mirror of
https://git.savannah.gnu.org/git/emacs.git
synced 2024-11-28 07:45:00 +00:00
822 lines
30 KiB
EmacsLisp
822 lines
30 KiB
EmacsLisp
;;; calc-nlfit.el --- nonlinear curve fitting for Calc
|
|
|
|
;; Copyright (C) 2007, 2008, 2009 Free Software Foundation, Inc.
|
|
|
|
;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
|
|
|
|
;; This file is part of GNU Emacs.
|
|
|
|
;; GNU Emacs is free software: you can redistribute it and/or modify
|
|
;; it under the terms of the GNU General Public License as published by
|
|
;; the Free Software Foundation, either version 3 of the License, or
|
|
;; (at your option) any later version.
|
|
|
|
;; GNU Emacs is distributed in the hope that it will be useful,
|
|
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
|
;; GNU General Public License for more details.
|
|
|
|
;; You should have received a copy of the GNU General Public License
|
|
;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>.
|
|
|
|
;;; Commentary:
|
|
|
|
;; This code uses the Levenberg-Marquardt method, as described in
|
|
;; _Numerical Analysis_ by H. R. Schwarz, to fit data to
|
|
;; nonlinear curves. Currently, the only the following curves are
|
|
;; supported:
|
|
;; The logistic S curve, y=a/(1+exp(b*(t-c)))
|
|
;; Here, y is usually interpreted as the population of some
|
|
;; quantity at time t. So we will think of the data as consisting
|
|
;; of quantities q0, q1, ..., qn and their respective times
|
|
;; t0, t1, ..., tn.
|
|
|
|
;; The logistic bell curve, y=A*exp(B*(t-C))/(1+exp(B*(t-C)))^2
|
|
;; Note that this is the derivative of the formula for the S curve.
|
|
;; We get A=-a*b, B=b and C=c. Here, y is interpreted as the rate
|
|
;; of growth of a population at time t. So we will think of the
|
|
;; data as consisting of rates p0, p1, ..., pn and their
|
|
;; respective times t0, t1, ..., tn.
|
|
|
|
;; The Hubbert Linearization, y/x=A*(1-x/B)
|
|
;; Here, y is thought of as the rate of growth of a population
|
|
;; and x represents the actual population. This is essentially
|
|
;; the differential equation describing the actual population.
|
|
|
|
;; The Levenberg-Marquardt method is an iterative process: it takes
|
|
;; an initial guess for the parameters and refines them. To get an
|
|
;; initial guess for the parameters, we'll use a method described by
|
|
;; Luis de Sousa in "Hubbert's Peak Mathematics". The idea is that
|
|
;; given quantities Q and the corresponding rates P, they should
|
|
;; satisfy P/Q= mQ+a. We can use the parameter a for an
|
|
;; approximation for the parameter a in the S curve, and
|
|
;; approximations for b and c are found using least squares on the
|
|
;; linearization log((a/y)-1) = log(bb) + cc*t of
|
|
;; y=a/(1+bb*exp(cc*t)), which is equivalent to the above s curve
|
|
;; formula, and then tranlating it to b and c. From this, we can
|
|
;; also get approximations for the bell curve parameters.
|
|
|
|
;;; Code:
|
|
|
|
(require 'calc-arith)
|
|
(require 'calcalg3)
|
|
|
|
;; Declare functions which are defined elsewhere.
|
|
(declare-function calc-get-fit-variables "calcalg3" (nv nc &optional defv defc with-y homog))
|
|
(declare-function math-map-binop "calcalg3" (binop args1 args2))
|
|
|
|
(defun math-nlfit-least-squares (xdata ydata &optional sdata sigmas)
|
|
"Return the parameters A and B for the best least squares fit y=a+bx."
|
|
(let* ((n (length xdata))
|
|
(s2data (if sdata
|
|
(mapcar 'calcFunc-sqr sdata)
|
|
(make-list n 1)))
|
|
(S (if sdata 0 n))
|
|
(Sx 0)
|
|
(Sy 0)
|
|
(Sxx 0)
|
|
(Sxy 0)
|
|
D)
|
|
(while xdata
|
|
(let ((x (car xdata))
|
|
(y (car ydata))
|
|
(s (car s2data)))
|
|
(setq Sx (math-add Sx (if s (math-div x s) x)))
|
|
(setq Sy (math-add Sy (if s (math-div y s) y)))
|
|
(setq Sxx (math-add Sxx (if s (math-div (math-mul x x) s)
|
|
(math-mul x x))))
|
|
(setq Sxy (math-add Sxy (if s (math-div (math-mul x y) s)
|
|
(math-mul x y))))
|
|
(if sdata
|
|
(setq S (math-add S (math-div 1 s)))))
|
|
(setq xdata (cdr xdata))
|
|
(setq ydata (cdr ydata))
|
|
(setq s2data (cdr s2data)))
|
|
(setq D (math-sub (math-mul S Sxx) (math-mul Sx Sx)))
|
|
(let ((A (math-div (math-sub (math-mul Sxx Sy) (math-mul Sx Sxy)) D))
|
|
(B (math-div (math-sub (math-mul S Sxy) (math-mul Sx Sy)) D)))
|
|
(if sigmas
|
|
(let ((C11 (math-div Sxx D))
|
|
(C12 (math-neg (math-div Sx D)))
|
|
(C22 (math-div S D)))
|
|
(list (list 'sdev A (calcFunc-sqrt C11))
|
|
(list 'sdev B (calcFunc-sqrt C22))
|
|
(list 'vec
|
|
(list 'vec C11 C12)
|
|
(list 'vec C12 C22))))
|
|
(list A B)))))
|
|
|
|
;;; The methods described by de Sousa require the cumulative data qdata
|
|
;;; and the rates pdata. We will assume that we are given either
|
|
;;; qdata and the corresponding times tdata, or pdata and the corresponding
|
|
;;; tdata. The following two functions will find pdata or qdata,
|
|
;;; given the other..
|
|
|
|
;;; First, given two lists; one of values q0, q1, ..., qn and one of
|
|
;;; corresponding times t0, t1, ..., tn; return a list
|
|
;;; p0, p1, ..., pn of the rates of change of the qi with respect to t.
|
|
;;; p0 is the right hand derivative (q1 - q0)/(t1 - t0).
|
|
;;; pn is the left hand derivative (qn - q(n-1))/(tn - t(n-1)).
|
|
;;; The other pis are the averages of the two:
|
|
;;; (1/2)((qi - q(i-1))/(ti - t(i-1)) + (q(i+1) - qi)/(t(i+1) - ti)).
|
|
|
|
(defun math-nlfit-get-rates-from-cumul (tdata qdata)
|
|
(let ((pdata (list
|
|
(math-div
|
|
(math-sub (nth 1 qdata)
|
|
(nth 0 qdata))
|
|
(math-sub (nth 1 tdata)
|
|
(nth 0 tdata))))))
|
|
(while (> (length qdata) 2)
|
|
(setq pdata
|
|
(cons
|
|
(math-mul
|
|
'(float 5 -1)
|
|
(math-add
|
|
(math-div
|
|
(math-sub (nth 2 qdata)
|
|
(nth 1 qdata))
|
|
(math-sub (nth 2 tdata)
|
|
(nth 1 tdata)))
|
|
(math-div
|
|
(math-sub (nth 1 qdata)
|
|
(nth 0 qdata))
|
|
(math-sub (nth 1 tdata)
|
|
(nth 0 tdata)))))
|
|
pdata))
|
|
(setq qdata (cdr qdata)))
|
|
(setq pdata
|
|
(cons
|
|
(math-div
|
|
(math-sub (nth 1 qdata)
|
|
(nth 0 qdata))
|
|
(math-sub (nth 1 tdata)
|
|
(nth 0 tdata)))
|
|
pdata))
|
|
(reverse pdata)))
|
|
|
|
;;; Next, given two lists -- one of rates p0, p1, ..., pn and one of
|
|
;;; corresponding times t0, t1, ..., tn -- and an initial values q0,
|
|
;;; return a list q0, q1, ..., qn of the cumulative values.
|
|
;;; q0 is the initial value given.
|
|
;;; For i>0, qi is computed using the trapezoid rule:
|
|
;;; qi = q(i-1) + (1/2)(pi + p(i-1))(ti - t(i-1))
|
|
|
|
(defun math-nlfit-get-cumul-from-rates (tdata pdata q0)
|
|
(let* ((qdata (list q0)))
|
|
(while (cdr pdata)
|
|
(setq qdata
|
|
(cons
|
|
(math-add (car qdata)
|
|
(math-mul
|
|
(math-mul
|
|
'(float 5 -1)
|
|
(math-add (nth 1 pdata) (nth 0 pdata)))
|
|
(math-sub (nth 1 tdata)
|
|
(nth 0 tdata))))
|
|
qdata))
|
|
(setq pdata (cdr pdata))
|
|
(setq tdata (cdr tdata)))
|
|
(reverse qdata)))
|
|
|
|
;;; Given the qdata, pdata and tdata, find the parameters
|
|
;;; a, b and c that fit q = a/(1+b*exp(c*t)).
|
|
;;; a is found using the method described by de Sousa.
|
|
;;; b and c are found using least squares on the linearization
|
|
;;; log((a/q)-1) = log(b) + c*t
|
|
;;; In some cases (where the logistic curve may well be the wrong
|
|
;;; model), the computed a will be less than or equal to the maximum
|
|
;;; value of q in qdata; in which case the above linearization won't work.
|
|
;;; In this case, a will be replaced by a number slightly above
|
|
;;; the maximum value of q.
|
|
|
|
(defun math-nlfit-find-qmax (qdata pdata tdata)
|
|
(let* ((ratios (math-map-binop 'math-div pdata qdata))
|
|
(lsdata (math-nlfit-least-squares ratios tdata))
|
|
(qmax (math-max-list (car qdata) (cdr qdata)))
|
|
(a (math-neg (math-div (nth 1 lsdata) (nth 0 lsdata)))))
|
|
(if (math-lessp a qmax)
|
|
(math-add '(float 5 -1) qmax)
|
|
a)))
|
|
|
|
(defun math-nlfit-find-logistic-parameters (qdata pdata tdata)
|
|
(let* ((a (math-nlfit-find-qmax qdata pdata tdata))
|
|
(newqdata
|
|
(mapcar (lambda (q) (calcFunc-ln (math-sub (math-div a q) 1)))
|
|
qdata))
|
|
(bandc (math-nlfit-least-squares tdata newqdata)))
|
|
(list
|
|
a
|
|
(calcFunc-exp (nth 0 bandc))
|
|
(nth 1 bandc))))
|
|
|
|
;;; Next, given the pdata and tdata, we can find the qdata if we know q0.
|
|
;;; We first try to find q0, using the fact that when p takes on its largest
|
|
;;; value, q is half of its maximum value. So we'll find the maximum value
|
|
;;; of q given various q0, and use bisection to approximate the correct q0.
|
|
|
|
;;; First, given pdata and tdata, find what half of qmax would be if q0=0.
|
|
|
|
(defun math-nlfit-find-qmaxhalf (pdata tdata)
|
|
(let ((pmax (math-max-list (car pdata) (cdr pdata)))
|
|
(qmh 0))
|
|
(while (math-lessp (car pdata) pmax)
|
|
(setq qmh
|
|
(math-add qmh
|
|
(math-mul
|
|
(math-mul
|
|
'(float 5 -1)
|
|
(math-add (nth 1 pdata) (nth 0 pdata)))
|
|
(math-sub (nth 1 tdata)
|
|
(nth 0 tdata)))))
|
|
(setq pdata (cdr pdata))
|
|
(setq tdata (cdr tdata)))
|
|
qmh))
|
|
|
|
;;; Next, given pdata and tdata, approximate q0.
|
|
|
|
(defun math-nlfit-find-q0 (pdata tdata)
|
|
(let* ((qhalf (math-nlfit-find-qmaxhalf pdata tdata))
|
|
(q0 (math-mul 2 qhalf))
|
|
(qdata (math-nlfit-get-cumul-from-rates tdata pdata q0)))
|
|
(while (math-lessp (math-nlfit-find-qmax
|
|
(mapcar
|
|
(lambda (q) (math-add q0 q))
|
|
qdata)
|
|
pdata tdata)
|
|
(math-mul
|
|
'(float 5 -1)
|
|
(math-add
|
|
q0
|
|
qhalf)))
|
|
(setq q0 (math-add q0 qhalf)))
|
|
(let* ((qmin (math-sub q0 qhalf))
|
|
(qmax q0)
|
|
(qt (math-nlfit-find-qmax
|
|
(mapcar
|
|
(lambda (q) (math-add q0 q))
|
|
qdata)
|
|
pdata tdata))
|
|
(i 0))
|
|
(while (< i 10)
|
|
(setq q0 (math-mul '(float 5 -1) (math-add qmin qmax)))
|
|
(if (math-lessp
|
|
(math-nlfit-find-qmax
|
|
(mapcar
|
|
(lambda (q) (math-add q0 q))
|
|
qdata)
|
|
pdata tdata)
|
|
(math-mul '(float 5 -1) (math-add qhalf q0)))
|
|
(setq qmin q0)
|
|
(setq qmax q0))
|
|
(setq i (1+ i)))
|
|
(math-mul '(float 5 -1) (math-add qmin qmax)))))
|
|
|
|
;;; To improve the approximations to the parameters, we can use
|
|
;;; Marquardt method as described in Schwarz's book.
|
|
|
|
;;; Small numbers used in the Givens algorithm
|
|
(defvar math-nlfit-delta '(float 1 -8))
|
|
|
|
(defvar math-nlfit-epsilon '(float 1 -5))
|
|
|
|
;;; Maximum number of iterations
|
|
(defvar math-nlfit-max-its 100)
|
|
|
|
;;; Next, we need some functions for dealing with vectors and
|
|
;;; matrices. For convenience, we'll work with Emacs lists
|
|
;;; as vectors, rather than Calc's vectors.
|
|
|
|
(defun math-nlfit-set-elt (vec i x)
|
|
(setcar (nthcdr (1- i) vec) x))
|
|
|
|
(defun math-nlfit-get-elt (vec i)
|
|
(nth (1- i) vec))
|
|
|
|
(defun math-nlfit-make-matrix (i j)
|
|
(let ((row (make-list j 0))
|
|
(mat nil)
|
|
(k 0))
|
|
(while (< k i)
|
|
(setq mat (cons (copy-sequence row) mat))
|
|
(setq k (1+ k)))
|
|
mat))
|
|
|
|
(defun math-nlfit-set-matx-elt (mat i j x)
|
|
(setcar (nthcdr (1- j) (nth (1- i) mat)) x))
|
|
|
|
(defun math-nlfit-get-matx-elt (mat i j)
|
|
(nth (1- j) (nth (1- i) mat)))
|
|
|
|
;;; For solving the linearized system.
|
|
;;; (The Givens method, from Schwarz.)
|
|
|
|
(defun math-nlfit-givens (C d)
|
|
(let* ((C (copy-tree C))
|
|
(d (copy-tree d))
|
|
(n (length (car C)))
|
|
(N (length C))
|
|
(j 1)
|
|
(r (make-list N 0))
|
|
(x (make-list N 0))
|
|
w
|
|
gamma
|
|
sigma
|
|
rho)
|
|
(while (<= j n)
|
|
(let ((i (1+ j)))
|
|
(while (<= i N)
|
|
(let ((cij (math-nlfit-get-matx-elt C i j))
|
|
(cjj (math-nlfit-get-matx-elt C j j)))
|
|
(when (not (math-equal 0 cij))
|
|
(if (math-lessp (calcFunc-abs cjj)
|
|
(math-mul math-nlfit-delta (calcFunc-abs cij)))
|
|
(setq w (math-neg cij)
|
|
gamma 0
|
|
sigma 1
|
|
rho 1)
|
|
(setq w (math-mul
|
|
(calcFunc-sign cjj)
|
|
(calcFunc-sqrt
|
|
(math-add
|
|
(math-mul cjj cjj)
|
|
(math-mul cij cij))))
|
|
gamma (math-div cjj w)
|
|
sigma (math-neg (math-div cij w)))
|
|
(if (math-lessp (calcFunc-abs sigma) gamma)
|
|
(setq rho sigma)
|
|
(setq rho (math-div (calcFunc-sign sigma) gamma))))
|
|
(setq cjj w
|
|
cij rho)
|
|
(math-nlfit-set-matx-elt C j j w)
|
|
(math-nlfit-set-matx-elt C i j rho)
|
|
(let ((k (1+ j)))
|
|
(while (<= k n)
|
|
(let* ((cjk (math-nlfit-get-matx-elt C j k))
|
|
(cik (math-nlfit-get-matx-elt C i k))
|
|
(h (math-sub
|
|
(math-mul gamma cjk) (math-mul sigma cik))))
|
|
(setq cik (math-add
|
|
(math-mul sigma cjk)
|
|
(math-mul gamma cik)))
|
|
(setq cjk h)
|
|
(math-nlfit-set-matx-elt C i k cik)
|
|
(math-nlfit-set-matx-elt C j k cjk)
|
|
(setq k (1+ k)))))
|
|
(let* ((di (math-nlfit-get-elt d i))
|
|
(dj (math-nlfit-get-elt d j))
|
|
(h (math-sub
|
|
(math-mul gamma dj)
|
|
(math-mul sigma di))))
|
|
(setq di (math-add
|
|
(math-mul sigma dj)
|
|
(math-mul gamma di)))
|
|
(setq dj h)
|
|
(math-nlfit-set-elt d i di)
|
|
(math-nlfit-set-elt d j dj))))
|
|
(setq i (1+ i))))
|
|
(setq j (1+ j)))
|
|
(let ((i n)
|
|
s)
|
|
(while (>= i 1)
|
|
(math-nlfit-set-elt r i 0)
|
|
(setq s (math-nlfit-get-elt d i))
|
|
(let ((k (1+ i)))
|
|
(while (<= k n)
|
|
(setq s (math-add s (math-mul (math-nlfit-get-matx-elt C i k)
|
|
(math-nlfit-get-elt x k))))
|
|
(setq k (1+ k))))
|
|
(math-nlfit-set-elt x i
|
|
(math-neg
|
|
(math-div s
|
|
(math-nlfit-get-matx-elt C i i))))
|
|
(setq i (1- i))))
|
|
(let ((i (1+ n)))
|
|
(while (<= i N)
|
|
(math-nlfit-set-elt r i (math-nlfit-get-elt d i))
|
|
(setq i (1+ i))))
|
|
(let ((j n))
|
|
(while (>= j 1)
|
|
(let ((i N))
|
|
(while (>= i (1+ j))
|
|
(setq rho (math-nlfit-get-matx-elt C i j))
|
|
(if (math-equal rho 1)
|
|
(setq gamma 0
|
|
sigma 1)
|
|
(if (math-lessp (calcFunc-abs rho) 1)
|
|
(setq sigma rho
|
|
gamma (calcFunc-sqrt
|
|
(math-sub 1 (math-mul sigma sigma))))
|
|
(setq gamma (math-div 1 (calcFunc-abs rho))
|
|
sigma (math-mul (calcFunc-sign rho)
|
|
(calcFunc-sqrt
|
|
(math-sub 1 (math-mul gamma gamma)))))))
|
|
(let ((ri (math-nlfit-get-elt r i))
|
|
(rj (math-nlfit-get-elt r j))
|
|
h)
|
|
(setq h (math-add (math-mul gamma rj)
|
|
(math-mul sigma ri)))
|
|
(setq ri (math-sub
|
|
(math-mul gamma ri)
|
|
(math-mul sigma rj)))
|
|
(setq rj h)
|
|
(math-nlfit-set-elt r i ri)
|
|
(math-nlfit-set-elt r j rj))
|
|
(setq i (1- i))))
|
|
(setq j (1- j))))
|
|
|
|
x))
|
|
|
|
(defun math-nlfit-jacobian (grad xlist parms &optional slist)
|
|
(let ((j nil))
|
|
(while xlist
|
|
(let ((row (apply grad (car xlist) parms)))
|
|
(setq j
|
|
(cons
|
|
(if slist
|
|
(mapcar (lambda (x) (math-div x (car slist))) row)
|
|
row)
|
|
j)))
|
|
(setq slist (cdr slist))
|
|
(setq xlist (cdr xlist)))
|
|
(reverse j)))
|
|
|
|
(defun math-nlfit-make-ident (l n)
|
|
(let ((m (math-nlfit-make-matrix n n))
|
|
(i 1))
|
|
(while (<= i n)
|
|
(math-nlfit-set-matx-elt m i i l)
|
|
(setq i (1+ i)))
|
|
m))
|
|
|
|
(defun math-nlfit-chi-sq (xlist ylist parms fn &optional slist)
|
|
(let ((cs 0))
|
|
(while xlist
|
|
(let ((c
|
|
(math-sub
|
|
(apply fn (car xlist) parms)
|
|
(car ylist))))
|
|
(if slist
|
|
(setq c (math-div c (car slist))))
|
|
(setq cs
|
|
(math-add cs
|
|
(math-mul c c))))
|
|
(setq xlist (cdr xlist))
|
|
(setq ylist (cdr ylist))
|
|
(setq slist (cdr slist)))
|
|
cs))
|
|
|
|
(defun math-nlfit-init-lambda (C)
|
|
(let ((l 0)
|
|
(n (length (car C)))
|
|
(N (length C)))
|
|
(while C
|
|
(let ((row (car C)))
|
|
(while row
|
|
(setq l (math-add l (math-mul (car row) (car row))))
|
|
(setq row (cdr row))))
|
|
(setq C (cdr C)))
|
|
(calcFunc-sqrt (math-div l (math-mul n N)))))
|
|
|
|
(defun math-nlfit-make-Ctilda (C l)
|
|
(let* ((n (length (car C)))
|
|
(bot (math-nlfit-make-ident l n)))
|
|
(append C bot)))
|
|
|
|
(defun math-nlfit-make-d (fn xdata ydata parms &optional sdata)
|
|
(let ((d nil))
|
|
(while xdata
|
|
(setq d (cons
|
|
(let ((dd (math-sub (apply fn (car xdata) parms)
|
|
(car ydata))))
|
|
(if sdata (math-div dd (car sdata)) dd))
|
|
d))
|
|
(setq xdata (cdr xdata))
|
|
(setq ydata (cdr ydata))
|
|
(setq sdata (cdr sdata)))
|
|
(reverse d)))
|
|
|
|
(defun math-nlfit-make-dtilda (d n)
|
|
(append d (make-list n 0)))
|
|
|
|
(defun math-nlfit-fit (xlist ylist parms fn grad &optional slist)
|
|
(let*
|
|
((C (math-nlfit-jacobian grad xlist parms slist))
|
|
(d (math-nlfit-make-d fn xlist ylist parms slist))
|
|
(chisq (math-nlfit-chi-sq xlist ylist parms fn slist))
|
|
(lambda (math-nlfit-init-lambda C))
|
|
(really-done nil)
|
|
(iters 0))
|
|
(while (and
|
|
(not really-done)
|
|
(< iters math-nlfit-max-its))
|
|
(setq iters (1+ iters))
|
|
(let ((done nil))
|
|
(while (not done)
|
|
(let* ((Ctilda (math-nlfit-make-Ctilda C lambda))
|
|
(dtilda (math-nlfit-make-dtilda d (length (car C))))
|
|
(zeta (math-nlfit-givens Ctilda dtilda))
|
|
(newparms (math-map-binop 'math-add (copy-tree parms) zeta))
|
|
(newchisq (math-nlfit-chi-sq xlist ylist newparms fn slist)))
|
|
(if (math-lessp newchisq chisq)
|
|
(progn
|
|
(if (math-lessp
|
|
(math-div
|
|
(math-sub chisq newchisq) newchisq) math-nlfit-epsilon)
|
|
(setq really-done t))
|
|
(setq lambda (math-div lambda 10))
|
|
(setq chisq newchisq)
|
|
(setq parms newparms)
|
|
(setq done t))
|
|
(setq lambda (math-mul lambda 10)))))
|
|
(setq C (math-nlfit-jacobian grad xlist parms slist))
|
|
(setq d (math-nlfit-make-d fn xlist ylist parms slist))))
|
|
(list chisq parms)))
|
|
|
|
;;; The functions that describe our models, and their gradients.
|
|
|
|
(defun math-nlfit-s-logistic-fn (x a b c)
|
|
(math-div a (math-add 1 (math-mul b (calcFunc-exp (math-mul c x))))))
|
|
|
|
(defun math-nlfit-s-logistic-grad (x a b c)
|
|
(let* ((ep (calcFunc-exp (math-mul c x)))
|
|
(d (math-add 1 (math-mul b ep)))
|
|
(d2 (math-mul d d)))
|
|
(list
|
|
(math-div 1 d)
|
|
(math-neg (math-div (math-mul a ep) d2))
|
|
(math-neg (math-div (math-mul a (math-mul b (math-mul x ep))) d2)))))
|
|
|
|
(defun math-nlfit-b-logistic-fn (x a c d)
|
|
(let ((ex (calcFunc-exp (math-mul c (math-sub x d)))))
|
|
(math-div
|
|
(math-mul a ex)
|
|
(math-sqr
|
|
(math-add
|
|
1 ex)))))
|
|
|
|
(defun math-nlfit-b-logistic-grad (x a c d)
|
|
(let* ((ex (calcFunc-exp (math-mul c (math-sub x d))))
|
|
(ex1 (math-add 1 ex))
|
|
(xd (math-sub x d)))
|
|
(list
|
|
(math-div
|
|
ex
|
|
(math-sqr ex1))
|
|
(math-sub
|
|
(math-div
|
|
(math-mul a (math-mul xd ex))
|
|
(math-sqr ex1))
|
|
(math-div
|
|
(math-mul 2 (math-mul a (math-mul xd (math-sqr ex))))
|
|
(math-pow ex1 3)))
|
|
(math-sub
|
|
(math-div
|
|
(math-mul 2 (math-mul a (math-mul c (math-sqr ex))))
|
|
(math-pow ex1 3))
|
|
(math-div
|
|
(math-mul a (math-mul c ex))
|
|
(math-sqr ex1))))))
|
|
|
|
;;; Functions to get the final covariance matrix and the sdevs
|
|
|
|
(defun math-nlfit-find-covar (grad xlist pparms)
|
|
(let ((j nil))
|
|
(while xlist
|
|
(setq j (cons (cons 'vec (apply grad (car xlist) pparms)) j))
|
|
(setq xlist (cdr xlist)))
|
|
(setq j (cons 'vec (reverse j)))
|
|
(setq j
|
|
(math-mul
|
|
(calcFunc-trn j) j))
|
|
(calcFunc-inv j)))
|
|
|
|
(defun math-nlfit-get-sigmas (grad xlist pparms chisq)
|
|
(let* ((sgs nil)
|
|
(covar (math-nlfit-find-covar grad xlist pparms))
|
|
(n (1- (length covar)))
|
|
(N (length xlist))
|
|
(i 1))
|
|
(when (> N n)
|
|
(while (<= i n)
|
|
(setq sgs (cons (calcFunc-sqrt (nth i (nth i covar))) sgs))
|
|
(setq i (1+ i)))
|
|
(setq sgs (reverse sgs)))
|
|
(list sgs covar)))
|
|
|
|
;;; Now the Calc functions
|
|
|
|
(defun math-nlfit-s-logistic-params (xdata ydata)
|
|
(let ((pdata (math-nlfit-get-rates-from-cumul xdata ydata)))
|
|
(math-nlfit-find-logistic-parameters ydata pdata xdata)))
|
|
|
|
(defun math-nlfit-b-logistic-params (xdata ydata)
|
|
(let* ((q0 (math-nlfit-find-q0 ydata xdata))
|
|
(qdata (math-nlfit-get-cumul-from-rates xdata ydata q0))
|
|
(abc (math-nlfit-find-logistic-parameters qdata ydata xdata))
|
|
(B (nth 1 abc))
|
|
(C (nth 2 abc))
|
|
(A (math-neg
|
|
(math-mul
|
|
(nth 0 abc)
|
|
(math-mul B C))))
|
|
(D (math-neg (math-div (calcFunc-ln B) C)))
|
|
(A (math-div A B)))
|
|
(list A C D)))
|
|
|
|
;;; Some functions to turn the parameter lists and variables
|
|
;;; into the appropriate functions.
|
|
|
|
(defun math-nlfit-s-logistic-solnexpr (pms var)
|
|
(let ((a (nth 0 pms))
|
|
(b (nth 1 pms))
|
|
(c (nth 2 pms)))
|
|
(list '/ a
|
|
(list '+
|
|
1
|
|
(list '*
|
|
b
|
|
(calcFunc-exp
|
|
(list '*
|
|
c
|
|
var)))))))
|
|
|
|
(defun math-nlfit-b-logistic-solnexpr (pms var)
|
|
(let ((a (nth 0 pms))
|
|
(c (nth 1 pms))
|
|
(d (nth 2 pms)))
|
|
(list '/
|
|
(list '*
|
|
a
|
|
(calcFunc-exp
|
|
(list '*
|
|
c
|
|
(list '- var d))))
|
|
(list '^
|
|
(list '+
|
|
1
|
|
(calcFunc-exp
|
|
(list '*
|
|
c
|
|
(list '- var d))))
|
|
2))))
|
|
|
|
(defun math-nlfit-enter-result (n prefix vals)
|
|
(setq calc-aborted-prefix prefix)
|
|
(calc-pop-push-record-list n prefix vals)
|
|
(calc-handle-whys))
|
|
|
|
(defun math-nlfit-fit-curve (fn grad solnexpr initparms &optional sdv)
|
|
(calc-slow-wrapper
|
|
(let* ((sdevv (or (eq sdv 'calcFunc-efit) (eq sdv 'calcFunc-xfit)))
|
|
(calc-display-working-message nil)
|
|
(data (calc-top 1))
|
|
(xdata (cdr (car (cdr data))))
|
|
(ydata (cdr (car (cdr (cdr data)))))
|
|
(sdata (if (math-contains-sdev-p ydata)
|
|
(mapcar (lambda (x) (math-get-sdev x t)) ydata)
|
|
nil))
|
|
(ydata (mapcar (lambda (x) (math-get-value x)) ydata))
|
|
(calc-curve-varnames nil)
|
|
(calc-curve-coefnames nil)
|
|
(calc-curve-nvars 1)
|
|
(fitvars (calc-get-fit-variables 1 3))
|
|
(var (nth 1 calc-curve-varnames))
|
|
(parms (cdr calc-curve-coefnames))
|
|
(parmguess
|
|
(funcall initparms xdata ydata))
|
|
(fit (math-nlfit-fit xdata ydata parmguess fn grad sdata))
|
|
(finalparms (nth 1 fit))
|
|
(sigmacovar
|
|
(if sdevv
|
|
(math-nlfit-get-sigmas grad xdata finalparms (nth 0 fit))))
|
|
(sigmas
|
|
(if sdevv
|
|
(nth 0 sigmacovar)))
|
|
(finalparms
|
|
(if sigmas
|
|
(math-map-binop
|
|
(lambda (x y) (list 'sdev x y)) finalparms sigmas)
|
|
finalparms))
|
|
(soln (funcall solnexpr finalparms var)))
|
|
(let ((calc-fit-to-trail t)
|
|
(traillist nil))
|
|
(while parms
|
|
(setq traillist (cons (list 'calcFunc-eq (car parms) (car finalparms))
|
|
traillist))
|
|
(setq finalparms (cdr finalparms))
|
|
(setq parms (cdr parms)))
|
|
(setq traillist (calc-normalize (cons 'vec (nreverse traillist))))
|
|
(cond ((eq sdv 'calcFunc-efit)
|
|
(math-nlfit-enter-result 1 "efit" soln))
|
|
((eq sdv 'calcFunc-xfit)
|
|
(let (sln)
|
|
(setq sln
|
|
(list 'vec
|
|
soln
|
|
traillist
|
|
(nth 1 sigmacovar)
|
|
'(vec)
|
|
(nth 0 fit)
|
|
(let ((n (length xdata))
|
|
(m (length finalparms)))
|
|
(if (and sdata (> n m))
|
|
(calcFunc-utpc (nth 0 fit)
|
|
(- n m))
|
|
'(var nan var-nan)))))
|
|
(math-nlfit-enter-result 1 "xfit" sln)))
|
|
(t
|
|
(math-nlfit-enter-result 1 "fit" soln)))
|
|
(calc-record traillist "parm")))))
|
|
|
|
(defun calc-fit-s-shaped-logistic-curve (arg)
|
|
(interactive "P")
|
|
(math-nlfit-fit-curve 'math-nlfit-s-logistic-fn
|
|
'math-nlfit-s-logistic-grad
|
|
'math-nlfit-s-logistic-solnexpr
|
|
'math-nlfit-s-logistic-params
|
|
arg))
|
|
|
|
(defun calc-fit-bell-shaped-logistic-curve (arg)
|
|
(interactive "P")
|
|
(math-nlfit-fit-curve 'math-nlfit-b-logistic-fn
|
|
'math-nlfit-b-logistic-grad
|
|
'math-nlfit-b-logistic-solnexpr
|
|
'math-nlfit-b-logistic-params
|
|
arg))
|
|
|
|
(defun calc-fit-hubbert-linear-curve (&optional sdv)
|
|
(calc-slow-wrapper
|
|
(let* ((sdevv (or (eq sdv 'calcFunc-efit) (eq sdv 'calcFunc-xfit)))
|
|
(calc-display-working-message nil)
|
|
(data (calc-top 1))
|
|
(qdata (cdr (car (cdr data))))
|
|
(pdata (cdr (car (cdr (cdr data)))))
|
|
(sdata (if (math-contains-sdev-p pdata)
|
|
(mapcar (lambda (x) (math-get-sdev x t)) pdata)
|
|
nil))
|
|
(pdata (mapcar (lambda (x) (math-get-value x)) pdata))
|
|
(poverqdata (math-map-binop 'math-div pdata qdata))
|
|
(parmvals (math-nlfit-least-squares qdata poverqdata sdata sdevv))
|
|
(finalparms (list (nth 0 parmvals)
|
|
(math-neg
|
|
(math-div (nth 0 parmvals)
|
|
(nth 1 parmvals)))))
|
|
(calc-curve-varnames nil)
|
|
(calc-curve-coefnames nil)
|
|
(calc-curve-nvars 1)
|
|
(fitvars (calc-get-fit-variables 1 2))
|
|
(var (nth 1 calc-curve-varnames))
|
|
(parms (cdr calc-curve-coefnames))
|
|
(soln (list '* (nth 0 finalparms)
|
|
(list '- 1
|
|
(list '/ var (nth 1 finalparms))))))
|
|
(let ((calc-fit-to-trail t)
|
|
(traillist nil))
|
|
(setq traillist
|
|
(list 'vec
|
|
(list 'calcFunc-eq (nth 0 parms) (nth 0 finalparms))
|
|
(list 'calcFunc-eq (nth 1 parms) (nth 1 finalparms))))
|
|
(cond ((eq sdv 'calcFunc-efit)
|
|
(math-nlfit-enter-result 1 "efit" soln))
|
|
((eq sdv 'calcFunc-xfit)
|
|
(let (sln
|
|
(chisq
|
|
(math-nlfit-chi-sq
|
|
qdata poverqdata
|
|
(list (nth 1 (nth 0 finalparms))
|
|
(nth 1 (nth 1 finalparms)))
|
|
(lambda (x a b)
|
|
(math-mul a
|
|
(math-sub
|
|
1
|
|
(math-div x b))))
|
|
sdata)))
|
|
(setq sln
|
|
(list 'vec
|
|
soln
|
|
traillist
|
|
(nth 2 parmvals)
|
|
(list
|
|
'vec
|
|
'(calcFunc-fitdummy 1)
|
|
(list 'calcFunc-neg
|
|
(list '/
|
|
'(calcFunc-fitdummy 1)
|
|
'(calcFunc-fitdummy 2))))
|
|
chisq
|
|
(let ((n (length qdata)))
|
|
(if (and sdata (> n 2))
|
|
(calcFunc-utpc
|
|
chisq
|
|
(- n 2))
|
|
'(var nan var-nan)))))
|
|
(math-nlfit-enter-result 1 "xfit" sln)))
|
|
(t
|
|
(math-nlfit-enter-result 1 "fit" soln)))
|
|
(calc-record traillist "parm")))))
|
|
|
|
(provide 'calc-nlfit)
|
|
|
|
;; arch-tag: 6eba3cd6-f48b-4a84-8174-10c15a024928
|