Multi-dimensional splines

## Project description     This is a Python package for multivariate B-splines with performant NumPy and C (via Cython) implementations. For a mathematical overview of tensor product B-splines, see the Splines page of the documentation.

The primary goal of this package is to provide a unified API for tensor product splines of arbitrary input and output dimension. For a list of related packages see the Comparisons page.

## Installation

Install ndsplines with pip:

```\$ pip install ndsplines
```

or from source:

```\$ git clone https://github.com/kb-press/ndsplines
\$ cd ndsplines
\$ pip install .
```

Reasonably recent versions of pip will automatically pull Cython and NumPy to build the compiled implementation. If you need to install ndsplines without compiling, try python setup.py install instead of using pip install.

## Usage

The easiest way to use ndsplines is to use one of the make_* functions: make_interp_spline, make_interp_spline_from_tidy, or make_lsq_spline, which return an NDSpline object which can be used to evaluate the spline. For example, suppose we have data over a two-dimensional mesh.

```import ndsplines
import numpy as np

# generate grid of independent variables
x = np.array([-1, -7/8, -3/4, -1/2, -1/4, -1/8, 0, 1/8, 1/4, 1/2, 3/4, 7/8, 1])*np.pi
y = np.array([-1, -1/2, 0, 1/2, 1])
meshx, meshy = np.meshgrid(x, y, indexing='ij')
gridxy = np.stack((meshx, meshy), axis=-1)

# evaluate a function to interpolate over input grid
meshf = np.sin(meshx) * (meshy-3/8)**2 + 2
```

We can then use make_interp_spline to create an interpolating spline and evaluate it over a denser mesh.

```# create the interpolating splane
interp = ndsplines.make_interp_spline(gridxy, meshf)

# generate denser grid of independent variables to interpolate
sparse_dense = 2**7
xx = np.concatenate([np.linspace(x[i], x[i+1], sparse_dense) for i in range(x.size-1)])
yy = np.concatenate([np.linspace(y[i], y[i+1], sparse_dense) for i in range(y.size-1)])
gridxxyy = np.stack(np.meshgrid(xx, yy, indexing='ij'), axis=-1)

# evaluate spline over denser grid
meshff = interp(gridxxyy)
```

Generally, we construct data so that the first ndim axes index the independent variables and the remaining axes index output. This is a generalization of using rows to index time and columns to index output variables for time-series data.

We can also create an interpolating spline from a tidy data format:

```tidy_data = np.dstack((gridxy, meshf)).reshape((-1,3))
tidy_interp = ndsplines.make_interp_spline_from_tidy(
tidy_data,
[0,1], # columns to use as independent variable data
    # columns to use as dependent variable data
)

print("\nCoefficients all same?",
np.all(tidy_interp.coefficients == interp.coefficients))
print("Knots all same?",
np.all([np.all(k0 == k1) for k0, k1 in zip(tidy_interp.knots, interp.knots)]))
```

Note however, that the tidy dataset must be over a structured rectangular grid equivalent to the N-dimensional tensor product representation. Also note that Pandas dataframes can be used, in which case lists of column names can be used instead of lists of column indices.

To see examples for creating least-squares regression splines with make_lsq_spline, see the 1D example and 2D example.

Derivatives of constructed splines can be evaluated in two ways: (1) by using the nus parameter while calling the interpolator or (2) by creating a new spline with the derivative method. In this codeblock, we show both ways of evaluating derivatives in each direction.

```# two ways to evaluate derivatives x-direction: create a derivative spline or call with nus:
deriv_interp = interp.derivative(0)
deriv1 = deriv_interp(gridxxy)
deriv2 = interp(gridxy, nus=np.array([1,0]))

# two ways to evaluate derivative - y direction
deriv_interp = interp.derivative(1)
deriv1 = deriv_interp(gridxy)
deriv2 = interp(gridxxyy, nus=np.array([0,1]))
```

The NDSpline class also has an antiderivative method for creating a spline representative of the anti-derivative in the specified direction.

```# Calculus demonstration
interp1 = deriv_interp.antiderivative(0)
coeff_diff = interp1.coefficients - interp.coefficients
print("\nAntiderivative of derivative:\n","Coefficients differ by constant?",
np.allclose(interp1.coefficients+2.0, interp.coefficients))
print("Knots all same?",
np.all([np.all(k0 == k1) for k0, k1 in zip(interp1.knots, interp.knots)]))

antideriv_interp = interp.antiderivative(0)
interp2 = antideriv_interp.derivative(0)
print("\nDerivative of antiderivative:\n","Coefficients the same?",
np.allclose(interp2.coefficients, interp.coefficients))
print("Knots all same?",
np.all([np.all(k0 == k1) for k0, k1 in zip(interp2.knots, interp.knots)]))
```

## Contributing

Please feel free to share any thoughts or opinions about the design and implementation of this software by opening an issue on GitHub. Constructive feedback is welcomed and appreciated.

Bug fix pull requests are always welcome. For feature additions, breaking changes, etc. check if there is an open issue discussing the change and reference it in the pull request. If there isn’t one, it is recommended to open one with your rationale for the change before spending significant time preparing the pull request.

Ideally, new/changed functionality should come with tests and documentation. If you are new to contributing, it is perfectly fine to open a work-in-progress pull request and have it iteratively reviewed.

## Testing

To test, install the developer requirements and use pytest:

```\$ pip install -r requirements-dev.txt
\$ pip install -e .
\$ pytest
```

## Documentation

To build the docs, install the docs feature requirements (a subset of the developer requirements above):

```\$ pip install -e .[docs]
\$ cd docs
\$ make html
```

## Project details

This version 0.1.4 0.1.3 0.1.2 0.1.1 0.1.0

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