Message ID: 2578
Entry time: Sun May 31 11:44:20 2020
In reply to: 2577
Reply to this: 2579 2580

Author:

Anchal

Type:

DailyProgress

Category:

NoiseBudget

Subject:

Bayesian Analysis Finalized

I've implemented all the proper analysis norms that Jon suggested and are mentioned in the previous post. Following is the gist of the analysis:

All measurements taken to date are sifted through and the sum of PSD bins between 70 Hz to 600 Hz (excluding 60 Hz harmonics and region between 260 Hz to 290 Hz (Known bad region)) is summed. The least noise measurement is chosen then.

If time-series data is available (which at the moment is available for lowest noise measurement of May 29^{th} taken at 1 am), following is done:

Following steps are repeated for the frequency range 70 Hz to 100 Hz and 100 Hz to 600 Hz with timeSegement values 5s and 0.5s respectively.

The time series data is divided into pieces of length timeSegment with half overlap.

For each timeSegment welch function is run with npersegment equal to length of time series data. So each welch function returns PSD for corresponding timeSegement.

In each array of such PSD, rebining is done by taking median of 5 consecutive frequency bins. This makes the PSD data with bin widths of 1 Hz and 10 Hz respectively.

The PSD data for each segement is then reduced by using only the bins in the frequency range and removing 60 Hz harmonics and the above mentioned bad region.

Logarithm of this welch data is taken.

It was found that this logarithm of PSD data is close to Gaussian distributed with a skewness towards lower values. Since this is logarithm of PSD, it can take both positive and negative values and is a known practice to do to reach to normally distributed data.

A skew-normal distribution is fitted to each frequency bin across different timeSegments.

The fitted parameters of the skew-normal distribution are stored for each frequency bin in a list and passed for further analysis.

Prior distribution of Bulk Loss Angle is taken to be uniform. Shear loss angle is fixed to 5.2 x 10^{-7} from Penn et al..

The Log Likelihood function is calculated in the following manner:

For each frequency bin in the PSD distribution list, the estimated total noise is calculated for the given value of bulk loss angle.

Probability of this total estimated noise is calculated with the skew-normal function fitted for each frequency bin and logarithm is taken.

Each frequency bin is supposed to be independent now since we have rebinned, so the log-likelihood of each frequency bin is added to get total log-likelihood value for that bulk loss angle.

Bayesian probability distribution is calculated from sum of log-likelihood and log-prior distribution.

Maximum of the Bayesian probability distribution is taken as the most likely estimate.

The upper and lower limits are calculated by going away from most likely estimate in equal amounts on both sides until 90% of the Bayesian probability is covered.

Final result of CTN experiment as of now:

with shear loss angle taken from Penn et al. which is 5.2 x 10^{-7}. The limits are 90% confidence interval.

The analysis is attached. This result will be displayed in upcoming DAMOP conference and would be updated in paper if any lower measurement is made.