1-D interpolation#

Piecewise linear interpolation#

If all you need is a linear (a.k.a. broken line) interpolation, you can use the numpy.interp routine. It takes two arrays of data to interpolate, x, and y, and a third array, xnew, of points to evaluate the interpolation on:

>>> import numpy as np
>>> x = np.linspace(0, 10, num=11)
>>> y = np.cos(-x**2 / 9.0)

Construct the interpolation

>>> xnew = np.linspace(0, 10, num=1001)
>>> ynew = np.interp(xnew, x, y)

And plot it

>>> import matplotlib.pyplot as plt
>>> plt.plot(xnew, ynew, '-', label='linear interp')
>>> plt.plot(x, y, 'o', label='data')
>>> plt.legend(loc='best')
>>> plt.show()
../../_images/1D-1.png

One limitation of numpy.interp is that it does not allow controlling the extrapolation. See the interpolation with B-Splines section section for alternative routines which provide this kind of functionality.

Cubic splines#

Of course, piecewise linear interpolation produces corners at data points, where linear pieces join. To produce a smoother curve, you can use cubic splines, where the interpolating curve is made of cubic pieces with matching first and second derivatives. In code, these objects are represented via the CubicSpline class instances. An instance is constructed with the x and y arrays of data, and then it can be evaluated using the target xnew values:

>>> from scipy.interpolate import CubicSpline
>>> spl = CubicSpline([1, 2, 3, 4, 5, 6], [1, 4, 8, 16, 25, 36])
>>> spl(2.5)
5.57

A CubicSpline object’s __call__ method accepts both scalar values and arrays. It also accepts a second argument, nu, to evaluate the derivative of order nu. As an example, we plot the derivatives of a spline:

>>> from scipy.interpolate import CubicSpline
>>> x = np.linspace(0, 10, num=11)
>>> y = np.cos(-x**2 / 9.)
>>> spl = CubicSpline(x, y)

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(4, 1, figsize=(5, 7))
>>> xnew = np.linspace(0, 10, num=1001)
>>> ax[0].plot(xnew, spl(xnew))
>>> ax[0].plot(x, y, 'o', label='data')
>>> ax[1].plot(xnew, spl(xnew, nu=1), '--', label='1st derivative')
>>> ax[2].plot(xnew, spl(xnew, nu=2), '--', label='2nd derivative')
>>> ax[3].plot(xnew, spl(xnew, nu=3), '--', label='3rd derivative')
>>> for j in range(4):
...     ax[j].legend(loc='best')
>>> plt.tight_layout()
>>> plt.show()
../../_images/1D-2.png

Note that the first and second derivatives are continuous by construction, and the third derivative jumps at data points.

Monotone interpolants#

Cubic splines are by construction twice continuously differentiable. This may lead to the spline function oscillating and ‘’overshooting’’ in between the data points. In these situations, an alternative is to use the so-called monotone cubic interpolants: these are constructed to be only once continuously differentiable, and attempt to preserve the local shape implied by the data. scipy.interpolate provides two objects of this kind: PchipInterpolator and Akima1DInterpolator . To illustrate, let’s consider data with an outlier:

>>> from scipy.interpolate import CubicSpline, PchipInterpolator, Akima1DInterpolator
>>> x = np.array([1., 2., 3., 4., 4.5, 5., 6., 7., 8])
>>> y = x**2
>>> y[4] += 101

>>> import matplotlib.pyplot as plt
>>> xx = np.linspace(1, 8, 51)
>>> plt.plot(xx, CubicSpline(x, y)(xx), '--', label='spline')
>>> plt.plot(xx, Akima1DInterpolator(x, y)(xx), '-', label='Akima1D')
>>> plt.plot(xx, PchipInterpolator(x, y)(xx), '-', label='pchip')
>>> plt.plot(x, y, 'o')
>>> plt.legend()
>>> plt.show()
../../_images/1D-3.png

Interpolation with B-splines#

B-splines form an alternative (if formally equivalent) representation of piecewise polynomials. This basis is generally more computationally stable than the power basis and is useful for a variety of applications which include interpolation, regression and curve representation. Details are given in the piecewise polynomials section, and here we illustrate their usage by constructing the interpolation of a sine function:

>>> x = np.linspace(0, 3/2, 7)
>>> y = np.sin(np.pi*x)

To construct the interpolating objects given data arrays, x and y, we use the make_interp_spline function:

>>> from scipy.interpolate import make_interp_spline
>>> bspl = make_interp_spline(x, y, k=3)

This function returns an object which has an interface similar to that of the CubicSpline objects. In particular, it can be evaluated at a data point and differentiated:

>>> der = bspl.derivative()      # a BSpline representing the derivative
>>> import matplotlib.pyplot as plt
>>> xx = np.linspace(0, 3/2, 51)
>>> plt.plot(xx, bspl(xx), '--', label=r'$\sin(\pi x)$ approx')
>>> plt.plot(x, y, 'o', label='data')
>>> plt.plot(xx, der(xx)/np.pi, '--', label=r'$d \sin(\pi x)/dx / \pi$ approx')
>>> plt.legend()
>>> plt.show()
../../_images/1D-4.png

Note that by specifying k=3 in the make_interp_spline call, we requested a cubic spline (this is the default, so k=3 could have been omitted); the derivative of a cubic is a quadratic:

>>> bspl.k, der.k
(3, 2)

By default, the result of make_interp_spline(x, y) is equivalent to CubicSpline(x, y). The difference is that the former allows several optional capabilities: it can construct splines of various degrees (via the optional argument k) and predefined knots (via the optional argument t).

Boundary conditions for the spline interpolation can be controlled by the bc_type argument to make_interp_spline function and CubicSpline constructor. By default, both use the ‘not-a-knot’ boundary condition.

Non-cubic splines#

One use of make_interp_spline is constructing a linear interpolant with linear extrapolation since make_interp_spline extrapolates by default. Consider

>>> from scipy.interpolate import make_interp_spline
>>> x = np.linspace(0, 5, 11)
>>> y = 2*x
>>> spl = make_interp_spline(x, y, k=1)  # k=1: linear
>>> spl([-1, 6])
[-2., 12.]
>>> np.interp([-1, 6], x, y)
[0., 10.]

See the extrapolation section for more details and discussion.

Batches of y#

Univariate interpolators accept not only one-dimensional y arrays, but also y.ndim > 1. The interpretation is that y is a batch of 1D data arrays: by default, the zeroth dimension of y is the interpolation axis, and the trailing dimensions are batch dimensions. Consider a collection (a batch) of functions \(f_j\) sampled at the points \(x_i\). We can instantiate a single interpolator for all of these functions by providing a two-dimensional array y such that y[i, j] records \(f_j(x_i)\).

To illustrate:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import make_interp_spline
>>> n = 11
>>> x = 2 * np.pi * np.arange(n) / n
>>> x.shape
(11,)
>>> y = np.stack((np.sin(x)**2, np.cos(x)), axis=1)
>>> y.shape
(11, 2)
>>> spl = make_interp_spline(x, y)
>>> xv = np.linspace(0, 2*np.pi, 51)
>>> plt.plot(x, y, 'o')
>>> plt.plot(xv, spl(xv), '-')
>>> plt.show()

Several notes are in order. First and foremost, the behavior here looks similar to NumPy’s broadcasting, but differs in two respects:

  1. The x array is expected to be 1D even if the y array is not: x.ndim == 1 while y.ndim >= 1. There is no broadcasting of x vs y.

  2. By default, the trailing dimensions are used as batch dimensions, in contrast to the NumPy convention of using the leading dimensions as batch dimensions.

Second, the interpolation axis can be controlled by an optional axis argument. The example above uses the default value of axis=0. For a non-default values, the following is true:

  • y.shape[axis] == x.size (otherwise en error is raised)

  • the shape of spl(xv) is y.shape[axis:] + xv.shape + y.shape[:axis]

While we demonstrated the batching behavior with make_interp_spline, in fact the majority of univariate interpolators support this functionality: PchipInterpolator and Akima1DInterpolator, CubicSpline; low-level polynomial representation classes, PPoly, BPoly and BSpline; as well as least-squares fit and spline smoothing functions, make_lsq_spline and make_smoothing_spline.

Parametric spline curves#

So far we considered spline functions, where the data, y, is expected to depend explicitly on the independent variable x—so that the interpolating function satisfies \(f(x_j) = y_j\). Spline curves treat the x and y arrays as coordinates of points, \(\mathbf{p}_j\) on a plane, and an interpolating curve which passes through these points is parameterized by some additional parameter (typically called u). Note that this construction readily generalizes to higher dimensions where \(\mathbf{p}_j\) are points in an N-dimensional space.

Spline curves can be easily constructed using the fact that interpolation functions handle multidimensional data arrays, as discussed in the previous section. The values of the parameter, u, corresponding to the data points, need to be separately supplied by the user.

The choice of parametrization is problem-dependent and different parametrizations may produce vastly different curves. As an example, we consider three parametrizations of (a somewhat difficult) dataset, which we take from Chapter 6 of Ref [1] listed in the BSpline docstring:

>>> x = [0, 1, 2, 3, 4, 5, 6]
>>> y = [0, 0, 0, 9, 0, 0, 0]
>>> p = np.stack((x, y))
>>> p
array([[0, 1, 2, 3, 4, 5, 6],
       [0, 0, 0, 9, 0, 0, 0]])

We take elements of the p array as coordinates of seven points on the plane, where p[:, j] gives the coordinates of the point \(\mathbf{p}_j\).

First, consider the uniform parametrization, \(u_j = j\):

>>> u_unif = x

Second, we consider the so-called cord length parametrization, which is nothing but a cumulative length of straight line segments connecting the data points:

\[u_j = u_{j-1} + |\mathbf{p}_j - \mathbf{p}_{j-1}|\]

for \(j=1, 2, \dots\) and \(u_0 = 0\). Here \(| \cdots |\) is the length between the consecutive points \(p_j\) on the plane.

>>> dp = p[:, 1:] - p[:, :-1]      # 2-vector distances between points
>>> l = (dp**2).sum(axis=0)        # squares of lengths of 2-vectors between points
>>> u_cord = np.sqrt(l).cumsum()   # cumulative sums of 2-norms
>>> u_cord = np.r_[0, u_cord]      # the first point is parameterized at zero

Finally, we consider what is sometimes called the centripetal parametrization: \(u_j = u_{j-1} + |\mathbf{p}_j - \mathbf{p}_{j-1}|^{1/2}\). Due to the extra square root, the difference between consecutive values \(u_j - u_{j-1}\) will be smaller than for the cord length parametrization:

>>> u_c = np.r_[0, np.cumsum((dp**2).sum(axis=0)**0.25)]

Now plot the resulting curves:

>>> from scipy.interpolate import make_interp_spline
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 3, figsize=(8, 3))
>>> parametrizations = ['uniform', 'cord length', 'centripetal']
>>>
>>> for j, u in enumerate([u_unif, u_cord, u_c]):
...    spl = make_interp_spline(u, p, axis=1)    # note p is a 2D array
...
...    uu = np.linspace(u[0], u[-1], 51)
...    xx, yy = spl(uu)
...
...    ax[j].plot(xx, yy, '--')
...    ax[j].plot(p[0, :], p[1, :], 'o')
...    ax[j].set_title(parametrizations[j])
>>> plt.show()
../../_images/1D-5.png

Missing data#

We note that scipy.interpolate does not support interpolation with missing data. Two popular ways of representing missing data are using masked arrays of the numpy.ma library, and encoding missing values as not-a-number, NaN.

Neither of these two approaches is directly supported in scipy.interpolate. Individual routines may offer partial support, and/or workarounds, but in general, the library firmly adheres to the IEEE 754 semantics where a NaN means not-a-number, i.e. a result of an illegal mathematical operation (e.g., division by zero), not missing.

Legacy interface for 1-D interpolation (interp1d)#

Note

interp1d is considered legacy API and is not recommended for use in new code. Consider using more specific interpolators instead.

The interp1d class in scipy.interpolate is a convenient method to create a function based on fixed data points, which can be evaluated anywhere within the domain defined by the given data using linear interpolation. An instance of this class is created by passing the 1-D vectors comprising the data. The instance of this class defines a __call__ method and can therefore be treated like a function which interpolates between known data values to obtain unknown values. Behavior at the boundary can be specified at instantiation time. The following example demonstrates its use, for linear and cubic spline interpolation:

>>> from scipy.interpolate import interp1d

>>> x = np.linspace(0, 10, num=11, endpoint=True)
>>> y = np.cos(-x**2/9.0)
>>> f = interp1d(x, y)
>>> f2 = interp1d(x, y, kind='cubic')

>>> xnew = np.linspace(0, 10, num=41, endpoint=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'o', xnew, f(xnew), '-', xnew, f2(xnew), '--')
>>> plt.legend(['data', 'linear', 'cubic'], loc='best')
>>> plt.show()
"This code generates an X-Y plot of a time-series with amplitude on the Y axis and time on the X axis. The original time-series is shown as a series of blue markers roughly defining some kind of oscillation. An orange trace showing the linear interpolation is drawn atop the data forming a jagged representation of the original signal. A dotted green cubic interpolation is also drawn that appears to smoothly represent the source data."

The ‘cubic’ kind of interp1d is equivalent to make_interp_spline, and the ‘linear’ kind is equivalent to numpy.interp while also allowing N-dimensional y arrays.

Another set of interpolations in interp1d is nearest, previous, and next, where they return the nearest, previous, or next point along the x-axis. Nearest and next can be thought of as a special case of a causal interpolating filter. The following example demonstrates their use, using the same data as in the previous example:

>>> from scipy.interpolate import interp1d

>>> x = np.linspace(0, 10, num=11, endpoint=True)
>>> y = np.cos(-x**2/9.0)
>>> f1 = interp1d(x, y, kind='nearest')
>>> f2 = interp1d(x, y, kind='previous')
>>> f3 = interp1d(x, y, kind='next')

>>> xnew = np.linspace(0, 10, num=1001, endpoint=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'o')
>>> plt.plot(xnew, f1(xnew), '-', xnew, f2(xnew), '--', xnew, f3(xnew), ':')
>>> plt.legend(['data', 'nearest', 'previous', 'next'], loc='best')
>>> plt.show()
"This code generates an X-Y plot of a time-series with amplitude on the Y axis and time on the X axis. The original time-series is shown as a series of blue markers roughly defining some kind of oscillation. An orange trace showing the nearest neighbor interpolation is drawn atop the original with a stair-like appearance where the original data is right in the middle of each stair step. A green trace showing the previous neighbor interpolation looks similar to the orange trace but the original data is at the back of each stair step. Similarly a dotted red trace showing the next neighbor interpolation goes through each of the previous points, but it is centered at the front edge of each stair."