【Basic】Regression Forecast Model: MATLAB Polynomial Regression

# 2.1 Polynomial Regression Model

The polynomial regression model is a type of regression model used to fit nonlinear relationships. It represents the relationship between the dependent variable and independent variables using polynomial functions, thereby capturing more complex nonlinear patterns.

The general form of the polynomial regression model is:

y = β0 + β1x + β2x^2 + ... + βnx^n + ε

Where:

- y is the dependent variable
- x is the independent variable
- β0, β1, ..., βn are the model parameters
- n is the order of the polynomial
- ε is the error term

The polynomial regression model estimates model parameters by minimizing the sum of squared errors. This can be achieved using the principle of least squares.

# 2. Polynomial Regression Theoretical Foundation

### 2.1 Polynomial Regression Model

Polynomial regression is a nonlinear regression model that models the dependent variable as a polynomial function of the independent variables. The general form is as follows:

y = β0 + β1x + β2x^2 + ... + βnx^n + ε

Where:

- y is the dependent variable
- x is the independent variable
- β0, β1, ..., βn are the model parameters
- n is the order of the polynomial
- ε is the error term, representing the variation in the dependent variable that the model cannot explain

### 2.2 Principle of Least Squares

The parameter estimation of the polynomial regression model usually employs the principle of least squares. This principle aims to find a set of parameters that minimize the squared differences between the model's predicted values and actual observed values.

Mathematically, the least squares objective function is:

S = Σ(yi - ŷi)^2

Where:

* yi is the actual observed value
* ŷi is the model's predicted value

Minimizing S means finding a set of parameters that result in the best fit between the model and the observed data.

**Parameter Estimation Process**

The least squares parameter estimation process involves the following steps:

1. Construct the Vandermonde matrix X, with elements:

Xij = xi^j

***pute the pseudo-inverse matrix X+:

X+ = (X'X)^-1X'

3. Solve for the parameter vector β:

β = X+y

Where y is the observed value vector of the dependent variable.

**Code Example**

The following MATLAB code demonstrates how to estimate the parameters of a polynomial regression model using least squares:

% Data preparation
x = [1, 2, 3, 4, 5];
y = [2, 4, 8, 16, 32];

% Model fitting
n = 2; % Polynomial order
X = vander(x, n + 1);
beta = pinv(X) * y;

% Model prediction
x_new = 6;
y_pred = beta(1) + beta(2) * x_new + beta(3) * x_new^2;

% Print results
fprintf('Parameter estimates:\n');
disp(beta);
fprintf('Prediction value: %.2f\n', y_pred);

**Code Logic Analysis**

* The `vander` function creates the Vandermonde matrix X.
* The `pinv` function computes the pseudo-inverse matrix X+.
* The `beta` variable stores the estimated parameter vector.
* The `y_pred` variable stores the predicted value for the new data point x_new.

# 3.1 Data Preparation and Import

Before performing polynomial regression, data preparation and import are necessary. Data preparation includes the following steps:

- **Data Collection:** Gather historical data related to the regression variables. Ensure the data is accurate and complete.
- **Data Cleaning:** Handle missing values, outliers, and inconsistent data. Missing values can be dealt with using interpolation or deletion, and outliers can be managed using thresholds or transformations.
- **Data Transformation:** Convert data into a format suitable for regression analysis. For example, convert categorical variables into dummy variables.
- **Data Standardization:** Standardize or normalize the data to ensure all variables are on the same scale. This helps improve the stability and accuracy of the regression model.

Data im