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# Interpolation (`scipy.interpolate`)

There are several general facilities available in SciPy for interpolation and smoothing for data in 1, 2, and higher dimensions. The choice of a specific interpolation routine depends on the data: whether it is one-dimensional, is given on a structured grid, or is unstructured. One other factor is the desired smoothness of the interpolator. In short, routines recommended for interpolation can be summarized as follows:

| kind | routine | continuity | comment
--- | --- | --- | --- | ---
1D | linear | `numpy.interp` | piecewise continuous | Alternatively, `make_interp_spline(..., k=1)`
cubic spline | `CubicSpline` | 2nd derivative |
monotone cubic spline | `PchipInterpolator` | 1st derivative | non-overshooting
non-cubic spline | `make_interp_spline` | (k-1)th derivative | `k=3` is equivalent to `CubicSpline`
nearest | `interp1d` |  | kind=’nearest’, ‘previous’, ‘next’
N-D curve | nearest, linear, spline | `make_interp_spline` | (k-1)th derivative | use N-dim y array
N-D regular (rectilinear) grid | nearest | `RegularGridInterpolator` |  | method=’nearest’
linear | method=’linear’
splines | 2nd derivatives | method=’cubic’, ‘quintic’
monotone splines | 1st derivatives | method=’pchip’
N-D scattered | nearest | `NearestNDInterpolator` |  | alias: `griddata`
linear | `LinearNDInterpolator`
cubic (2D only) | `CloughTocher2DInterpolator` | 1st derivatives
radial basis function | `RBFInterpolator` |  |

## Smoothing and approximation of data

1D spline functions | `make_smoothing_spline` | classic smoothing splines, GCV penalty
`make_splrep` | automated/semi-automated knot selection
spine curves in N-D | `make_splprep` |
unconstrained least squares spline fit | `make_lsq_spline` |
2D smoothing surfaces | `bisplrep` | scattered data
| `RectBivariateSpline` | gridded data
Radial basis functions in N-D | `RBFInterpolator` |

Further details are given in the links below

- 1-D interpolation
  - Piecewise linear interpolation
  - Cubic splines
  - Monotone interpolants
  - Interpolation with B-splines
    - Non-cubic splines
  - Batches of `y`
  - Parametric spline curves
  - Missing data
  - Out-of-bounds behavior
    - The `bc_type` parameter
    - The `extrapolate` constructor argument
    - Overriding `extrapolate` per call
  - Legacy interface for 1-D interpolation (`interp1d`)
    - Recommended replacements for `interp1d` modes
- Piecewise polynomials and splines
  - Manipulating `PPoly` objects
  - B-splines: knots and coefficients
    - B-spline basis elements
    - Design matrices in the B-spline basis
  - Bernstein polynomials, `BPoly`
  - Conversion between bases
- Smoothing splines
  - Spline smoothing in 1D
    - “Classic” smoothing splines and generalized cross-validation (GCV) criterion
    - Smoothing splines with automatic knot selection
    - Smoothing spline curves in \(d>1\)
    - Batching of `y` arrays
    - Legacy routines for spline smoothing in 1-D
  - 2-D smoothing splines
    - Bivariate spline fitting of scattered data
    - Bivariate spline fitting of data on a grid
    - Bivariate spline fitting of data in spherical coordinates
- Multivariate data interpolation on a regular grid (`RegularGridInterpolator`)
  - Batch dimensions of `values`
  - Uniformly spaced data
- Scattered data interpolation (`griddata`)
- Using radial basis functions for smoothing/interpolation
  - 1-D Example
  - 2-D Example
- Extrapolation tips and tricks
  - Replicate `numpy.interp` left and right fill values
  - CubicSpline extend the boundary conditions
  - Manually implement the asymptotics
    - The setup
    - Use the known asymptotics
  - Extrapolation in `D > 1`
- `interp2d` transition guide
  - 1. How to transition away from using `interp2d`
    - 1.1 `interp2d` on a regular grid
    - 1.2. `interp2d` with full coordinates of points (scattered interpolation)
  - 2. Alternative to `interp2d`: regular grid
  - 3. Scattered 2D linear interpolation: prefer `LinearNDInterpolator` to `SmoothBivariateSpline` or `bisplrep`
- Filling an image from a border
  - Analytic solutions
  - Simple unstructured interpolators
    - Simplex tolerance
  - Radial basis functions
  - Multi-linear regression
    - Unconstrained polynomials
    - Restricted polynomials
  - Other packages
  - Closing notes