MATLAB Gaussian Fitting Parameter Estimation: From Theory to Practice, Mastering the Fitting Parameters

# Chapter 1: Theoretical Foundation of Gaussian Distribution**

The Gaussian distribution, also known as the normal distribution, is a continuous probability distribution characterized by its probability density function:

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

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

The Gaussian distribution has the following properties:

***Symmetry:** The distribution is symmetric around the mean.
***Bell-shaped curve:** The distribution is shaped like a bell, peaking at the mean.
***Asymptotic tails:** The tails of the distribution fall off exponentially.

# Chapter 2: Gaussian Fitting Parameter Estimation Methods in MATLAB

### 2.1 Fitting Based on Normal Distribution Probability Density Function

#### 2.1.1 Normal Distribution Probability Density Function Formula

The normal distribution, also known as the Gaussian distribution, has the probability density function:

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

Where:

* x: Random variable
* μ: Mean
* σ: Standard deviation

#### 2.1.2 Parameter Estimation Methods

The parameter estimation methods based on the normal distribution probability density function are as follows:

1. **Maximum Likelihood Estimation:**

Maximum likelihood estimation estimates parameters by maximizing the likelihood function. The likelihood function is:

    L(μ, σ) = ∏[f(x_i)]

Where, x_i represents the data samples.

2. **Moment Estimation:**

Moment estimation estimates parameters by matching sample moments with theoretical moments. The sample mean and sample variance are:

    μ_s = (1 / n) * ∑x_i
    σ_s^2 = (1 / n) * ∑(x_i - μ_s)^2

Where, n is the sample size.

### 2.2 Fitting Based on Maximum Likelihood Estimation

#### 2.2.1 Principle of Maximum Likelihood Estimation

Maximum likelihood estimation assumes that the data samples come from a population with a specific parameter distribution. By maximizing the likelihood function, the parameter values that are most likely to produce the observed data can be found.

#### 2.2.2 Parameter Estimation Methods

Based on maximum likelihood estimation, parameter estimation methods are as follows:

1. **Log-Likelihood Function:**

To simplify calculations, the log-likelihood function is used:

    log L(μ, σ) = -n log(σ) - (1 / (2σ^2)) * ∑(x_i - μ)^2

2. **Differentiate and Set to Zero:**

Differentiate μ and σ and set the results to zero to obtain the estimated parameter values:

    μ_mle = (1 / n) * ∑x_i
    σ_mle^2 = (1 / n) * ∑(x_i - μ_mle)^2

### 2.3 Fitting Based on Least Squares Method

#### 2.3.1 Principle of Least Squares Method

The least squares method estimates parameters by minimizing the sum of the squared errors between the fitted curve and the data points.

#### 2.3.2 Parameter Estimation Methods

Based on the least squares method, parameter estimation methods are as follows:

1. **Error Function:**

The error function is:

    E(μ, σ) = ∑(y_i - f(x_i))^2

Where, y_i are the observed values, and f(x_i) are the fitted curve values.

2. **Differentiate and Set to Zero:**

Differentiate μ and σ and set the results to zero to obtain the estimated parameter values:

    μ_ls = (∑x_i * y_i) / ∑x_i^2
    σ_ls^2 = (∑y_i^2 - μ_ls * ∑y_i) / (n - 1)

# Chapter 3: Practice of Gaussian Fitting Parameter Estimation in MATLAB

### 3.1 Data Preparation and Visualization

#### 3.1.1 Data Generation

To demonstrate Gaussian fitting parameter estim