[Advanced] Implementing Partial Least Squares Regression (PLSR) Mathematical Modeling Algorithm in MATLAB

# [Advanced篇] Implementing Partial Least Squares Regression (PLSR) in MATLAB for Mathematical Modeling

## 2.1 Mathematical Principles of PLSR Algorithm

### 2.1.1 Derivation of PLSR Algorithm

Partial Least Squares Regression (PLSR) is a supervised dimensionality reduction regression algorithm aimed at establishing a linear relationship between predictive variables (X) and response variables (Y). The derivation process of PLSR algorithm is as follows:

1. **Centering and Scaling:** Centering and scaling X and Y to eliminate the effects of variable units and dimensions.
2. **Singular Value Decomposition (SVD):** Conducting SVD on the centered X to obtain its left singular vectors U and right singular vectors V.
3. **Projection:** Projecting X onto the subspace of U to obtain new predictive variables T: T = XU.
4. **Regression:** Performing regression on T and Y to obtain regression coefficients b: b = (T'T)^-1T'Y.
5. **Prediction:** Using regression coefficients b and new predictive variables T to predict response variables Y: Y' = Tb.

### 2.1.2 Advantages and Limitations of PLSR Algorithm

**Advantages:**

* Can handle high-dimensional data and automatically perform dimensionality reduction.
* Robust to collinear variables.
* Can consider multiple response variables simultaneously.

**Limitations:**

* Prediction accuracy may be affected by noise and outliers.
* PLSR algorithm's predictive ability is limited for nonlinear relationships.
* The algorithm has high complexity and may require a longer computation time.

## 2. Implementing PLSR Algorithm in MATLAB

### 2.1 Mathematical Principles of PLSR Algorithm

#### 2.1.1 Derivation of PLSR Algorithm

Partial Least Squares Regression (PLSR) algorithm is a multivariate statistical regression technique used to analyze datasets with multicollinearity. Its goal is to find a set of linear regression equations that predict response variables (Y) as a linear combination of independent variables (X).

The derivation process of PLSR algorithm is as follows:

1. **Centering and Standardizing Data:** Centering and standardizing X and Y to eliminate the effects of scale differences.
2. **Calculating Covariance Matrix:** Calculating the covariance matrix C between X and Y.
3. **Singular Value Decomposition:** Performing SVD on C to obtain three matrices U, S, and V.
4. **Extracting Eigenvectors:** Selecting the first k eigenvectors of U as the eigenvectors of X, denoted as P.
5. **Calculating Regression Coefficients:** Calculating regression coefficients B to minimize the sum of squared residuals between the predicted values Y^ and the linear combination of P.

#### 2.1.2 Advantages and Limitations of PLSR Algorithm

**Advantages:**

* Able to handle multicollinearity data
* Able to extract important features from the dataset
* Has good predictive performance

**Limitations:**

* Sensitive to outliers
* Difficult to interpret the model
* May perform poorly when the relationship between independent variables and response variables is nonlinear

### 2.2 Function Implementation of PLSR Algorithm in MATLAB

#### 2.2.1 Basic Usage of plsregress Function

The `plsregress` function in MATLAB is used to implement the PLSR algorithm. Its basic syntax is as follows:

[B, FitInfo] = plsregress(Y, X, ncomp)

Where:

* `Y`: Response variable matrix
* `X`: Independent variable matrix
* `ncomp`: Number of eigenvectors to be extracted

Returns:

* `B`: Regression coefficient matrix
* `FitInfo`: Fitting information structure, containing fit quality, predictive power, and other indicators

#### 2.2.2 Advanced Options of plsregress Function

The `plsregress` function also provides many advanced options to control the behavior of the algorithm. These options include:

* `Validation`: Specifies cross-validation methods
* `Method`: Specifies the method for solving regression coefficients
* `Scale`: Specifies whether to center and standardize the data
* `WMode`: Specifies the weight mode

### 2.3 Model Evaluation of PLSR Algorithm

#### 2.3.1 Model Fit Evaluation Indicators

***R^2:** Coefficient of determination, measures the fit of the model
***RMSE:** Root Mean Square Error, measures the average value of prediction errors
***MAE:** Mean Absolute Error, measures the average absolute value of prediction errors

#### 2.3.2 Model Predictive Ability Evaluation Indicators

***Q^2:** Predictive sum of squares, measures the predictive ability of the model
***RMSEP:** Root Mean Square Error of Prediction, measures the root mean square value of prediction errors
***MAPE:** Mean Absolute Percentage Error, measures the average absolute percentage of prediction errors

# 3. Application Examples of PLSR Algorithm in MATLAB

### 3.1 Application of PLSR in Spectral Data Analysis

#### 3.1.1 Pretreatment of Spectral Data

In spectral data analysis, the application of PLSR algorithm is mainly used to extract useful information from spectral data and establish prediction models between spectra and target variables. Before applying the PLSR algorithm, spectral data needs to be preprocessed to remove noise and interference information, ***mon spectral data preprocessing methods include:

- **Standard Normal Variate Transformation (SNV):** Subtracting the mean from each spectral value at each wavelength, then dividing by the standard deviation, to eliminate the effects of spectral intensity differences.
- **Multiplicative Scatter Correction (MSC):** Correcting baseline drift and path length variations in spectra due to scattering through a multiplicative factor.
- **First Derivative and Second Derivative:** Enhancing spectral features and removing background noise by calculating the first or second derivative of the spectrum.

#### 3.1.2 Establishment and Validation of PLSR Model

After spectral data preprocessing, the PLSR model can be established. The `plsregress` function in MATLAB is used to create the PLSR model.

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