Issues-with-covaraince-matrix.md

@@ -51,7 +51,7 @@ The spectrum of the eigenvalues is also plotted.
 
 <img src="uploads/eca6880d8670d2050f72374008fd8d24/beta_lambda_all_2_.png"  width="440" > 
 
-We can see that the median $`\beta`$ (black trace) saturates after a certain number of the top basis vectors have been used to approximate the covariance matrix.  If we use the top 250 basis vectors (rank = 250, instead of the full rank), the Frobenius norm of the residual is $`1.4\times 10^{-8}`$, indicating the extent to which we capture the features in the covariance matrix. The blue trace is for the relative magnitude of the eigenvalues. If we include the eigenvectors with relative importance less than few times 1e-10, you can see that the variance of the $`\beta`$ values starts flaring up again due to errors in the inverse of the covariance matrix. So our suggestion is to use top-p = 250 eigenvectors for the pseudo inverse of the covariance matrix for this event. We feel that this should be a tunable number based on plots like this.
+We can see that the median $`\beta`$ (black trace) saturates after a certain number of the top basis vectors have been used to approximate the covariance matrix.  If we use the top 150 basis vectors (rank = 150, instead of the full rank), the Frobenius norm of the residual is $`6.9\times 10^{-8}`$, indicating the extent to which we capture the features in the covariance matrix. The blue trace is for the relative magnitude of the eigenvalues. If we include the eigenvectors with relative importance less than few times $`3.3 \times 10^{-9}`$, you can see that the variance of the $`\beta`$ values starts flaring up again due to errors in the inverse of the covariance matrix. So our suggestion is to use top-p = 150 eigenvectors for the pseudo inverse of the covariance matrix for this event. We feel that this should be a tunable number based on plots like this.
 
 ## GW190412
 
@@ -65,4 +65,8 @@ A similar plot for the GW190412 event (top 5 [PE samples](https://git.ligo.org/p
 | 31.1000 | 10.2668 | 0.0572   |  0.6907 |  -48630.0520   | 6.7  |
 | 33.7223 | 9.7035  | 0.3080   | -0.0899 |  -48630.0727   | 7.0  |
 
-<img src="uploads/b950108077a19ca098f0b79b9949e030/beta_lambda_all.png"  width="440" > 
+<img src="uploads/c7df73e51383e0485310fe22cd0f536c/beta_lambda_all_3_.png"  width="440" > 
\ No newline at end of file
+
+The effective-rank of the covariance matrix needs to be tuned for each event using a diagnostic plot such as those shown above. 
+
+