hesseflux.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:
cost_abs(p, func, x, y) |
General cost function for robust optimising func(x,p) vs. |
cost_square(p, func, x, y) |
General cost function for optimising func(x,p) vs. |
arrhenius(T, E) |
Arrhenius temperature dependence of rates. |
arrhenius_p(T, p) |
Arrhenius temperature dependence of rates. |
cost_arrhenius(p, T, rate) |
Sum of absolute deviations of obs and arrhenius function. |
cost2_arrhenius(p, T, rate) |
Sum of squared deviations of obs and arrhenius. |
f1x(x, a, b) |
General 1/x function: a + b/x |
f1x_p(x, p) |
General 1/x function: a + b/x |
cost_f1x(p, x, y) |
Sum of absolute deviations of obs and general 1/x function: a + b/x |
cost2_f1x(p, x, y) |
Sum of squared deviations of obs and general 1/x function: a + b/x |
fexp(x, a, b, c) |
General exponential function: a + b * exp(c*x) |
fexp_p(x, p) |
General exponential function: a + b * exp(c*x) |
cost_fexp(p, x, y) |
Sum of absolute deviations of obs and general exponential function: a + b * exp(c*x) |
cost2_fexp(p, x, y) |
Sum of squared deviations of obs and general exponential function: a + b * exp(c*x) |
gauss(x, mu, sig) |
Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
gauss_p(x, p) |
Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
cost_gauss(p, x, y) |
Sum of absolute deviations of obs and Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
cost2_gauss(p, x, y) |
Sum of squared deviations of obs and Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) ) |
lasslop(Rg, et, VPD, alpha, beta0, k, Rref) |
Lasslop et al. |
lasslop_p(Rg, et, VPD, p) |
Lasslop et al. |
cost_lasslop(p, Rg, et, VPD, NEE) |
Sum of absolute deviations of obs and Lasslop. |
cost2_lasslop(p, Rg, et, VPD, NEE) |
Sum of squared deviations of obs and Lasslop. |
line(x, a, b) |
Straight line: a + b*x |
line_p(x, p) |
Straight line: a + b*x |
cost_line(p, x, y) |
Sum of absolute deviations of obs and straight line: a + b*x |
cost2_line(p, x, y) |
Sum of squared deviations of obs and straight line: a + b*x |
line0(x, a) |
Straight line through origin: a*x |
line0_p(x, p) |
Straight line through origin: a*x |
cost_line0(p, x, y) |
Sum of absolute deviations of obs and straight line through origin: a*x |
cost2_line0(p, x, y) |
Sum of squared deviations of obs and straight line through origin: a*x |
lloyd_fix(T, Rref, E0) |
Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC |
lloyd_fix_p(T, p) |
Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC |
cost_lloyd_fix(p, T, resp) |
Sum of absolute deviations of obs and Lloyd & Taylor (1994) Arrhenius type. |
cost2_lloyd_fix(p, T, resp) |
Sum of squared deviations of obs and Lloyd & Taylor (1994) Arrhenius type. |
lloyd_only_rref(et, Rref) |
If E0 is know in Lloyd & Taylor (1994) then one can calc the exponential term outside the routine and the fitting becomes linear. |
lloyd_only_rref_p(et, p) |
If E0 is know in Lloyd & Taylor (1994) then one can calc the exponential term outside the routine and the fitting becomes linear. |
cost_lloyd_only_rref(p, et, resp) |
Sum of absolute deviations of obs and Lloyd & Taylor with known exponential term. |
cost2_lloyd_only_rref(p, et, resp) |
Sum of squared deviations of obs and Lloyd & Taylor with known exponential term. |
sabx(x, a, b) |
Square root of general 1/x function: sqrt(a + b/x) |
sabx_p(x, p) |
Square root of general 1/x function: sqrt(a + b/x) |
cost_sabx(p, x, y) |
Sum of absolute deviations of obs and square root of general 1/x function: sqrt(a + b/x) |
cost2_sabx(p, x, y) |
Sum of squared deviations of obs and square root of general 1/x function: sqrt(a + b/x) |
poly(x, *args) |
General polynomial: c0 + c1*x + c2*x**2 + … |
poly_p(x, p) |
General polynomial: c0 + c1*x + c2*x**2 + … |
cost_poly(p, x, y) |
Sum of absolute deviations of obs and general polynomial: c0 + c1*x + c2*x**2 + … |
cost2_poly(p, x, y) |
Sum of squared deviations of obs and general polynomial: c0 + c1*x + c2*x**2 + … |
cost_logistic(p, x, y) |
Sum of absolute deviations of obs and logistic function L/(1+exp(-k(x-x0))) |
cost2_logistic(p, x, y) |
Sum of squared deviations of obs and logistic function L/(1+exp(-k(x-x0))) |
cost_logistic_offset(p, x, y) |
Sum of absolute deviations of obs and logistic function 1/x function: L/(1+exp(-k(x-x0))) + a |
cost2_logistic_offset(p, x, y) |
Sum of squared deviations of obs and logistic function 1/x function: L/(1+exp(-k(x-x0))) + a |
cost_logistic2_offset(p, x, y) |
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 |
cost2_logistic2_offset(p, x, y) |
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 |
see(x, a, b, c) |
Fit function of Sequential Elementary Effects: a * (x-b)**c |
see_p(x, p) |
Fit function of Sequential Elementary Effects: a * (x-b)**c |
cost_see(p, x, y) |
Sum of absolute deviations of obs and fit function of Sequential Elementary Effects: a * (x-b)**c |
cost2_see(p, x, y) |
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.
Parameters: Returns: sum of absolute deviations
Return type:
-
cost_square(p, func, x, y)[source]¶ General cost function for optimising func(x,p) vs. y with sum of square deviations.
Parameters: Returns: sum of squared deviations
Return type:
-
arrhenius(T, E)[source]¶ Arrhenius temperature dependence of rates.
Parameters: Returns: function value(s)
Return type:
-
arrhenius_p(T, p)[source]¶ Arrhenius temperature dependence of rates.
Parameters: - T (float or array_like of floats) – temperature [degC]
- p (iterable) – p[0] is activation energy [J]
Returns: function value(s)
Return type:
-
cost_arrhenius(p, T, rate)[source]¶ Sum of absolute deviations of obs and arrhenius function.
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_arrhenius(p, T, rate)[source]¶ Sum of squared deviations of obs and arrhenius.
Parameters: Returns: sum of squared deviations
Return type:
-
f1x(x, a, b)[source]¶ General 1/x function: a + b/x
Parameters: Returns: function value(s)
Return type:
-
f1x_p(x, p)[source]¶ General 1/x function: a + b/x
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) –
parameters (len(p)=2)
p[0] a
p[1] b
Returns: function value(s)
Return type:
-
cost_f1x(p, x, y)[source]¶ Sum of absolute deviations of obs and general 1/x function: a + b/x
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_f1x(p, x, y)[source]¶ Sum of squared deviations of obs and general 1/x function: a + b/x
Parameters: Returns: sum of squared deviations
Return type:
-
fexp(x, a, b, c)[source]¶ General exponential function: a + b * exp(c*x)
Parameters: Returns: function value(s)
Return type:
-
fexp_p(x, p)[source]¶ General exponential function: a + b * exp(c*x)
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) –
parameters (len(p)=3)
p[0] a
p[1] b
p[2] c
Returns: function value(s)
Return type:
-
cost_fexp(p, x, y)[source]¶ Sum of absolute deviations of obs and general exponential function: a + b * exp(c*x)
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_fexp(p, x, y)[source]¶ Sum of squared deviations of obs and general exponential function: a + b * exp(c*x)
Parameters: Returns: sum of squared deviations
Return type:
-
gauss(x, mu, sig)[source]¶ Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )
Parameters: Returns: function value(s)
Return type:
-
gauss_p(x, p)[source]¶ Gauss function: 1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) –
parameters (len(p)=2)
p[0] mean
p[1] width
Returns: function value(s)
Return type:
-
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) )
Parameters: Returns: sum of absolute deviations
Return type:
-
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) )
Parameters: Returns: sum of squared deviations
Return type:
-
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:
-
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: - p (iterable of floats) –
-
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: - p (iterable of floats) –
-
line_p(x, p)[source]¶ Straight line: a + b*x
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) –
parameters (len(p)=2)
p[0] a
p[1] b
Returns: function value(s)
Return type:
-
cost_line(p, x, y)[source]¶ Sum of absolute deviations of obs and straight line: a + b*x
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_line(p, x, y)[source]¶ Sum of squared deviations of obs and straight line: a + b*x
Parameters: Returns: sum of squared deviations
Return type:
-
line0(x, a)[source]¶ Straight line through origin: a*x
Parameters: Returns: function value(s)
Return type:
-
line0_p(x, p)[source]¶ Straight line through origin: a*x
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) – p[0] is a
Returns: function value(s)
Return type:
-
cost_line0(p, x, y)[source]¶ Sum of absolute deviations of obs and straight line through origin: a*x
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_line0(p, x, y)[source]¶ Sum of squared deviations of obs and straight line through origin: a*x
Parameters: Returns: sum of squared deviations
Return type:
-
lloyd_fix(T, Rref, E0)[source]¶ Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
Parameters: Returns: Respiration [umol(C) m-2 s-1]
Return type:
-
lloyd_fix_p(T, p)[source]¶ Lloyd & Taylor (1994) Arrhenius type with T0=-46.02 degC and Tref=10 degC
Parameters: - T (float or array_like of floats) – Temperature [K]
- p (iterable of floats) –
parameters (len(p)=2)
p[0] Respiration at Tref=10 degC [umol(C) m-2 s-1]
p[1] Activation energy [K]
Returns: Respiration [umol(C) m-2 s-1]
Return type:
-
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:
-
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:
-
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.
Parameters: Returns: Respiration [umol(C) m-2 s-1]
Return type:
-
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.
Parameters: - et (float or array_like of floats) – exp-term in Lloyd & Taylor
- p (iterable of floats) – p[0] is respiration at Tref=10 degC [umol(C) m-2 s-1]
Returns: Respiration [umol(C) m-2 s-1]
Return type:
-
cost_lloyd_only_rref(p, et, resp)[source]¶ Sum of absolute deviations of obs and Lloyd & Taylor with known exponential term.
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_lloyd_only_rref(p, et, resp)[source]¶ Sum of squared deviations of obs and Lloyd & Taylor with known exponential term.
Parameters: Returns: sum of squared deviations
Return type:
-
sabx(x, a, b)[source]¶ Square root of general 1/x function: sqrt(a + b/x)
Parameters: Returns: function value(s)
Return type:
-
sabx_p(x, p)[source]¶ Square root of general 1/x function: sqrt(a + b/x)
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) –
parameters (len(p)=2)
p[0] a
p[1] b
Returns: function value(s)
Return type:
-
cost_sabx(p, x, y)[source]¶ Sum of absolute deviations of obs and square root of general 1/x function: sqrt(a + b/x)
Parameters: Returns: sum of absolute deviations
Return type:
-
cost2_sabx(p, x, y)[source]¶ Sum of squared deviations of obs and square root of general 1/x function: sqrt(a + b/x)
Parameters: Returns: sum of squared deviations
Return type:
-
poly(x, *args)[source]¶ General polynomial: c0 + c1*x + c2*x**2 + … + cn*x**n
Parameters: Returns: function value(s)
Return type:
-
poly_p(x, p)[source]¶ General polynomial: c0 + c1*x + c2*x**2 + … + cn*x**n
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) – parameters (len(p)=n+1)
Returns: function value(s)
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
Parameters: Returns: sum of absolute 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
Parameters: Returns: sum of squared deviations
Return type:
-
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:
-
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:
-
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: - p (iterable of floats) –
-
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: - p (iterable of floats) –
-
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: - p (iterable of floats) –
-
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: - p (iterable of floats) –
-
see(x, a, b, c)[source]¶ Fit function of Sequential Elementary Effects: a * (x-b)**c
Parameters: Returns: function value(s)
Return type:
-
see_p(x, p)[source]¶ Fit function of Sequential Elementary Effects: a * (x-b)**c
Parameters: - x (float or array_like of floats) – independent variable
- p (iterable of floats) –
parameters (len(p)=3)
p[0] a
p[1] b
p[2] c
Returns: function value(s)
Return type:
-
cost_see(p, x, y)[source]¶ Sum of absolute deviations of obs and fit function of Sequential Elementary Effects: a * (x-b)**c
Parameters: Returns: sum of absolute deviations
Return type: