Bias on r from Band Center Errors
Steve Palladino, Justin Willmert, Colin Bischoff -- 2017-09-13
In this posting, I looked at the systematic bias on r from band center errors using the data challenge 02.00 which has nine observing frequencies. The simulated maps were processed through to BB bandpowers by Justin. First, I applied a quadratic estimator for r to these bandpowers with 0% band center error. I then held each of the band center errors at 0% except for allowing one observing frequency band center to vary from -3% to 3% (Figure 1) while subtracting off estimate of r from the 0% case (let’s call this r0). As can be seen, the r bias appears to be quadratic and asymmetric about the 0% band center error. It is also interesting to note that the largest biases on r come from errors in the 20 GHz and 270 GHz band centers.
Figure 1
I next looked at how the r bias changed with allowing all the band center errors to vary. I looked at two cases of band center error (uncorrelated and correlated). Again, I constructed a quadratic estimator for r using the case with 0% band center error which gave an estimate for r (r0) and applied this estimator to the bandpowers. For the uncorrelated case, I used an uncorrelated nine-dimensional Gaussian distribution centered about 0 with a σ of 1% for the band center errors with 10,000 realizations. From these sets of 9 Gaussian random band centers, I calculated the bandpower expectation values, applied the quadratic estimator, subtracted off r0 and took the magnitude. This gave a distribution, Figure 2, for the r bias from which I calculated <r> and <|r|>.
Figure 2
Lensing | Mean r | Std r | Mean mag(r) | Std mag(r) |
---|---|---|---|---|
AL=1.0 | 1.49e-4 | 2.05e-4 | 1.81e-4 | 1.77e-4 |
AL=0.3 | 1.35e-4 | 1.61e-4 | 1.48e-4 | 1.50e-4 |
AL=0.1 | 1.18e-4 | 1.34e-4 | 1.24e-4 | 1.28e-4 |
AL=0.03 | 1.05e-4 | 1.19e-4 | 1.10e-4 | 1.14e-4 |
I repeated this process using a band center σ of 0.5%, 1.5%, and 2%. The mean of the magnitude of these distributions are shown in Figure 3 below. Also seen in Figure 3 is a quadratic fit to the band center errors.
Figure 3
Band Center Error | Mean (AL=1.0) | Mean (AL=0.3) | Mean (AL=0.1) | Mean (AL=0.03) |
---|---|---|---|---|
0.5% | 0.64e-4 | 0.45e-4 | 0.36e-4 | 0.31e-4 |
1% | 1.81e-4 | 1.48e-4 | 1.24e-4 | 1.10e-4 |
1.5% | 3.70e-4 | 3.18e-4 | 2.73e-4 | 2.41e-4 |
2% | 6.35e-4 | 5.58e-4 | 4.82e-4 | 4.27e-4 |
Benchmark ( r = 1e-4 ) | 0.70% | 0.81% | 0.89% | 0.95% |
For the correlated band center error case, I varied the band center errors for all of nine observing frequencies simultaneously and by the same amount. From here, it was a similar procedure of calculating the bias on r; I calculated the bandpower expectation values, applied the quadratic estimator, subtracted off r0 and took the magnitude. The <|r|> bias can be seen in Figure 4 below with a quadratic fit.
Figure 4
Band Center Error | Mean (AL=1.0) | Mean (AL=0.3) | Mean (AL=0.1) | Mean (AL=0.03) |
---|---|---|---|---|
0.5% | 0.15e-4 | 0.11e-4 | 0.09e-4 | 0.08e-4 |
1% | 0.44e-4 | 0.35e-4 | 0.29e-4 | 0.25e-4 |
1.5% | 0.87e-4 | 0.72e-4 | 0.61e-4 | 0.54e-4 |
2% | 1.46e-4 | 1.23e-4 | 1.05e-4 | 0.94e-4 |
Benchmark ( r = 1e-4 ) | -2.61%, 1.55% | -2.58%, 1.72% | -2.65%, 1.89% | -2.74%, 2.03% |
The bias on r for the correlated case also seems to scale as a quadratic and is asymmetric about the 0% band center error. However, these biases are less significant when compared to the uncorrelated case.