Radial Basis Function Interpolation
Radial Basis Function (RBF) interpolation is a method of interpolation that uses radial basis functions to approximate the underlying data. Unlike polynomial interpolation, which fits a single polynomial to the entire dataset, RBF interpolation uses a combination of radial basis functions centered at each data point to construct the interpolating function.
Implementation
- This code demonstrates Radial Basis Function (RBF) interpolation using `RBFInterpolator` from SciPy.
- It generates random data points in a 2D space and calculates corresponding y-values based on a predefined function.
- A grid is then created for visualization purposes.
- The `RBFInterpolator` function constructs an interpolation function based on the random data points.
- Finally, it plots the interpolated surface and scatter plot of the original data points to visualize the interpolation result.
import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import RBFInterpolator
# Generate random data points
rng = np.random.default_rng()
x_data = rng.uniform(-1, 1, size=(100, 2))
y_data = np.sum(x_data, axis=1) * np.exp(-6 * np.sum(x_data**2, axis=1))
# Generate a grid for visualization
x_grid = np.mgrid[-1:1:50j, -1:1:50j]
x_flat = np.column_stack((x_grid[0].flatten(), x_grid[1].flatten()))
# Perform RBF interpolation
rbf_interpolator = RBFInterpolator(x_data, y_data)
y_flat = rbf_interpolator(x_flat)
y_grid = y_flat.reshape(50, 50)
# Plot the interpolated surface and scatter plot of original points
fig, ax = plt.subplots()
ax.pcolormesh(x_grid[0], x_grid[1], y_grid)
p = ax.scatter(x_data[:,0], x_data[:,1], c=y_data, s=50, ec='k')
fig.colorbar(p)
plt.title('RBF Interpolation with Random Data')
plt.xlabel('X1')
plt.ylabel('X2')
plt.show()
Output:
Interpolation in Machine Learning
In machine learning, interpolation refers to the process of estimating unknown values that fall between known data points. This can be useful in various scenarios, such as filling in missing values in a dataset or generating new data points to smooth out a curve. In this article, we are going to explore fundamentals and implementation of different types of interpolation along with it’s application in machine learning.
In machine learning, interpolation is an essential method for estimating values within a range of known data points. Forecasting values at intermediate points entails building a function that roughly mimics the behavior of the underlying data.