In the electronic measurement of the Boltzmann constant based on Johnson noise thermometry, the ratio of the power spectral densities of thermal noise across a resistor at the triple point of water, and pseudo-random noise synthetically generated by a quantum-accurate voltage-noise source is constant to within 1 part in a billion for frequencies up to 1 GHz. Given knowledge of this ratio, and the values of other parameters that are known or measured, one can determine the Boltzmann constant. Due, in part, to mismatch between transmission lines, the experimental ratio spectrum varies with frequency. We model this spectrum as an even polynomial function of frequency where the constant term in the polynomial determines the Boltzmann constant. When determining this constant (offset) from experimental data, the assumed complexity of the ratio spectrum model and the maximum frequency analyzed (fitting bandwidth) dramatically affects results. Here, we select the complexity of the model by cross-validation - a data-driven statistical learning method. For each of many fitting bandwidths, we determine the component of uncertainty of the offset term that accounts for random and systematic effects associated with imperfect knowledge of model complexity. We select the fitting bandwidth that minimizes this uncertainty. In the most recent measurement of the Boltzmann constant, results were determined, in part, by application of an earlier version of the method described here. Here, we extend the earlier analysis by considering a broader range of fitting bandwidths and quantify an additional component of uncertainty that accounts for imperfect performance of our fitting bandwidth selection method. For idealized simulated data with additive noise similar to experimental data, our method correctly selects the true complexity of the ratio spectrum model for all cases considered. A new analysis of data from the recent experiment yields evidence for a temporal trend in the offset parameters.

中文翻译：

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约翰逊噪声测温法电子测量玻尔兹曼常数的光谱模型选择

在基于约翰逊噪声测温法的玻尔兹曼常数电子测量中，水三相点处电阻两端的热噪声功率谱密度与由量子精确电压噪声源合成的伪随机噪声之比对于高达 1 GHz 的频率，恒定在十亿分之一以内。已知这一比率，以及已知或测量的其他参数的值，就可以确定玻尔兹曼常数。部分由于传输线之间的失配，实验比率谱随频率变化。我们将此频谱建模为频率的偶数多项式函数，其中多项式中的常数项决定了玻尔兹曼常数。当从实验数据确定这个常数（偏移量）时，比率谱模型的假设复杂性和分析的最大频率（拟合带宽）极大地影响了结果。在这里，我们通过交叉验证来选择模型的复杂度——一种数据驱动的统计学习方法。对于许多拟合带宽中的每一个，我们确定偏移项的不确定性分量，该分量解释了与模型复杂性的不完善知识相关的随机和系统效应。我们选择最小化这种不确定性的拟合带宽。在最近的玻尔兹曼常数测量中，部分结果是通过应用这里描述的方法的早期版本来确定的。这里，我们通过考虑更广泛的拟合带宽来扩展先前的分析，并量化导致我们的拟合带宽选择方法不完美的不确定性的额外分量。对于具有与实验数据相似的加性噪声​​的理想化模拟数据，我们的方法正确选择了所有考虑情况下的比率谱模型的真实复杂性。对最近实验数据的新分析为偏移参数的时间趋势提供了证据。