Coefficients of Multivariate Polynomial Regression
• polyfitc(X, Y, n/"terms"/M, [conf])—Returns the regression coefficients for a multivariate polynomial regression surface fitting the results recorded in matrix Y to the data found in matrix X. You can define the polynomial regression equation by its polynomial order n or by its terms as specified in the string “terms” or in matrix M. Use matrix M when you do not want to include the intercept in the polynomial fit. Use the optional argument conf to specify a confidence interval other than the default 95%.
The matrix returned by polyfitc has the following columns:
|
Column
|
Description
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|---|---|
|
0
|
Labels for each term reported upon
|
|
1
|
Regression coefficient for each term
|
|
2
|
Standard error for the regression coefficient
|
|
3, 4
|
Lower and upper boundary for the confidence interval of the regression coefficient
|
|
5
|
Variance Inflation Factor—Measure of the inflation of the regression coefficient due to multicollinearity
|
|
6
|
Student’s t test statistic to test if the term is significant
|
|
7
|
P-value—Probability of rejecting the term based on its t-statistic when in fact it is significant
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Arguments
• X is a design matrix or a matrix in which each column represents an independent variable. Each column of X must have compatible units.
• Y is a vector or a matrix of measured or simulated results with each row containing the results for each run or data point defined in X. When the rows do not all contain the same number of replicates, you must pad the empty elements of Y with NaNs. The elements of Y must have compatible units.
• n is an integer specifying the polynomial order. It must be smaller than the total number of data points: 1 ≤ n ≤ length(Y) − 1 . Otherwise, the problem is under constrained with no unique solution.
• “terms” is a string specifying the terms, or the factors and interactions, to include in the polynomial regression. “A B AB AA BB” means that the polynomial contains the following terms:
c0 + c1∙A + c2∙B + c3∙A∙B + c4∙A2 + c5∙B2
For the separators, you can use a space, a comma, a colon, or a semicolon.
• M is a matrix specifying a polynomial with guess values for the coefficients in the first column and the power of the independent variables for each term in the remaining columns. For the polynomial described above, define M as follows:
• conf (optional) is the desired confidence limit, a percentage expressed as a number between 0 and 1, inclusive. By default, conf = 0.95 for a 95% confidence interval.