functions.fit_functions¶
Module defines common functions that are used in curve_fit or fmin parameter estimations.
For all fit functions, it defines the functions in two forms (ex. of 3 params):
func(x, p1, p2, p3)
func_p(x, p) with p[0:3]
The first form can be used, for example, with scipy.optimize.curve_fit (ex. function f1x=a+b/x):
p, cov = scipy.optimize.curve_fit(functions.f1x, x, y, p0=[p0,p1])
It also defines two cost functions along with the fit functions, one with the absolute sum, one with the squared sum of the deviations:
cost_func sum(abs(obs-func))
cost2_func sum((obs-func)**2)
These cost functions can be used, for example, with scipy.optimize.minimize:
p = scipy.optimize.minimize(jams.functions.cost_f1x, np.array([p1,p2]), args=(x,y), method=’Nelder-Mead’, options={‘disp’:False})
Note the different argument orders:
curvefit needs f(x,*args) with the independent variable as the first argument and the parameters to fit as separate remaining arguments.
minimize is a general minimiser with respect to the first argument, i.e. func(p,*args).
The module provides also two common cost functions (absolute and squared deviations) where any function in the form func(x, p) can be used as second argument:
cost_abs(p, func, x, y)
cost_square(p, func, x, y)
This means, for example cost_f1x(p, x, y) is the same as cost_abs(p, functions.f1x_p, x, y). For example:
p = scipy.optimize.minimize(jams.functions.cost_abs, np.array([p1,p2]), args=(functions.f1x_p,x,y), method=’Nelder-Mead’, options={‘disp’:False})
The current functions are (the functions have the name in the first column. The seond form has a ‘_p’ appended to the name. The cost functions, which have ‘cost_’ and ‘cost2_’ prepended to the name.):
arrhenius 1 param: Arrhenius temperature dependence of biochemical rates: exp((T-TC25)*E/(T25*R*(T+T0))), parameter: E
f1x 2 params: General 1/x function: a + b/x
fexp 3 params: General exponential function: a + b * exp(c*x)
gauss 2 params: Gauss function: 1/(sig*sqrt(2*pi)) *exp(-(x-mu)**2/(2*sig**2)), parameter: mu, sig
lasslop 6 params: Lasslop et al. (2010) a rectangular, hyperbolic light-response GPP with Lloyd & Taylor (1994) respiration and the maximum canopy uptake rate at light saturation decreases exponentially with VPD as in Koerner (1995)
line0 1 params: Straight line: a*x
line 2 params: Straight line: a + b*x
lloyd_fix 2 params: Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
lloyd_only_rref 1 param: Lloyd & Taylor (1994) Arrhenius type with fixed exponential term
logistic 3 params: Logistic function: a/(1+exp(-b(x-c)))
logistic_offset 4 params: Logistic function with offset: a/(1+exp(-b(x-c))) + d
logistic2_offset 7 params: Double logistic function with offset L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a
poly n params: General polynomial: c0 + c1*x + c2*x**2 + … + cn*x**n
sabx 2 params: sqrt(f1x), i.e. general sqrt(1/x) function: sqrt(a + b/x)
see 3 params: Sequential Elementary Effects fitting function: a*(x-b)**c
This module was written by Matthias Cuntz while at Department of Computational Hydrosystems, Helmholtz Centre for Environmental Research - UFZ, Leipzig, Germany, and continued while at Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Nancy, France.
Copyright (c) 2012-2020 Matthias Cuntz - mc (at) macu (dot) de Released under the MIT License; see LICENSE file for details.
Written Dec 2012 by Matthias Cuntz (mc (at) macu (dot) de)
Ported to Python 3, Feb 2013, Matthias Cuntz
Added general cost functions cost_abs and cost_square, May 2013, Matthias Cuntz
Added line0, Feb 2014, Matthias Cuntz
Removed multiline_p, May 2020, Matthias Cuntz
Changed to Sphinx docstring and numpydoc, May 2020, Matthias Cuntz
The following functions are provided:
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General cost function for robust optimising func(x,p) vs. |
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General cost function for optimising func(x,p) vs. |
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Arrhenius temperature dependence of rates. |
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Arrhenius temperature dependence of rates. |
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Sum of absolute deviations of obs and arrhenius function. |
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Sum of squared deviations of obs and arrhenius. |
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General 1/x function: a + b/x |
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General 1/x function: a + b/x |
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Sum of absolute deviations of obs and general 1/x function: a + b/x |
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Sum of squared deviations of obs and general 1/x function: a + b/x |
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General exponential function: a + b * exp(c*x) |
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General exponential function: a + b * exp(c*x) |
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Sum of absolute deviations of obs and general exponential function: a + b * exp(c*x) |
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Sum of squared deviations of obs and general exponential function: a + b * exp(c*x) |
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Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
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Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
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Sum of absolute deviations of obs and Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
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Sum of squared deviations of obs and Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
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Lasslop et al. (2010) is the rectangular, hyperbolic light-response of NEE as by Falge et al. (2001), where the respiration is calculated with Lloyd & Taylor (1994), and the maximum canopy uptake rate at light saturation decreases exponentially with VPD as in Koerner (1995). |
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Lasslop et al. (2010) is the rectangular, hyperbolic light-response of NEE as by Falge et al. (2001), where the respiration is calculated with Lloyd & Taylor (1994), and the maximum canopy uptake rate at light saturation decreases exponentially with VPD as in Koerner (1995). |
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Sum of absolute deviations of obs and Lasslop. |
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Sum of squared deviations of obs and Lasslop. |
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Straight line: a + b*x |
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Straight line: a + b*x |
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Sum of absolute deviations of obs and straight line: a + b*x |
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Sum of squared deviations of obs and straight line: a + b*x |
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Straight line through origin: a*x |
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Straight line through origin: a*x |
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Sum of absolute deviations of obs and straight line through origin: a*x |
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Sum of squared deviations of obs and straight line through origin: a*x |
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Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC |
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Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC |
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Sum of absolute deviations of obs and Lloyd & Taylor (1994) Arrhenius type. |
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Sum of squared deviations of obs and Lloyd & Taylor (1994) Arrhenius type. |
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If E0 is know in Lloyd & Taylor (1994) then one can calc the exponential term outside the routine and the fitting becomes linear. |
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If E0 is know in Lloyd & Taylor (1994) then one can calc the exponential term outside the routine and the fitting becomes linear. |
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Sum of absolute deviations of obs and Lloyd & Taylor with known exponential term. |
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Sum of squared deviations of obs and Lloyd & Taylor with known exponential term. |
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Square root of general 1/x function: sqrt(a + b/x) |
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Square root of general 1/x function: sqrt(a + b/x) |
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Sum of absolute deviations of obs and square root of general 1/x function: sqrt(a + b/x) |
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Sum of squared deviations of obs and square root of general 1/x function: sqrt(a + b/x) |
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General polynomial: c0 + c1*x + c2*x**2 + . |
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General polynomial: c0 + c1*x + c2*x**2 + . |
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Sum of absolute deviations of obs and general polynomial: c0 + c1*x + c2*x**2 + . |
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Sum of squared deviations of obs and general polynomial: c0 + c1*x + c2*x**2 + . |
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Sum of absolute deviations of obs and logistic function L/(1+exp(-k(x-x0))) |
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Sum of squared deviations of obs and logistic function L/(1+exp(-k(x-x0))) |
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Sum of absolute deviations of obs and logistic function 1/x function: L/(1+exp(-k(x-x0))) + a |
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Sum of squared deviations of obs and logistic function 1/x function: L/(1+exp(-k(x-x0))) + a |
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Sum of absolute deviations of obs and double logistic function with offset: L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a |
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Sum of squared deviations of obs and double logistic function with offset: L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a |
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Fit function of Sequential Elementary Effects: a * (x-b)**c |
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Fit function of Sequential Elementary Effects: a * (x-b)**c |
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Sum of absolute deviations of obs and fit function of Sequential Elementary Effects: a * (x-b)**c |
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Sum of squared deviations of obs and fit function of Sequential Elementary Effects: a * (x-b)**c |
- cost2_fexp(p, x, y)[source]¶
Sum of squared deviations of obs and general exponential function: a + b * exp(c*x)
- cost2_gauss(p, x, y)[source]¶
Sum of squared deviations of obs and Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )
- cost2_lasslop(p, Rg, et, VPD, NEE)[source]¶
Sum of squared deviations of obs and Lasslop.
- Parameters
p (iterable of floats) –
parameters (len(p)=4)
p[0] Light use efficiency, i.e. initial slope of light response curve [umol(C) J-1]
p[1] Maximum CO2 uptake rate at VPD0=10 hPa [umol(C) m-2 s-1]
p[2] e-folding of exponential decrease of maximum CO2 uptake with VPD increase [Pa-1]
p[3] Respiration at Tref (10 degC) [umol(C) m-2 s-1]
Rg (float or array_like of floats) – Global radiation [W m-2]
et (float or array_like of floats) – Exponential in Lloyd & Taylor: np.exp(E0*(1./(Tref-T0)-1./(T-T0))) []
VPD (float or array_like of floats) – Vapour Pressure Deficit [Pa]
NEE (float or array_like of floats) – Observed net ecosystem exchange [umol(CO2) m-2 s-1]
- Returns
sum of squared deviations
- Return type
- cost2_line0(p, x, y)[source]¶
Sum of squared deviations of obs and straight line through origin: a*x
- cost2_lloyd_fix(p, T, resp)[source]¶
Sum of squared deviations of obs and Lloyd & Taylor (1994) Arrhenius type.
- Parameters
- Returns
sum of squared deviations
- Return type
- cost2_lloyd_only_rref(p, et, resp)[source]¶
Sum of squared deviations of obs and Lloyd & Taylor with known exponential term.
- cost2_logistic(p, x, y)[source]¶
Sum of squared deviations of obs and logistic function L/(1+exp(-k(x-x0)))
- Parameters
- Returns
sum of squared deviations
- Return type
- cost2_logistic2_offset(p, x, y)[source]¶
Sum of squared deviations of obs and double logistic function with offset: L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a
- Parameters
p (iterable of floats) –
parameters (len(p)=7)
p[0] L1 - Maximum of first logistic function
p[1] k1 - Steepness of first logistic function
p[2] x01 - Inflection point of first logistic function
p[3] L2 - Maximum of second logistic function
p[4] k2 - Steepness of second logistic function
p[5] x02 - Inflection point of second logistic function
p[6] a - Offset of double logistic function
x (float or array_like of floats) – independent variable
y (float or array_like of floats) – dependent variable, observations
- Returns
sum of squared deviations
- Return type
- cost2_logistic_offset(p, x, y)[source]¶
Sum of squared deviations of obs and logistic function 1/x function: L/(1+exp(-k(x-x0))) + a
- Parameters
p (iterable of floats) –
parameters (len(p)=4)
p[0] L - Maximum of logistic function
p[1] k - Steepness of logistic function
p[2] x0 - Inflection point of logistic function
p[3] a - Offset of logistic function
x (float or array_like of floats) – independent variable
y (float or array_like of floats) – dependent variable, observations
- Returns
sum of squared deviations
- Return type
- cost2_poly(p, x, y)[source]¶
Sum of squared deviations of obs and general polynomial: c0 + c1*x + c2*x**2 + … + cn*x**n
- cost2_sabx(p, x, y)[source]¶
Sum of squared deviations of obs and square root of general 1/x function: sqrt(a + b/x)
- cost2_see(p, x, y)[source]¶
Sum of squared deviations of obs and fit function of Sequential Elementary Effects: a * (x-b)**c
- cost_abs(p, func, x, y)[source]¶
General cost function for robust optimising func(x,p) vs. y with sum of absolute deviations.
- cost_fexp(p, x, y)[source]¶
Sum of absolute deviations of obs and general exponential function: a + b * exp(c*x)
- cost_gauss(p, x, y)[source]¶
Sum of absolute deviations of obs and Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )
- cost_lasslop(p, Rg, et, VPD, NEE)[source]¶
Sum of absolute deviations of obs and Lasslop.
- Parameters
p (iterable of floats) –
parameters (len(p)=4)
p[0] Light use efficiency, i.e. initial slope of light response curve [umol(C) J-1]
p[1] Maximum CO2 uptake rate at VPD0=10 hPa [umol(C) m-2 s-1]
p[2] e-folding of exponential decrease of maximum CO2 uptake with VPD increase [Pa-1]
p[3] Respiration at Tref (10 degC) [umol(C) m-2 s-1]
Rg (float or array_like of floats) – Global radiation [W m-2]
et (float or array_like of floats) – Exponential in Lloyd & Taylor: np.exp(E0*(1./(Tref-T0)-1./(T-T0))) []
VPD (float or array_like of floats) – Vapour Pressure Deficit [Pa]
NEE (float or array_like of floats) – Observed net ecosystem exchange [umol(CO2) m-2 s-1]
- Returns
sum of absolute deviations
- Return type
- cost_line0(p, x, y)[source]¶
Sum of absolute deviations of obs and straight line through origin: a*x
- cost_lloyd_fix(p, T, resp)[source]¶
Sum of absolute deviations of obs and Lloyd & Taylor (1994) Arrhenius type.
- Parameters
- Returns
sum of absolute deviations
- Return type
- cost_lloyd_only_rref(p, et, resp)[source]¶
Sum of absolute deviations of obs and Lloyd & Taylor with known exponential term.
- cost_logistic(p, x, y)[source]¶
Sum of absolute deviations of obs and logistic function L/(1+exp(-k(x-x0)))
- Parameters
- Returns
sum of absolute deviations
- Return type
- cost_logistic2_offset(p, x, y)[source]¶
Sum of absolute deviations of obs and double logistic function with offset: L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a
- Parameters
p (iterable of floats) –
parameters (len(p)=7)
p[0] L1 - Maximum of first logistic function
p[1] k1 - Steepness of first logistic function
p[2] x01 - Inflection point of first logistic function
p[3] L2 - Maximum of second logistic function
p[4] k2 - Steepness of second logistic function
p[5] x02 - Inflection point of second logistic function
p[6] a - Offset of double logistic function
x (float or array_like of floats) – independent variable
y (float or array_like of floats) – dependent variable, observations
- Returns
sum of absolute deviations
- Return type
- cost_logistic_offset(p, x, y)[source]¶
Sum of absolute deviations of obs and logistic function 1/x function: L/(1+exp(-k(x-x0))) + a
- Parameters
p (iterable of floats) –
parameters (len(p)=4)
p[0] L - Maximum of logistic function
p[1] k - Steepness of logistic function
p[2] x0 - Inflection point of logistic function
p[3] a - Offset of logistic function
x (float or array_like of floats) – independent variable
y (float or array_like of floats) – dependent variable, observations
- Returns
sum of absolute deviations
- Return type
- cost_poly(p, x, y)[source]¶
Sum of absolute deviations of obs and general polynomial: c0 + c1*x + c2*x**2 + … + cn*x**n
- cost_sabx(p, x, y)[source]¶
Sum of absolute deviations of obs and square root of general 1/x function: sqrt(a + b/x)
- cost_see(p, x, y)[source]¶
Sum of absolute deviations of obs and fit function of Sequential Elementary Effects: a * (x-b)**c
- cost_square(p, func, x, y)[source]¶
General cost function for optimising func(x,p) vs. y with sum of square deviations.
- lasslop(Rg, et, VPD, alpha, beta0, k, Rref)[source]¶
Lasslop et al. (2010) is the rectangular, hyperbolic light-response of NEE as by Falge et al. (2001), where the respiration is calculated with Lloyd & Taylor (1994), and the maximum canopy uptake rate at light saturation decreases exponentially with VPD as in Koerner (1995).
- Parameters
Rg (float or array_like of floats) – Global radiation [W m-2]
et (float or array_like of floats) – Exponential in Lloyd & Taylor: np.exp(E0*(1./(Tref-T0)-1./(T-T0))) []
VPD (float or array_like of floats) – Vapour Pressure Deficit [Pa]
alpha (float) – Light use efficiency, i.e. initial slope of light response curve [umol(C) J-1]
beta0 (float) – Maximum CO2 uptake rate at VPD0=10 hPa [umol(C) m-2 s-1]
k (float) – e-folding of exponential decrease of maximum CO2 uptake with VPD increase [Pa-1]
Rref (float) – Respiration at Tref (10 degC) [umol(C) m-2 s-1]
- Returns
net ecosystem exchange [umol(CO2) m-2 s-1]
- Return type
- lasslop_p(Rg, et, VPD, p)[source]¶
Lasslop et al. (2010) is the rectangular, hyperbolic light-response of NEE as by Falge et al. (2001), where the respiration is calculated with Lloyd & Taylor (1994), and the maximum canopy uptake rate at light saturation decreases exponentially with VPD as in Koerner (1995).
- Parameters
Rg (float or array_like of floats) – Global radiation [W m-2]
et (float or array_like of floats) – Exponential in Lloyd & Taylor: np.exp(E0*(1./(Tref-T0)-1./(T-T0))) []
VPD (float or array_like of floats) – Vapour Pressure Deficit [Pa]
p (iterable of floats) –
parameters (len(p)=4)
p[0] Light use efficiency, i.e. initial slope of light response curve [umol(C) J-1]
p[1] Maximum CO2 uptake rate at VPD0=10 hPa [umol(C) m-2 s-1]
p[2] e-folding of exponential decrease of maximum CO2 uptake with VPD increase [Pa-1]
p[3] Respiration at Tref (10 degC) [umol(C) m-2 s-1]
- Returns
net ecosystem exchange [umol(CO2) m-2 s-1]
- Return type
- lloyd_fix(T, Rref, E0)[source]¶
Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
- lloyd_fix_p(T, p)[source]¶
Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
- lloyd_only_rref(et, Rref)[source]¶
If E0 is know in Lloyd & Taylor (1994) then one can calc the exponential term outside the routine and the fitting becomes linear. One could also use functions.line0.
- lloyd_only_rref_p(et, p)[source]¶
If E0 is know in Lloyd & Taylor (1994) then one can calc the exponential term outside the routine and the fitting becomes linear. One could also use functions.line0.