The least squares Gaussian approximation method is an approach for fitting a Gaussian function to a set of data points by minimizing the error between the actual data and the fitted Gaussian curve. This method is commonly used in signal processing, curve fitting, and data analysis where Gaussian distributions are expected or assumed.

Gaussian Function Overview:

A Gaussian function can be represented mathematically as:

$$ f(x) = a \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) $$

where:

• $a$ is the amplitude (peak height),
• $\mu$ is the mean (center of the peak),
• $\sigma$ is the standard deviation (controls the width of the peak).

Least Squares Method:

The least squares method aims to find the parameters $a$ , $\mu$ , and $\sigma$ such that the sum of the squared differences between the data points $y_i$ and the values predicted by the Gaussian function $f(x_i)$ is minimized:

$$ \text{Objective:} \quad \min_{a, \mu, \sigma} \sum_{i=1}^N \left(y_i - f(x_i)\right)^2 $$

where $N$ is the number of data points, $x_i$ are the data points' positions, and $y_i$ are the observed values.

Steps to Perform Least Squares Gaussian Approximation:

1. Data Preparation:
   • Collect the data points $(x_i, y_i)$ .
   • Ensure the data reflects a distribution that could be approximated by a Gaussian function.
2. Initial Parameter Estimation:
   • Use a heuristic or a simpler method to make an initial guess for $a$ , $\mu$ , and $\sigma$ .
   • Common initial guesses:
     • $\mu$ : The mean of the $x_i$ values.
     • $\sigma$ : The standard deviation of the $x_i$ values.
     • $a$ : The maximum $y_i$ value.
3. Set Up the Objective Function:
   • Define the objective function to compute the sum of squared differences: $E(a, \mu, \sigma) = \sum_{i=1}^N \left(y_i - a \exp\left(-\frac{(x_i - \mu)^2}{2\sigma^2}\right)\right)^2$
4. Optimization:
   • Use optimization algorithms like gradient descent, Levenberg-Marquardt, or built-in functions (e.g., scipy.optimize.curve_fit in Python) to adjust $a$ , $\mu$ , and $\sigma$ and minimize $E(a, \mu, \sigma)$ .
   • These algorithms iteratively refine the parameters by computing the gradient of $E$ and updating the parameters in the direction that reduces the error.
5. Check for Convergence:
   • Ensure that the optimization converges to a solution where the sum of squared errors no longer decreases significantly.
6. Evaluate Fit Quality:
   • Analyze the residuals (differences between the actual data and the fitted curve) to ensure the fit is appropriate.
   • Calculate metrics such as R-squared or root mean square error (RMSE) to quantify the fit quality.

Practical Implementation: