Abstract
The nonlinear energy response of cryogenic microcalorimeters is usually corrected through an empirical calibration. X-ray or gamma-ray emission lines of known shape and energy anchor a smooth function that generalizes the calibration data and converts detector measurements to energies. We argue that this function should be an approximating spline. The theory of Gaussian process regression makes a case for this functional form. It also provides an important benefit previously absent from our calibration method: a quantitative uncertainty estimate for the calibrated energies, with lower uncertainty near the best-constrained calibration points.
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Notes
When the data can be exactly interpolated by a line, that line is found for any value of \(\lambda\).
Defining curvature as the integral of the kth derivative squared yields [6] splines of degree (\(2k-1\)).
Estimates of the uncertainty on the measurements are also required, for which we use the simplest possible model: that the noise is independent and Gaussian-distributed with mean zero and variance \(\sigma _i^2\).
Here the covariance is simplified by assuming the domain is transformed to \([\mathrm {min}\,x_i,\mathrm {max}\,x_i]=[0,1]\).
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Acknowledgements
This work was supported by NIST’s Innovations in Measurement Science program. We thank Dan Becker, Michael Frey, and two anonymous reviewers for many helpful suggestions. The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Fowler, J.W., Alpert, B.K., O’Neil, G.C. et al. Energy Calibration of Nonlinear Microcalorimeters with Uncertainty Estimates from Gaussian Process Regression. J Low Temp Phys 209, 1047–1054 (2022). https://doi.org/10.1007/s10909-022-02740-w
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DOI: https://doi.org/10.1007/s10909-022-02740-w