Gaussian Curve Fitting Project

Gaussian Function

$$ G(x, A, \mu, \sigma) = A \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) $$

Gradient Descent

$$ \text{grad}_A = -2 \sum (y - G(x, A, \mu, \sigma)) \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) $$

$$ \text{grad}_\mu = -2 \sum (y - G(x, A, \mu, \sigma)) \frac{A(x - \mu)}{\sigma^2} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) $$

$$ \text{grad}_\sigma = -2 \sum (y - G(x, A, \mu, \sigma)) \frac{A(x - \mu)^2}{\sigma^3} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) $$

Levenberg-Marquardt Algorithm

$$ \mathbf{J} = \frac{\partial r}{\partial \mathbf{p}} $$

$$ \mathbf{H} = \mathbf{J}^T \mathbf{J} $$

$$ \mathbf{g} = \mathbf{J}^T \mathbf{r} $$

$$ \Delta \mathbf{p} = (\mathbf{H} + \lambda \mathbf{I})^{-1} \mathbf{g} $$

Bayesian Inference

$$ P(A, \mu, \sigma \mid x, y) \propto P(y \mid x, A, \mu, \sigma) P(A) P(\mu) P(\sigma) $$

Important Mathematical Concepts

Least Squares and Gauss' Contribution

The method of least squares minimizes the sum of squared errors:

$$ \sum_{i=1}^n \epsilon_i^2 $$

$$ \Omega = \prod_{i=1}^n \phi(\epsilon_i) $$

Gaussian Distribution

$$ \phi(\epsilon_i) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{\epsilon_i^2}{2 \sigma^2}\right) $$

Maximizing Likelihood

$$ \Omega = \prod_{i=1}^n \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left(-\frac{\epsilon_i^2}{2 \sigma^2}\right) $$

$$ \text{maximize} \quad -\sum_{i=1}^n \epsilon_i^2 $$