# partial correlation coefficient for a set of random variables

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. 