Question3: Systematic Uncertainties

We fit a power-law function to the corrected and uncorrected spectra, to allow us to estimate the effects of the two corrections from Question 1 & Question 2

Power Law Function

Power Law Function

We use R’s fit_power_law function from the igraph package, to estimate the alpha value for the uncorrected spectrum:

continuous alpha xmin logLik KS.stat KS.p
1 TRUE 1.265252 0.00055 21.30521 0.1609785 0.8891033

An explanation of the parameters obtained:

Continuous

Indicates whether the power law that was fit was continuous, in this case it was.

Alpha

This is the estimate for ‘a’.

Xmin

The minimum x value for which the power law was fitted

LogLik

The log-likelihood of the parameters

KS.Stat

A test statistic from a Kolmogorov-Smirnov test. Smaller scores denote better fit.

KS.p

The p-value of the above test. A value of p < 0.5 would indicate that the test rejects the null hypothesis that original data points could have been sampled from this power-law distribution

Next, we need to fit a power law function to a corrected spectrum, to compare the goodness of fit of the model

We consider the spectrum corrected for momentum resolution and detector efficiency, from question 1

continuous alpha xmin logLik KS.stat KS.p
1 TRUE 1.365833 0.0068458 5.279106 0.1615924 0.9728557

According to this estimate the alpha value for the corrected distribution seems more likely to be from the expected power-law distribution:

  • Smaller KS-stat value (better fit)
  • Larger KS.p value (more likely to come from estimated power-law distribution)

We need a bit more to go on, in order to quantify the uncertainty in our estimate

The more comprehensive poweRlaw package allows us to investigating the uncertainty of our alpha estimates

Fitting a powerlaw function to the original spectrum, can be assessed, using the plots produced below

Note that the estimate for alpha using this algorithm on the ORIGINAL SPECTRUM is alpha = 1.287356

## [1] 1.287356

## [1] "LET'S ANALYZE THE UNCERTAINTY OF OUR ALPHA ESTIMATE FOR THIS SPECTRUM:"
## [1] "1: The standard deviation of parameter uncertainty for alpha is: 0.207707365151886"
## [1] "Let's look at the 95% confidence intervals for the mean estimates of the parameters"
## [1] "AS well as the 95% CI for the standard deviation of the parameters"
## [1] "In the 4 plots shown below:"

## [1] "The p value for whether this sistribution follows a power-law is: 0.311"
## [1] "NOTE that a p-value of p < 0.05 would indicate that this distributon is likely not a power-law function"

We do the same for the spectrum corrected for momentum-spectrum and detector efficiency:

Note that the estimate for alpha using this algorithm on the SPECTRUM CORRECTED FOR DETECTOR EFFECTS is alpha = 1.296308

## [1] 1.411562

## [1] "LET'S ANALYZE THE UNCERTAINTY OF OUR ALPHA ESTIMATE FOR THIS SPECTRUM:"
## [1] "1: The standard deviation of parameter uncertainty for alpha is: 0.20873806867068"
## [1] "Let's look at the 95% confidence intervals for the mean estimates of the parameters"
## [1] "AS well as the 95% CI for the standard deviation of the parameters"
## [1] "In the 4 plots shown below:"

## [1] "The p value for whether this sistribution follows a power-law is: 0.486"
## [1] "NOTE that a p-value of p < 0.05 would indicate that this distributon is likely not a power-law function"

Lastly, we look at the spectrum corrected for bin-width effects:

Because only the x-values were shifted, we end up with the same range of y-values as the original spectrum, and therefore the same alpha value

My best estimate for alpha:

For both spectra, we cannot rule out the null hypothesis that the data was generated by a power-law process

From the second algorithm’s analysis of the corrected spectrum, an alpha of:

alpha = 1.29 +/- 0.22 is my best guess ###This is proportionally more uncertainty than was estimated for the original spectrum:

Divding the uncertainty by the estimate for the corrected spectrum, gives:

## [1] 0.1705426

And for the original spectrum, a mean value is obtained as follows:

## [1] 0.0617087

Final comment: This difference in stated uncertainty: 6% uncertainty in the original spectrum vs. 17% uncertainty in the corrected spectrum, is also influenced by what one defines as the function to estimate the uncertainty with…