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I am trying to analyze how different independent variables are contributing to a dependent variable. I want to understand how important each independent variable is to the output. Is there any statistical processes and matlab function can do that? possibly compute the percentage of the variance of the dependent variable is explained by the other independent variables.
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If you have the latest version of the Statistics Toolbox, then it sounds like you want the anova method of LinearModel.
Unfortunately, the problem as you described it isn't uniquely determined. That's why there are various "types" of sums of squares in anova. For example, if x1 and x2 are highly correlated with each other and with y, then it could turn out that each x variable is important individually, but once you have either one, the other is less important.
I'm assuming you already have data. If not, and you have the opportunity to collect data, then you may be able to make the x variables orthogonal so that you can accurately attribute variance percentages to each x variable.