gstools.covmodel.Rational
- class gstools.covmodel.Rational(dim=3, var=1.0, len_scale=1.0, nugget=0.0, anis=1.0, angles=0.0, *, integral_scale=None, rescale=None, latlon=False, geo_scale=1.0, temporal=False, spatial_dim=None, var_raw=None, hankel_kw=None, **opt_arg)[source]
Bases:
CovModel
The rational quadratic covariance model.
Notes
This model is given by the following correlation function [Rasmussen2003]:
\[\rho(r) = \left(1 + \frac{1}{\alpha} \cdot \left(s\cdot\frac{r}{\ell}\right)^2\right)^{-\alpha}\]Where the standard rescale factor is \(s=1\). \(\alpha\) is a shape parameter and should be > 0.5.
For \(\alpha\to\infty\) this model converges to the Gaussian model:
\[\rho(r)= \exp\left(-\left(s\cdot\frac{r}{\ell}\right)^{2}\right)\]References
- Rasmussen2003
Rasmussen, C. E., “Gaussian processes in machine learning.” Summer school on machine learning. Springer, Berlin, Heidelberg, (2003)
- Other Parameters
alpha (
float
, optional) – Shape parameter. Standard range:[0.5, 50]
Default:1.0
- Parameters
dim (
int
, optional) – dimension of the model. Includes the temporal dimension if temporal is true. To specify only the spatial dimension in that case, use spatial_dim. Default:3
var (
float
, optional) – variance of the model (the nugget is not included in “this” variance) Default:1.0
len_scale (
float
orlist
, optional) – length scale of the model. If a single value is given, the same length-scale will be used for every direction. If multiple values (for main and transversal directions) are given, anis will be recalculated accordingly. If only two values are given in 3D, the latter one will be used for both transversal directions. Default:1.0
nugget (
float
, optional) – nugget of the model. Default:0.0
anis (
float
orlist
, optional) –anisotropy ratios in the transversal directions [e_y, e_z].
e_y = l_y / l_x
e_z = l_z / l_x
If only one value is given in 3D, e_y will be set to 1. This value will be ignored, if multiple len_scales are given. Default:
1.0
angles (
float
orlist
, optional) –angles of rotation (given in rad):
in 2D: given as rotation around z-axis
in 3D: given by yaw, pitch, and roll (known as Tait–Bryan angles)
Default:
0.0
integral_scale (
float
orlist
orNone
, optional) – If given,len_scale
will be ignored and recalculated, so that the integral scale of the model matches the given one. Default:None
rescale (
float
orNone
, optional) – Optional rescaling factor to divide the length scale with. This could be used for unit conversion or rescaling the length scale to coincide with e.g. the integral scale. Will be set by each model individually. Default:None
latlon (
bool
, optional) – Whether the model is describing 2D fields on earths surface described by latitude and longitude. When using this, the model will internally use the associated ‘Yadrenko’ model to represent a valid model. This means, the spatial distance \(r\) will be replaced by \(2\sin(\alpha/2)\), where \(\alpha\) is the great-circle distance, which is equal to the spatial distance of two points in 3D. As a consequence, dim will be set to 3 and anisotropy will be disabled. geo_scale can be set to e.g. earth’s radius, to have a meaningful len_scale parameter. Default: Falsegeo_scale (
float
, optional) – Geographic unit scaling in case of latlon coordinates to get a meaningful length scale unit. By default, len_scale is assumed to be in radians with latlon=True. Can be set toKM_SCALE
to have len_scale in km orDEGREE_SCALE
to have len_scale in degrees. Default:RADIAN_SCALE
temporal (
bool
, optional) – Create a metric spatio-temporal covariance model. Setting this to true will increase dim and field_dim by 1. spatial_dim will be field_dim - 1. The time-dimension is appended, meaning the pos tuple is (x,y,z,…,t). Default: Falsespatial_dim (
int
, optional) – spatial dimension of the model. If given, the model dimension will be determined from this spatial dimension and the possible temporal dimension if temporal is ture. Default: Nonevar_raw (
float
orNone
, optional) – raw variance of the model which will be multiplied withCovModel.var_factor
to result in the actual variance. If given,var
will be ignored. (This is just for models that overrideCovModel.var_factor
) Default:None
hankel_kw (
dict
orNone
, optional) – Modify the init-arguments ofhankel.SymmetricFourierTransform
used for the spectrum calculation. Use with caution (Better: Don’t!).None
is equivalent to{"a": -1, "b": 1, "N": 1000, "h": 0.001}
. Default:None
**opt_arg – Optional arguments are covered by these keyword arguments. If present, they are described in the section Other Parameters.
- Attributes
angles
numpy.ndarray
: Rotation angles (in rad) of the model.anis
numpy.ndarray
: The anisotropy factors of the model.anis_bounds
list
: Bounds for the nugget.arg
arg_bounds
dict
: Bounds for all parameters.arg_list
dim
int
: The dimension of the model.dist_func
do_rotation
bool
: State if a rotation is performed.field_dim
int
: The (parametric) field dimension of the model (with time).geo_scale
float
: Geographic scaling for geographical coords.hankel_kw
dict
:hankel.SymmetricFourierTransform
kwargs.has_cdf
bool
: State if a cdf is defined by the user.has_ppf
bool
: State if a ppf is defined by the user.integral_scale
float
: The main integral scale of the model.integral_scale_vec
numpy.ndarray
: The integral scales in each direction.is_isotropic
bool
: State if a model is isotropic.iso_arg
iso_arg_list
latlon
bool
: Whether the model depends on geographical coords.len_rescaled
float
: The rescaled main length scale of the model.len_scale
float
: The main length scale of the model.len_scale_bounds
list
: Bounds for the length scale.len_scale_vec
numpy.ndarray
: The length scales in each direction.name
str
: The name of the CovModel class.nugget
float
: The nugget of the model.nugget_bounds
list
: Bounds for the nugget.opt_arg
opt_arg_bounds
dict
: Bounds for the optional arguments.pykrige_angle
2D rotation angle for pykrige.
pykrige_angle_x
3D rotation angle around x for pykrige.
pykrige_angle_y
3D rotation angle around y for pykrige.
pykrige_angle_z
3D rotation angle around z for pykrige.
pykrige_anis
2D anisotropy ratio for pykrige.
pykrige_anis_y
3D anisotropy ratio in y direction for pykrige.
pykrige_anis_z
3D anisotropy ratio in z direction for pykrige.
pykrige_kwargs
Keyword arguments for pykrige routines.
rescale
float
: Rescale factor for the length scale of the model.sill
float
: The sill of the variogram.spatial_dim
int
: The spatial field dimension of the model (without time).temporal
bool
: Whether the model is a metric spatio-temporal one.var
float
: The variance of the model.var_bounds
list
: Bounds for the variance.var_raw
float
: The raw variance of the model without factor.
Methods
anisometrize
(pos)Bring a position tuple into the anisotropic coordinate-system.
Calculate the integral scale of the isotrope model.
Check arguments to be within their given bounds.
check_dim
(dim)Check the given dimension.
Run checks for the optional arguments.
cor
(h)Rational normalized correlation function.
cor_axis
(r[, axis])Correlation along axis of anisotropy.
cor_spatial
(pos)Spatial correlation respecting anisotropy and rotation.
cor_yadrenko
(zeta)Yadrenko correlation for great-circle distance from latlon-pos.
correlation
(r)Correlation function of the model.
cov_axis
(r[, axis])Covariance along axis of anisotropy.
cov_nugget
(r)Isotropic covariance of the model respecting the nugget at r=0.
cov_spatial
(pos)Spatial covariance respecting anisotropy and rotation.
cov_yadrenko
(zeta)Yadrenko covariance for great-circle distance from latlon-pos.
covariance
(r)Covariance of the model.
Provide default boundaries for arguments.
Defaults for the optional arguments.
Defaults for boundaries of the optional arguments.
Provide default rescaling factor.
fit_variogram
(x_data, y_data[, anis, sill, ...])Fitting the variogram-model to an empirical variogram.
fix_dim
()Set a fix dimension for the model.
isometrize
(pos)Make a position tuple ready for isotropic operations.
Log radial spectral density of the model.
Axes of the rotated coordinate-system.
percentile_scale
([per])Calculate the percentile scale of the isotrope model.
plot
([func])Plot a function of a the CovModel.
pykrige_vario
([args, r])Isotropic variogram of the model for pykrige.
set_arg_bounds
([check_args])Set bounds for the parameters of the model.
Spectral density of the covariance model.
Radial spectral density of the model.
spectrum
(k)Spectrum of the covariance model.
Factor for the variance.
vario_axis
(r[, axis])Variogram along axis of anisotropy.
vario_nugget
(r)Isotropic variogram of the model respecting the nugget at r=0.
vario_spatial
(pos)Spatial variogram respecting anisotropy and rotation.
vario_yadrenko
(zeta)Yadrenko variogram for great-circle distance from latlon-pos.
variogram
(r)Isotropic variogram of the model.
- anisometrize(pos)
Bring a position tuple into the anisotropic coordinate-system.
- check_arg_bounds()
Check arguments to be within their given bounds.
- check_dim(dim)
Check the given dimension.
- check_opt_arg()
Run checks for the optional arguments.
This is in addition to the bound-checks
Notes
You can use this to raise a ValueError/warning
Any return value will be ignored
This method will only be run once, when the class is initialized
- cor_axis(r, axis=0)
Correlation along axis of anisotropy.
- cor_spatial(pos)
Spatial correlation respecting anisotropy and rotation.
- cor_yadrenko(zeta)
Yadrenko correlation for great-circle distance from latlon-pos.
- correlation(r)
Correlation function of the model.
- cov_axis(r, axis=0)
Covariance along axis of anisotropy.
- cov_nugget(r)
Isotropic covariance of the model respecting the nugget at r=0.
- cov_spatial(pos)
Spatial covariance respecting anisotropy and rotation.
- cov_yadrenko(zeta)
Yadrenko covariance for great-circle distance from latlon-pos.
- covariance(r)
Covariance of the model.
- default_arg_bounds()
Provide default boundaries for arguments.
Given as a dictionary.
- default_opt_arg()[source]
Defaults for the optional arguments.
{"alpha": 1.0}
- Returns
Defaults for optional arguments
- Return type
- default_opt_arg_bounds()[source]
Defaults for boundaries of the optional arguments.
{"alpha": [0.5, 50.0]}
- Returns
Boundaries for optional arguments
- Return type
- default_rescale()
Provide default rescaling factor.
- fit_variogram(x_data, y_data, anis=True, sill=None, init_guess='default', weights=None, method='trf', loss='soft_l1', max_eval=None, return_r2=False, curve_fit_kwargs=None, **para_select)
Fitting the variogram-model to an empirical variogram.
- Parameters
x_data (
numpy.ndarray
) – The bin-centers of the empirical variogram.y_data (
numpy.ndarray
) – The measured variogram If multiple are given, they are interpreted as the directional variograms along the main axis of the associated rotated coordinate system. Anisotropy ratios will be estimated in that case.anis (
bool
, optional) – In case of a directional variogram, you can control anisotropy by this argument. Deselect the parameter from fitting, by setting it “False”. You could also pass a fixed value to be set in the model. Then the anisotropy ratios wont be altered during fitting. Default: Truesill (
float
orbool
, optional) – Here you can provide a fixed sill for the variogram. It needs to be in a fitting range for the var and nugget bounds. If variance or nugget are not selected for estimation, the nugget will be recalculated to fulfill:sill = var + nugget
if the variance is bigger than the sill, nugget will bet set to its lower bound and the variance will be set to the fitting partial sill.
If variance is deselected, it needs to be less than the sill, otherwise a ValueError comes up. Same for nugget. If sill=False, it will be deselected from estimation and set to the current sill of the model. Then, the procedure above is applied. Default: None
init_guess (
str
ordict
, optional) –Initial guess for the estimation. Either:
“default”: using the default values of the covariance model (“len_scale” will be mean of given bin centers; “var” and “nugget” will be mean of given variogram values (if in given bounds))
“current”: using the current values of the covariance model
dict: dictionary with parameter names and given value (separate “default” can bet set to “default” or “current” for unspecified values to get same behavior as given above (“default” by default)) Example:
{"len_scale": 10, "default": "current"}
Default: “default”
weights (
str
,numpy.ndarray
,callable
, optional) –Weights applied to each point in the estimation. Either:
‘inv’: inverse distance
1 / (x_data + 1)
list: weights given per bin
callable: function applied to x_data
If callable, it must take a 1-d ndarray. Then
weights = f(x_data)
. Default: Nonemethod ({‘trf’, ‘dogbox’}, optional) –
Algorithm to perform minimization.
‘trf’ : Trust Region Reflective algorithm, particularly suitable for large sparse problems with bounds. Generally robust method.
‘dogbox’ : dogleg algorithm with rectangular trust regions, typical use case is small problems with bounds. Not recommended for problems with rank-deficient Jacobian.
Default: ‘trf’
loss (
str
orcallable
, optional) – Determines the loss function in scipys curve_fit. The following keyword values are allowed:‘linear’ (default) :
rho(z) = z
. Gives a standard least-squares problem.‘soft_l1’ :
rho(z) = 2 * ((1 + z)**0.5 - 1)
. The smooth approximation of l1 (absolute value) loss. Usually a good choice for robust least squares.‘huber’ :
rho(z) = z if z <= 1 else 2*z**0.5 - 1
. Works similarly to ‘soft_l1’.‘cauchy’ :
rho(z) = ln(1 + z)
. Severely weakens outliers influence, but may cause difficulties in optimization process.‘arctan’ :
rho(z) = arctan(z)
. Limits a maximum loss on a single residual, has properties similar to ‘cauchy’.
If callable, it must take a 1-d ndarray
z=f**2
and return an array_like with shape (3, m) where row 0 contains function values, row 1 contains first derivatives and row 2 contains second derivatives. Default: ‘soft_l1’max_eval (
int
orNone
, optional) – Maximum number of function evaluations before the termination. If None (default), the value is chosen automatically: 100 * n.return_r2 (
bool
, optional) – Whether to return the r2 score of the estimation. Default: Falsecurve_fit_kwargs (
dict
, optional) – Other keyword arguments passed to scipys curve_fit. Default: None**para_select – You can deselect parameters from fitting, by setting them “False” using their names as keywords. You could also pass fixed values for each parameter. Then these values will be applied and the involved parameters wont be fitted. By default, all parameters are fitted.
- Returns
fit_para (
dict
) – Dictionary with the fitted parameter valuespcov (
numpy.ndarray
) – The estimated covariance of popt fromscipy.optimize.curve_fit
. To compute one standard deviation errors on the parameters useperr = np.sqrt(np.diag(pcov))
.r2_score (
float
, optional) – r2 score of the curve fitting results. Only if return_r2 is True.
Notes
You can set the bounds for each parameter by accessing
CovModel.set_arg_bounds
.The fitted parameters will be instantly set in the model.
- fix_dim()
Set a fix dimension for the model.
- isometrize(pos)
Make a position tuple ready for isotropic operations.
- ln_spectral_rad_pdf(r)
Log radial spectral density of the model.
- main_axes()
Axes of the rotated coordinate-system.
- percentile_scale(per=0.9)
Calculate the percentile scale of the isotrope model.
This is the distance, where the given percentile of the variance is reached by the variogram
- plot(func='variogram', **kwargs)
Plot a function of a the CovModel.
- Parameters
func (
str
, optional) –Function to be plotted. Could be one of:
“variogram”
“covariance”
“correlation”
“vario_spatial”
“cov_spatial”
“cor_spatial”
“vario_yadrenko”
“cov_yadrenko”
“cor_yadrenko”
“vario_axis”
“cov_axis”
“cor_axis”
“spectrum”
“spectral_density”
“spectral_rad_pdf”
**kwargs – Keyword arguments forwarded to the plotting function “plot_” + func in
gstools.covmodel.plot
.
See also
- pykrige_vario(args=None, r=0)
Isotropic variogram of the model for pykrige.
- set_arg_bounds(check_args=True, **kwargs)
Set bounds for the parameters of the model.
- Parameters
check_args (bool, optional) – Whether to check if the arguments are in their valid bounds. In case not, a proper default value will be determined. Default: True
**kwargs – Parameter name as keyword (“var”, “len_scale”, “nugget”, <opt_arg>) and a list of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).
- spectral_density(k)
Spectral density of the covariance model.
This is given by:
\[\tilde{S}(k) = \frac{S(k)}{\sigma^2}\]Where \(S(k)\) is the spectrum of the covariance model.
- Parameters
k (
float
) – Radius of the phase: \(k=\left\Vert\mathbf{k}\right\Vert\)
- spectral_rad_pdf(r)
Radial spectral density of the model.
- spectrum(k)
Spectrum of the covariance model.
This is given by:
\[S(\mathbf{k}) = \left(\frac{1}{2\pi}\right)^n \int C(r) e^{i \mathbf{k}\cdot\mathbf{r}} d^n\mathbf{r}\]Internally, this is calculated by the hankel transformation:
\[S(k) = \left(\frac{1}{2\pi}\right)^n \cdot \frac{(2\pi)^{n/2}}{k^{n/2-1}} \int_0^\infty r^{n/2} C(r) J_{n/2-1}(kr) dr\]Where \(C(r)\) is the covariance function of the model.
- Parameters
k (
float
) – Radius of the phase: \(k=\left\Vert\mathbf{k}\right\Vert\)
- var_factor()
Factor for the variance.
- vario_axis(r, axis=0)
Variogram along axis of anisotropy.
- vario_nugget(r)
Isotropic variogram of the model respecting the nugget at r=0.
- vario_spatial(pos)
Spatial variogram respecting anisotropy and rotation.
- vario_yadrenko(zeta)
Yadrenko variogram for great-circle distance from latlon-pos.
- variogram(r)
Isotropic variogram of the model.
- property angles
Rotation angles (in rad) of the model.
- Type
- property anis
The anisotropy factors of the model.
- Type
- property anis_bounds
Bounds for the nugget.
Notes
Is a list of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).- Type
- property arg_bounds
Bounds for all parameters.
Notes
Keys are the arg names and values are lists of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).- Type
- property dist_func
pdf, cdf and ppf.
Spectral distribution info from the model.
- property hankel_kw
hankel.SymmetricFourierTransform
kwargs.- Type
- property integral_scale
The main integral scale of the model.
- Raises
ValueError – If integral scale is not setable.
- Type
- property integral_scale_vec
The integral scales in each direction.
Notes
- This is calculated by:
integral_scale_vec[0] = integral_scale
integral_scale_vec[1] = integral_scale*anis[0]
integral_scale_vec[2] = integral_scale*anis[1]
- Type
- property len_scale_bounds
Bounds for the length scale.
Notes
Is a list of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).- Type
- property len_scale_vec
The length scales in each direction.
Notes
- This is calculated by:
len_scale_vec[0] = len_scale
len_scale_vec[1] = len_scale*anis[0]
len_scale_vec[2] = len_scale*anis[1]
- Type
- property nugget_bounds
Bounds for the nugget.
Notes
Is a list of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).- Type
- property opt_arg_bounds
Bounds for the optional arguments.
Notes
Keys are the opt-arg names and values are lists of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).- Type
- property pykrige_angle
2D rotation angle for pykrige.
- property pykrige_angle_x
3D rotation angle around x for pykrige.
- property pykrige_angle_y
3D rotation angle around y for pykrige.
- property pykrige_angle_z
3D rotation angle around z for pykrige.
- property pykrige_anis
2D anisotropy ratio for pykrige.
- property pykrige_anis_y
3D anisotropy ratio in y direction for pykrige.
- property pykrige_anis_z
3D anisotropy ratio in z direction for pykrige.
- property pykrige_kwargs
Keyword arguments for pykrige routines.
- property sill
The sill of the variogram.
Notes
- This is calculated by:
sill = variance + nugget
- Type
- property var_bounds
Bounds for the variance.
Notes
Is a list of 2 or 3 values:
[a, b]
or[a, b, <type>]
where <type> is one of"oo"
,"cc"
,"oc"
or"co"
to define if the bounds are open (“o”) or closed (“c”).- Type