Stability of calibrations and determination of global calibration values

For each of the calibration parameters, values for each individual run can be determined. Variation of these values was relatively small over the course of the experiment. To reduce the analytical complexity introduced by individual-run calibration values, single values for the calibrations were adopted when aggre- gating the experimental data. Representative values for light yield at the241_{Am peak and the integral of SPE}

signals were found to be 29.88±0.39 PE/keVee and 1502+22₋₂₆ ADC/SPE, respectively, and the distribution of the values as a function of time are shown in Figs. 4.11 and 4.12. Subsequent discussion focuses on the method by which these global values are determined.

To determine a representative “global” value for calibration parameters, consideration of the philosophy behind such an approach is important. The fundamental question is: what single value and error for, or (similarly) distribution in, a given parameter accurately describes the individual observed values taken in aggregate? A global value or distribution serving this end will necessarily have larger errors, or be wider,

2 − 10 1 − 10 1 10 2 10 3 10 4 10 Events / ( 100 ) Total model Pedestal Noise Single PE Double PE Triple PE 4PE 5PE 0 2000 4000 6000 8000 10000 12000

Scatterer Integral (ADC units)

150 − 100 − 50 − 0 50 100 150 200 Residual percent / ( 100 )

Figure 4.10: The distribution of integrals of the CsI[Na] waveform over the signal region, shown in ADC units, for a single hour-long run zoomed in to the few-photoelectron level. The data is fit with a model consisting of an additive combination of: Gaussian pedestal; exponential noise convolved with the pedestal; and Gamma-distributed models of nPE populations for n = 1, . . . ,5. A shaded region around the total model, shown in hot pink, indicates the 1σuncertainty band of the fit. The bottom panel shows the residual distribution of the fit, in percent of each data point, with a dashed gray line drawn at 0 residual. The model does not accurately fit a low-integral feature of the pedestal but describes the data well over the broad region with ample statistics. For a more complete description of the fitting procedure see Section 4.3.3 of the text.

04 Feb 00:0004 Feb 06:0004 Feb 12:0004 Feb 18:0005 Feb 00:0005 Feb 06:0005 Feb 12:0005 Feb 18:0006 Feb 00:0006 Feb 06:00 Run date 28.5 29.0 29.5 30.0 30.5 31.0 31.5

PE yield at Am peak (PE/keVee)

0.000.250.500.751.00

Intensity (a.u.)

Figure 4.11: Measured “yield” for the CsI[Na] detector used in the QF measurements, in integrated digitizer units, for the full-energy peak of 241Am, E = 59.54 keV. Source measurements were taken at several times through the experiment. The panel at right shows a kernel density estimate of the “global” value for the calibration parameter, determined using the method described in 4.3.4, finding a value of 29.88± 0.39 PE/keVee.

than the errors on individual data points.

Temptation to perform a fit through the measured parameter values should be avoided. Though the result of a fit of a constant value through a series of parameter data will indeed yield a single value with (speciously) appropriate error and quite probably acceptable results from any chosen goodness of fit tests, the underlying question that is answered by a fit is different and not compatible with the goal of determining a global, representative parameter value. Interpreted through the more-intuitive Bayesian view of probabilities and statistics, a fit responds to a scenario posed thusly: a single, “true” value of the parameter exists and the data are the result of different measurements of this single value; given the data, what is the underlying value of the parameter? With this philosophical underpinning, it follows naturally that the fitted value of a parameter will generally have smaller error than the individual data points2.

To establish global parameter values which more accurately reflect the real variation of calibration values reflected in the observed data, an approach similar to a Gaussian kernel density estimate is used [231]. For each individual calibration parameter data pointβi with error δi, where i = 1, . . . , N with N the number

of calibration values measured, 1000 samples are drawn from a Gaussian PDF with µ = βi and σ = δi.

The collection of N*1000 values are then taken to provide a non-parametric model of the desired “global” distribution of the parameter.

To utilize these global calibration parameter values in the analysis chain, we seek simplified representa-

2_{In the context of distributions or Bayesian approaches, the result of a fit will yield a more narrow posterior distribution on}

04 Feb 00:0004 Feb 06:0004 Feb 12:0004 Feb 18:0005 Feb 00:0005 Feb 06:0005 Feb 12:0005 Feb 18:0006 Feb 00:0006 Feb 06:00 Run date 1420 1440 1460 1480 1500 1520 1540 1560 1580

SPE mean calibration (ADC/SPE)

0.00 0.05

Intensity (a.u.)

Figure 4.12: Average integrated charge (ADC units) per single photoelectron (SPE) for each hour-long run in the CsI[Na] QF experiment. The data analyzed to inform these calibrations is drawn from the integrals of the signal regions in waveforms from each individual run. The distribution of the “global” calibration parameter, found to be 1502+22₋₂₆ ADC/SPE, is shown at right with the mean and a band corresponding to the central 68% of the distribution indicated. For a full discussion of the procedure for determining the SPE charge, see Section 4.3.3 of the text; for a discussion of the production of the global distribution, see Section 4.3.4.

tions of these distributions in the form of a single value with asymmetric errors,

β=βµ +δ+

−δ−. (4.3)

We take βµ to be given by the mean of the distribution, δ− to be the value below which 16% of the distribution falls, and δ+ to be the value above which 16% of the distribution falls3. The distributions of

“global” parameters, along with central values and central 68% confidence intervals, for the energy calibration and mean charge of SPEs can be seen on the right panels of Figures 4.11 and 4.12, respectively.

Note that the SPE calibration value used in subsequent analysis is 1486+20₋₁₈ADC/SPE, corresponding to a value derived from an earlier, less-constrained fit to the SPE shape. This is a∼1% difference from the value derived using the model like that shown in Fig. 4.10, and the calibration values agree within uncertainty.

3_{In a fully Bayesian analysis chain, the aggregate numerical distributions resulting from Gaussian sampling of each data point}