Hiroshi Sato, Kensei Ehara
DETERMINATION OF UNCERTAINTY ASSOCIATED WITH QUANTIZATION ERRORS USING THE BAYESIAN APPROACH
In practice, quantization of a measured quantity often significantly influences observation values. A typical example is found in measurements using digital instruments. In some cases, due to the quantization, no dispersion is observed among repeated measurements. The type A evaluation then gives zero standard uncertainty. In such a case, the most common practice is to assume, as an a priori distribution in type B evaluation, a uniform distribution, the width of which is given by the quantization interval, and take the width divided by square root of 12 as the standard uncertainty.
This practice, however, is justified only when the population standard deviation is exactly zero. But generally this condition does not hold true even if the sample standard deviation appears to be zero. In the present study, we use the Bayesian approach to evaluate the uncertainty of a measurement based on quantized data with due consideration to the difference between the standard deviation of the apparent sample and the population standard deviation.
We assume that the quantity before quantization obeys a normal distribution having the average µ and standard deviation σ. A measurement data corresponds to a value of the quantity after quantization. Based on a specific combination of n repeated measurements, we can construct the probability density p(µ, σ) using the Bayesian method. The standard deviation of the function, p~(µ) = ∫p(µ, σ)dσ , in terms of µ gives the uncertainty of the measurement result. We have shown that when all of the measurement data take the same value, the conventional type B evaluation described the above results in an underestimate of the uncertainty, if the number of data is less than five. Analysis is also conducted in cases in which not all of the data take the same value.
