Please excuse if my use of terms is not correct. I have only very basic theoretical background in statistics. I tried to search for my answer here on stackexchange and also in other places with no success. In fact I don't know exactly for what terms to search for.

First I will explain the task I am currently solving. I have some measured data which depends on time variable. So time "t" is independent variable, and the measured data is dependent variable. I am trying to find an appropriate mathematical model to describe these data as accurately as possible. As mathematical method I have choosen linear regression. My software is written to be as universal as possible but my current model is as follows:

Y = b1 * 1 + b2 * t + b3 * t^2 + b4 * sin(wt) + b5 * cos(wt)

where Y is dependent variable (measured data), t (time) is independent variable and w (greek letter omega) = 2 * PI (greek letter pi) * f (frequency) is circular frequency.

"w" is parameter in model. In every iteration of computation I use another replacement value (constant) instead of it. I am able to compute values of regression coefficients b1 to b5. As method of computation I choosed Singular Value Decomposition (or SVD) implemented in LAPACK software package. My problem is related to computation of p-values for particular regression coefficients. I want to know if some part of model (for example linear relationship represented by "b2" coefficient) is really contributing to it in significant degree. I don't want to use some theoretical formulas because they have some requirements bind to them (for example residuals should have normal distribution described by bell curve). Therefore I decided to use permutation test for calculating p-values. The test statistic used in permutation test is partial correlation coefficient. The exact definition is described in this article:


Basically the partial correlation coefficient (or square of it if we want to use two tailed test) is Pearson correlation coefficient between two residuals:

r^2 = (sum(Ryx * Rzx))^2 / (sum(Ryx^2)*sum(Rzx^2))

where Ryx are residuals when approximating measured data by reduced set of functions in set (let it be set containing functions "1", "t", "sin(wt)" and "cos(wt)" - "t^2" is missing because we are investigating what influence have the addition of "t^2" term to the model) and Rzx are residuals when approximating "t^2" function by a set of other functions in model (i.e. "1", "t", "sin(wt)" and "cos(wt)"). So far so good. What I don't is how to apply this method when considering periodic component in model. I need to get one joined p-value for both "sin" and "cos" component. I am not interested in knowing how much significant is one these components. I need to know answer to question "is periodic (cyclic) component in signal significant and to which degree" ? Should I calculate partial correlation coefficients for both "sin" and "cos" component separately and then "join" them using some formula ? Or should I calculate from beginning one joined partial correlation coefficient including both "sin" and "cos" component ? But how to do that ? Its no problem to calculate Ryx in above term (It is sufficient to approximate measured data only by terms "1", "t" and "t^2" something like this Y = b'1 * 1 + b'2 * t + b'3 * t^2 + Ryx, calculate regression coefficients b'1, b'2 and b'3 and then calculating residuals by subtracting part explained by model from measured data), but how to calculate value Rzx ? What would you recommend ? Thanks for your patience to reading to this place and for your answers.


1 Answer 1


After studying some chapters from book "Applied Regression Analysis and Other Multivariable Methods" by David G. Kleinbaum (Author), Lawrence L. Kupper (Author), Azhar Nizam (Author) and Eli S. Rosenberg (Author) (pages 199 - 214) I understood that what I need to calculate is named multiple partial correlation coefficient. Multiple partial correlation coefficient can be calculated by means of residuals sum of squares as follows:

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The value of squared multiple partial correlation coefficient lies between 0 and 1. Two extreme cases are as following:

r2 = 0 means that Residual SS (smaller model) = Residual SS (larger model) i.e. adding sin, cos terms to model doesn't lower energy of residual function.

r2 = 1 means that residual SS (larger model) = 0 i.e. new model completely describes measured data.

If some pages are not available in google book service you can use another browser. I was successfull by using combination of Google Chrome and Firefox. The pages that were not available in Google Chrome I can access in Firefox and vice versa.


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