1D interpolation

Scope

Finite number $N$ of data points are available: $P_i = (x_i, y_i)$ , $i \in \lbrace 0, \ldots, N \rbrace$
Interpolation is about filling the gaps by building back the function $y(x)$

https://en.wikipedia.org/wiki/Interpolation



In [1]:

    
# Setup
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import matplotlib
params = {'font.size'     : 14,
          'figure.figsize':(15.0, 8.0),
          'lines.linewidth': 2.,
          'lines.markersize': 15,}
matplotlib.rcParams.update(params)

Let's do it with Python



In [2]:

    
N = 10
xmin, xmax = 0., 1.5
xi = np.linspace(xmin, xmax, N)
yi = np.random.rand(N)

plt.plot(xi,yi, 'o', label = "$Pi$")
plt.grid()
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.show()

Nearest (aka. piecewise) interpolation

Function $y(x)$ takes the value $y_i$ of the nearest point $P_i$ on the $x$ direction.



In [3]:

    
from scipy import interpolate
x = np.linspace(xmin, xmax, 1000)
interp = interpolate.interp1d(xi, yi, kind = "nearest")
y_nearest = interp(x)

plt.plot(xi,yi, 'o', label = "$Pi$")
plt.plot(x, y_nearest, "-", label = "Nearest")
plt.grid()
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.show()

Pros

$y(x)$ only takes values of existing $y_i$.

Cons

Discontinuous

Linear interpolation

Function $y(x)$ depends linearly on its closest neighbours.



In [4]:

    
from scipy import interpolate
x = np.linspace(xmin, xmax, 1000)
interp = interpolate.interp1d(xi, yi, kind = "linear")
y_linear = interp(x)

plt.plot(xi,yi, 'o', label = "$Pi$")
plt.plot(x, y_nearest, "-", label = "Nearest")
plt.plot(x, y_linear, "-", label = "Linear")
plt.grid()
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.show()

Pros

$y(x)$ stays in the limits of $y_i$
Continuous

Cons

Discontinuous first derivative.

Spline interpolation



In [5]:

    
from scipy import interpolate
x = np.linspace(xmin, xmax, 1000)
interp2 = interpolate.interp1d(xi, yi, kind = "quadratic")
interp3 = interpolate.interp1d(xi, yi, kind = "cubic")
y_quad = interp2(x)
y_cubic = interp3(x)

plt.plot(xi,yi, 'o', label = "$Pi$")
plt.plot(x, y_nearest, "-", label = "Nearest")
plt.plot(x, y_linear,  "-", label = "Linear")
plt.plot(x, y_quad,    "-", label = "Quadratic")
plt.plot(x, y_cubic,   "-", label = "Cubic")
plt.grid()
plt.xlabel("x")
plt.ylabel("y")
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.show()

Pros

Smoother
Cubic generally more reliable that quadratic

Cons

Less predictable values between points.