Fit functions#
Common functions used in scipy’s curve_fit or minimize 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.cost2_f1x, np.array([p1, p2]), args=(x, y),
method='Nelder-Mead', options={'disp': False})
- Note the different argument orders:
curvefit needs f(x, *p) 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})
There are the following functions in four forms. They have the name in the
first column. The second form has a _p appended to the name. The cost
functions have cost_ or cost2_ prepended to the name, for example
gauss(), gauss_p(), cost_gauss(), cost2_gauss():
Function - number of paramters |
Description |
|---|---|
Arrhenius temperature dependence of biochemical rates: \(\exp((T-TC25)*E/(T25*R*(T+T0)))\) |
|
General 1/x function: \(a + b/x\) |
|
General exponential function: \(a + b * \exp(c*x)\) |
|
Gauss function: \(1/(\sigma*\sqrt(2*\pi)) * \exp(-(x-\mu)^2/(2*\sigma^2))\) |
|
NEE of Lasslop et al. (2010) |
|
Straight line: \(a*x\) |
|
Straight line: \(a + b*x\) |
|
Heterotrophic respiration of Lloyd & Taylor (1994) |
|
Heterotrophic respiration of Lloyd & Taylor (1994) with fixed E0 |
|
|
Logistic function: \(a/(1+\exp(-b(x-c)))\) |
|
Logistic function with offset: \(a/(1+\exp(-b(x-c))) + d\) |
|
Double logistic function with offset \(L1/(1+\exp(-k1(x-x01))) - L2/(1+\exp(-k2(x-x02))) + a\) |
General polynomial: \(c_0 + c_1*x + c_2*x^2 + ... + c_n*x^n\) |
|
sqrt(f1x), i.e. general sqrt(1/x) function: \(\sqrt(a + b/x)\) |
|
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:
Copyright 2012-2022 Matthias Cuntz, see AUTHORS.rst for details.
- license:
MIT License, see LICENSE for details.
Functions:
|
General cost function for robust optimising func(x, p) vs y with sum of absolute deviations. |
|
General cost function for optimising func(x, p) vs y with sum of square deviations. |
|
Arrhenius temperature dependence of rates |
|
Arrhenius temperature dependence of rates |
|
Sum of absolute deviations of obs and arrhenius function |
|
Sum of squared deviations of obs and arrhenius function |
|
General 1/x function: \(a + b/x\) |
|
General 1/x function: \(a + b/x\) |
|
Sum of absolute deviations of obs and general 1/x function: \(a + b/x\) |
|
Sum of squared deviations of obs and general 1/x function: \(a + b/x\) |
|
General exponential function: \(a + b * exp(c*x)\) |
|
General exponential function: \(a + b * exp(c*x)\) |
|
Sum of absolute deviations of obs and general exponential function: \(a + b * exp(c*x)\) |
|
Sum of squared deviations of obs and general exponential function: \(a + b * exp(c*x)\) |
|
Gauss function: \(1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )\) |
|
Gauss function: \(1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )\) |
|
Sum of absolute deviations of obs and Gauss function: \(1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )\) |
|
Sum of squared deviations of obs and Gauss function: \(1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )\) |
|
NEE of Lasslop et al (2010) |
|
NEE of Lasslop et al (2010) |
|
Sum of absolute deviations of obs and Lasslop et al (2010) |
|
Sum of squared deviations of obs and Lasslop et al (2010) |
|
Straight line: \(a + b*x\) |
|
Straight line: \(a + b*x\) |
|
Sum of absolute deviations of obs and straight line: \(a + b*x\) |
|
Sum of squared deviations of obs and straight line: \(a + b*x\) |
|
Straight line through origin: \(a*x\) |
|
Straight line through origin: \(a*x\) |
|
Sum of absolute deviations of obs and straight line through origin: \(a*x\) |
|
Sum of squared deviations of obs and straight line through origin: \(a*x\) |
|
Soil respiration of Lloyd & Taylor (1994) |
|
Soil respiration of Lloyd & Taylor (1994) |
|
Sum of absolute deviations of obs and Lloyd & Taylor (1994) Arrhenius type. |
|
Sum of squared deviations of obs and Lloyd & Taylor (1994) Arrhenius type. |
|
Soil respiration of Lloyd & Taylor (1994) with fix E0 |
|
Soil respiration of Lloyd & Taylor (1994) with fix E0 |
|
Sum of absolute deviations of obs and Lloyd & Taylor with fixed E0 |
|
Sum of squared deviations of obs and Lloyd & Taylor with fixed E0 |
|
Square root of general 1/x function: \(sqrt(a + b/x)\) |
|
Square root of general 1/x function: \(sqrt(a + b/x)\) |
|
Sum of absolute deviations of obs and square root of general 1/x function: \(sqrt(a + b/x)\) |
|
Sum of squared deviations of obs and square root of general 1/x function: \(sqrt(a + b/x)\) |
|
General polynomial: :math:`c0 + c1*x + c2*x**2 + . |
|
General polynomial: :math:`c0 + c1*x + c2*x**2 + . |
|
Sum of absolute deviations of obs and general polynomial: :math:`c0 + c1*x + c2*x**2 + . |
|
Sum of squared deviations of obs and general polynomial: :math:`c0 + c1*x + c2*x**2 + . |
|
Sum of absolute deviations of obs and logistic function \(L/(1+exp(-k(x-x0)))\) |
|
Sum of squared deviations of obs and logistic function \(L/(1+exp(-k(x-x0)))\) |
|
Sum of absolute deviations of obs and logistic function 1/x function: \(L/(1+exp(-k(x-x0))) + a\) |
|
Sum of squared deviations of obs and logistic function 1/x function: \(L/(1+exp(-k(x-x0))) + a\) |
|
Sum of absolute deviations of obs and double logistic function with offset: \(L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\) |
|
Sum of squared deviations of obs and double logistic function with offset: \(L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\) |
|
Fit function of Sequential Elementary Effects: \(a * (x-b)**c\) |
|
Fit function of Sequential Elementary Effects: \(a * (x-b)**c\) |
|
Sum of absolute deviations of obs and fit function of Sequential Elementary Effects: \(a * (x-b)**c\) |
|
Sum of squared deviations of obs and fit function of Sequential Elementary Effects: \(a * (x-b)**c\) |
- History
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
More consistent docstrings, Jan 2022, Matthias Cuntz
- 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 et al (2010)
- 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 fixed E0
- cost2_logistic(p, x, y)[source]#
Sum of squared deviations of obs and logistic function \(L/(1+exp(-k(x-x0)))\)
- Parameters:
p (iterable of floats) –
- parameters (len(p)=3)
p[0] = L = Maximum of logistic function
p[1] = k = Steepness of logistic function
p[2] = x0 = Inflection point 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_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 et al (2010)
- 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 fixed E0
- cost_logistic(p, x, y)[source]#
Sum of absolute deviations of obs and logistic function \(L/(1+exp(-k(x-x0)))\)
- Parameters:
p (iterable of floats) –
- parameters (len(p)=3)
p[0] = L = Maximum of logistic function
p[1] = k = Steepness of logistic function
p[2] = x0 = Inflection point 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_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.
- gauss(x, mu, sig)[source]#
Gauss function: \(1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )\)
- lasslop(Rg, et, VPD, alpha, beta0, k, Rref)[source]#
NEE of Lasslop et al (2010)
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]#
NEE of Lasslop et al (2010)
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]#
Soil respiration of Lloyd & Taylor (1994)
Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
- lloyd_fix_p(T, p)[source]#
Soil respiration of Lloyd & Taylor (1994)
Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
- lloyd_only_rref(et, Rref)[source]#
Soil respiration of Lloyd & Taylor (1994) with fix E0
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]#
Soil respiration of Lloyd & Taylor (1994) with fix E0
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.