Assigning a probability density function for the value of a quantity based on discrete data: the resolution problem

Abstract

It often happens that knowledge about a particular quantity has to be reached by processing a series of resolution-limited indications. It is a well-established fact that if the variability of the data is large compared with the resolution interval, the effect of discretization can be ignored. Otherwise, it needs to be taken into account since it can then be an important source of uncertainty, sometimes more significant than randomness itself. The objective of this paper is to derive a probability density function (pdf) for the value of a quantity based on discretized data. This pdf allows the standard uncertainty associated with the best estimate of the quantity to be computed and, perhaps more importantly, it can be used as an input to evaluate a measurement model in which the quantity is involved. Bayesian concepts are used towards this goal. Although reaching an appropriate pdf has been attempted before, limited success has been attained, as the pdfs that have been obtained exhibit some undesirable characteristics. Herein a new approach is proposed. Unlike previous efforts, this time the quantity of interest is modelled as a sum of two other quantities, one that can only assume discrete values and the other that takes values within the resolution interval centred on zero. The resulting pdf exhibits a satisfactory behaviour, but further work would be required to provide firmer theoretical grounds for the employed prior.

Non-parametric estimation of probability density functions via a simple interpolation filter

“…A comparison between the three estimators (histogram, interpolationfilter-based and kernel-based estimators) was carried out using Montecarlo simulations based on 1000 records of 500 samples each. The value of ∆ = 0.1529 obtained by using the Silverman's rule was chosen because it minimizes the histogram RMSE, as also verified through simulations based on the PDF (8). The mean values of the obtained estimators are plotted in Fig.…”

“…To support this statement consider for instance the arcsine distribution, whose PDF has the following expression: . Arcsine PDF given by (8): mean values of the three considered estimators (histogram, interpolation-filter-based and kernel-based estimators) obtained using Montecarlo simulations based on 1000 records of 500 samples each and ∆ = 0.1529. All estimators poorly follow the discontinuous behavior of the true PDF at x = −1, shown using a solid line.…”

“…Thus, he/she might be interested in automating data processing so to mitigate the occurrence of this risk. The issue of automatically selecting the optimal estimator parameters was analyzed for instance in [7], [8] with respect to optimal bin width selection and in [9] for the optimization of kernel-based estimation. In [6] a different method was proposed that is blind and adaptive, so that a smooth estimate is achieved.…”

Finite resolution and serial correlations in mass metrology

“…Different methods including conventional frequentist [1,[13][14][15][16], Bayesian [17][18][19][20], and fiducial inference [21,22] have been developed for the estimation of the measurand and the uncertainty of repeated measurements affected by finite resolution. For the theoretical modelling of the problem Monte Carlo simulation [23][24][25][26][27] has been used.…”

“…For the theoretical modelling of the problem Monte Carlo simulation [23][24][25][26][27] has been used. Two significant factors contributing to the measurement outcome and uncertainty are always assumed in [13][14][15][16][17][18][19][20][21][22][23][24][25][26] to be present: finite resolution and Gaussian noise. In the majority of these publications, it is generally concluded that if the sample standard deviation s is sufficiently large in comparison to the resolution or quantization step size q of the measurement instrument (about 0.5 ), s q ≥ the effect of the finite resolution will be insignificant, and conventional statistical inference will be valid.…”