With an argument
186 NEW-X sets the x matrix to NEW-X and recomputes the estimates."
187 (when (and new-x
(matrixp new-x
))
188 (setf (slot-value 'x
) new-x
)
189 (send self
:needs-computing t
))
192 (defmeth regression-model-proto
:y
(&optional new-y
)
193 "Message args: (&optional new-y)
194 With no argument returns the y sequence as supplied to m. With an argument
195 NEW-Y sets the y sequence to NEW-Y and recomputes the estimates."
196 (when (and new-y
(or (matrixp new-y
) (sequencep new-y
)))
197 (setf (slot-value 'y
) new-y
)
198 (send self
:needs-computing t
))
201 (defmeth regression-model-proto
:intercept
(&optional
(val nil set
))
202 "Message args: (&optional new-intercept)
203 With no argument returns T if the model includes an intercept term, nil if
204 not. With an argument NEW-INTERCEPT the model is changed to include or
205 exclude an intercept, according to the value of NEW-INTERCEPT."
207 (setf (slot-value 'intercept
) val
)
208 (send self
:needs-computing t
))
209 (slot-value 'intercept
))
211 (defmeth regression-model-proto
:weights
(&optional
(new-w nil set
))
212 "Message args: (&optional new-w)
213 With no argument returns the weight sequence as supplied to m; NIL means
214 an unweighted model. NEW-W sets the weights sequence to NEW-W and
215 recomputes the estimates."
217 (setf (slot-value 'weights
) new-w
)
218 (send self
:needs-computing t
))
219 (slot-value 'weights
))
221 (defmeth regression-model-proto
:total-sum-of-squares
()
223 Returns the total sum of squares around the mean."
224 (if (send self
:needs-computing
) (send self
:compute
))
225 (slot-value 'total-sum-of-squares
))
227 (defmeth regression-model-proto
:residual-sum-of-squares
()
229 Returns the residual sum of squares for the model."
230 (if (send self
:needs-computing
) (send self
:compute
))
231 (slot-value 'residual-sum-of-squares
))
233 (defmeth regression-model-proto
:basis
()
235 Returns the indices of the variables used in fitting the model."
236 (if (send self
:needs-computing
) (send self
:compute
))
239 (defmeth regression-model-proto
:sweep-matrix
()
241 Returns the swept sweep matrix. For internal use"
242 (if (send self
:needs-computing
) (send self
:compute
))
243 (slot-value 'sweep-matrix
))
245 (defmeth regression-model-proto
:included
(&optional new-included
)
246 "Message args: (&optional new-included)
247 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."
248 (when (and new-included
249 (= (length new-included
) (send self
:num-cases
)))
250 (setf (slot-value 'included
) (copy-seq new-included
))
251 (send self
:needs-computing t
))
252 (if (slot-value 'included
)
253 (slot-value 'included
)
254 (repeat t
(send self
:num-cases
))))
256 (defmeth regression-model-proto
:predictor-names
(&optional
(names nil set
))
257 "Message args: (&optional (names nil set))
258 With no argument returns the predictor names. NAMES sets the names."
259 (if set
(setf (slot-value 'predictor-names
) (mapcar #'string names
)))
260 (let ((p (array-dimension (send self
:x
) 1))
261 (p-names (slot-value 'predictor-names
)))
262 (if (not (and p-names
(= (length p-names
) p
)))
263 (setf (slot-value 'predictor-names
)
264 (mapcar #'(lambda (a) (format nil
"Variable ~a" a
))
266 (slot-value 'predictor-names
))
268 (defmeth regression-model-proto
:response-name
(&optional
(name "Y" set
))
269 "Message args: (&optional name)
270 With no argument returns the response name. NAME sets the name."
271 (if set
(setf (slot-value 'response-name
) (if name
(string name
) "Y")))
272 (slot-value 'response-name
))
274 (defmeth regression-model-proto
:case-labels
(&optional
(labels nil set
))
275 "Message args: (&optional labels)
276 With no argument returns the case-labels. LABELS sets the labels."
277 (if set
(setf (slot-value 'case-labels
)
279 (mapcar #'string labels
)
280 (mapcar #'(lambda (x) (format nil
"~d" x
))
281 (iseq 0 (- (send self
:num-cases
) 1))))))
282 (slot-value 'case-labels
))
286 ;;; None of these methods access any slots directly.
289 (defmeth regression-model-proto
:num-cases
()
291 Returns the number of cases in the model."
292 (length (send self
:y
)))
294 (defmeth regression-model-proto
:num-included
()
296 Returns the number of cases used in the computations."
297 (sum (if-else (send self
:included
) 1 0)))
299 (defmeth regression-model-proto
:num-coefs
()
301 Returns the number of coefficients in the fit model (including the
302 intercept if the model includes one)."
303 (if (send self
:intercept
)
304 (+ 1 (length (send self
:basis
)))
305 (length (send self
:basis
))))
307 (defmeth regression-model-proto
:df
()
309 Returns the number of degrees of freedom in the model."
310 (- (send self
:num-included
) (send self
:num-coefs
)))
312 (defmeth regression-model-proto
:x-matrix
()
314 Returns the X matrix for the model, including a column of 1's, if
315 appropriate. Columns of X matrix correspond to entries in basis."
316 (let ((m (select (send self
:x
)
317 (iseq 0 (- (send self
:num-cases
) 1))
318 (send self
:basis
))))
319 (if (send self
:intercept
)
320 (bind-columns (repeat 1 (send self
:num-cases
)) m
)
323 (defmeth regression-model-proto
:leverages
()
325 Returns the diagonal elements of the hat matrix."
326 (let* ((weights (send self
:weights
))
327 (x (send self
:x-matrix
))
329 (matmult (* (matmult x
(send self
:xtxinv
)) x
)
330 (repeat 1 (send self
:num-coefs
)))))
331 (if weights
(* weights raw-levs
) raw-levs
)))
333 (defmeth regression-model-proto
:fit-values
()
335 Returns the fitted values for the model."
336 (matmult (send self
:x-matrix
) (send self
:coef-estimates
)))
338 (defmeth regression-model-proto
:raw-residuals
()
340 Returns the raw residuals for a model."
341 (- (send self
:y
) (send self
:fit-values
)))
343 (defmeth regression-model-proto
:residuals
()
345 Returns the raw residuals for a model without weights. If the model
346 includes weights the raw residuals times the square roots of the weights
348 (let ((raw-residuals (send self
:raw-residuals
))
349 (weights (send self
:weights
)))
350 (if weights
(* (sqrt weights
) raw-residuals
) raw-residuals
)))
352 (defmeth regression-model-proto
:sum-of-squares
()
354 Returns the error sum of squares for the model."
355 (send self
:residual-sum-of-squares
))
357 (defmeth regression-model-proto
:sigma-hat
()
359 Returns the estimated standard deviation of the deviations about the
361 (let ((ss (send self
:sum-of-squares
))
362 (df (send self
:df
)))
363 (if (/= df
0) (sqrt (/ ss df
)))))
365 ;; for models without an intercept the 'usual' formula for R^2 can give
366 ;; negative results; hence the max.
367 (defmeth regression-model-proto
:r-squared
()
369 Returns the sample squared multiple correlation coefficient, R squared, for
371 (max (- 1 (/ (send self
:sum-of-squares
) (send self
:total-sum-of-squares
)))
374 (defmeth regression-model-proto
:coef-estimates
()
376 Returns the OLS (ordinary least squares) estimates of the regression
377 coefficients. Entries beyond the intercept correspond to entries in basis."
378 (let ((n (array-dimension (send self
:x
) 1))
379 (indices (if (send self
:intercept
)
380 (cons 0 (+ 1 (send self
:basis
)))
381 (+ 1 (send self
:basis
))))
382 (m (send self
:sweep-matrix
)))
383 (coerce (compound-data-seq (select m
(+ 1 n
) indices
)) 'list
)))
385 (defmeth regression-model-proto
:xtxinv
()
387 Returns ((X^T) X)^(-1) or ((X^T) W X)^(-1)."
388 (let ((indices (if (send self
:intercept
)
389 (cons 0 (1+ (send self
:basis
)))
390 (1+ (send self
:basis
)))))
391 (select (send self
:sweep-matrix
) indices indices
)))
393 (defmeth regression-model-proto
:coef-standard-errors
()
395 Returns estimated standard errors of coefficients. Entries beyond the
396 intercept correspond to entries in basis."
397 (let ((s (send self
:sigma-hat
)))
398 (if s
(* (send self
:sigma-hat
) (sqrt (diagonal (send self
:xtxinv
)))))))
400 (defmeth regression-model-proto
:studentized-residuals
()
402 Computes the internally studentized residuals for included cases and externally studentized residuals for excluded cases."
403 (let ((res (send self
:residuals
))
404 (lev (send self
:leverages
))
405 (sig (send self
:sigma-hat
))
406 (inc (send self
:included
)))
408 (/ res
(* sig
(sqrt (pmax .00001 (- 1 lev
)))))
409 (/ res
(* sig
(sqrt (+ 1 lev
)))))))
411 (defmeth regression-model-proto
:externally-studentized-residuals
()
413 Computes the externally studentized residuals."
414 (let* ((res (send self
:studentized-residuals
))
415 (df (send self
:df
)))
416 (if-else (send self
:included
)
417 (* res
(sqrt (/ (- df
1) (- df
(^ res
2)))))
420 (defmeth regression-model-proto
:cooks-distances
()
422 Computes Cook's distances."
423 (let ((lev (send self
:leverages
))
424 (res (/ (^
(send self
:studentized-residuals
) 2)
425 (send self
:num-coefs
))))
426 (if-else (send self
:included
) (* res
(/ lev
(- 1 lev
) )) (* res lev
))))
428 ;; Can not plot points yet!!
429 ;;(defmeth regression-model-proto :plot-residuals (&optional x-values)
430 ;;"Message args: (&optional x-values)
431 ;;Opens a window with a plot of the residuals. If X-VALUES are not supplied
432 ;;the fitted values are used. The plot can be linked to other plots with the
433 ;;link-views function. Returns a plot object."
434 ;; (plot-points (if x-values x-values (send self :fit-values))
435 ;; (send self :residuals)
436 ;; :title "Residual Plot"
437 ;; :point-labels (send self :case-labels)))
439 (defmeth regression-model-proto
:plot-bayes-residuals
441 "Message args: (&optional x-values)
442 Opens a window with a plot of the standardized residuals and two standard
443 error bars for the posterior distribution of the actual deviations from the
444 line. See Chaloner and Brant. If X-VALUES are not supplied the fitted values
445 are used. The plot can be linked to other plots with the link-views function.
446 Returns a plot object."
447 (let* ((r (/ (send self
:residuals
) (send self
:sigma-hat
)))
448 (d (* 2 (sqrt (send self
:leverages
))))
451 (x-values (if x-values x-values
(send self
:fit-values
)))
452 (p (plot-points x-values r
453 :title
"Bayes Residual Plot"
454 :point-labels
(send self
:case-labels
))))
456 ;; the lambda needs to be something that fits into list
458 ;; #'(lambda (a b c d) (send p :plotline a b c d nil))
459 ;; x-values low x-values high)
460 (send p
:adjust-to-data
)