__all__ = ['RegularGridInterpolator', 'interpn'] import itertools import numpy as np from .interpnd import _ndim_coords_from_arrays from ._cubic import PchipInterpolator from ._rgi_cython import evaluate_linear_2d, find_indices from ._bsplines import make_interp_spline from ._fitpack2 import RectBivariateSpline def _check_points(points): descending_dimensions = [] grid = [] for i, p in enumerate(points): # early make points float # see https://github.com/scipy/scipy/pull/17230 p = np.asarray(p, dtype=float) if not np.all(p[1:] > p[:-1]): if np.all(p[1:] < p[:-1]): # input is descending, so make it ascending descending_dimensions.append(i) p = np.flip(p) else: raise ValueError( "The points in dimension %d must be strictly " "ascending or descending" % i) # see https://github.com/scipy/scipy/issues/17716 p = np.ascontiguousarray(p) grid.append(p) return tuple(grid), tuple(descending_dimensions) def _check_dimensionality(points, values): if len(points) > values.ndim: raise ValueError("There are %d point arrays, but values has %d " "dimensions" % (len(points), values.ndim)) for i, p in enumerate(points): if not np.asarray(p).ndim == 1: raise ValueError("The points in dimension %d must be " "1-dimensional" % i) if not values.shape[i] == len(p): raise ValueError("There are %d points and %d values in " "dimension %d" % (len(p), values.shape[i], i)) class RegularGridInterpolator: """ Interpolation on a regular or rectilinear grid in arbitrary dimensions. The data must be defined on a rectilinear grid; that is, a rectangular grid with even or uneven spacing. Linear, nearest-neighbor, spline interpolations are supported. After setting up the interpolator object, the interpolation method may be chosen at each evaluation. Parameters ---------- points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) The points defining the regular grid in n dimensions. The points in each dimension (i.e. every elements of the points tuple) must be strictly ascending or descending. values : array_like, shape (m1, ..., mn, ...) The data on the regular grid in n dimensions. Complex data can be acceptable. method : str, optional The method of interpolation to perform. Supported are "linear", "nearest", "slinear", "cubic", "quintic" and "pchip". This parameter will become the default for the object's ``__call__`` method. Default is "linear". bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data, a ValueError is raised. If False, then `fill_value` is used. Default is True. fill_value : float or None, optional The value to use for points outside of the interpolation domain. If None, values outside the domain are extrapolated. Default is ``np.nan``. Methods ------- __call__ Attributes ---------- grid : tuple of ndarrays The points defining the regular grid in n dimensions. This tuple defines the full grid via ``np.meshgrid(*grid, indexing='ij')`` values : ndarray Data values at the grid. method : str Interpolation method. fill_value : float or ``None`` Use this value for out-of-bounds arguments to `__call__`. bounds_error : bool If ``True``, out-of-bounds argument raise a ``ValueError``. Notes ----- Contrary to `LinearNDInterpolator` and `NearestNDInterpolator`, this class avoids expensive triangulation of the input data by taking advantage of the regular grid structure. In other words, this class assumes that the data is defined on a *rectilinear* grid. .. versionadded:: 0.14 The 'slinear'(k=1), 'cubic'(k=3), and 'quintic'(k=5) methods are tensor-product spline interpolators, where `k` is the spline degree, If any dimension has fewer points than `k` + 1, an error will be raised. .. versionadded:: 1.9 If the input data is such that dimensions have incommensurate units and differ by many orders of magnitude, the interpolant may have numerical artifacts. Consider rescaling the data before interpolating. Examples -------- **Evaluate a function on the points of a 3-D grid** As a first example, we evaluate a simple example function on the points of a 3-D grid: >>> from scipy.interpolate import RegularGridInterpolator >>> import numpy as np >>> def f(x, y, z): ... return 2 * x**3 + 3 * y**2 - z >>> x = np.linspace(1, 4, 11) >>> y = np.linspace(4, 7, 22) >>> z = np.linspace(7, 9, 33) >>> xg, yg ,zg = np.meshgrid(x, y, z, indexing='ij', sparse=True) >>> data = f(xg, yg, zg) ``data`` is now a 3-D array with ``data[i, j, k] = f(x[i], y[j], z[k])``. Next, define an interpolating function from this data: >>> interp = RegularGridInterpolator((x, y, z), data) Evaluate the interpolating function at the two points ``(x,y,z) = (2.1, 6.2, 8.3)`` and ``(3.3, 5.2, 7.1)``: >>> pts = np.array([[2.1, 6.2, 8.3], ... [3.3, 5.2, 7.1]]) >>> interp(pts) array([ 125.80469388, 146.30069388]) which is indeed a close approximation to >>> f(2.1, 6.2, 8.3), f(3.3, 5.2, 7.1) (125.54200000000002, 145.894) **Interpolate and extrapolate a 2D dataset** As a second example, we interpolate and extrapolate a 2D data set: >>> x, y = np.array([-2, 0, 4]), np.array([-2, 0, 2, 5]) >>> def ff(x, y): ... return x**2 + y**2 >>> xg, yg = np.meshgrid(x, y, indexing='ij') >>> data = ff(xg, yg) >>> interp = RegularGridInterpolator((x, y), data, ... bounds_error=False, fill_value=None) >>> import matplotlib.pyplot as plt >>> fig = plt.figure() >>> ax = fig.add_subplot(projection='3d') >>> ax.scatter(xg.ravel(), yg.ravel(), data.ravel(), ... s=60, c='k', label='data') Evaluate and plot the interpolator on a finer grid >>> xx = np.linspace(-4, 9, 31) >>> yy = np.linspace(-4, 9, 31) >>> X, Y = np.meshgrid(xx, yy, indexing='ij') >>> # interpolator >>> ax.plot_wireframe(X, Y, interp((X, Y)), rstride=3, cstride=3, ... alpha=0.4, color='m', label='linear interp') >>> # ground truth >>> ax.plot_wireframe(X, Y, ff(X, Y), rstride=3, cstride=3, ... alpha=0.4, label='ground truth') >>> plt.legend() >>> plt.show() Other examples are given :ref:`in the tutorial `. See Also -------- NearestNDInterpolator : Nearest neighbor interpolation on *unstructured* data in N dimensions LinearNDInterpolator : Piecewise linear interpolant on *unstructured* data in N dimensions interpn : a convenience function which wraps `RegularGridInterpolator` scipy.ndimage.map_coordinates : interpolation on grids with equal spacing (suitable for e.g., N-D image resampling) References ---------- .. [1] Python package *regulargrid* by Johannes Buchner, see https://pypi.python.org/pypi/regulargrid/ .. [2] Wikipedia, "Trilinear interpolation", https://en.wikipedia.org/wiki/Trilinear_interpolation .. [3] Weiser, Alan, and Sergio E. Zarantonello. "A note on piecewise linear and multilinear table interpolation in many dimensions." MATH. COMPUT. 50.181 (1988): 189-196. https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917826-0/S0025-5718-1988-0917826-0.pdf :doi:`10.1090/S0025-5718-1988-0917826-0` """ # this class is based on code originally programmed by Johannes Buchner, # see https://github.com/JohannesBuchner/regulargrid _SPLINE_DEGREE_MAP = {"slinear": 1, "cubic": 3, "quintic": 5, 'pchip': 3} _SPLINE_METHODS = list(_SPLINE_DEGREE_MAP.keys()) _ALL_METHODS = ["linear", "nearest"] + _SPLINE_METHODS def __init__(self, points, values, method="linear", bounds_error=True, fill_value=np.nan): if method not in self._ALL_METHODS: raise ValueError("Method '%s' is not defined" % method) elif method in self._SPLINE_METHODS: self._validate_grid_dimensions(points, method) self.method = method self.bounds_error = bounds_error self.grid, self._descending_dimensions = _check_points(points) self.values = self._check_values(values) self._check_dimensionality(self.grid, self.values) self.fill_value = self._check_fill_value(self.values, fill_value) if self._descending_dimensions: self.values = np.flip(values, axis=self._descending_dimensions) def _check_dimensionality(self, grid, values): _check_dimensionality(grid, values) def _check_points(self, points): return _check_points(points) def _check_values(self, values): if not hasattr(values, 'ndim'): # allow reasonable duck-typed values values = np.asarray(values) if hasattr(values, 'dtype') and hasattr(values, 'astype'): if not np.issubdtype(values.dtype, np.inexact): values = values.astype(float) return values def _check_fill_value(self, values, fill_value): if fill_value is not None: fill_value_dtype = np.asarray(fill_value).dtype if (hasattr(values, 'dtype') and not np.can_cast(fill_value_dtype, values.dtype, casting='same_kind')): raise ValueError("fill_value must be either 'None' or " "of a type compatible with values") return fill_value def __call__(self, xi, method=None): """ Interpolation at coordinates. Parameters ---------- xi : ndarray of shape (..., ndim) The coordinates to evaluate the interpolator at. method : str, optional The method of interpolation to perform. Supported are "linear", "nearest", "slinear", "cubic", "quintic" and "pchip". Default is the method chosen when the interpolator was created. Returns ------- values_x : ndarray, shape xi.shape[:-1] + values.shape[ndim:] Interpolated values at `xi`. See notes for behaviour when ``xi.ndim == 1``. Notes ----- In the case that ``xi.ndim == 1`` a new axis is inserted into the 0 position of the returned array, values_x, so its shape is instead ``(1,) + values.shape[ndim:]``. Examples -------- Here we define a nearest-neighbor interpolator of a simple function >>> import numpy as np >>> x, y = np.array([0, 1, 2]), np.array([1, 3, 7]) >>> def f(x, y): ... return x**2 + y**2 >>> data = f(*np.meshgrid(x, y, indexing='ij', sparse=True)) >>> from scipy.interpolate import RegularGridInterpolator >>> interp = RegularGridInterpolator((x, y), data, method='nearest') By construction, the interpolator uses the nearest-neighbor interpolation >>> interp([[1.5, 1.3], [0.3, 4.5]]) array([2., 9.]) We can however evaluate the linear interpolant by overriding the `method` parameter >>> interp([[1.5, 1.3], [0.3, 4.5]], method='linear') array([ 4.7, 24.3]) """ is_method_changed = self.method != method method = self.method if method is None else method if method not in self._ALL_METHODS: raise ValueError("Method '%s' is not defined" % method) xi, xi_shape, ndim, nans, out_of_bounds = self._prepare_xi(xi) if method == "linear": indices, norm_distances = self._find_indices(xi.T) if (ndim == 2 and hasattr(self.values, 'dtype') and self.values.ndim == 2 and self.values.flags.writeable and self.values.dtype in (np.float64, np.complex128) and self.values.dtype.byteorder == '='): # until cython supports const fused types, the fast path # cannot support non-writeable values # a fast path out = np.empty(indices.shape[1], dtype=self.values.dtype) result = evaluate_linear_2d(self.values, indices, norm_distances, self.grid, out) else: result = self._evaluate_linear(indices, norm_distances) elif method == "nearest": indices, norm_distances = self._find_indices(xi.T) result = self._evaluate_nearest(indices, norm_distances) elif method in self._SPLINE_METHODS: if is_method_changed: self._validate_grid_dimensions(self.grid, method) result = self._evaluate_spline(xi, method) if not self.bounds_error and self.fill_value is not None: result[out_of_bounds] = self.fill_value # f(nan) = nan, if any if np.any(nans): result[nans] = np.nan return result.reshape(xi_shape[:-1] + self.values.shape[ndim:]) def _prepare_xi(self, xi): ndim = len(self.grid) xi = _ndim_coords_from_arrays(xi, ndim=ndim) if xi.shape[-1] != len(self.grid): raise ValueError("The requested sample points xi have dimension " f"{xi.shape[-1]} but this " f"RegularGridInterpolator has dimension {ndim}") xi_shape = xi.shape xi = xi.reshape(-1, xi_shape[-1]) xi = np.asarray(xi, dtype=float) # find nans in input nans = np.any(np.isnan(xi), axis=-1) if self.bounds_error: for i, p in enumerate(xi.T): if not np.logical_and(np.all(self.grid[i][0] <= p), np.all(p <= self.grid[i][-1])): raise ValueError("One of the requested xi is out of bounds " "in dimension %d" % i) out_of_bounds = None else: out_of_bounds = self._find_out_of_bounds(xi.T) return xi, xi_shape, ndim, nans, out_of_bounds def _evaluate_linear(self, indices, norm_distances): # slice for broadcasting over trailing dimensions in self.values vslice = (slice(None),) + (None,)*(self.values.ndim - len(indices)) # Compute shifting up front before zipping everything together shift_norm_distances = [1 - yi for yi in norm_distances] shift_indices = [i + 1 for i in indices] # The formula for linear interpolation in 2d takes the form: # values = self.values[(i0, i1)] * (1 - y0) * (1 - y1) + \ # self.values[(i0, i1 + 1)] * (1 - y0) * y1 + \ # self.values[(i0 + 1, i1)] * y0 * (1 - y1) + \ # self.values[(i0 + 1, i1 + 1)] * y0 * y1 # We pair i with 1 - yi (zipped1) and i + 1 with yi (zipped2) zipped1 = zip(indices, shift_norm_distances) zipped2 = zip(shift_indices, norm_distances) # Take all products of zipped1 and zipped2 and iterate over them # to get the terms in the above formula. This corresponds to iterating # over the vertices of a hypercube. hypercube = itertools.product(*zip(zipped1, zipped2)) value = np.array([0.]) for h in hypercube: edge_indices, weights = zip(*h) weight = np.array([1.]) for w in weights: weight = weight * w term = np.asarray(self.values[edge_indices]) * weight[vslice] value = value + term # cannot use += because broadcasting return value def _evaluate_nearest(self, indices, norm_distances): idx_res = [np.where(yi <= .5, i, i + 1) for i, yi in zip(indices, norm_distances)] return self.values[tuple(idx_res)] def _validate_grid_dimensions(self, points, method): k = self._SPLINE_DEGREE_MAP[method] for i, point in enumerate(points): ndim = len(np.atleast_1d(point)) if ndim <= k: raise ValueError(f"There are {ndim} points in dimension {i}," f" but method {method} requires at least " f" {k+1} points per dimension.") def _evaluate_spline(self, xi, method): # ensure xi is 2D list of points to evaluate (`m` is the number of # points and `n` is the number of interpolation dimensions, # ``n == len(self.grid)``.) if xi.ndim == 1: xi = xi.reshape((1, xi.size)) m, n = xi.shape # Reorder the axes: n-dimensional process iterates over the # interpolation axes from the last axis downwards: E.g. for a 4D grid # the order of axes is 3, 2, 1, 0. Each 1D interpolation works along # the 0th axis of its argument array (for 1D routine it's its ``y`` # array). Thus permute the interpolation axes of `values` *and keep # trailing dimensions trailing*. axes = tuple(range(self.values.ndim)) axx = axes[:n][::-1] + axes[n:] values = self.values.transpose(axx) if method == 'pchip': _eval_func = self._do_pchip else: _eval_func = self._do_spline_fit k = self._SPLINE_DEGREE_MAP[method] # Non-stationary procedure: difficult to vectorize this part entirely # into numpy-level operations. Unfortunately this requires explicit # looping over each point in xi. # can at least vectorize the first pass across all points in the # last variable of xi. last_dim = n - 1 first_values = _eval_func(self.grid[last_dim], values, xi[:, last_dim], k) # the rest of the dimensions have to be on a per point-in-xi basis shape = (m, *self.values.shape[n:]) result = np.empty(shape, dtype=self.values.dtype) for j in range(m): # Main process: Apply 1D interpolate in each dimension # sequentially, starting with the last dimension. # These are then "folded" into the next dimension in-place. folded_values = first_values[j, ...] for i in range(last_dim-1, -1, -1): # Interpolate for each 1D from the last dimensions. # This collapses each 1D sequence into a scalar. folded_values = _eval_func(self.grid[i], folded_values, xi[j, i], k) result[j, ...] = folded_values return result @staticmethod def _do_spline_fit(x, y, pt, k): local_interp = make_interp_spline(x, y, k=k, axis=0) values = local_interp(pt) return values @staticmethod def _do_pchip(x, y, pt, k): local_interp = PchipInterpolator(x, y, axis=0) values = local_interp(pt) return values def _find_indices(self, xi): return find_indices(self.grid, xi) def _find_out_of_bounds(self, xi): # check for out of bounds xi out_of_bounds = np.zeros((xi.shape[1]), dtype=bool) # iterate through dimensions for x, grid in zip(xi, self.grid): out_of_bounds += x < grid[0] out_of_bounds += x > grid[-1] return out_of_bounds def interpn(points, values, xi, method="linear", bounds_error=True, fill_value=np.nan): """ Multidimensional interpolation on regular or rectilinear grids. Strictly speaking, not all regular grids are supported - this function works on *rectilinear* grids, that is, a rectangular grid with even or uneven spacing. Parameters ---------- points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) The points defining the regular grid in n dimensions. The points in each dimension (i.e. every elements of the points tuple) must be strictly ascending or descending. values : array_like, shape (m1, ..., mn, ...) The data on the regular grid in n dimensions. Complex data can be acceptable. xi : ndarray of shape (..., ndim) The coordinates to sample the gridded data at method : str, optional The method of interpolation to perform. Supported are "linear", "nearest", "slinear", "cubic", "quintic", "pchip", and "splinef2d". "splinef2d" is only supported for 2-dimensional data. bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data, a ValueError is raised. If False, then `fill_value` is used. fill_value : number, optional If provided, the value to use for points outside of the interpolation domain. If None, values outside the domain are extrapolated. Extrapolation is not supported by method "splinef2d". Returns ------- values_x : ndarray, shape xi.shape[:-1] + values.shape[ndim:] Interpolated values at `xi`. See notes for behaviour when ``xi.ndim == 1``. Notes ----- .. versionadded:: 0.14 In the case that ``xi.ndim == 1`` a new axis is inserted into the 0 position of the returned array, values_x, so its shape is instead ``(1,) + values.shape[ndim:]``. If the input data is such that input dimensions have incommensurate units and differ by many orders of magnitude, the interpolant may have numerical artifacts. Consider rescaling the data before interpolation. Examples -------- Evaluate a simple example function on the points of a regular 3-D grid: >>> import numpy as np >>> from scipy.interpolate import interpn >>> def value_func_3d(x, y, z): ... return 2 * x + 3 * y - z >>> x = np.linspace(0, 4, 5) >>> y = np.linspace(0, 5, 6) >>> z = np.linspace(0, 6, 7) >>> points = (x, y, z) >>> values = value_func_3d(*np.meshgrid(*points, indexing='ij')) Evaluate the interpolating function at a point >>> point = np.array([2.21, 3.12, 1.15]) >>> print(interpn(points, values, point)) [12.63] See Also -------- NearestNDInterpolator : Nearest neighbor interpolation on unstructured data in N dimensions LinearNDInterpolator : Piecewise linear interpolant on unstructured data in N dimensions RegularGridInterpolator : interpolation on a regular or rectilinear grid in arbitrary dimensions (`interpn` wraps this class). RectBivariateSpline : Bivariate spline approximation over a rectangular mesh scipy.ndimage.map_coordinates : interpolation on grids with equal spacing (suitable for e.g., N-D image resampling) """ # sanity check 'method' kwarg if method not in ["linear", "nearest", "cubic", "quintic", "pchip", "splinef2d", "slinear"]: raise ValueError("interpn only understands the methods 'linear', " "'nearest', 'slinear', 'cubic', 'quintic', 'pchip', " f"and 'splinef2d'. You provided {method}.") if not hasattr(values, 'ndim'): values = np.asarray(values) ndim = values.ndim if ndim > 2 and method == "splinef2d": raise ValueError("The method splinef2d can only be used for " "2-dimensional input data") if not bounds_error and fill_value is None and method == "splinef2d": raise ValueError("The method splinef2d does not support extrapolation.") # sanity check consistency of input dimensions if len(points) > ndim: raise ValueError("There are %d point arrays, but values has %d " "dimensions" % (len(points), ndim)) if len(points) != ndim and method == 'splinef2d': raise ValueError("The method splinef2d can only be used for " "scalar data with one point per coordinate") grid, descending_dimensions = _check_points(points) _check_dimensionality(grid, values) # sanity check requested xi xi = _ndim_coords_from_arrays(xi, ndim=len(grid)) if xi.shape[-1] != len(grid): raise ValueError("The requested sample points xi have dimension " "%d, but this RegularGridInterpolator has " "dimension %d" % (xi.shape[-1], len(grid))) if bounds_error: for i, p in enumerate(xi.T): if not np.logical_and(np.all(grid[i][0] <= p), np.all(p <= grid[i][-1])): raise ValueError("One of the requested xi is out of bounds " "in dimension %d" % i) # perform interpolation if method in ["linear", "nearest", "slinear", "cubic", "quintic", "pchip"]: interp = RegularGridInterpolator(points, values, method=method, bounds_error=bounds_error, fill_value=fill_value) return interp(xi) elif method == "splinef2d": xi_shape = xi.shape xi = xi.reshape(-1, xi.shape[-1]) # RectBivariateSpline doesn't support fill_value; we need to wrap here idx_valid = np.all((grid[0][0] <= xi[:, 0], xi[:, 0] <= grid[0][-1], grid[1][0] <= xi[:, 1], xi[:, 1] <= grid[1][-1]), axis=0) result = np.empty_like(xi[:, 0]) # make a copy of values for RectBivariateSpline interp = RectBivariateSpline(points[0], points[1], values[:]) result[idx_valid] = interp.ev(xi[idx_valid, 0], xi[idx_valid, 1]) result[np.logical_not(idx_valid)] = fill_value return result.reshape(xi_shape[:-1])