So, I want to simulate data based on given means, standard deviation and correlation. What i have done thus far is random simulation of the three variables based on normal distribution, means and standard deviation. For the correlation I'm trying to use Eigenvectors / Eigenvalues, as i have understood (maybe wrong?) that this is an alternative to the Cholesky Method. However, when i simulate the data (50 000 + datapoints per series) two of the three correlations are within acceptable range (ie. 0.175 vs. 0.16 and 0.22 vs 0.24) but the last one is way off (0.44 vs 0.58). I have run the simulation many times and the values fluctuates around these numbers. What i have done:
Means and Std.Dev
| | A | B | C |
|---------|-------:|------:|------:|
| Mean | 0.045 | 0.07 | 0.095 |
| Std.Dev | 0.0155 | 0.128 | 0.131 |
Correlation Matrix:
| | A | B | C |
|---|-----:|-----:|-----:|
| A | 1 | 0.16 | 0.24 |
| B | 0.16 | 1 | 0.58 |
| C | 0.24 | 0.58 | 1 |
Eigenvectors / Eigenvalues
| E1 | E2 | E3 |
|-------:|-------:|------:|
| 0.105 | 0.920 | 0.378 |
| 0.685 | -0.343 | 0.643 |
| -0.721 | -0.192 | 0.666 |
| λ1 | λ2 | λ3 |
|------:|-------:|------:|
| 0.418 | 0.8904 | 0.378 |
$$V = E_{i}*Diag(\sqrt{\lambda _{i}})$$
And then used $$R_{C} = R*V^{T}$$ Where $R_{C}$ is the correlated random numbers and $R$ is the uncorrelated random numbers. I can't seem to get it right. Anyone?
Edit: I have also tried using the Cholesky Method but it gives the same results but slightly worse. I was also under the impression that the Eigensystem is better as the Cholesky Method needs a PD correlation Matrix. Since I want to expand later on that may not be the case anymore (PD that is).