Home.md

@@ -37,7 +37,7 @@ This is the home page for the review of the detection of non-quadrupole modes us
 *  [Nathan K. Johnson-McDaniel](https://git.ligo.org/nathan-johnson-mcdaniel) 
 
 
-## Review statements
+# Review statements
 
 We have reviewed the implementation of 'search of higher modes in the gravitational wave strain assuming the detection of 22 mode' using the time-frequency map of the data. The method was introduced in (https://arxiv.org/abs/1910.04565) and presented in [R&D call 2019-08-19](https://dcc.ligo.org/G1901496). The brief summary of analysis can be found [here](https://git.ligo.org/soumen.roy/o3inspiralhom/-/wikis/Home):
 
@@ -61,7 +61,7 @@ The python implementation of the analysis can be found [here](https://git.ligo.o
 
 We performed a set of investigations on the analysis to check its accuracy.  Below is the summary of investigations performed to test the pipeline.
 
-# Investigations for code review
+## Investigations for code review
 
 | Reviewer| Task  | Status | Link to review material | Persons Responsible | Comment | final sign-off |
 | ------ | ------ |--------|-------------------------|---------------------|---------------|----------------|
@@ -71,7 +71,7 @@ We performed a set of investigations on the analysis to check its accuracy.  Bel
 | [Haris K](https://git.ligo.org/haris-k), [Nathan K. Johnson-McDaniel](https://git.ligo.org/nathan-johnson-mcdaniel) | Study on $`p-`$value  |  Done |      [Link](https://git.ligo.org/soumen.roy/o3inspiralhom/-/wikis/Study-on-p-value) | Soumen  |  Extrapolation of $`P(\beta\|H_0)`$ doesn't seem to be a good idea. Will use background samples to quote an upper bound on *p*-value |  :heavy_check_mark: |
 
 
-# GW190412
+## GW190412
 | Script | Short description | Status | git hash | Comment | final sign-off |
 | ------ | ----------------- | ------ | -------- | ------- | -------------- |
 | [SEOBNRv4HM_test_insphom_GW190412_C01_60Hz_Fs4096.sh](https://git.ligo.org/soumen.roy/o3inspiralhom/-/blob/master/GW190412/analysis/C01_60HzCut/SEOBNRv4HM/test_insphom_GW190412_C01_60Hz_Fs4096.sh) | Bash script to produce results (h5 file) for the event GW190412 from SEOBNRv4HM PE result | Done | c7caff8ea86c90fbbb6ebef932bcfa18c9acac7a |  In future, the choice of `track-max-length` can be optimized to get maximum SNR. | :heavy_check_mark: |
@@ -95,7 +95,7 @@ Below is the summary of investigations to test the S190412m results.
  
 
 
-# GW190814
+## GW190814
 
 | Script | Short description | Status | git hash | Comment | final sign-off |
 | ------ | ----------------- | ------ | -------- | ------- | -------------- |
@@ -113,7 +113,7 @@ Investigations for GW190814 result  review
 | [Haris K](https://git.ligo.org/haris-k), [Nathan K. Johnson-McDaniel](https://git.ligo.org/nathan-johnson-mcdaniel) | Analysis of GW190814 with PE samples as templates. For this analysis data quality flags are not applied.  |  Done |      [Link](https://git.ligo.org/soumen.roy/o3inspiralhom/-/wikis/Distribution-of-on-source-beta) |    Soumen |  This analysis is done without data quality flags. |   :heavy_check_mark:          |
 | [Haris K](https://git.ligo.org/haris-k), [Nathan K. Johnson-McDaniel](https://git.ligo.org/nathan-johnson-mcdaniel) | Injection study for GW190814 event using C00 data   |  Done |      [Link](https://git.ligo.org/soumen.roy/o3inspiralhom/wikis/injection-Study-for-S190814bv-event) |    Soumen |   This exercise is done on C01 data with the final version of code. We note that the event $`\beta`$ value is on the higher tail of the distribution of $`\beta`$s obtained from the injection campaign. This might come from using the same maxL PE results regardless of the background noise. We suggest investigating this in the future. Further, we observe that the distribution of $\beta$ for h_22+h_33 injections and h_33(residual) injections are consistent.  |   :heavy_check_mark: |
 
-## Updated results:
+# **Updated results:**
   After the sign-off of the results, we identified a bug in the analysis.  The $\beta$ value computed in time-frequency track analysis seems to be highly sensitive to very small changes in the masses and spins. Unfortunately, we didn't notice it earlier (though we knew already that the distribution of $\beta_33$  computed using a subset of PE samples as templates is not very narrow). Soumen and Anand did some investigation to understand the problem.  The noise covariance matrix $\sigma(\alpha,\alpha')$ computed from the ensemble of noise vectors $N(\alpha)$ is rank deficient (ill-conditioned). This leads to an unstable inverse matrix (see Supplementary Material:B of https://arxiv.org/pdf/1910.04565.pdf). Replacing the normal inverse function with the pseudo inverse (Penrose Moore pseudo inverse) seems to have solved the problem. The summary of the investigations and fixes are described in [link](https://git.ligo.org/soumen.roy/o3inspiralhom/-/wikis/Issues-with-covaraince-matrix).
 
 To fix the bug the following scripts are added/updated in the pipeline. 
@@ -124,11 +124,11 @@ To fix the bug the following scripts are added/updated in the pipeline.
 | [read_insphom_results.py](https://git.ligo.org/soumen.roy/o3inspiralhom/-/blob/master/scripts/read_insphom_results.py) | Script for making plots and displaying results. | Done| 67846be99ba1a8fa7501be1f659b8f7dd6c0c5c7 |  |  :X: |
 | [covmat_rank_analysis.py](https://git.ligo.org/soumen.roy/o3inspiralhom/-/blob/master/scripts/covmat_rank_analysis.py) | Stand-alone python script to obtain final results after fixing the covariance matrix inverse | Done | 67846be99ba1a8fa7501be1f659b8f7dd6c0c5c7 |  |  :X: |
 
-# GW190412
+## GW190412
 
 
 
-# GW190814
+## GW190814
 
 | Script | Short description | Status | git hash | Comment | final sign-off |
 | ------ | ----------------- | ------ | -------- | ------- | -------------- |