MSE Computation for Least Squares Algorithm and Minimum Mean Square Error Algorithms

Least squares method and Minimum Mean Square Error (MSE) are core concepts in regression analysis, primarily used to evaluate the deviation between predicted values and actual values. The least squares method fits optimal parameters by minimizing the sum of squared errors, while MSE serves as a crucial metric for measuring model prediction performance.

The MSE calculation formula represents the average of squared differences between predicted and true values. Its mathematical expression is:

[ MSE = \frac{1}{n} \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 ]

Where ( y_i ) represents actual values, ( \hat{y}_i ) denotes predicted values, and ( n ) indicates the number of data samples. In least squares optimization, the objective is to minimize MSE, ensuring minimal deviation between predicted and actual values.

In MATLAB implementation, MSE computation can be achieved either manually or using built-in functions. Manual implementation typically involves calculating squared differences between predicted and actual values followed by averaging. When utilizing regression toolbox or statistical functions, developers can directly call relevant functions like `mse()` or `regression` functions for efficient computation, significantly improving processing efficiency.

The MATLAB implementation of least squares method commonly involves matrix operations, such as using the left division operator (backslash) to solve linear regression problems. For model evaluation, MSE serves as a key indicator to analyze model fitting quality. Lower MSE values generally indicate better prediction performance, while higher MSE may suggest significant model bias or underfitting issues. The algorithm typically employs gradient descent or normal equations for parameter optimization, with MATLAB's `lscov` function providing weighted least squares solutions for more complex scenarios.