sphered f-statistic: 0 warning: regression design matrix is rank deficient to within machine precision. in matlab

This warning indicates that the design matrix used in the regression analysis is rank deficient, meaning that some of the predictor variables are linearly dependent on each other. As a result, it can cause numerical instability and affect the accuracy of the F-statistic calculation.

To address this issue, you can use a technique called "sphering" to orthogonalize the predictor variables and remove the linear dependency. Here's how you can do it in MATLAB:

1. Identify the set of linearly dependent predictor variables in your design matrix. You can use MATLAB's rank function to check the rank of the matrix.

main.m
rank(X)
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Replace X with your design matrix.

2. Once you've identified the linearly dependent variables, create a new design matrix that includes only the linearly independent variables. You can use MATLAB's rref function to obtain a reduced row-echelon form of the matrix and then select the independent rows.

main.m
rref_X = rref(X);
new_X = X(any(rref_X, 2), :);
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3. Now, you can calculate the F-statistic using the new_design matrix.

main.m
[p_value, f_statistic] = f_test(new_X, Y);
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Replace Y with your response variable, and f_test is a function that calculates the F-statistic. Depending on your specific regression model, you may need to adjust the code accordingly.

This procedure will help avoid the rank deficiency issue and provide a valid F-statistic for your regression analysis.