Usability of the Benford's law for the results of least square estimation

Abstract

Benford's law (BL), also known as the first-digit or significant-digit law, is an intriguing pattern in data sets, considers the frequency of occurrence of the first digits, which are not uniformly distributed as might be expected, conversely follow a specified theoretical distribution. According to BL, the occurrence of first non-zero digit in a numerical data, which is generated or found in nature, depends on a logarithmic distribution. Least square estimation (LSE) method is mostly preferred for the estimation of the unknown parameters from different types of geodetic data. The residuals and the normalized residuals of the LSE method, which follow normal distribution and expected values of them are zero are used in outlier detection problem. In this study, BL is investigated for residuals and the normalized residuals estimated from LSE method. Three types of geodetic data are used: (1) simulated regression models, (2) global positioning system (GPS) data, (3) leveling network. The first group data sets are simulated based on linear regression and univariate models and each simulated group is generated for a number of 100, 1000, and 10,000 samples. To generate second group, an international global navigation satellite system (GNSS) service (IGS) station data (ISTA) is processed by kinematic PPP approach using GIPSY OASIS II v6.4 software. Here, the observation duration of GPS data is 4 days. For the last data, a leveling network with 55 points involving 110 observations of height differences is simulated. BL has been applied to the residuals (v) and normalized residuals (w) estimated from LSE method. Goodness-of-fit test has been implemented to determine whether a population has a specified BL distribution or not. This test is based on how good a fit we have between the frequency of occurrence of residuals and normalized residuals in an observed sample and the expected frequencies obtained from the hypothesized distribution. The results depending on the statistical test show that each data set (residuals and normalized residuals) used in this study follows BL.