Function module#
Provides a variety of special functions, including common test functions for
parameter estimations such as Rosenbrock and Griewank, test functions for
parameter sensitivity analysis such as the Ishigami and Homma function, several
forms of the logistic function and its first and second derivatives, and a
variety of other functions together with robust and square cost functions to
use with the scipy.optimize package.
- copyright:
Copyright 2014-2021 Matthias Cuntz, see AUTHORS.rst for details.
- license:
MIT License, see LICENSE for details.
Subpackages#
General functions that are not specialised for fitting, optimisation, sensitivity analysis, etc. |
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Common functions used in scipy's curve_fit or minimize parameter estimations. |
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Several forms of the logistic function and their first and second derivatives |
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Common test functions for parameter estimation and optimisation algorithms. |
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Test functions for parameter sensitivity analysis. |
- B(X)[source]#
B function
Saltelli et al. (2010) Comp. Phys. Comm. 181, p. 259-270
- Parameters:
X (array_like) – (nX,) or (nX,npoints) array of floats
- Returns:
B – float or (npoints,) floats of B function values at X
- Return type:
float or ndarray
- G(X, a)[source]#
G-function
Sobol’ (1990) Matematicheskoe Modelirovanie 2, 112-118 (in Russian)
Sobol’ (1993) Mathematical Modelling and Computational Experiment 1, 407-414 (English translation)
- Parameters:
X (array_like) – (nX,) or (nX,npoints) array of floats
a (array_like) – (nX,) array of floats
- Returns:
g – float or (npoints,) floats of G function values at X with parameters a
- Return type:
float or ndarray
- Gstar(X, alpha, delta, a)[source]#
G* example
Saltelli et al. (2010) Comp. Phys. Comm., 181, p. 259-270
- Parameters:
X (array_like) – (nX,) or (nX,npoints) array of floats
alpha (array_like) – (nX,) array of floats
delta (array_like) – (nX,) array of floats
a (array_like) – (nX,) array of floats
- Returns:
G* – float or (npoints,) floats of G* function values at X with parameters alpha, delta and a
- Return type:
float or ndarray
- K(X)[source]#
K example
Saltelli et al. (2010) Comp. Phys. Comm., 181, p. 259-270
- Parameters:
X (array_like) – (nX,) or (nX,npoints) array of floats
- Returns:
K – float or (npoints,) floats of K function values at X
- Return type:
float or ndarray
- ackley(x)[source]#
Ackley function (>= 2-D).
Global Optimum: 0.0 at origin.
- Parameters:
x (array) – multi-dimensional x-values (len(x) >= 2)
- Returns:
Value of Ackley function.
- Return type:
- bratley(*args)[source]#
K example
Saltelli et al. (2010) Comp. Phys. Comm., 181, p. 259-270
- Parameters:
X (array_like) – (nX,) or (nX,npoints) array of floats
- Returns:
bratley – float or (npoints,) floats of K function values at X
- Return type:
float or ndarray
- 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.
- curvature(x, dfunc, d2func, *args, **kwargs)[source]#
Curvature of a function f
\[f''/(1+f'^2)^{3/2}\]- Parameters:
x (array_like) – Independent variable to evalute curvature
dfunc (callable) – Function giving first derivative of function f: f’, to be called dfunc(x, *args, **kwargs)
d2func (callable) – Function giving second derivative of function f: f’’, to be called d2func(x, *args, **kwargs)
args (iterable) – Arguments passed to dfunc and d2func
kwargs (dict) – Keyword arguments passed to dfunc and d2func
- Returns:
Curvature of function f at x
- Return type:
float or ndarray
Examples
from pyjams.functions import dlogistic_offset, d2logistic_offset curvature(1., dlogistic_offset, d2logistic_offset, [1., 2., 2., 1.])
- d2logistic(x, L, k, x0)[source]#
Second derivative of
logistic()function\[L/(1+exp(-k(x-x0)))\]which is
\[-k^2.L.sinh(k(x-x0))/(2(cosh(k(x-x0))+1)^2)\]- Parameters:
- Returns:
Second derivative of logistic function at x with maximum L, steepness k and inflection point x0
- Return type:
float or ndarray
- d2logistic2_offset(x, L1, k1, x01, L2, k2, x02, a)[source]#
Second derivative of
logistic_offset()function\[L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\]which is
\[- k1^2.L1.sinh(k1(x-x01)) / (2(cosh(k1(x-x01))+1)^2) + k2^2.L2.sinh(k2(x-x02)) / (2(cosh(k2(x-x02))+1)^2)\]- Parameters:
x (array_like) – Independent variable to evalute logistic function
L1 (float) – Maximum of first logistic function
k1 (float) – Steepness of first logistic function
x01 (float) – Inflection point of first logistic function
L2 (float) – Maximum of second logistic function
k2 (float) – Steepness of second logistic function
x02 (float) – Inflection point of second logistic function
a (float) – Offset of double logistic function
- Returns:
Second derivative of double logistic function with offset at x
- Return type:
float or ndarray
- d2logistic2_offset_p(x, p)[source]#
Wrapper function for
d2logistic2_offset(): d2logistic2_offset(x, *p)
- d2logistic_offset(x, L, k, x0, a)[source]#
Second derivative of
logistic_offset()function\[L/(1+exp(-k(x-x0))) + a\]which is
\[-k^2.L.sinh(k(x-x0))/(2(cosh(k(x-x0))+1)^2)\]- Parameters:
- Returns:
Second derivative of logistic function at x with maximum L, steepness k, inflection point x0, and offset a
- Return type:
float or ndarray
- d2logistic_offset_p(x, p)[source]#
Wrapper function for
d2logistic_offset(): d2logistic_offset(x, *p)
- d2logistic_p(x, p)[source]#
Wrapper function for
d2logistic(): d2logistic(x, *p)
- dlogistic(x, L, k, x0)[source]#
First derivative of
logistic()function\[L/(1+exp(-k(x-x0)))\]which is
\[k.L/(2(cosh(k(x-x0))+1))\]- Parameters:
- Returns:
First derivative of logistic function at x with maximum L, steepness k and inflection point x0
- Return type:
float or ndarray
- dlogistic2_offset(x, L1, k1, x01, L2, k2, x02, a)[source]#
First derivative of
logistic2_offset()function\[L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\]which is
\[k1.L1/(2(cosh(k1(x-x01))+1)) - k2.L2/(2(cosh(k2(x-x02))+1))\]- Parameters:
x (array_like) – Independent variable to evalute logistic function
L1 (float) – Maximum of first logistic function
k1 (float) – Steepness of first logistic function
x01 (float) – Inflection point of first logistic function
L2 (float) – Maximum of second logistic function
k2 (float) – Steepness of second logistic function
x02 (float) – Inflection point of second logistic function
a (float) – Offset of double logistic function
- Returns:
First derivative of double logistic function with offset at x
- Return type:
float or ndarray
- dlogistic2_offset_p(x, p)[source]#
Wrapper function for
dlogistic2_offset(): dlogistic2_offset(x, *p)
- dlogistic_offset(x, L, k, x0, a)[source]#
First derivative of
logistic_offset()function\[L/(1+exp(-k(x-x0))) + a\]which is
\[k.L/(2(cosh(k(x-x0))+1))\]- Parameters:
- Returns:
First derivative of logistic function with offset at x with maximum L, steepness k, inflection point x0, and offset a
- Return type:
float or ndarray
- dlogistic_offset_p(x, p)[source]#
Wrapper function for
dlogistic_offset(): dlogistic_offset(x, *p)
- dlogistic_p(x, p)[source]#
Wrapper function for
dlogistic(): dlogistic(x, *p)
- fmorris(X, beta0, beta1, beta2, beta3, beta4)[source]#
Morris-function
Morris (1991) Technometrics 33, 161-174
- Parameters:
X (array_like) – (20,) or (20,npoints) array of floats
beta0 (float) – float
beta1 (array_like) – (20,) array of floats
beta2 (array_like) – (20,20) array of floats
beta3 (array_like) – (20,20,20) array of floats
beta4 (array_like) – (20,20,20,20) array of floats
- Returns:
fmorris – float or (npoints,) floats of Morris function values at X with parameters beta0-beta4
- Return type:
float or ndarray
- g(X, a)[source]#
G-function
Sobol’ (1990) Matematicheskoe Modelirovanie 2, 112-118 (in Russian)
Sobol’ (1993) Mathematical Modelling and Computational Experiment 1, 407-414 (English translation)
- Parameters:
X (array_like) – (nX,) or (nX,npoints) array of floats
a (array_like) – (nX,) array of floats
- Returns:
G – float or (npoints,) floats of G function values at X with parameters a
- Return type:
float or ndarray
- gauss(x, mu, sig)[source]#
Gauss function: \(1 / (sqrt(2*pi)*sig) * exp( -(x-mu)**2 / (2*sig**2) )\)
- goldstein_price(x)[source]#
Goldstein-Price function (2-D).
Global Optimum: 3.0 at (0.0, -1.0).
- Parameters:
x (array) – 2 x-values. len(x)=2. x[i] bound to [-2, 2] for i=1 and 2.
- Returns:
Value of Goldstein-Price function.
- Return type:
- griewank(x)[source]#
Griewank function (2-D or 10-D).
Global Optimum: 0.0 at origin.
- Parameters:
x (array) – multi-dimensional x-values. len(x)=2 or len(x)=10. x[i] bound to [-600, 600] for all i.
- Returns:
Value of Griewank function.
- Return type:
- ishigami_homma(X, a, b)[source]#
Ishigami and Homma (1990)
given by Saltelli et al. (2008, page 179)
- Parameters:
X (array_like) – (3,) or (3,npoints) array of floats
a (array_like) – float or (npoints,) array of floats
b (array_like) – float or (npoints,) array of floats
- Returns:
ishigami_homma – float or (npoints,) floats of Ishigami and Homma function values at X with parameters a and b
- Return type:
float or ndarray
- ishigami_homma_easy(X)[source]#
Simplified Ishigami and Homma function
\[Y = sin(X[0]) + X[1]\]with X[0], X[1] ~ Uniform[-Pi, Pi]
- Parameters:
X (array_like) – (2,) or (2,npoints) array of floats
- Returns:
ishigami_homma_easy – float or (npoints,) floats of simplified Ishigami and Homma function values at X
- Return type:
float or ndarray
- 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:
- linear(X, a, b)[source]#
Linear test function
\[Y = a*X + b\]- Parameters:
X (array_like) – (1,) or (1,npoints) array of floats
a (array_like) – float or (npoints,) array of floats
b (array_like) – float or (npoints,) array of floats
- Returns:
linear – float or (npoints,) floats of linear function values at X with parameters a and b
- Return type:
float or ndarray
- 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.
- logistic(x, L, k, x0)[source]#
Logistic function
\[L/(1+exp(-k(x-x0)))\]- Parameters:
- Returns:
Logistic function at x with maximum L, steepness k and inflection point x0
- Return type:
float or ndarray
- logistic2_offset(x, L1, k1, x01, L2, k2, x02, a)[source]#
Double logistic function with offset
\[L1/(1+exp(-k1(x-x01))) - L2/(1+exp(-k2(x-x02))) + a\]- Parameters:
x (array_like) – Independent variable to evalute logistic function
L1 (float) – Maximum of first logistic function
k1 (float) – Steepness of first logistic function
x01 (float) – Inflection point of first logistic function
L2 (float) – Maximum of second logistic function
k2 (float) – Steepness of second logistic function
x02 (float) – Inflection point of second logistic function
a (float) – Offset of double logistic function
- Returns:
Double Logistic function at x
- Return type:
float or ndarray
- logistic2_offset_p(x, p)[source]#
Wrapper function for
logistic2_offset(): logistic2_offset(x, *p)
- logistic_offset(x, L, k, x0, a)[source]#
Logistic function with offset
\[L/(1+exp(-k(x-x0))) + a\]- Parameters:
- Returns:
Logistic function at x with maximum L, steepness k, inflection point x0 and offset a
- Return type:
float or ndarray
- logistic_offset_p(x, p)[source]#
Wrapper function for
logistic_offset(): logistic_offset(x, *p)
- logistic_p(x, p)[source]#
Wrapper function for
logistic(): logistic(x, *p)
- morris(*args)[source]#
Morris-function
Morris (1991) Technometrics 33, 161-174
- Parameters:
X (array_like) – (20,) or (20, npoints) array of floats
beta0 (float) – float
beta1 (array_like) – (20,) array of floats
beta2 (array_like) – (20, 20) array of floats
beta3 (array_like) – (20, 20, 20) array of floats
beta4 (array_like) – (20, 20, 20, 20) array of floats
- Returns:
morris – float or (npoints,) floats of Morris function values at X with parameters beta0-beta4
- Return type:
float or ndarray
- oakley_ohagan(X)[source]#
Oakley and O’Hagan (2004)
Statist. Soc. B 66, Part 3, 751-769
- Parameters:
X (array_like) – (15,) or (15, npoints) array of floats
- Returns:
oakley_ohagan – float or (npoints,) floats of Oakley and O’Hagan function values at X
- Return type:
float or ndarray
- product(X)[source]#
Product test function
\[Y = X[0] * X[1]\]- Parameters:
X (array_like) – (2,) or (2,npoints) array of floats
- Returns:
product – float or (npoints,) floats of product function values at X
- Return type:
float or ndarray
- rastrigin(x)[source]#
Rastrigin function (2-D).
Global Optimum: -2.0 at origin.
- Parameters:
x (array) – 2 x-values. len(x)=2. x[i] bound to [-1, 1] for i=1 and 2.
- Returns:
Value of Rastrigin function.
- Return type:
- ratio(X)[source]#
Ratio test function
\[Y = X[0] / X[1]\]Simple nonlinear model proposed by Liu et al. (2006):
Liu, H., Sudjianto, A., Chen, W., 2006. Relative entropy based method for probabilistic sensitivity analysis in engineering design. J. Mech. Des. 128, 326-336.
Used by Pianosi & Wagener, Environmental Modelling & Software (2015)
Pianosi, F. & Wagener T., 2015 A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling & Software 67, 1-11.
- Parameters:
X (array_like) – (2,) or (2,npoints) array of floats
- Returns:
ratio – float or (npoints,) floats of ratio function values at X
- Return type:
float or ndarray
- rosenbrock(x)[source]#
Rosenbrock function (2-D).
Global Optimum: 0.0 at (1.0, 1.0).
- Parameters:
x (array) – 2 x-values. len(x)=2. x[1] bound to [-5, 5]. x[2] bound to [-2, 8].
- Returns:
Value of Rosenbrock function.
- Return type: