# verde.Spline#

class verde.Spline(mindist=None, damping=None, force_coords=None, engine='auto')[source]#

Biharmonic spline interpolation using Green’s functions.

This gridder assumes Cartesian coordinates.

Implements the 2D splines of [Sandwell1987]. The Green’s function for the spline corresponds to the elastic deflection of a thin sheet subject to a vertical force. For an observation point at the origin and a force at the coordinates given by the vector $$\mathbf{x}$$, the Green’s function is:

$g(\mathbf{x}) = \|\mathbf{x}\|^2 \left(\log \|\mathbf{x}\| - 1\right)$

In practice, this function is not defined for data points that coincide with a force. To prevent this, a fudge factor is added to $$\|\mathbf{x}\|$$.

The interpolation is performed by estimating forces that produce deflections that fit the observed data (using least-squares). Then, the interpolated points can be evaluated at any location.

By default, the forces will be placed at the same points as the input data given to fit. This configuration provides an exact solution on top of the data points. However, this solution can be unstable for certain configurations of data points.

Approximate (and more stable) solutions can be obtained by applying damping regularization to smooth the estimated forces (and interpolated values) or by not using the data coordinates to position the forces (use the force_coords parameter).

Data weights can be used during fitting but only have an any effect when using the approximate solutions.

Before fitting, the Jacobian (design, sensitivity, feature, etc) matrix for the spline is normalized using sklearn.preprocessing.StandardScaler without centering the mean so that the transformation can be undone in the estimated forces.

Parameters:
mindist

DEPRECATED: This argument is no longer needed and will be removed in Verde 2.0.0. This fudge factor is no longer required since a new formulation eliminates the singularity at zero distance. Use the default value of mindist=None to get the future behaviour. A minimum distance between the point forces and data points.

damping

The positive damping regularization parameter. Controls how much smoothness is imposed on the estimated forces. If None, no regularization is used.

force_coordsNone or tuple of arrays

The easting and northing coordinates of the point forces. If None (default), then will be set to the data coordinates used to fit the spline.

enginestr

DEPRECATED: This option is deprecated and will be removed in Verde v2.0.0. The numba engine will be the only option. Computation engine for the Jacobian matrix and prediction. Can be 'auto', 'numba', or 'numpy'. If 'auto', will use numba if it is installed or numpy otherwise. The numba version is multi-threaded and usually faster, which makes fitting and predicting faster.

SplineCV

Cross-validated version of the bi-harmonic spline

Attributes:
force_array

The estimated forces that fit the observed data.

force_coords_tuple of arrays

The easting and northing coordinates of the point forces. Same as force_coords if it is not None. Otherwise, same as the data locations used to fit the spline.

region_tuple

The boundaries ([W, E, S, N]) of the data used to fit the interpolator. Used as the default region for the grid and scatter methods.

Methods

 filter(coordinates, data[, weights]) Filter the data through the gridder and produce residuals. fit(coordinates, data[, weights]) Fit the biharmonic spline to the given data. Get metadata routing of this object. get_params([deep]) Get parameters for this estimator. grid([region, shape, spacing, dims, ...]) Interpolate the data onto a regular grid. jacobian(coordinates, force_coords[, dtype]) Make the Jacobian matrix for the 2D biharmonic spline. predict(coordinates) Evaluate the estimated spline on the given set of points. profile(point1, point2, size[, dims, ...]) Interpolate data along a profile between two points. scatter([region, size, random_state, dims, ...]) Interpolate values onto a random scatter of points. score(coordinates, data[, weights]) Score the gridder predictions against the given data. set_fit_request(*[, coordinates, data, weights]) Request metadata passed to the fit method. set_params(**params) Set the parameters of this estimator. set_predict_request(*[, coordinates]) Request metadata passed to the predict method. set_score_request(*[, coordinates, data, ...]) Request metadata passed to the score method.

## Attributes#

Spline.data_names_defaults = [('scalars',), ('east_component', 'north_component'), ('east_component', 'north_component', 'vertical_component')]#
Spline.dims = ('northing', 'easting')#
Spline.extra_coords_name = 'extra_coord'#

## Methods#

Spline.filter(coordinates, data, weights=None)#

Filter the data through the gridder and produce residuals.

Calls fit on the data, evaluates the residuals (data - predicted data), and returns the coordinates, residuals, and weights.

Not very useful by itself but this interface makes gridders compatible with other processing operations and is used by verde.Chain to join them together (for example, so you can fit a spline on the residuals of a trend).

Parameters:
coordinatestuple of arrays

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). For the specific definition of coordinate systems and what these names mean, see the class docstring.

dataarray or tuple of arrays

The data values of each data point. If the data has more than one component, data must be a tuple of arrays (one for each component).

weightsNone or array or tuple of arrays

If not None, then the weights assigned to each data point. If more than one data component is provided, you must provide a weights array for each data component (if not None).

Returns:
coordinates, residuals, weights

The coordinates and weights are same as the input. Residuals are the input data minus the predicted data.

Spline.fit(coordinates, data, weights=None)[source]#

Fit the biharmonic spline to the given data.

The data region is captured and used as default for the grid and scatter methods.

All input arrays must have the same shape.

Parameters:
coordinatestuple of arrays

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). Only easting and northing will be used, all subsequent coordinates will be ignored.

dataarray

The data values of each data point.

weights

If not None, then the weights assigned to each data point. Typically, this should be 1 over the data uncertainty squared.

Returns:
self

Returns this estimator instance for chaining operations.

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

Spline.get_params(deep=True)#

Get parameters for this estimator.

Parameters:
deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:
paramsdict

Parameter names mapped to their values.

Spline.grid(region=None, shape=None, spacing=None, dims=None, data_names=None, projection=None, coordinates=None, **kwargs)#

Interpolate the data onto a regular grid.

The grid can be specified by two methods:

• Pass the actual coordinates of the grid points, as generated by verde.grid_coordinates or from an existing xarray.Dataset grid.

• Let the method define a new grid by either passing the number of points in each dimension (the shape) or by the grid node spacing. If the interpolator collected the input data region, then it will be used if region=None. Otherwise, you must specify the grid region. See verde.grid_coordinates for details. Other arguments for verde.grid_coordinates can be passed as extra keyword arguments (kwargs) to this method.

Use the dims and data_names arguments to set custom names for the dimensions and the data field(s) in the output xarray.Dataset. Default names will be provided if none are given.

Parameters:
regionlist = [W, E, S, N]

The west, east, south, and north boundaries of a given region. Use only if coordinates is None.

shapetuple = (n_north, n_east) or None

The number of points in the South-North and West-East directions, respectively. Use only if coordinates is None.

spacingtuple = (s_north, s_east) or None

The grid spacing in the South-North and West-East directions, respectively. Use only if coordinates is None.

dimslist or None

The names of the northing and easting data dimensions, respectively, in the output grid. Default is determined from the dims attribute of the class. Must be defined in the following order: northing dimension, easting dimension. NOTE: This is an exception to the “easting” then “northing” pattern but is required for compatibility with xarray.

data_namesstr, list or None

The name(s) of the data variables in the output grid. Defaults to 'scalars' for scalar data, ['east_component', 'north_component'] for 2D vector data, and ['east_component', 'north_component', 'vertical_component'] for 3D vector data.

projection

If not None, then should be a callable object projection(easting, northing) -> (proj_easting, proj_northing) that takes in easting and northing coordinate arrays and returns projected northing and easting coordinate arrays. This function will be used to project the generated grid coordinates before passing them into predict. For example, you can use this to generate a geographic grid from a Cartesian gridder.

coordinatestuple of arrays

Tuple of arrays containing the coordinates of the grid in the following order: (easting, northing, vertical, …). The easting and northing arrays could be 1d or 2d arrays, if they are 2d they must be part of a meshgrid. If coordinates are passed, region, shape, and spacing are ignored.

Returns:
gridxarray.Dataset

The interpolated grid. Metadata about the interpolator is written to the attrs attribute.

verde.grid_coordinates

Generate the coordinate values for the grid.

Spline.jacobian(coordinates, force_coords, dtype='float64')[source]#

Make the Jacobian matrix for the 2D biharmonic spline.

Each column of the Jacobian is the Green’s function for a single force evaluated on all observation points [Sandwell1987].

Parameters:
coordinatestuple of arrays

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). Only easting and northing will be used, all subsequent coordinates will be ignored.

force_coordstuple of arrays

Arrays with the coordinates for the forces. Should be in the same order as the coordinate arrays.

dtypestr or numpy dtype

The type of the Jacobian array.

Returns:
jacobian2D array

The (n_data, n_forces) Jacobian matrix.

Spline.predict(coordinates)[source]#

Evaluate the estimated spline on the given set of points.

Requires a fitted estimator (see fit).

Parameters:
coordinatestuple of arrays

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). Only easting and northing will be used, all subsequent coordinates will be ignored.

Returns:
dataarray

The data values evaluated on the given points.

Spline.profile(point1, point2, size, dims=None, data_names=None, projection=None, **kwargs)#

Interpolate data along a profile between two points.

Generates the profile along a straight line assuming Cartesian distances. Point coordinates are generated by verde.profile_coordinates. Other arguments for this function can be passed as extra keyword arguments (kwargs) to this method.

Use the dims and data_names arguments to set custom names for the dimensions and the data field(s) in the output pandas.DataFrame. Default names are provided.

Includes the calculated Cartesian distance from point1 for each data point in the profile.

To specify point1 and point2 in a coordinate system that would require projection to Cartesian (geographic longitude and latitude, for example), use the projection argument. With this option, the input points will be projected using the given projection function prior to computations. The generated Cartesian profile coordinates will be projected back to the original coordinate system. Note that the profile points are evenly spaced in projected coordinates, not the original system (e.g., geographic).

Warning

The profile calculation method with a projection has changed in Verde 1.4.0. Previous versions generated coordinates (assuming they were Cartesian) and projected them afterwards. This led to “distances” being incorrectly handled and returned in unprojected coordinates. For example, if projection is from geographic to Mercator, the distances would be “angles” (incorrectly calculated as if they were Cartesian). After 1.4.0, point1 and point2 are projected prior to generating coordinates for the profile, guaranteeing that distances are properly handled in a Cartesian system. With this change, the profile points are now evenly spaced in projected coordinates and the distances are returned in projected coordinates as well.

Parameters:
point1tuple

The easting and northing coordinates, respectively, of the first point.

point2tuple

The easting and northing coordinates, respectively, of the second point.

sizeint

The number of points to generate.

dimslist or None

The names of the northing and easting data dimensions, respectively, in the output dataframe. Default is determined from the dims attribute of the class. Must be defined in the following order: northing dimension, easting dimension. NOTE: This is an exception to the “easting” then “northing” pattern but is required for compatibility with xarray.

data_namesstr, list or None

The name(s) of the data variables in the output dataframe. Defaults to 'scalars' for scalar data, ['east_component', 'north_component'] for 2D vector data, and ['east_component', 'north_component', 'vertical_component'] for 3D vector data.

projection

If not None, then should be a callable object projection(easting, northing, inverse=False) -> (proj_easting, proj_northing) that takes in easting and northing coordinate arrays and returns projected northing and easting coordinate arrays. Should also take an optional keyword argument inverse (default to False) that if True will calculate the inverse transform instead. This function will be used to project the profile end points before generating coordinates and passing them into predict. It will also be used to undo the projection of the coordinates before returning the results.

Returns:
tablepandas.DataFrame

The interpolated values along the profile.

Spline.scatter(region=None, size=300, random_state=0, dims=None, data_names=None, projection=None, **kwargs)#

Interpolate values onto a random scatter of points.

Point coordinates are generated by verde.scatter_points. Other arguments for this function can be passed as extra keyword arguments (kwargs) to this method.

If the interpolator collected the input data region, then it will be used if region=None. Otherwise, you must specify the grid region.

Use the dims and data_names arguments to set custom names for the dimensions and the data field(s) in the output pandas.DataFrame. Default names are provided.

Warning

The scatter method is deprecated and will be removed in Verde 2.0.0. Use verde.scatter_points and the predict method instead.

Parameters:
regionlist = [W, E, S, N]

The west, east, south, and north boundaries of a given region.

sizeint

The number of points to generate.

random_statenumpy.random.RandomState or an int seed

A random number generator used to define the state of the random permutations. Use a fixed seed to make sure computations are reproducible. Use None to choose a seed automatically (resulting in different numbers with each run).

dimslist or None

The names of the northing and easting data dimensions, respectively, in the output dataframe. Default is determined from the dims attribute of the class. Must be defined in the following order: northing dimension, easting dimension. NOTE: This is an exception to the “easting” then “northing” pattern but is required for compatibility with xarray.

data_namesstr, list or None

The name(s) of the data variables in the output dataframe. Defaults to 'scalars' for scalar data, ['east_component', 'north_component'] for 2D vector data, and ['east_component', 'north_component', 'vertical_component'] for 3D vector data.

projection

If not None, then should be a callable object projection(easting, northing) -> (proj_easting, proj_northing) that takes in easting and northing coordinate arrays and returns projected northing and easting coordinate arrays. This function will be used to project the generated scatter coordinates before passing them into predict. For example, you can use this to generate a geographic scatter from a Cartesian gridder.

Returns:
tablepandas.DataFrame

The interpolated values on a random set of points.

Spline.score(coordinates, data, weights=None)#

Score the gridder predictions against the given data.

Calculates the R^2 coefficient of determination of between the predicted values and the given data values. A maximum score of 1 means a perfect fit. The score can be negative.

Warning

The default scoring will change from R² to negative root mean squared error (RMSE) in Verde 2.0.0. This may change model selection results slightly. The negative version will be used to maintain the behaviour of larger scores being better, which is more compatible with current model selection code.

If the data has more than 1 component, the scores of each component will be averaged.

Parameters:
coordinatestuple of arrays

Arrays with the coordinates of each data point. Should be in the following order: (easting, northing, vertical, …). For the specific definition of coordinate systems and what these names mean, see the class docstring.

dataarray or tuple of arrays

The data values of each data point. If the data has more than one component, data must be a tuple of arrays (one for each component).

weightsNone or array or tuple of arrays

If not None, then the weights assigned to each data point. If more than one data component is provided, you must provide a weights array for each data component (if not None).

Returns:
scorefloat

The R^2 score

Spline.set_fit_request(*, coordinates: = '$UNCHANGED$', data: = '$UNCHANGED$', weights: = '$UNCHANGED$') #

Request metadata passed to the fit method.

Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to fit.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

New in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.

Parameters:
coordinatesstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for coordinates parameter in fit.

datastr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for data parameter in fit.

weightsstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for weights parameter in fit.

Returns:
selfobject

The updated object.

Spline.set_params(**params)#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:
**paramsdict

Estimator parameters.

Returns:
selfestimator instance

Estimator instance.

Spline.set_predict_request(*, coordinates: = '$UNCHANGED$') #

Request metadata passed to the predict method.

Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to predict if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to predict.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

New in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.

Parameters:
coordinatesstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for coordinates parameter in predict.

Returns:
selfobject

The updated object.

Spline.set_score_request(*, coordinates: = '$UNCHANGED$', data: = '$UNCHANGED$', weights: = '$UNCHANGED$') #

Request metadata passed to the score method.

Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to score.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

New in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.

Parameters:
coordinatesstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for coordinates parameter in score.

datastr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for data parameter in score.

weightsstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED

Metadata routing for weights parameter in score.

Returns:
selfobject

The updated object.

### Examples using verde.Spline#

Gridding with splines

Gridding with splines

Gridding with splines (cross-validated)

Gridding with splines (cross-validated)

Gridding with splines and weights

Gridding with splines and weights

Gridding 2D vectors

Gridding 2D vectors

Chaining Operations

Chaining Operations

Evaluating Performance

Evaluating Performance

Model Selection

Model Selection

Geographic Coordinates

Geographic Coordinates

Trend Estimation

Trend Estimation

Vector Data

Vector Data

Using Weights

Using Weights