Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> A recursive testing procedure for use in integrated navigation systems is introduced. It can be used in conjunction with the Kalman filter algorithm. Diagnostic tools for inferring the detectability of particular model errors are given. The testing procedure consists of three steps: detection, identification, and adaptation. It can accommodate model errors in both the measurement model and the dynamic model of the integrated navigation system. The tests proposed are optimal in the uniformly-most-powerful-invariant sense. The procedure can accommodate slippages in the mean of the predicted residuals caused by outliers in the data, sensor failures, or switches in the dynamic model. The method is therefore also applicable to the important problem of GPS (Global Positioning System) failure detection and integrity checking. > <s> BIB001 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> Following the pioneering work by W. Baarda, surveying engineers routinely inspect the Studentized residuals after an adjustment when high reliability is crucial. To detect outliers among uncorrelated observations, the relative magnitude between the cofactor of the residual and the corresponding (unadjusted) observation has to be checked. These ratios are commonly taken from a so-called “reliability matrix.” As other researchers have pointed out recently, the traditional approach breaks down in the case of correlated observations, and new measures of reliability have been proposed that, however, are not necessarily bounded. Therefore, we introduce a standardization procedure that guarantees our new reliability measures to fall between 0 to 1. We then show their behavior in a few simple examples, followed by a (simulated) global positioning system (GPS) application that allows these conclusions: (1) the “traditional” redundancy numbers give much too optimistic results for correlated observations; and (2) the ranking of observations according to previously recorded reliability measures may well be reversed after the standardization, making the “least reliable” observations moderately reliable, and vice versa. <s> BIB002 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> In this contribution closed-form expressions are given for the minimal detectable biases of single- and dual-frequency pseudo-range and carrier-phase data. They are given for three different single-baseline models. These are the geometry-free model and two variants of the geometry-based model, namely the roving and stationary variants. The baselines are considered to be sufficiently short such that orbital uncertainties in the fixed orbits and residual ionospheric and tropospheric delays can be assumed absent. The stochastic model used is one that permits cross-correlation and the use of different variances for individual GPS observables, including the possibility to weigh the observables in dependence on which satellite is tracked. <s> BIB003 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> The application of the theory of reliability has become a fundamental part of measurement analysis, whether in order to optimise measurement systems so that they are resistant to the influence of outliers or in the post-analysis identification of outliers. However, the current theory of reliability is based on the assumption of a single outlier—an assumption that may not necessarily be the case. This paper extends reliability theory so that it can be applied to multiple outliers through the derivation of appropriate measures of reliability for multiple outliers. The measures of reliability covered include minimal detectable biases, reliability numbers, controllability, and external reliability. <s> BIB004 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> With the increasing automation of measurement, adjustment, outlier detection and the consequential use of the results for real-time applications, reliable methods to detect and mitigate an outlier are required. However, it is frequently advised that the outlier test should not be used as a means to automatically reject an outlier. One of the reasons for this is that the outlier detection test at times can identify a wrong measurement. To address this issue, this paper proposes a new outlier separability test to confirm that the identified outlier, by the outlier test, can be confidently rejected as the outlier. In addition, the Minimally Separable Bias and the separability multiplying factor are also obtained for the proposed outlier separability test. With the initial comparisons between the proposed method with the reapplication of the global model test method and multiple hypothesis method, the advantages of the new outlier separability test are demonstrated. <s> BIB005 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> Outlier separability analysis is a fundamental component of modern geodetic measurement analysis, positioning, navigation, and many other applications. The current theory of outlier separability is based on using two alternative hypotheses—an assumption that may not necessarily be valid. In this paper, the current theory of outlier separability is statistically analysed and then extended to the general case, where there are multiple alternative hypotheses. Taking into consideration the complexity of the critical region and the probability density function of the outlier test, the bounds of the associated statistical decision probabilities are then developed. With this theory, the probabilities of committing type I, II, and III errors can be controlled so that the probability of successful identification of an outlier can be guaranteed when performing data snooping. The theoretical findings are then demonstrated using a simulated GPS point positioning example. Detailed analysis shows that the larger the correlation coefficient, between the outlier statistics, the smaller the probability of committing a type II error and the greater the probability of committing a type III error. When the correlation coefficient is greater than 0.8, there is a far greater chance of committing a type III error than committing a type II error. In addition, to guarantee successful identification of an outlier with a set probability, the minimal detectable size of the outlier (often called the Minimal Detectable Bias or MDB) should dramatically increase with the correlation coefficient. <s> BIB006 </s> Illustrative presentation of some basic concepts of network internal reliability with comments as regards engineering surveys <s> Distinction between the concepts of detectability and identifiability <s> The concept of minimal detectable bias (MDB) as initiated by Baarda (Publ Geod New Ser 2(5), 1968) and later developed by Wang and Chen (Acta Geodaet et Cartograph Sin Engl Edn 42–51, 1994), Schaffrin (J Eng Surv 123:126–137, 1997), Teunissen (IEEE Aerosp Electron Syst Mag 5(7):35–41, 1990, J Geod 72:236–244 1998, Testing theory: an introduction. Delft University Press, Delft, 2000) and others, refers to the issue of outlier detectability. A supplementation of the concept is proposed for the case of correlated observations contaminated with a single gross error. The supplementation consists mainly of an outlier identifiability index assigned to each individual observation in a network and a mis-identifiability index being the maximum probability of identifying a wrong observation. To those indices there can also be added the MDB multiplying factor to increase the identifiability index to a satisfactory level. As auxiliary measures there are indices of partial identifiability concerning pairs of observations. The indices were derived assuming the generalized outlier identification procedure as in Knight et al. (J Geod. doi: 10.1007/s00190-010-0392-4, 2010), which with one outlier case being assumed is similar to Baarda’s w-test (Baarda in Publ Geod New Ser 2(5), 1968). The following two options of identifiability indices and partial identifiability indices are distinguished: I. the indices related to identification of a contaminated observation within a set of observations suspected of containing a gross error (identifiability), II. the indices related to identification of a contaminated observation within a whole set of observations (pseudo-identifiability). To characterize the proposed approach in the context of the existing solutions of similar topic being the separability testing, the properties of both types of identifiability indices are discussed with reference to the concept of Minimal Separable Bias (Wang and Knight in J Glob Position Syst 11(1):46–57, 2012) and a general approach in Yang et al. (J Geod 87(6):591–604, 2013). Numerical examples are provided to verify the proposed approach. <s> BIB007
In the subsequent explanations we concentrate on a single outlier case in networks with uncorrelated or correlated observations. In some publications dealing with data processing procedures the terms detection and identification are used interchangeably for describing the final effect of finding an observation contaminated with a gross error. Following rigorous distinction between these concepts made by  , we emphasize here their direct assignment to the global model test and the outlier tests correspondingly. As shown in Fig. 1 , when the test statistic T in a global model test is greater than its critical value crit T and we may assume that the model itself is correct, we have a basis to state that we found (i.e. detected) the presence of a gross error in a network. Obviously, we do not know its location in a network. In the outlier tests [see  ] for uncorrelated observations and (Knight et al., 2010) for correlated observations), by finding the observations for which the absolute value of outlier test statistic i.e. i w (i = 1, …, s) exceeds the critical value crit w , we obtain a set of s observations suspected of containing a gross error. The element of the set with the maximal w , is identified as the observation containing a gross error (k-th observation in Fig.1 ). In the phase of a priori analysis of network internal reliability we deal with the corresponding concepts detectability and identifiability. Detectability is expressed by the minimal magnitude of gross error (MDB) in a particular observation, the presence of which in a system can be detected in a global model test. The concept of MDB initiated by  was further analyzed by  , BIB002 BIB001 BIB003  . It should be noted that the MDB is a stochastic quantity that depends on the assumed probabilities α and β of the I-type and II-type error respectively. Identifiability relates to the chances that the observation found in the identification process is the observation actually contaminated with a gross error. The chances are expressed in a form of identifiability index ID for a particular observation as presented in BIB007 , where coordination between the global test and the outlier tests was assumed with respect to non-centrality effect (Knight et al., 2010) . The index is a probability that the i-th observation contaminated with a gross error of the i MDB magnitude will have i w dominating in a set of suspected observations. The value of ID i can be determined on the basis of numerical simulation of observation random errors with the non-centrality of i MDB magnitude for the i-th observation Another measure of identifiability for the i-th observation can be the maximal of mis-identifiability indices MID indices is a probability of type III error j/i  , for j ≠ i, i.e. a probability that with a gross error residing in the i-th observation, j w will be dominating in a set of suspected observations. The concept of type III error was first presented in BIB004 Foerstner, 1983) . The term identifiability has much in common with the more and more widely used term separability (eg. BIB005 BIB006 . However, the latter term seems to have a broader usability especially for distinguishing between different types of alternative hypotheses. It is necessary to mention also the concept of Minimal Separable Bias (MSB) (eg. BIB005 , which is the MDB magnified so as to be separable in the outlier test with a desired level of probability. For the error of MSB magnitude the ID index should satisfy the correspondingly high level of probability. Figure 2 shows classification of gross errors in a form of a tree-diagram. It has hierarchical 4-level structure which should be analyzed from top to bottom, i.e. to be detectable a gross error must be perceptible and to be identfiable the error must be detectable. Each classification level corresponds to a specified criterion as will be presented further in this Section.  . The formal definition of this space is as follows: