【Advanced】Regression Analysis Using Gaussian Processes in MATLAB

# 2.1 Gaussian Process Model

A Gaussian Process (GP) is a non-parametric Bayesian model that treats functions as a Gaussian distribution. The GP model assumes that any finite-dimensional subset of functions follows a multivariate Gaussian distribution, as follows:

f(x_1), ..., f(x_n) ~ N(μ, K)

where:

* `f(x)` is the value of the function at input `x`
* `μ` is the mean vector of the function
* `K` is the covariance matrix, where `K(x_i, x_j)` denotes the covariance between the function values at inputs `x_i` and `x_j`

# 2. Foundations of Gaussian Process Regression Theory

### 2.1 Gaussian Process Model

A Gaussian Process (GP) is a non-parametric Bayesian model that treats functions as random variables. Within the context of GP, each finite-dimensional subset of functions follows a multivariate normal distribution. The mathematical definition of GP is as follows:

f(x) ~ GP(m(x), k(x, x'))

where:

- `f(x)` is the function defined by the GP
- `m(x)` is the mean function of the function
- `k(x, x')` is the covariance function (also known as the kernel function)

***mon covariance functions include:

- Squared Exponential Kernel
- Linear Kernel
- Periodic Kernel

### 2.2 Kernel Functions

The kernel function is ***mon kernel functions include:

| Kernel Function | Expression | Features |
|---|---|---|
| Squared Exponential Kernel | `k(x, x') = exp(-γ ||x - x'||^2)` | Smooth, suitable for stationary functions |
| Linear Kernel | `k(x, x') = x^T x'` | Linear, suitable for linear functions |
| Periodic Kernel | `k(x, x') = exp(-2 sin^2(π(x - x') / p))` | Periodic, suitable for functions with periodic patterns |

### 2.3 Prior Distribution and Posterior Distribution

In the GP model, the prior distribution of the function `f(x)` is defined by the mean function `m(x)` and the covariance function `k(x, x')`. When data `D = {(x_1, y_1), ..., (x_n, y_n)}` is observed, the posterior distribution of the function `f(x)` is calculated using Bayes' theorem as follows:

p(f(x) | D) ∝ p(D | f(x)) p(f(x))

where:

- `p(D | f(x))` is the likelihood function, representing the probability of observing the data given the function `f(x)`
- `p(f(x))` is the prior distribution of the function `f(x)`

The posterior distribution provides the distribution of the function `f(x)` given the observed data. It can be used to predict the function value for a new input `x*`, as shown below:

p(f(x*) | D) = ∫ p(f(x*) | f(x), D) p(f(x) | D) df(x)

where:

- `p(f(x*) | f(x), D)` is the predictive distribution, representing the probability of predicting `f(x*)` given the function `f(x)` and observed data `D`
- `p(f(x) | D)` is the posterior distribution of the function `f(x)`

The posterior distribution and predictive distribution are key components of GP regression, providing a model of the uncertainty in the function `f(x)`.

# 3. Practical Application of Gaussian Process Regression

### 3.1 Data Preparation and Preprocessing

Before applying Gaussian Process Regression, it is necessary to appropriately prepare and preprocess the data to ensure the accuracy and robustness of the model. The main steps of data preparation and preprocessing include:

- **Data Cleaning:** Remove missing values, outliers, and noisy data.
- **Feature Engineering:** Select and transform features to improve model performance. For instance, features can be scaled through standardization or normalization, or non-linear features can be extracted by creating polynomial features or performing Principal Component Analysis.
- **Data Splitting:** Divide the dataset into training sets, validation sets, and test sets. The training set is used for model trainin