Author | Joris Brouwer and Jan Tukker |

Title | Random uncertainty of variance of finite length measurement signals |

Conference/Journal | The 5th international conference on advanced model measurements technology (AMT '17) 11/13-10-2017 |

Month | October |

Year | 2017 |

Abstract

When considering stationary measurements, the finite length of any practical measurement imposes

a random uncertainty component to statistical quantities being researched. In other words, repeating the same experiment will result in a slightly different answer. This happens for example when the limiting factor is facility length (e.g. performing resistance measurements in a towing tank) or when the limiting factor is time (e.g. offshore platform motions in a wave basin).

This paper is the fourth in a series of papers considering the analytical derivation and practical

estimation of such statistical uncertainties. The previous three papers considered mean values of

random processes. This fourth paper considers variance instead and is analogous to the first paper

of the series. Both random and periodic process classes are considered. The analytical derivation of the statistical uncertainty of signal variance is given for random, finite bandwidth noise processes and periodic processes. The former is a general solution while the latter is only valid for the trivial case of sinusoid signals. The reason to include periodic solution is its significant deviation from the finite bandwidth solution. The analytical solutions are verified by means of artificially generated signals. In general, the uncertainty of variance for finite bandwidth processes reduce with the square root of the signal length once measurement length exceeds the inverse bandwidth of the process. Measuring too short may result in a standard uncertainty equal to the variance itself. For periodic signals, the uncertainty of variance reduces with signal length itself. Estimating methods to find the statistical uncertainty of signal variance from a single measurement are given for both classes of processes. The analytic solutions and estimating methods are verified with artificially created signals. The presented uncertainty estimators are able to yield reliable and accurate estimates of the 95% confidence intervals.

When considering stationary measurements, the finite length of any practical measurement imposes

a random uncertainty component to statistical quantities being researched. In other words, repeating the same experiment will result in a slightly different answer. This happens for example when the limiting factor is facility length (e.g. performing resistance measurements in a towing tank) or when the limiting factor is time (e.g. offshore platform motions in a wave basin).

This paper is the fourth in a series of papers considering the analytical derivation and practical

estimation of such statistical uncertainties. The previous three papers considered mean values of

random processes. This fourth paper considers variance instead and is analogous to the first paper

of the series. Both random and periodic process classes are considered. The analytical derivation of the statistical uncertainty of signal variance is given for random, finite bandwidth noise processes and periodic processes. The former is a general solution while the latter is only valid for the trivial case of sinusoid signals. The reason to include periodic solution is its significant deviation from the finite bandwidth solution. The analytical solutions are verified by means of artificially generated signals. In general, the uncertainty of variance for finite bandwidth processes reduce with the square root of the signal length once measurement length exceeds the inverse bandwidth of the process. Measuring too short may result in a standard uncertainty equal to the variance itself. For periodic signals, the uncertainty of variance reduces with signal length itself. Estimating methods to find the statistical uncertainty of signal variance from a single measurement are given for both classes of processes. The analytic solutions and estimating methods are verified with artificially created signals. The presented uncertainty estimators are able to yield reliable and accurate estimates of the 95% confidence intervals.