principal component of x and use the orthogonal score vectors in a linear regression with y-variable then do fitlm in matlab

To use the principal components of variable X in a linear regression with a y-variable and to perform the linear regression using fitlm function in MATLAB, you would follow the steps below:

1. Compute the principal components of X using the pca function in MATLAB:

main.m
[coeff, score, ~, ~, explained] = pca(X);
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Here, X is the input matrix (each row represents an observation, and each column represents a variable). The output variables coeff and score represent the principal component loadings and scores, respectively.

2. Select the desired number of principal components based on the explained variance. You can decide how many components to keep based on the amount of variance explained by each component. For example, you can choose to keep the first k components that explain a certain percentage (e.g., 95%) of the total variance. The explained variable from the pca function provides the explained variance for each component.

3. Use the selected principal component score vectors as predictors in a linear regression model.

main.m
X_pca = score(:, 1:k);   % Select the desired number of principal components
mdl = fitlm(X_pca, y);   % Perform linear regression with the principal components and y-variable
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Here, k is the number of principal components to keep, y is the response variable, and mdl is the fitted linear regression model.

Note: It's always a good idea to normalize or standardize your variables before performing principal component analysis and linear regression.

Keep in mind that the interpretation of the principal component loadings and the regression coefficients may differ. The principal component loadings represent the relationships between the original variables and the principal components, while the regression coefficients represent the relationships between the principal components and the response variable.