Determinant of auto correlation matrix

[vaibhavtewari]

Hello everyone, I have a question..

if I have a data stream
X=some thousand random numbers with mean close to 0 and standard deviation close to 1 and then I construct my autocorrelation matrix from these numbers

Q=|R_xx{0} R_xx{1}...R_xx{L}|
| R_xx{1} R_xx{0} ...R_xx{L-1}|
|.........|
|R_xx{L}......R_xx{0}|

I normalize the matrix(getting 1 on diagonal) and calculate its determinant. As I am using a gaussian random number distribution for x, in theory I would get 0 for off-diagonal terms and 1 for diagonal terms. But in reality I have 1 on diagonal(because of normalization) and very small numbers for off-diagonal. The determinant turns out to be less than 1. Also the determinant is dependent on L, for small L its close to 1, as I increase L determinant starts decreasing. My question is is there a way I can normalize the determinant of this matrix so that I always get 1 for uncorrelated data ?

Thank You for reading this long description.

 

[mmwave]

Hello, thank you for your question. It is interesting that the determinant of your autocorrelation matrix is less than 1, even though you are using a Gaussian random number distribution with a mean close to 0 and standard deviation close to 1. This suggests that there may be some correlation present in your data, even though it may not be obvious from the mean and standard deviation.

To answer your question, there is not a way to normalize the determinant of the autocorrelation matrix to always equal 1 for uncorrelated data. The determinant is a measure of the overall correlation present in the data, and if there is any correlation at all, the determinant will be less than 1.

However, you can compare the determinant of your autocorrelation matrix to the determinant of an uncorrelated data set with the same mean and standard deviation. If the determinant of your autocorrelation matrix is significantly smaller, it suggests that there is some correlation present in your data.

Additionally, you can use other statistical tests, such as the Pearson correlation coefficient, to further investigate the correlation in your data. I would recommend consulting with a statistician for further guidance on analyzing and interpreting your data.

I hope this helps answer your question. Best of luck with your research.