gstools.covmodel.Exponential

class gstools.covmodel.Exponential(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 Exponential covariance model.

Notes

This model is given by the following variogram [Webster2007]:

\[\gamma(r)= \sigma^{2} \left(1-\exp\left(-s\cdot\frac{r}{\ell}\right)\right)+n\]

Where the standard rescale factor is \(s=1\).

References

Webster2007: Webster, R. and Oliver, M. A. “Geostatistics for environmental scientists.”, John Wiley & Sons. (2007)

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 or list, 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 or list, 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 or list, 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 or list or None, 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 or None, 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: False
geo_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 to KM_SCALE to have len_scale in km or DEGREE_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: False
spatial_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: None
var_raw (float or None, optional) – raw variance of the model which will be multiplied with CovModel.var_factor to result in the actual variance. If given, var will be ignored. (This is just for models that override CovModel.var_factor) Default: None
hankel_kw (dict or None, optional) – Modify the init-arguments of hankel.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: list of str: Names of all arguments.
arg_bounds: dict: Bounds for all parameters.
arg_list: list of float: Values of all arguments.
dim: int: The dimension of the model.
dist_func: tuple of callable: pdf, cdf and ppf.
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: list of str: Names of isotropic arguments.
iso_arg_list: list of float: Values of isotropic arguments.
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: list of str: Names of the optional arguments.
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.
`calc_integral_scale`()	Calculate the integral scale of the isotrope model.
`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.
`cor`(h)	Exponential 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.
`default_arg_bounds`()	Provide default boundaries for arguments.
`default_opt_arg`()	Provide default optional arguments by the user.
`default_opt_arg_bounds`()	Provide default boundaries for optional arguments.
`default_rescale`()	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.
`ln_spectral_rad_pdf`(r)	Log radial spectral density of the model.
`main_axes`()	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`(k)	Spectral density of the covariance model.
`spectral_rad_cdf`(r)	Exponential radial spectral cdf.
`spectral_rad_pdf`(r)	Radial spectral density of the model.
`spectral_rad_ppf`(u)	Exponential radial spectral ppf.
`spectrum`(k)	Spectrum of the covariance model.
`var_factor`()	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.

calc_integral_scale()[source]: Calculate the integral scale of the isotrope model.

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(h)[source]: Exponential normalized correlation function.

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()

Provide default optional arguments by the user.

Should be given as a dictionary when overridden.

default_opt_arg_bounds(): Provide default boundaries for optional arguments.

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: True
sill (float or bool, 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 or dict, 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: None
method ({‘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 or callable, 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 or None, 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: False
curve_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 values
pcov (numpy.ndarray) – The estimated covariance of popt from scipy.optimize.curve_fit. To compute one standard deviation errors on the parameters use perr = 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.