Gaussian function: Difference between revisions

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For the general form of the equation the coefficient ''A'' is the height of the peak and {{math|(''x''₀, ''y''₀)}} is the center of the blob.

If we set
<math display="block">

\begin{align}

a &= \frac{\cos^2\theta}{2\sigma_X^2} + \frac{\sin^2\theta}{2\sigma_Y^2}, \\

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\end{align}

</math>then we rotate the blob by a positive, counter-clockwise angle <math>\theta</math> (for negative, clockwise rotation, invert the signs in the ''b'' coefficient).<ref>{{cite web |last1=Nawri |first1=Nikolai |title=Berechnung von Kovarianzellipsen |url=http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf |access-date=14 August 2019 |url-status=dead |archive-url=https://web.archive.org/web/20190814081830/http://imkbemu.physik.uni-karlsruhe.de/~eisatlas/covariance_ellipses.pdf |archive-date=2019-08-14}}</ref>

To get back the coefficients <math>\theta</math>, <math>\sigma_X</math> and <math>\sigma_Y</math> from <math>a</math>, <math>b</math> and <math>c</math> use

<math display="block">\begin{align}

\theta &= \frac{1}{2}\arctan\left(\frac{2b}{a-c}\right), \quad \theta \in [-45, 45], \\

\sigma_X^2 &= \frac{1}{2 (a \cdot \cos^2\theta + 2 b \cdot \cos\theta\sin\theta + c \cdot \sin^2\theta)}, \\

\sigma_Y^2 &= \frac{1}{2 (a \cdot \sin^2\theta - 2 b \cdot \cos\theta\sin\theta + c \cdot \cos^2\theta)}.

\end{align}</math>

Example rotations of Gaussian blobs can be seen in the following examples:

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Also see [[multivariate normal distribution]].

=== Higher-order Gaussian or super-Gaussian function or generalized Gaussian function ===

A more general formulation of a Gaussian function with a flat-top and Gaussian fall-off can be taken by raising the content of the exponent to a power <math>P</math>:

<math display="block">f(x) = A \exp\left(-\left(\frac{(x - x_0)^2}{2\sigma_X^2}\right)^P\right).</math>

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[[Category:Articles containing proofs]]

[[Category:Articles with example MATLAB/Octave code]]

]

import numpy as np

import matplotlib.pyplot as plt

def gaussian_weight(x, x_i, tau):

return np.exp(-(x - x_i)**2 / (2 * tau**2)) # Use regular minus and correct exponentiation

def locally_weighted_regression(X, y, tau):

X = np.array(X)

y = np.array(y)

n = len(X)

y_pred = []

for query_point in X:

weights = np.array([gaussian_weight(query_point, x_i, tau) for x_i in X])

W = np.diag(weights)

X_mat = np.c_[np.ones(n), X] # Add intercept (column of ones)

theta = np.linalg.pinv(X_mat.T @ W @ X_mat) @ X_mat.T @ W @ y # Solving for theta

y_pred.append(theta[0] + theta[1] * query_point)

return np.array(y_pred)

# Generate data

np.random.seed(42)

X = np.linspace(-3, 3, 50)

y = np.sin(X) + np.random.normal(0, 0.2, X.shape)

tau = 0.5

# Perform locally weighted regression

y_pred = locally_weighted_regression(X, y, tau)

# Plot results

plt.figure(figsize=(10, 6))

plt.scatter(X, y, color='red', label='Data Points')

plt.plot(X, y_pred, color='blue', label='LWR Fitted Curve')

plt.title('Locally Weighted Regression')

plt.xlabel('X')

plt.ylabel('y')

plt.legend()

plt.show()