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;;; calc-nlfit.el --- nonlinear curve fitting for Calc
;; Copyright (C) 2007-2015 Free Software Foundation, Inc.
;; Maintainer: Jay Belanger <>
;; 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
;; 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 <>.
;;; 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 translating 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)
(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-sub (nth 1 qdata)
(nth 0 qdata))
(math-sub (nth 1 tdata)
(nth 0 tdata))))))
(while (> (length qdata) 2)
(setq pdata
'(float 5 -1)
(math-sub (nth 2 qdata)
(nth 1 qdata))
(math-sub (nth 2 tdata)
(nth 1 tdata)))
(math-sub (nth 1 qdata)
(nth 0 qdata))
(math-sub (nth 1 tdata)
(nth 0 tdata)))))
(setq qdata (cdr qdata)))
(setq pdata
(math-sub (nth 1 qdata)
(nth 0 qdata))
(math-sub (nth 1 tdata)
(nth 0 tdata)))
(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
(math-add (car qdata)
'(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)))
(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)
(defun math-nlfit-find-logistic-parameters (qdata pdata tdata)
(let* ((a (math-nlfit-find-qmax qdata pdata tdata))
(mapcar (lambda (q) (calcFunc-ln (math-sub (math-div a q) 1)))
(bandc (math-nlfit-least-squares tdata newqdata)))
(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
'(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)))
;;; 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
(lambda (q) (math-add q0 q))
pdata tdata)
'(float 5 -1)
(setq q0 (math-add q0 qhalf)))
(let* ((qmin (math-sub q0 qhalf))
(qmax q0)
(qt (math-nlfit-find-qmax
(lambda (q) (math-add q0 q))
pdata tdata))
(i 0))
(while (< i 10)
(setq q0 (math-mul '(float 5 -1) (math-add qmin qmax)))
(if (math-lessp
(lambda (q) (math-add q0 q))
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)))
(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))
(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)
(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)
(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-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)
(math-sub 1 (math-mul gamma gamma)))))))
(let ((ri (math-nlfit-get-elt r i))
(rj (math-nlfit-get-elt r j))
(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))))
(defun math-nlfit-jacobian (grad xlist parms &optional slist)
(let ((j nil))
(while xlist
(let ((row (apply grad (car xlist) parms)))
(setq j
(if slist
(mapcar (lambda (x) (math-div x (car slist))) row)
(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)))
(defun math-nlfit-chi-sq (xlist ylist parms fn &optional slist)
(let ((cs 0))
(while xlist
(let ((c
(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)))
(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))
(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)
((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)
(if (math-lessp
(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)))
(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-mul a ex)
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)))
(math-sqr ex1))
(math-mul a (math-mul xd ex))
(math-sqr ex1))
(math-mul 2 (math-mul a (math-mul xd (math-sqr ex))))
(math-pow ex1 3)))
(math-mul 2 (math-mul a (math-mul c (math-sqr ex))))
(math-pow ex1 3))
(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
(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
(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 '+
(list '*
(list '*
(defun math-nlfit-b-logistic-solnexpr (pms var)
(let ((a (nth 0 pms))
(c (nth 1 pms))
(d (nth 2 pms)))
(list '/
(list '*
(list '*
(list '- var d))))
(list '^
(list '+
(list '*
(list '- var d))))
(defun math-nlfit-enter-result (n prefix vals)
(setq calc-aborted-prefix prefix)
(calc-pop-push-record-list n prefix vals)
(defun math-nlfit-fit-curve (fn grad solnexpr initparms &optional sdv)
(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)
(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))
(funcall initparms xdata ydata))
(fit (math-nlfit-fit xdata ydata parmguess fn grad sdata))
(finalparms (nth 1 fit))
(if sdevv
(math-nlfit-get-sigmas grad xdata finalparms (nth 0 fit))))
(if sdevv
(nth 0 sigmacovar)))
(if sigmas
(lambda (x y) (list 'sdev x y)) finalparms sigmas)
(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))
(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
(nth 1 sigmacovar)
(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)))
(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
(defun calc-fit-bell-shaped-logistic-curve (arg)
(interactive "P")
(math-nlfit-fit-curve 'math-nlfit-b-logistic-fn
(defun calc-fit-hubbert-linear-curve (&optional sdv)
(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)
(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-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
qdata poverqdata
(list (nth 1 (nth 0 finalparms))
(nth 1 (nth 1 finalparms)))
(lambda (x a b)
(math-mul a
(math-div x b))))
(setq sln
(list 'vec
(nth 2 parmvals)
'(calcFunc-fitdummy 1)
(list 'calcFunc-neg
(list '/
'(calcFunc-fitdummy 1)
'(calcFunc-fitdummy 2))))
(let ((n (length qdata)))
(if (and sdata (> n 2))
(- n 2))
'(var nan var-nan)))))
(math-nlfit-enter-result 1 "xfit" sln)))
(math-nlfit-enter-result 1 "fit" soln)))
(calc-record traillist "parm")))))
(provide 'calc-nlfit)
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