Unveiling the Latest Advances in MATLAB Gaussian Fitting: Exploring Cutting-edge Techniques and Mastering the New Trends in Fitting

# Decoding the Latest Advancements in MATLAB Gaussian Fitting: Exploring Cutting-edge Techniques and Mastering New Trends

## 1. Basics of Gaussian Fitting

Gaussian fitting is a statistical modeling technique used to fit a Gaussian distribution curve to data points. The Gaussian distribution, also known as the normal distribution, is a common probability distribution characterized by a bell-shaped curve. Gaussian fitting is widely used in fields such as science, engineering, and finance for data analysis and interpretation.

In MATLAB, Gaussian fitting can be achieved through various methods. One common approach is using the least squares method, which minimizes the squared errors between data points and the fitted curve. Another method is based on the maximum likelihood estimation, which maximizes the likelihood function of data points belonging to the fitted curve.

## 2. MATLAB Gaussian Fitting Methods

### 2.1 Fitting Based on the Least Squares Method

#### 2.1.1 Linear Least Squares Method

**Principle:**

The linear least squares method is a classic fitting method aiming to find a straight line that minimizes the sum of the vertical distances to a set of data points. For Gaussian fitting, we assume that the data points follow a normal distribution, and thus their probability density function is:

f(x) = (1 / (σ√(2π))) * exp(-((x - μ)² / (2σ²)))

Where μ is the mean and σ is the standard deviation.

**MATLAB Code:**

% Data points
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

% Fitting parameters
params = polyfit(data, data, 1);

% Plotting the fitted curve
plot(data, data, 'o');
hold on;
plot(data, polyval(params, data), 'r-');
legend('Data Points', 'Fitted Line');
xlabel('X');
ylabel('Y');
title('Linear Least Squares Fitting');

**Code Logic Analysis:**

* The `polyfit` function uses the linear least squares method to fit the data points and returns the fitting parameters `params`.
* The `polyval` function calculates points on the fitted curve based on the fitting parameters.
* The `plot` function draws the data points and the fitted curve.

#### 2.1.2 Nonlinear Least Squares Method

**Principle:**

The nonlinear least squares method is used for fitting nonlinear functions. For Gaussian fitting, we use the following nonlinear function:

f(x) = a * exp(-((x - b)² / (2c²)))

Where a is the amplitude, b is the mean, and c is the standard deviation.

**MATLAB Code:**

% Data points
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

% Initial fitting parameters
initialParams = [1, 5, 1];

% Fitting parameters
params = nlinfit(data, @gaussFunction, initialParams);

% Plotting the fitted curve
plot(data, data, 'o');
hold on;
plot(data, gaussFunction(params, data), 'r-');
legend('Data Points', 'Fitted Curve');
xlabel('X');
ylabel('Y');
title('Nonlinear Least Squares Fitting');

% Defining the Gaussian function
function y = gaussFunction(params, x)
     a = params(1);
     b = params(2);
     c = params(3);
     y = a * exp(-((x - b)² / (2c²)));
end

**Code Logic Analysis:**

* The `nlinfit` function uses the nonlinear least squares method to fit the data points and returns the fitting parameters `params`.
* The `gaussFunction` function defines the Gaussian function, where `params` are the fitting parameters.
* The `plot` function draws the data points and the fitted curve.

### 2.2 Fitting Based on Maximum Likelihood Estimation

#### 2.2.1 Principle of Maximum Likelihood Estimation

**Principle:**

Maximum likelihood estimation is a statistical method aimed at finding a set of parameters that maximize the likelihood function given the data. For Gaussian fitting, the likelihood function is:

L(μ, σ) = (1 / (σ√(2π)))^n * exp(-(1 / (2σ²)) * Σ(x - μ)²)

Where n is the number of data points.

#### 2.2.2 Likelihood Function for Gaussian Fitting

**MATLAB Code:**

% Data points
data = [1, 2, 3, 4, 5, 6, 7, 8, 9