Determination of start values, parameter limits and possible out- out-liersout-liers

In this study, the fits of the individual histograms are therefore based on pre-defined values for the system noise and offset. These were either given by the simulation truth or determined by application of the error function method to corresponding dark frame jk-grids as presented in sec. 5.2.

With focus on the fit of the “gain”-parameter and the scaling parameters, this section will evaluate the achievable accuracy of the gain determination method.

5.3.6 Determination of start values, parameter limits and possible

5. Calibration methods that can be calculated by

xnoise =µ=^X

i

Xipi (5.29)

withp_i being the bin-wise relative probability of the distribution according to the number of countsCi in each bin, withpi=Ci/P

i

Ci. The system noise or standard deviation σnoise of this distribution is then the square root of the variance:

σ_noise= q

Var(X) = s

X

i

(X_i−µ)²p_i (5.30)

• Perform a Gaussian fit to a range of a predefined number of bins adjacent to the highest bin.

In [35] the advantage of a Gaussian fit with integration over the “basic” calculation of mean and standard of a binned distribution has been studied in detail. The accuracy of both the (Gaussian) fit and the calculation ofµandσare impeded by the characteristic low-energy through between noise peak and calibration lines. Depending on the combination of count rate, system gain and noise one of the three proposed methods is suited best for the determination of the respective start values. For highly irregular spectra (e.g. caused by high DNL), the usage of a predefined default noise value prevents erroneous start values that impede the subsequent process steps. For spectra with higher resolution and less adverse conditions, the results of a Gaussian fit to the noise peak can improve the stability and thereby reduce the run-time of the fit of the global fit function.

An estimate of the start value for the scaling parameter of the noise peak “noise_norm”

(N_noise in table 5.2) is given by the width of the highest bin multiplied by the bin content in units of “counts per LSB”, assuming that in most of the calibration line spectra the noise peak is dominated by a single bin.

2. In the second step, the fit range is determined. The low boundary of the fit range is set to be the lower bin boundary of the first bin below the spectrum that has no entries. As the global fit function is fitted with a likelihood fit that is based on the assumption of Poissonian bin-wise statistics, the information content of the empty bin is correctly taken into account, which would not be the case with standardχ²-fitting [35].

The upper boundary of the fit range is more difficult to determine due to possible “outliers”

from the main spectral distribution, that can be caused e.g. by signal pile-up as can be seen for example in fig. 5.41. The global fit function in the version given in sec. 5.3.2 does not support this spectral feature. Therefore the fit range has to be limited to the main portion of the spectrum. The approach used in this study is illustrated by the large gray arrow in fig. 5.38:

The bins are examined consecutively with decreasing bin number, starting from the last bin that has entries. In each step, the median number of entries of all bins examined so far is calculated. If this median is higher than a certain threshold, e.g. theMAD(eq. A.6 in app. A) of the distribution of the bin entries examined so far, and in addition the next,

lower bin has a higher number of entries than the current bin B_i, the upper boundary of the fit range is set to the upper boundary of the binB_i+1.

3. The start value for the “gain”-parameter is then determined by separating the fit range in two halves (red dashed line in fig. 5.38): With the bin center of the highest binc_h in the upper half and the start value for the offsetx_noise from the first step, the “gain”-parameter start value can be calculated byD=c_h−x_noise.

Taking into account the normalization in eq. 5.18, the start value for the scaling parameter of the calibration line peak (N_i in table 5.2) is calculated byN_i=^√2π·σ_noise·C_Bi with C_Bi being the number of entries in bin c_h.

An example of the resulting fit function with applied start values is given in fig. 5.38 in green. In this example, the start value forσ_noise was given by the simulation truth. All other start values have been determined as described.

For additional stability of the fitting process, limits can be set for the fit parameters. This technique is usually discouraged in the usage of theMINUITfitter [51] due to numerical disadvan-tages but can be motivated by physical constraints: For example the scaling parameter of the calibration lineN_i cannot be negative and should also not be higher than the scaling parameter of the noise peakN_noise. Another motivation for such a constraint is the experience, that the correct result of the “gain”-parameter given by the simulation truth does not deviate more than one or two bin-widths from the start value that places the calibration line peak in the highest bin in the upper half of the fit range. The latter constraint will inhibit fit results in which the

“gain”-parameter “runs off” to very high values locating the calibration line peak out of the defined fit range. This would otherwise occur often, when the ratio of the calibration peak to the trough is low, as illustrated in an example in fig. 5.39.

The setting of parameter limits for the “gain”-parameter comes with the disadvantage, that the fit-based determination of this measure is bound strictly to the region near the highest bin in the upper half of the histogram. If the binning information of the ADC is erroneous, the occurrence of bins whose width is severely underestimated lead to “spikes” in the spectra that will cause a faulty determination of the start value of the “gain”-parameter, as is illustrated in fig. 5.40.

It has to be noted though, that this effect is caused predominantly by the chosen approach for the simulation of erroneous binning information that was used in this study: The “blurring”

of the bin-boundaries (sec. 5.2.5) by a Gaussian distribution can lead to very narrow bin widths due to statistical outliers. It is highly unlikely, that this effect occurs in the real detector. There, the bin-boundary determination algorithm [32] can be based on the assumption that the ADC bin-widths are cyclical (sec. 3.1.4), which prevents overly narrow bins. Up to now, no experimental determination of the binning of DSSC prototypes did result in overly narrow bins [62]. In the presentation of the results of this study the median and the median absolute deviation (MAD, see app. A eq. A.5 to A.7) are used in order minimize the impact of outliers due to overly narrow bins.

5. Calibration methods

24 26 28 30 32 34 36LSB

counts per LSB

1 10 102

103

104

105

results

Entries 563140

Mean 25.96 ±0.0004737 Std Dev 0.3555 ±0.0003349

Underflow 0

Overflow 0

/ ndf

χ2 2.14e+05 / 5

Prob 0

noise_norm 5.555e+05 ±7.480e+02 peak1_norm 660.5 ±29.3

gain 7.39 ±0.53

Figure 5.39: Fit result for a spectrum simulated with ideal binning. The start values (green) have been determined with the proposed algorithm. The parameter for the system noise was given by the simulation truth. In this case, no parameter limits have been set for the “gain”-parameter. The very highχ²value can help identify such a “misfit”. Here, the “gain”-parameter was determined to be 7.39 LSB. The simulation truth would have been 3.85 LSB – a relative deviation of nearly 100%.

40 60 80 100 120 140 LSB

counts per LSB

1 10 102 103 104 105 106

30 40 50 60 70 80 90 100 110 120LSB

counts per LSB

1 10 102 103 104 105 106

Figure 5.40: Illustration of faulty determination of the “gain”-parameter. In the depicted spectra individual bins show a very small width due to a combination of simulated intrinsic DNL with σDNL=0.3 LSB and additional bin-boundary blurring withσblur=0.3 LSB.

If one of these very narrow bins lies in the upper half of the fit range, its bin content, scaled to the bin width, can be very high in comparison to the other bins – even higher than the actualKα-line (green arrows). Due to the start-value algorithm and the parameter limits the fit is then limited to

the near region of this “spike”.