Consistency of the Empirical Distributions of Navigation Positioning System Errors with Theoretical Distributions—Comparative

sensors

Article Consistency of the Empirical Distributions of Navigation Positioning System Errors with Theoretical Distributions—Comparative Analysis of the DGPS and EGNOS Systems in the Years 2006 and 2014

Mariusz Specht

Department of Transport and Logistics, Gdynia Maritime University, Morska 81-87, 81-225 Gdynia, Poland; [email protected]

Abstract: Positioning systems are used to determine position coordinates in navigation (air, land and marine). The accuracy of an object’s position is described by the position error and a statistical analysis can determine its measures, which usually include: Root Mean Square (RMS), twice the Distance Root Mean Square (2DRMS), Circular Error Probable (CEP) and Spherical Probable Error (SEP). It is commonly assumed in navigation that position errors are random and that their distribution are consistent with the normal distribution. This assumption is based on the popularity of the Gauss distribution in science, the simplicity of calculating RMS values for 68% and 95% probabilities, as well as the intuitive perception of randomness in the statistics which this distribution reﬂects. It should be noted, however, that the necessary conditions for a random variable to be normally distributed include the independence of measurements and identical conditions of their realisation, which is not the case in the iterative method of determining successive positions, the ﬁltration of coordinates or the dependence of the position error on meteorological conditions. In the preface to this publication, examples are provided which indicate that position errors in some navigation systems may not be consistent with the normal distribution. The subsequent section describes basic statistical tests for

Citation: Specht, M. Consistency of assessing the fit between the empirical and theoretical distributions (Anderson-Darling, chi-square the Empirical Distributions of and Kolmogorov-Smirnov). Next, statistical tests of the position error distributions of very long Navigation Positioning System Errors Differential Global Positioning System (DGPS) and European Geostationary Navigation Overlay with Theoretical Distributions— Service (EGNOS) campaigns from different years (2006 and 2014) were performed with the number Comparative Analysis of the DGPS of measurements per session being 900’000 fixes. In addition, the paper discusses selected statistical and EGNOS Systems in the Years 2006 distributions that fit the empirical measurement results better than the normal distribution. Research and 2014. Sensors 2021, 21, 31. has shown that normal distribution is not the optimal statistical distribution to describe position https://dx.doi.org/10.3390/s21010031 errors of navigation systems. The distributions that describe navigation positioning system errors more accurately include: beta, gamma, logistic and lognormal distributions. Received: 17 November 2020 Accepted: 21 December 2020 Keywords: statistical distribution; navigation positioning system; position error; Differential Global Published: 23 December 2020 Positioning System (DGPS); European Geostationary Navigation Overlay Service (EGNOS)

Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and institutional affiliations. 1. Introduction The assumption that a position line error in navigation has a normal distribution is commonplace for book authors [1–3], as well as in monographs, regulations and standards directly related to the statistical analysis of position errors [4,5]. It should be noted, however, Copyright: © 2020 by the author. Li- that several scientific publications draw attention to existing differences between the censee MDPI, Basel, Switzerland. This article is an open access article distributed empirical and theoretical distributions. Global Positioning System (GPS) is the basic under the terms and conditions of the positioning system used in navigation. Its operational characteristics are periodically Creative Commons Attribution (CC BY) described in several technical standards, of which Global Positioning System Standard license (https://creativecommons.org/ Positioning Service Signal Specification versions have already been issued five times; licenses/by/4.0/). in 1993, 1995, 2001, 2008, and 2020. The first version of this document [6] states expressly

Sensors 2021, 21, 31. https://dx.doi.org/10.3390/s21010031 https://www.mdpi.com/journal/sensors Sensors 2021, 21, x FOR PEER REVIEW 2 of 21

Sensors 2021, 21, 31 2 of 23 cally described in several technical standards, of which Global Positioning System Standard Positioning Service Signal Specification versions have already been issued five times; in 1993, 1995, 2001, 2008, and 2020. The first version of this document [6] states thatexpressly the empirical that the error empirical distributions error distributions are overlaid are with overlaid Gauss with distributions, Gauss distributions, as a basis for as a comparisonbasis for comparison with theoretical with theoretical expectations expect (Figureations1). The(Figure theoretical 1). The distributionstheoretical distribu- were generatedtions were using generated the means using and the standardmeans and deviations standard of deviations the empirical of the datasets. empirical The datasets. error distributionsThe error distributions are based uponare based measured upon datameas fromured thedata GPS from Control the GPS Segment Control monitor Segment stationsmonitor recorded stations forrecorded three months. for three months.

(a)

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FigureFigure 1. 1.Comparison Comparison of of empirical empirical data data of of the the GPS GPS position position error error in in North North (a (),a), East East (b ()b and) and vertical vertical (c()c axes) axes with with the the theoretical theoretical normal normal distribution. distribution. Own Own study study based based on: on: [ 6[6].]

TheThe presented presented differences differences in in local local axes axes must must result result in in significant significant differences differences in in the the fit fit ofof 2D 2D position position error error with with the the chi-square chi-square distribution. distribution. Therefore, Therefore, in in the the same same document, document, FigureFigure2 presents2 presents the the empirical empirical (64 (64 m) m) and and theoretical theoretical (83 (83 m) m) values values of of twice twice the the Distance Distance RootRoot Mean Mean Square Square (2DRMS) (2DRMS) measure. measure. It It should should be be stressed stressed that that since since the the estimation estimation error error waswas as as high high as as 19 19 m, m, 95% 95% of of the th measurementse measurements will will be smallerbe smaller than than this. this. In thisIn this situation, situation, it isit difficult is difficult to support to support the use the of use normal of normal distribution distribution for the calculation for the calculation of the basic of quantitythe basic describingquantity describing the system the positioning system positioning accuracy accuracy (2DRMS). (2DRMS). Similar conclusions Similar conclusions concerning con- the inconsistency of the statistical distributions of Differential Global Positioning System (DGPS) and GPS position errors are raised by the Frank van Diggelen, but with much smaller discrepancies [5].

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Sensors 2021, 21, 31 cerning the inconsistency of the statistical distributions of Differential Global Positioning3 of 23 System (DGPS) and GPS position errors are raised by the Frank van Diggelen, but with much smaller discrepancies [5].

FigureFigure 2.2. NominalNominal GPSGPS SPSSPS horizontalhorizontal errorerror distribution.distribution. OwnOwn studystudy basedbased on:on: [[6].6]. In the authors’ research on the accuracy of various navigation positioning systems, two In the authors’ research on the accuracy of various navigation positioning systems, measures of accuracy were also often compared: 2DRMS and R95. The latter is an empirical two measures of accuracy were also often compared: 2DRMS and R95. The latter is an quantity calculated by sorting errors from the smallest to the largest. This value is higher empirical quantity calculated by sorting errors from the smallest to the largest. This value than 95% of the measurements made. Please note that if the empirical 2DRMS statistics is higher than 95% of the measurements made. Please note that if the empirical 2DRMS fit the chi-square distribution, these values should be almost identical. The author’s statistics fit the chi-square distribution, these values should be almost identical. The au- research into the accuracy of various Global Navigation Satellite Systems (GNSS), such thor’s research into the accuracy of various Global Navigation Satellite Systems (GNSS), as DGPS and European Geostationary Navigation Overlay Service (EGNOS) [7], GNSS such as DGPS and European Geostationary Navigation Overlay Service (EGNOS) [7], geodetic networks, as well as multi-GNSS solutions [8,9], has repeatedly shown significant GNSS geodetic networks, as well as multi-GNSS solutions [8,9], has repeatedly shown discrepancies between 2DRMS and R95 measures. significant discrepancies between 2DRMS and R95 measures. In order to quantify the discrepancy between the 2DRMS and R95 measures, let us In order to quantify the discrepancy between the 2DRMS and R95 measures, let us analyse the results of the position accuracy tests of six different mobile phones working inanalyse parallel, the whichresults were of the conducted position accuracy in 2017. tests The sameof six smartphonesdifferent mobile were phones tested working during bothin parallel, dynamic which [8] and were stationary conducted [9] measurementin 2017. The same campaigns. smartphones To compare were thetested fit ofduring both values,both dynamic the concept [8] and of Relativestationary Percent [9] measurement Error (RPE) campaigns. has been introduced, To compare according the fit of to both the relationship:values, the concept of Relative Percent Error (RPE) has been introduced, according to the relationship: R95(2D) − 2DRMS(2D) RPE = · 100% (1) R95(2D) RDDRMSD95() 2− 2 () 2 The obtained results areRPE presented=⋅ in the last rows of Tables100%1 and2. (1) RD95() 2 Table 1. Statistics of position errors of Samsung Galaxy phones during the dynamic measurement campaign.The obtained Own study results based are on: [presented8]. in the last rows of Tables 1 and 2.

TableStatistics 1. Statistics of of position errors of Samsung GaSamsunglaxy phones Galaxy during the dynamic measurement campaign.Position Own Error study basedY on: [8]. S3 Mini S4 S5 S6 S7 Number of Samsung Galaxy Statistics of Position6041 Error 3410 10’950 10’939 10’906 10’926 measurements Y S3 Mini S4 S5 S6 S7 2DRMS(2D) 9.47 m 5.23 m 6.59 m 6.72 m 8.32 m 10.54 m Number of measurements 6041 3410 10,950 10,939 10,906 10,926 R95(2D) 6.84 m 3.67 m 5.80 m 5.77 m 9.38 m 9.62 m 2DRMS(2D) 9.47 m 5.23 m 6.59 m 6.72 m 8.32 m 10.54 m RPE −38.45% −42.51% −13.62% −16.46% 11.30% −9.56% R95(2D) 6.84 m 3.67 m 5.80 m 5.77 m 9.38 m 9.62 m RPE −38.45% −42.51% −13.62% −16.46% 11.30% −9.56%

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Table 2. Statistics of position errors of Samsung Galaxy phones during the 24 h stationary measurement campaign. Own study based on: [9].

Statistics of Samsung Galaxy Position Error Y S3 Mini S4 S5 S6 S7 Number of 73’699 71’438 86’290 86’346 86’371 86’355 measurements 2DRMS(2D) 5.61 m 6.79 m 2.04 m 2.06 m 13.69 m 8.93 m R95(2D) 4.93 m 3.76 m 1.65 m 1.76 m 12.64 m 8.39 m RPE −13.79% −80.59% −23.64% −17.05% −8.31% −6.44%

The research results indicated that the differences between the values of 2DRMS and R95 may reach a dozen percent or so. Therefore, it can be assumed that there may be significant differences between empirical distributions of latitude (ϕ) and longitude (λ) errors and the normal distribution. This problem may concern various navigation positioning systems, so it is justified to undertake more research into testing the actual results obtained by positioning systems. This article examines the statistical fit between empirical distributions with theoretical position errors of two Differential Global Navigation Satellite Systems (DGNSS): marine DGPS and EGNOS. Two measurement campaigns of both systems were used for research purposes, during which more than 1–2 million fixes were recorded. Since there numerous measurements, the conclusions drawn from them can be considered representative. The research were carried out in the years 2006 and 2014. The aim of the publication is: • Determining the consistency of empirical distributions with the theoretical (normal and chi-square) for DGPS and EGNOS position errors. Latitude and longitude errors were referred to as the normal distribution and 2D position errors were referred to as the chi-square distribution. • Finding statistical distributions other than normal and chi-square distributions that present a better fit with DGPS and EGNOS empirical data. • Comparison of the statistical distributions of DGPS and EGNOS position errors from 2006 and 2014 will make it possible to answer the following question: do the statistical distributions of 1D and 2D position errors also change together with the evolution of the system and increases in its accuracy? The introduction of the article describes the premises for starting the discussion. The author refers to the works of selected authors and their own research which dis- cuss the discrepancies between empirical statistics of errors in the navigation positioning systems and their theoretical values. The materials and methods section presents selected statistical distribution measures together with the interpretation of the histogram, probability-probability (P-P) plots, as well as differences between the empirical and theoretical cumulative distribution functions. In addition, the three types of statistical tests used in the research were described (Anderson-Darling, chi-square and Kolmogorov-Smirnov). The main research results are shown in the results section and discussed in the discussion section. The publication ends with conclusions and suggestions for further research. This is the second article in a series of monothematic publications, the aim of which will be statistical distribution analysis of navigation positioning system errors.

2. Materials and Methods 2.1. Statistical Distribution Measures Statistical testing to assess the consistency of empirical with theoretical distributions should be preceded by the calculation of specific distribution measures to determine their asymmetry, central tendency, concentration and dispersion. It should be noted that there is no specific set of distribution measures for specific analyses or processes in navigation Sensors 2021, 21, 31 5 of 23

or statistics. This selection should result from the statistical nature of the variable under investigation and the research aim. For example, for normal distributions of ϕ and λ errors using GPS, it makes sense to determine both the mean and the median. If these values are similar, this may indicate the empirical distribution fitting the normal distribution. However, for a 2D position error distribution (which exhibits an asymmetric chi-square distribution), it is not justified to determine both of these values as this distribution is asymmetric by its nature. With this in mind, it is proposed to divide the assessment of the fit between error distributions in the navigation positioning systems and the theoretical distributions into two stages: • Stage I: Calculation of selected statistical distribution measures: asymmetry (skewness), central tendency (arithmetic mean and median), concentration (kurtosis) and dispersion (range, standard deviation and variance). • Stage II: Statistical testing using Anderson-Darling, chi-square and Kolmogorov- Smirnov tests. Table3 presents selected statistical distribution measures that will be used for empirical testing of 1D and 2D position errors. Their definitions, estimators, interpretations and properties are also given.

2.2. Analysis of the Histogram, P-P Plot, as Well as Differences Between the Empirical and Theoretical Cumulative Distribution Functions A histogram is a very important element in assessing the distribution of the studied population. It is one of the graphic methods of representing the empirical distribution of a characteristic. It is made up of a series of rectangles placed on the axis of coordinates. These rectangles are, on the one hand, determined by the class interval values of the characteristic, while their height is determined by the number (or frequency, or possibly also probability density) of elements included in a given class interval. If the histogram shows the number of elements and not the probability density, then the interval widths should be equal. In P-P plots, the empirical probability distribution function is plotted against the theoretical distribution. The observations are ﬁrst sorted in descending order. The i-th i observation is then plotted on one axis as n (i.e., the value of the observed cumulative distribution) and the other axis as F(xi), where F(xi) is the value of the theoretical probability distribution function for respective observation xi. If the theoretical cumulative distribution is a good approximation of the empirical distribution, then the points on the diagram should be close to the diagonal. Regarding the idea behind them, the Fn(x), F(x) and Fn(x)–F(x) graphs are based on a comparison of empirical and theoretical distributions, similarly to P-P plots. They present both functions simultaneously or their differences.

2.3. Testing Statistical Distributions of Navigation Positioning System Errors Testing statistical distributions of navigation positioning system errors is a key issue for assessing their distributions. This study tested the fit between 1D position errors (ϕ and λ) with the family of normal distributions. To this end, statistical hypotheses were verified, which means that any judgment on the population issued without detailed examination and verification was now tested. These allow determining whether the results obtained for the sample can be applied to the whole population [10]. Sensors 2021, 21, 31 6 of 23

Table 3. Selected statistical distribution measures, their deﬁnitions, estimators, interpretations and properties, used to study ϕ and λ error distributions (separately) of navigation positioning systems.

Distribution Measure Estimator Deﬁnition/Properties/Interpretation Deﬁnition:

n Arithmetic mean—the sum of numbers divided by their number. ∑ xi Properties: x1+x2+...+xn i=1 xn = = n n • The arithmetic mean of a sample is, regardless of the distribution, Arithmetic mean where: a consistent and unbiased estimator of the expected value of the (central tendency measure) xn—arithmetic mean from the sample, distribution from which the sample was drawn. x —subsequent values of a given random variable in the sample, i • The arithmetic mean is sensitive to the skewness and outlier n—sample size. observations.

Deﬁnition: If n is an even number, the median (m) is: Median—characteristic in ordered series, with an equal number of x n +x n + observations found above and below it. Median = 2 2 1 m 2 Properties: (central tendency measure) when n is an odd number, it m: x n + • It is a measure that is much more resistant to outliers than the = 2 1 m 2 arithmetic mean.

Deﬁnition: Range(x) = max(x) − min(x) Range—the difference between the maximum and minimum value. where: Properties: Range Range(x)—range, • (dispersion measure) Distorted by outliers. max(x)—maximum value of a given random variable in the sample, • Leaves most information out. min(x)—minimum value of a given random variable in the sample. • Not algebraically deﬁned. Sensors 2021, 21, 31 7 of 23

Table 3. Cont.

Distribution Measure Estimator Definition/Properties/Interpretation Definition: Variance—the arithmetic mean of the squares of the deviations (differences) between the indvidual values of a characteristic and the expected value. Unbiased variance estimator (s2): n Standard deviation—square root of the variance. 2 1 2 s = n−1 ∑ (xi − xn) Properties: Variance and standard deviation i=1 • (dispersion measures) Standard deviation of the sample (s): Easy to calculate, they are defined algebraically. s n 2 • They take into account all values of characteristic variants. ∑ (xi −xn ) • They are strongly influenced by outliers. s = i=1 n−1 • Distortion in the case of skewed distributions. • Difficult to compare for varied values.

Definition: Skewness—a measure of distribution’s asymmetry. Properties: • It illustrates to what extent the arithmetic mean reflects the actual central tendency of the distribution. • If its value is high, the arithmetic mean does not properly reflect the most typical measured value. In this case, the existence of outliers in the distribution may be suspected. As a result, data need correction of the

n application of non-parametric tests. 1 (x −x )3 n ∑ i n • This is very important when assessing the symmetry of a variable’s Skewness G = i=1 s3 distribution. (asymmetry measure) where: • It determines the degree and direction of asymmetry, with a negative G—skewness. value standing for distribution skewed to the left and a positive value standing for distribution skewed to the right. Interpretation: • |G| ∈ h 0, 0.4) very weak asymmetry, almost symmetrical distribution. • |G| ∈ h 0.4, 0.8) weak asymmetry. • |G| ∈ h 0.8, 1.2) moderate asymmetry. • |G| ∈ h 1.2, 1.6) strong asymmetry. • |G| ∈ h 1.6, ∞) very strong asymmetry. Sensors 2021, 21, 31 8 of 23

Table 3. Cont.

Distribution Measure Estimator Definition/Properties/Interpretation Definition: Kurtosis—a measure of distribution’s flattening. Properties: • It measures whether the distribution is “peaky”. • If the kurtosis value is clearly different from zero, then the distribution is either flatter or more pointed than the normal distribution. • The kurtosis for a normal distribution is zero.

( + ) n 4 ( − )2 Interpretation: = n n 1 xi −xn − 3 n 1 Kurt (n−1)(n−2)(n−3) ∑ s (n−2)(n−3) Kurtosis i=1 • Mesokurtic distributions (Kurt = 0 for a normal distribution)—the (concentration measure) where: kurtosis value is zero, the flattening distribution is similar to the Kurt—sample kurtosis. flattening of a normal distribution (for which the kurtosis is exactly zero). • Leptokurtic distributions (Kurt > 0 for a slender distribution)—the kurtosis value is positive, the characteristic values are more concentrated than in the normal distribution. • Platykurtic distributions (Kurt < 0 for a flattened distribution)—the kurtosis value is negative, the characteristic values are less concentrated than in the normal distribution. Sensors 2021, 21, 31 9 of 23

In the literature review, it was noted that in statistical studies a large sample is considered to be a set consisting of at least 30 or 40 elements. Other samples are considered small. Furthermore, the sample size affects the choice of the type of statistical test. For example, the Shapiro-Wilk test, as confirmed by the experience of other authors, should be used for samples of less than 20 or 30 elements [11]. Another popular test, the Lilliefors test, is used to test the normality of distribution for samples of similar size to the Shapiro-Wilk test [12]. Tests such as the Cramér-von Mises test or the D’Agostino-Pearson test are used for statistical studies with large samples [13–16]. With this multitude of statistical tests, it was decided to choose the three most frequently used tests for large samples: Anderson-Darling, chi-square and Kolmogorov-Smirnov [17–19]. As in the statistical analysis, since it was planned to use records from a navigation positioning system ranging from several hundred thousand to over two million measurements, it became necessary to determine the appropriate sample size [20–22]. Based on the literature [23–25], to obtain the desired test power (0.8) at a significance level of 5% for the most popular statistical distributions, such as log-normal, normal, Weibull, etc., the statistical hypotheses should be tested for a sample size of about 1000 elements. The approach to statistical testing presented above is based on a well-known statistical research theory. However, navigation positioning systems have a specific feature that distinguishes them from other measurement systems. This feature is the Position Random Walk (PRW). Its essence lies in the position coordinates “walking” around the reference coordinates. This issue has been described in detail in [26]. In this publication, a detailed analysis of this phenomenon was presented with the need to ensure a representative sample size highlighted based on empirical research. In addition, it shows that it is only selecting a representative sample size and 1000 measurements should be randomly drawn from this sample for statistical testing.

2.4. Statistical Tests Used in Research The following statistical tests were used in the research: • Anderson-Darling test: This test is based on the Cramér-von Mises weighted distance between the empirical Fn(x) and theoretical F(x) distributions with weights corre- sponding to the inverse of the empirical distribution variance (note that Fn(x) has a binomial distribution) [27]:

∞ Z (F (x) − F(x))2 d(F , F) = n n dF(x) (2) n F(x)(1 − F(x)) −∞

Test statistics based on the above distance for a simple random sample xi may be written as: A2 = −n − S (3) where: n 2i − 1 S = ∑ [ln(F(xi)) + ln(1 − F(xn+1−i))] (4) i=1 n • Chi-square test: This test is based on the χ2 statistic [28]:

n (O − E )2 χ2 = ∑ i i (5) i=1 Ei

2 which, for a true zero hypothesis, has an asymptotic distribution χ . The Ei symbol indicates the expected number of observations in the i class and Oi stands for the actual number of observations in the i class. • Kolmogorov-Smirnov test: The test is based on the supremum distance between the empirical Fn(x) and theoretical F(x) distribution functions [29,30]: Sensors 2021, 21, 31 10 of 23

d(Fn, F) = sup|Fn(x) − F(x)| (6) x Test statistics based on the above distance consist of counting the maximum module of probability distribution difference at the empirical distribution function step points:

D = max|Fn(xi) − F(xi)| (7) xi

2.5. Description of DGPS and EGNOS Measurement Campaigns Studies of the position determination accuracy of the DGPS and EGNOS systems have been conducted in Poland for many years [31–33]. Due to the changing values of GPS position errors, which resulted in the increasing accuracy of DGPS and EGNOS augmentation systems, such research were conducted regularly in the years 2006 and 2014. The paper analyses two long-term measurement campaigns: • The first measurement campaign took place in March 2006, in Gdynia (Poland). Dur- ing this campaign, 2’187’842 fixes for DGPS and 1’774’705 fixes for EGNOS were recorded respectively with a recording frequency of 1 Hz. • The second measurement campaign took place in May 2014, in Gdynia (Poland). During this campaign, 951’698 fixes for DGPS and 927’553 fixes for EGNOS were recorded respectively with a recording frequency of 1 Hz. Studies of both measurement campaigns included the installation of the DGPS and EG- NOS receivers always in the same place-on the radio beacon in the port of Gdynia (Figure3) . It was a reference point with ellipsoidal coordinates amounting to: ϕ = 54◦31.7560870 N, λ = 18◦33.5741380 E and h = 68.070 m. During the measurement campaigns, typical DGPS (Leica MX9212 + MX51R) and EGNOS (Septentrio PolaRx2e) receivers with the possibil- ity of saving data in the form of National Marine Electronics Association (NMEA) GGA messages were used. These measurements yielded text files with position coordinates which were compliant with the data transmission protocol described above. They contain geographic coordinates of designated points presented in angular (curvilinear) measure. To determine individual measurement errors, they were projected from the surface of Sensors 2021, 21, x FOR PEER REVIEWthe World Geodetic System 1984 (WGS 84) ellipsoid to a flat surface using Gauss-Krüger10 of 21

projection.

Figure 3. Measurement site-the radio beacon in the port of Gdynia. Figure 3. Measurement site-the radio beacon in the port of Gdynia. 2.6. Research Assumptions Basic assumptions for research and numerical analyses were as follows: • Preliminary analyses carried out in [26] showed that a representative sample for DGPS and EGNOS systems should consist of about 900’000 measurements. Only with this sample size, 1D and 2D position errors are representative. Therefore, each of the analysed campaigns was shortened so that all sessions consist of the same number of measurements (900’000 fixes). • Analysis of the campaigns (DGPS 2006, DGPS 2014, EGNOS 2006 and EGNOS 2014) with the same number of measurements makes it possible to compare their results and draw generalised conclusions. • Selected statistical distribution measures (asymmetry, central tendency, concentration and dispersion) were determined for the same sample size (900’000 fixes). • For statistical testing of fit between empirical distributions of 1D position errors (φ and λ) with the normal distribution, 1000 measurements were randomly selected and subjected to Anderson-Darling, chi-square and Kolmogorov-Smirnov tests. • In comparative analyses of empirical distributions (1D and 2D position errors), the most frequently used theoretical distributions were used: Beta, Cauchy, chi-square, exponential, gamma, Laplace, logistic, lognormal, normal, Pareto, Rayleigh, Stu- dent’s and Weibull. • Two values were used to assess position accuracy: the 2DRMS(2D) value, which was determined for the entire population (900’000 fixes) based on the relationship:

=+22 2(2)2DRMS D sφλ s (8)

where: sφ—standard deviation of geodetic (geographic) latitude, sλ—standard deviation of geodetic (geographic) longitude, and the R95 value, which was determined by sorting the data from the lowest to the highest value. • Easy Fit software was used for the analyses. To evaluate the fit of empirical with theoretical distributions, a significance level of 5% was assumed. The rankings of the best fit distributions were created based on the Kolmogorov-Smirnov statistic (D). • Mathcad software was used to calculate the values of 2DRMS(2D) and R95(2D) and plot graphs of the position error distribution.

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2.6. Research Assumptions Basic assumptions for research and numerical analyses were as follows: • Preliminary analyses carried out in [26] showed that a representative sample for DGPS and EGNOS systems should consist of about 900’000 measurements. Only with this sample size, 1D and 2D position errors are representative. Therefore, each of the analysed campaigns was shortened so that all sessions consist of the same number of measurements (900’000 fixes). • Analysis of the campaigns (DGPS 2006, DGPS 2014, EGNOS 2006 and EGNOS 2014) with the same number of measurements makes it possible to compare their results and draw generalised conclusions. • Selected statistical distribution measures (asymmetry, central tendency, concentration and dispersion) were determined for the same sample size (900’000 fixes). • For statistical testing of fit between empirical distributions of 1D position errors (ϕ and λ) with the normal distribution, 1000 measurements were randomly selected and subjected to Anderson-Darling, chi-square and Kolmogorov-Smirnov tests. • In comparative analyses of empirical distributions (1D and 2D position errors), the most frequently used theoretical distributions were used: Beta, Cauchy, chi-square, exponential, gamma, Laplace, logistic, lognormal, normal, Pareto, Rayleigh, Student’s and Weibull. • Two values were used to assess position accuracy: the 2DRMS(2D) value, which was determined for the entire population (900’000 fixes) based on the relationship: q 2 2 2DRMS(2D) = 2 sϕ + sλ (8) where:

sϕ—standard deviation of geodetic (geographic) latitude, sλ—standard deviation of geodetic (geographic) longitude, and the R95 value, which was determined by sorting the data from the lowest to the highest value. • Easy Fit software was used for the analyses. To evaluate the fit of empirical with theoretical distributions, a significance level of 5% was assumed. The rankings of the best fit distributions were created based on the Kolmogorov-Smirnov statistic (D). • Mathcad software was used to calculate the values of 2DRMS(2D) and R95(2D) and plot graphs of the position error distribution.

3. Results 3.1. DGPS 2006 Results The study started with analyses of ϕ and λ error distributions assessed individually. Table4 presents the results of statistical analysis and tests of ϕ and λ errors determined using the DGPS system in 2006. These include the evaluation of selected distribution measures and the results of testing the statistical ﬁt of ϕ and λ errors with the normal distribution. SensorsSensors 2021, 212021, x ,FOR 21, x PEER FOR PEERREVIEW REVIEW 11 of 2111 of 21

3. Results3. Results 3.1. DGPS3.1. DGPS 2006 2006Results Results The studyThe study started started with with analyses analyses of φ ofand φ λand error λ error distributions distributions assessed assessed individually. individually. Sensors 2021, 21, 31 12 of 23 TableTable 4 presents 4 presents the results the results of statistical of statistical analysis analysis and testsand testsof φ ofand φ λand errors λ errors determined determined

Sensors 2021Sensors, 21, x2021 FOR, 21 PEER, x FOR REVIEW PEERSensors REVIEW 2021using, 21 , xusing FOR the PEER DGPS the REVIEW DGPS system system in 2006. in 2006. These These include include the evaluation the evaluation11 of of21 11 selected of 21 selected distribution distribution11 of 21 Sensors 2021, 21, x FOR PEER REVIEW 11 of 21 measuresmeasures and andthe resultsthe results of testing of testing the statisticalthe statistical fit of fit φ ofand φ andλ errors λ errors with with the normalthe normal Table 4. Statistical analysis ofdistribution. distributiondistribution. measures and statistical tests of ϕ and λ errors using the DGPS system in 2006. 3. Results3. Results 3. Results 3. Results Distribution Probability Density Function Probability Density Function for λ TableTable 4. Statistical 4. Statistical analysisϕ Error3.1. analysis DGPS of3.1. distribution 2006 ofDGPSλ distributionError3.1. Results 2006DGPS me R esults2006asures3.1. me R DGPSasuresesults and 2006 statistical and R statisticalesults tests testsof φ andof φ λ and errors λ errors using using the DGPS the DGPS system system in 2006. in 2006. Measure for ϕ Error Error The studyThe started study startedwith analyses withThe analyses study of φ and started of λ φ error and with λdistributions analyseserror distributions of φassessed and λ assessed error individually. distributions individually. assessed individually. DistributionDistribution Measure Measure φ Errorφ Errorλ Errorλ ErrorTheProbability studyProbability started Density with Density analysesFunction Function of for φ φand forError λ φ error Error distributions ProbabilityProbability assessed Density Density individually.Function Function for λ Errorfor λ Error Table 4 Tablepresents 4Table presents the 4results presents theTable ofresults statistical the 4 presentsresultsof statistical analysis of the statistical resultsanalysis and tests ofanalysis and statistical of testsφ and of analysis λtests φ errors and of λφand determined errorsand tests λ determinederrors of φ and determined λ errors determined SampleSampleSample size size size 900 900’000’000900 ’000 using theusing DGPS theusing system DGPS the systemDGPS inusing 2006. system in the These 2006. DGPS in include These 2006. system include These the in evaluation 2006. include the evaluation These the of selectedevaluationinclude of selected the distribution of evaluation selected distribution distribution of selected distribution ArithmeticArithmeticArithmetic mean mean mean− −0.058− 0.058m m 0.005m 0.005 0.005m m m measuresmeasures and measuresthe and results the and ofresultsmeasures testingthe resultsof testing theand statisticalof the testingthe results statistical fitthe ofof statistical testingφ fitand of λtheφ fit errorsand statisticalof λφ with errorsand fittheλ with errorsof normal φ theand with normal λ theerrors normal with the normal MedianMedianMedian 0.0030.003 0.003m m −m0.007− 0.007− 0.007m m m distribution.distribution. distribution. distribution. RangeRangeRange 7.3957.395 7.395m m m6.167 6.167 6.167m m m VarianceTableVariance 4. TableStatisticalVariance 4.Table Statistical analysis 4. 0.626 0.626Statistical analysisof Tabledistribution 0.626m m analysis of 4. m 0.394 distributionStatistical 0.394 me of0.394masures distribution manalysis mme and asures statistical of me distribution andasures statistical tests and of mestatistical φ testsasures and of λ andφerrorstests and statistical of λusing φ errors and the testsλ using errorsDGPS of the φ usingsystem and DGPS λ the errorsin system DGPS2006. using insystem 2006. the DGPS in 2006. system in 2006. StandardStandard deviation deviation 0.791 0.791m m0.628 0.628m m StandardDistributionDistribution deviation MeasureDistribution Measureφ Error 0.791 Measure Distributionφ Errorλ m Error φ Error Measureλ0.628 ErrorProbability λm ErrorProbabilityφ ErrorDensity Probability Function λDensity Error FunctionDensityfor Probabilityφ Error Function for φ DensityErrorProbability for φ FunctionErrorProbability Density forProbability φ Function Density Error FunctionDensityfor Probabilityλ Error Function for λ ErrorDensity for λ ErrorFunction for λ Error SkewnessSkewness −0.435− 0.435 0.208 0.208 SkewnessSample sizeSample sizeSample − size9000.435 ’000Sample 900’000 size 900 0.208 ’000 900’000 KurtosisArithmeticKurtosisArithmetic Kurtosismean Arithmetic mean −0.058 0.7390.739mmean −0.058Arithmetic 0.005 0.739 m m −0.058 0.6840.005 mean m 0.684 m 0.684 0.005−0.058 m m 0.005 m Median Median Median0.003 m 0.003−0.007 mMedian m0.003 −0.007 m m −0.0070.003 m m Anderson−0.007Anderson m -Darling-Darling RejectReject AndersonAnderson-Darling-Darling No rejectNo reject Range 7.395 m 6.167 m Anderson-Darling Reject Anderson-Darling No reject Range 7.395Range m 6.167Range m7.395 m 6.1677.395 m m 6.167Chi -msquare Chi-square RejectReject Chi-squareChi-square No rejectNo reject Variance Variance Variance0.626 m 0.6260.394 Variancem m0.626 0.394 m m 0.3940.626 m m 0.394Chi-square m Reject Chi-square No reject KolmogorovKolmogorov-Smirnov-Smirnov RejectReject KolmogorovKolmogorov-Smirnov-Smirnov No rejectNo reject Standard StandarddeviationStandard deviation 0.791 deviation m 0.791Standard0.628 m m0.791 deviation 0.628 m m 0.6280.791 m mKolmogorov-Smirnov 0.628 m Reject Kolmogorov-Smirnov No reject SkewnessSkewness Skewness −0.435 −0.4350.208Skewness −0.4350.208 0.208−0.435 0.208 Kurtosis 0.739 0.684 Kurtosis Kurtosis0.739 0.684Kurtosis 0.739Next, Next, 0.684in 0.739Table in Table 50.684 the 5 fitthe between fit between empirical empirical data dataof φ ofand φ andλ errors λ errors and anddistributions distributions otherother than Anderson thanthe normal theAnderson-Darling normalAnderson -distributionDarling distribution -DarlingRejectAnderson forReject -theDarling forAndersonReject DGPS the Anderson-DGPSDarling systemRejectAnderson - Darlingsystem in - NoDarling2006 inrejectAnderson 2006 was No reject -wasassessed.DarlingNo rejectassessed. No reject Next, in Chi Table-square5Chi the-square ﬁtChi - betweensquareReject ChiReject empirical-square Reject Chi -square dataChi -Rejectsquare of Chi ϕ -andsquareNo reject λ errorsChi No- squarerejectandNo reject distributions No reject other thanKolmogorov the normalKolmogorov-SmirnovKolmogorov distribution -Smirnov- SmirnovRejectKolmogorov for Reject the-Kolmogorov Smirnov DGPSReject Kolmogorov- systemSmirnovKolmogorovReject -Smirnov in 2006No- Smirnov Kolmogorovreject was No reject assessed.-SmirnovNo reject No reject TableTable 5. Analysis 5. Analysis of fit ofbetween fit between empirical empirical data ofdata φ andof φ λ and errors λ errors and distributions and distributions other other than thanthe normal the normal distribution distribution for for Next, inNext, Table in Next,5 Tablethe infit 5 Table betweentheNext, fit 5 betweenthe empiricalin fitTable between empirical 5 data the empirical fitof databetweenφ and of data λφ errorsempiricaland of λφ and errorsand data distributions λ anderrors of φdistributions and and λ distributions errors and distributions the DGPSthe DGPS system system in 2006. in 2006. Table 5. Analysis of ﬁt betweenother thanother empirical the than othernormal the data than normaldistribution of theotherϕ normal anddistribution than λfor errorsdistribution thethe normal DGPSfor and the distributionssystem forDGPSdistribution the systeminDGPS 2006 for systemotherinwas the 2006 assessed. DGPS than inwas 2006 thesystemassessed. was normal assessed.in 2006 distribution was assessed. for the DGPS systemBest in 2006. BestFit Distribution Fit Distribution for φ for Error φ Error Best BestFit Distribution Fit Distribution for λ for Error λ Error Table 5. TableAnalysis 5.Table Analysis of fit 5. between Analysis of fitTable between empirical of 5.fit Analysis between empirical data of ofempirical φ fitdata and between of λ φerrorsdata and empirical of λand φ errors and distributions dataλ and errors ofdistributions φ and and other distributions λ errorsthan other the and thannormal otherdistributions the than distributionnormal the other distributionnormal for than distribution the for normal for distribution for the DGPSthe system DGPSthe in system DGPS2006. insystem 2006.the DGPS in 2006. system in 2006. Best Fit Distribution for ϕ Error Best Fit Distribution for λ Error Best FitBest Distribution FitBest Distribution Fit for Distribution φBest Error for Fit φ ErrorDistribution for φ Error for φ ErrorBest FitBest Distribution FitBest Distribution Fit for Distribution λ BestError for Fit λ Error Distribution for λ Error for λ Error

StatisticalStatistical analysis analysis of φ ofand φ andλ errors λ errors presented presented in Tables in Tables 4 and 4 5and allows 5 allows for the for fol-the fol- Statistical analysis of φ and λ errors presented in Tables 4 and 5 allows for the fol- Statisticallowinglowing conclusions:analysisStatistical conclusions: of φ and analysis Statistical λ errors of φ analysispresented and λ errors of in φ Tablesand presented λ errors4 and in 5presented Tables allows 4 for and in the Tables 5 allowsfol- 4 and for 5the allows fol- for the following lowingconclusions: lowingconclusions: conclusions:lowing conclusions: Statistical analysis of ϕ and λ errors presented in Tables4 and5 allows for the following  Central  Central Centraltendency CentralCentral tendency tendency measures tendencytendency measuresCentral: Themeasures measuresmean measures:tendency The values mean: :The The measures of:values meanmeanThe φ and meanvaluesof: valuesλThe φ errors and mean ofvalues λφ areof errors andvalues φvery ofλand areerrors closeφof veryλφand anderrorstoare close ze-λ veryλ errorserrors toare close ze- arevery are to very ze- veryclose close close to to ze-ze- to ze- conclusions:ro, whichro,ro, is which ro,indicativero, which iswhich is indicative indicative of is a indicativero,symmetricalindicative ofwhich a symmetricalof isof a indicative adistributionsymmetricalof symmetrical a symmetrical distribution of aof symmetrical distribution1D distribution position of distribution1D position ofdistributionerrors 1D of position in errors1D the of of positionN 1D in errors-1DS the positionposition N in- S errorsthe errorsN errors-S in in the the in N Nthe-S N-S •and ECentral-andWand directions. Eand--WandW tendency directions. Edirections. E- W-W directions. measures:and E-W directions. The mean values of ϕ and λ errors are very close to zero,  Dispersion Dispersion measures measures: The dispersionDispersion: The dispersion ofmeasures φ and of λ:φ Theerrors and dispersion λ ( serrors) is similar, ( sof) is φ similar, withand λa errors similarwith a( ssimilar) is similar, with a similar  whichDispersion  Dispersion isDispersion indicative measures measuresmeasures of a: symmetricalThe: The: dispersionThe dispersion dispersion distribution of of φ φand ofand λ φ errors ofandλ errors 1D (λs)position iserrors similar,(s) is ( ssimilar, errorswith) is similar,a similar in with the with N-Sa similarand a similar range value,range which value,range indicates whichvalue, rangeindicateswhich that value, thindicatese that use which thof e thatcircular use indicates th ofe circular usemeasures ofthat circular measures th (2DRMS)e use measures of (2DRMS) circular of 2D (2DRMS) posi- measuresof 2D posi- of (2DRMS)2D posi- of 2D position errorE-Wrangetion isdirections. errorrangejustified. value,tion is error justified.value, which is justified.tion which indicateserror indicatesis justified. that thatthe use the ofuse circular of circular measures measures (2DRMS) (2DRMS) of 2D of posi- 2D posi-  •Skewness: DispersiontionSkewness: The errortionSkewness: latitude error Theis measures: justified. latitude andis The Skewness:justified. longitude latitude and The longitude The and dispersionerrors latitudelongitude exhibiterrors and ofexhibitaerrors weaklongitudeϕ and exhibit aasymmetry, weakλ errorserrors a asymmetry, weak exhibit thus (asymmetry,s) isboth a thus similar,weak both asymmetry,thus with both a similarthus both distributions rangeSkewness: distributionsSkewness: value,candistributions be consideredThe whichcan be Thelatitudedistributions consideredcan indicates latitudetobe beconsidered and symmetrical. to can that beandlongitude besymmetrical. theto consideredlongitude be use symmetrical. oferrors circularto errorsbe exhibitsymmetrical. measuresexhibit a weak a (2DRMS)weak asymmetry, asymmetry, of 2D thus position thusboth both   Kurtosis: The latitude and longitude errors are leptokurtic (Kurt > 0), which means Kurtosis:errordistributions The is latitudeKurtosis: justified. andcan The Kurtosis:longitude belatitude considered The anderrors latitudelongitude are to leptoku beand errors symmetrical. longitudertic are (Kurt leptoku errors > 0), rtic whichare ( Kurtleptoku means > 0),rtic which (Kurt means> 0), which means that theydistributions are more concentratedthat can they be are considered aroundmore concentrated the mean to be value symmetrical.around than the the mean normal value distribution than the normal distribution •that they are morethat concentratedthey are more around concentrated the mean around value the than mean the valuenormal than distribution the normal distribution would Skewness:Kurtosis: suggest. wouldKurtosis: wouldsuggest. The The suggest. latitudeThe latitudewould latitude suggest.and and andlongitude longitude longitude errors errors errors are exhibit leptoku are aleptoku weakrtic (rtic asymmetry,Kurt ( Kurt> 0), >which 0), thus which means both means  Statistical distributionsthatStatistical testing: theythatStatistical testing:theyAllare cantests more are testing: All behaveStatistical more testsconcentrated considered shownAll have concentrated tests testing: thatshown have λ toAll erroraroundshownthat be tests symmetrical. λfitsaround errorhavethat the the λfitsnormalshown errormean the the fits meanthatnormaldistribution. value the λ error normal valuedistribution. than fits The distribution. than the normalnormal Thethe normal distribution.The distribution distribution The •inverseKurtosis:would inverserelationshipwouldinverse relationshipsuggest. The cansuggest. relationship latitude be inverse observedcan be and observedrelationshipcan for longitudebe φ observed error. for canφ error. befor errors observed φ error. are for leptokurtic φ error. (Kurt > 0), which means  Fit: StatisticalFit: Statistical distributions  distributions Fit: that Statistical fit that empirical fit distributions empirical data best data that are fitbest beta empirical are ( φ beta error) data (φ anderror) best are and beta (φ error) and  thatStatistical  they StatisticalFit: are Statisticaltesting: more testing: concentrated All distributions tests All testshave that aroundhave shown fit empiricalshown the that mean thatλ data error valueλ best error fits are than the fits beta normal thethe (φ normalnormal error) distribution. and distribution distribution. The The lognormallognormal (λ error) (λ distributions.error)lognormal distributions. These (λ error) distributions These distributions. distributions exhibit These exhibit a much distributions a better much fit better to exhibit fit to a much better fit to would suggest.lognormal (λ error) distributions. These distributions exhibit a much better fit to empiricalinverseempirical datainverse empiricalthan relationship data the relationshipthan normal data empiricalthe than normal distribution.can the bedatacan normal distribution. observed thanbe distribution.observed the normal for φ for distribution. error. φ error. • Statistical testing: All tests have shown that λ error fits the normal distribution. Similar Fit: Similaranalyses StatisticalFit:Similar analyses were Statistical analysescarried were distributionsSimilar carriedout distributionswere with analyses carriedout respect with thatwere out respectto with thatthecarried fit 2Drespect empiricalto fit positiontheout empirical 2D withto theposition error. respect data2D positionTheir error. data to best the results Their error.2D best are position results Their betaare results betaerror. (φ error)Their (φ error) results and and are presentedareThe presentedlognormal inare inverse Tableslognormal presented in 6Tables relationship (andλare in error)7 presented. (6Tables λ and error) 7 distributions.6. and canin distributions.Tables 7. be observed 6 and These 7. Thesefor distributionsϕ error. distributions exhibit exhibit a much a much better better fit to fit to • Fit:empirical Statisticalempirical data distributions datathan thanthe normal the that normal fit distribution. empirical distribution. data best are beta (ϕ error) and lognormal (λ error) distributions. These distributions exhibit a much better fit to empirical data SimilarSimilar analyses analyses were were carried carried out with out with respect respect to the to 2D the position 2D position error. error. Their Their results results than the normal distribution. are presentedare presented in Tables in Tables 6 and 6 7and. 7. Similar analyses were carried out with respect to the 2D position error. Their results are presented in Tables6 and7.

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SensorsSensorsTable 20212021 6.,, 2121Statistical,, xx FORFOR PEERPEER analysis REVIEWREVIEW ofdistribution measures of 2D position error using the DGPS system in 2006. 1212 ofof 2121

Probability Density Function 2D Position Error DistributionTable Measure 6. StatisticaTablel analysis 2D 6. Position Statistica of distribution Errorl analysis measures of distribution of 2D position measures error of 2Dusing position the DGPS error system using thein 2006. DGPS system in 2006. for 2D Position Error Distribution Distribution MeasureDistribution TableTable2D Position 6.6. Measure StatisticaStatistica Error ll 2Danalysisanalysis PositionProbability ofof Errordistributiondistribution Density Probability Function measuresmeasures for Density ofof 2D 2D2D Position positionposition Function Error errorerror for 2D usingusing Position2D thethe Position ErrorDGPSDGPS Error systemsystem Distribution2D inin Position 2006.2006. Error Distribution SampleSample sizesize Sample900 size’000 900’000900’000 ArithmeticArithmeticDistributionDistribution mean mean Arithmetic MeasureMeasure0.875 mean m2D2D PositionPosition 0.875 ErrorError m0.875 m ProbabilityProbability DensityDensity FunctionFunction forfor 2D2D PositionPosition ErrorError 2D2D PositionPosition ErrorError DistributionDistribution MedianMedian SampleSample sizesizeMedian 0.802 m 900900 0.802’’000000 m0.802 m RangeRangeArithmeticArithmetic meanmeanRange 5.993 m 0.8750.875 5.993 mm m5.993 m VarianceVariance MedianMedian Variance0.258 m 0.8020.802 0.258 mm m0.258 m StandardStandard deviation deviationRangeRange Standard 0.508deviation m 5.9935.993 0.508 mm m0.508 m SkewnessSkewness VarianceVarianceSkewness 1.192 0.2580.258 1.192 mm 1.192 KurtosisKurtosisStandardStandard deviationdeviationKurtosis 3.207 0.5080.508 3.207 mm 3.207 2DRMS(2D)Skewness Skewness2DRMS(2D) 2.013 m 1.1921.192 2.013 m 2DRMS(2D) 2.013 m R95(2D) KurtosisKurtosis R95(2D)1.815 m 3.2073.207 1.815 m R95(2D)2DRMS(2D)2DRMS(2D) 2.0132.013 1.815 mm m R95(2D)R95(2D) Table1.815 1.8157. Analysis mm Tableof fit between 7. Analysis empirical of fit between data of 2Dempirical position data error of and2D position distributions error andother distributions than the other than the

normal distributionnormal for the distribution DGPS system for thein 2006. DGPS system in 2006. Table 7. Analysis of ﬁt between empirical data of 2D position error and distributions other than the TableTable 7.7. AnalysisAnalysis ofof fitfit betweenbetween empiricalempirical datadata ofof 2D2D positionposition errorerror andand distributionsdistributions otherother thanthan thethe normal distribution for the DGPS system in 2006. normalnormal distributiondistributionBest forFitfor thetheDistribution DGPSDGPSBest systemsystem Fit for ininDistribution 2D 2006.2006. Position forError 2D 1 Position Error 1 Best Fit Distribution for 2D Position Error 1 BestBest FitFit DistributionDistribution forfor 2D2D PositionPosition ErrorError 11

1 The distribution 1parameters The distribution are generally parameters known are andgenerally will not known be described and will in not detail. be described in detail.

Tables11 TheThe 6 anddistributiondistribution 7 allowsTables parametersparameters for 6 and the 7following allowsareare generallygenerally for conclusions: the knownknown following andand willwill conclusions: notnot bebe describeddescribed inin detail.detail. 1 The distribution parameters are generally known and will not be described in detail.  Please noteTablesTables that 6Please6 thereandand 77 note areallowsallows no that outliersforfor there thethe followingfollowing inare the no measurements outliers conclusions:conclusions: in the under measurements analysis, under which analysis, which indicates the highindicates quality theof the high positioning quality of servicesthe positioning provided services by the provided DGPS system. by the DGPS system. Tables 6 and7 allows for the following conclusions:  The 2DRMSPleasePlease and noteThe note R95 2DRMS that thatvalues there there andare are are similar,R95 no no values outliers outliers which are inconfirms insimilar, the the measurements measurements whichthat the confirms φ and under under λthat error analysis, analysis,the dis- φ and which which λ error dis- • Please note that there are no outliers in the measurements under analysis, which tributionsindicatesindicates have similartributions thethe highhigh distributions. have qualityquality similar ofof thethe distributions. positioningpositioning servicesservices providedprovided byby thethe DGPSDGPS system.system. indicates the high quality of the positioning services provided by the DGPS system.  The distributionTheThe 2DRMS2DRMSThe of 2D distribution andand position R95R95 values valueserror of 2D is, are areposition by similar,similar, its nature, error whichwhich is,asymmetrical, by confirmsconfirms its nature, thatthat hence asymmetrical, thethe the φφ and andbest λ λfit henceerrorerror dis- dis-the best fit • The 2DRMS and R95 values are similar, which conﬁrms that the ϕ and λ error distri- distributionstributionstributions include:distributions havehave beta, similarsimilar gamma, include: distributions.distributions. lognormal, beta, gamma, Rayleigh lognormal, and Weibull Rayleigh distributions. and Weibull distributions. butions haveTheThe distributiondistribution similar distributions. ofof 2D2D positionposition errorerror is,is, byby itsits nature,nature, asymmetrical,asymmetrical, hencehence thethe bestbest fitfit • The distribution of 2D position error is, by its nature, asymmetrical, hence the best ﬁt 3.2. DGPS 2014distributionsdistributions R3.2.esults DGPS 2014 include:include: Results beta,beta, gamma,gamma, lognormal,lognormal, RRayleighayleigh andand WeibullWeibull distributions.distributions. distributions include: beta, gamma, lognormal, Rayleigh and Weibull distributions. Similarly to the Similarly2006 measurements, to the 2006 measurements,the results of the the 2014 results campaign of the 2014analysis campaign are analysis are 3.2.3.2. DGPSDGPS 20142014 RResultsesults presented3.2. DGPS i 2014n an Resultspresentedidentical tabular in an identical form in Ttabularables 8 form–11 below. in Tables 8–11 below. SimilarlySimilarly toto thethe 20062006 measurements,measurements, thethe resultsresults ofof thethe 20142014 campaigncampaign analysisanalysis areare Similarly to the 2006 measurements, the results of the 2014 campaign analysis are presentedpresented iinn anan identicalidentical tabulartabular formform inin TTablesables 88––1111 below.below. Table 8. StatisticalTable analysis 8. Statistical presentedof distribution analysis in anmeasures of identical distribution and tabular statistical measures form tests inand of Tables φstatistical and8 –λ11 errors tests below. usingof φ and the λ DGPS errors system using thein 2014. DGPS system in 2014. Distribution MeasureTableTableDistribution 8.8. StatisticalStatisticalφ Error Measure analysisanalysisλ Error ofof φ distribution distributionErrorProbability λ Error measuresmeasures Density Probability Function andand statisticalstatistical for Density φ Error teststests Function ofof φφProbability andand for λφλ errorsErrorerrors Density usingusingProbability Function thethe DGPSDGPS for Density systemλsystem Error Function inin 2014.2014. for λ Error Sample size Sample900 size’000 900’000 ArithmeticDistributionDistribution mean Arithmetic MeasureMeasure−0.001 m mean φφ−0.001 Error Error m − 0.001λλ ErrorError m −0.001ProbabilityProbability m DensityDensity FunctionFunction forfor φφ ErrorError ProbabilityProbability DensityDensity FunctionFunction forfor λλ ErrorError Median SampleSample sizesize−0.013Median m 0.010900 900m− ’0.013’000000 m 0.010 m RangeArithmeticArithmetic meanmean7.176Range m −−0.0010.0016.153 mm 7.176−−0.0010.001 m mm 6.153 m Variance MedianMedian 0.135 Variance m −−0.0130.0130.063 mm 0.1350.0100.010 m mm 0.063 m Standard deviationRangeRange Standard 0.368 deviationm 7.1767.1760.251 mm m 0.3686.1536.153 m mm 0.251 m Skewness VarianceVarianceSkewness − 0.114 0.1350.1350.216 mm −0.1140.0630.063 mm 0.216 KurtosisStandardStandard deviationdeviationKurtosis6.204 0.3680.36812.956 mm 6.2040.2510.251 mm 12.956 SkewnessSkewness −−0.1140.114 0.2160.216 Anderson -Darling AndersonNo-Darling reject NoAnderson reject -Darling Anderson-RejectDarling Reject KurtosisKurtosis 6.2046.204 12.95612.956 Chi-square Chi-squareReject RejectChi -square Chi-squareReject Reject KolmogorovAnderson-AndersonSmirnovKolmogorov --DarlingDarlingNo -Smirnov reject NoNo rejectrejectKolmogorovNo reject Anderson-AndersonSmirnovKolmogorov --DarlingDarling Reject-Smirnov RejectReject Reject ChiChi--squaresquare RejectReject ChiChi--squaresquare RejectReject KolmogorovKolmogorov--SmirnovSmirnov NoNo rejectreject KolmogorovKolmogorov--SmirnovSmirnov RejectReject

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Table 6.Table Statistica 6. Statistical analysisl analysis of distribution of distribution measures measures of 2D position of 2D position error using error the using DGPS the systemDGPS system in 2006. in 2006.

DistributionDistribution Measure Measure 2D Position 2D Position Error ErrorProbability Probability Density Density Function Function for 2D Positionfor 2D Position Error Error 2D Position2D Position Error Distribution Error Distribution Sample Samplesize size 900’000 900’000 ArithmeticArithmetic mean mean 0.875 m 0.875 m Median Median 0.802 m 0.802 m Range Range 5.993 m 5.993 m VarianceVariance 0.258 m 0.258 m StandardStandard deviation deviation 0.508 m 0.508 m SkewnessSkewness 1.192 1.192 KurtosisKurtosis 3.207 3.207 2DRMS(2D)2DRMS(2D) 2.013 m 2.013 m R95(2D)R95(2D) 1.815 m 1.815 m

Table 7.Table Analysis 7. Analysis of fit between of fit between empirical empirical data of data 2D position of 2D position error and error distributions and distributions other than other the than the normalnormal distribution distribution for the forDGPS the systemDGPS system in 2006. in 2006.

Best FitBest Distribution Fit Distribution for 2D for Position 2D Position Error 1Error 1

1 The distribution1 The distribution parameters parameters are generally are generally known known and will and not will be describednot be described in detail. in detail.

Sensors 2021, 21, x FORSensors PEER 2021 REVIEW, 21, x FOR PEER REVIEWTables Tables 6 and 76 allowsand 7 allows for the for following the following conclusions: conclusions: 13 of 21 13 of 21

 Please Please note that note there that arethere no are outliers no outliers in the in measurements the measurements under under analysis, analysis, which which indicatesindicates the high the quality high quality of the ofpositioning the positioning services services provided provided by the by DGPS the DGPSsystem. system.  The 2DRMSThe 2DRMS and R95 and values R95 values are similar, are similar, which which confirms confirms that the that φ andthe φλ anderror λ dis-error dis- SensorsTable2021, 219., Analysis 31 Table of fit 9.between Analysis empirical of fit between datatributions of empirical φtributions and have λ errors similardata have of similarand distributions. φ anddistributions distributions. λ errors and other distributions than the normal other distributionthan the normal for14 ofdistribution 23 for the DGPS systemthe in 2014.DGPS system in 2014. The distributionThe distribution of 2D positionof 2D position error is, error by itsis, nature,by its nature, asymmetrical, asymmetrical, hence hencethe best the fit best fit distributionsdistributions include: include: beta, gamma, beta, gamma, lognormal, lognormal, Rayleigh Rayleigh and Weibull and Weibull distributions. distributions. Best Fit DistributionBest Fitfor Distributionφ Error for φ Error Best Fit DistributionBest Fitfor Distributionλ Error for λ Error Table 8. Statistical analysis of distribution3.2. measures DGPS3.2. DGPS2014 and R 2014esults statistical R esults tests of ϕ and λ errors using the DGPS system in 2014. SimilarlySimilarly to the to2006 the measurements, 2006 measurements, the results the results of the of2014 the campaign 2014 campaign analysis analysis are are Distribution presentedpresentedProbability in an identical in an identical Density tabular tabular form Function in form Tables in T for8ables–11 below. 8–Probability11 below. Density Function ϕ Error λ Error Measure ϕ Error for λ Error Table 8.Table Statistical 8. Statistical analysis analysis of distribution of distribution measures measures and statistical and statistical tests of tests φ and of λφ errorsand λ usingerrors the using DGPS the systemDGPS system in 2014. in 2014.

Median Arithmetic−0.013Arithmetic mean m mean−0.0010.010 m− 0.001 m−0.001 m m− 0.001 m Median Median −0.013 m− 0.0130.010 m m 0.010 m RangeRange 7.176 Range m Statistical7.176 6.153 m 7.176 m 6.153analysis m m 6.153Statistical mof φ and analysis λ errors of φpresented and λ errors in Tables presented 8 and in 9 Tablesallows 8for and the 9 fol-allows for the fol- VarianceVariance 0.135Variancelowing m 0.135conclusions: 0.063 m 0.135 m0.063lowing m m 0.063 conclusions: m Standard deviationStandard 0.368Standard deviation m deviation 0.368 0.251 m 0.368 m0.251 m m 0.251 m SkewnessSkewness Central −0.114 −tendency0.1140.216 Central0.216 measures tendency: The meanmeasures values: The of mean φ and values λ errors of φare and very λ errorsclose to are ze- very close to ze- Skewness Kurtosis−0.114Kurtosis 6.204 0.216 6.20412.956 12.956 Kurtosis 6.204ro, which 12.956 is indicativero, whichAnderson of isaAnderson symmetricalindicative-Darling- Darling of No distributiona rejectsymmetricalNo reject Andersonof 1DdistributionAnderson position-Darling- Darling errorsof 1DReject positionin theReject N -errorsS in the N-S and E -W directions.and E -WChi directions.-squareChi -square Reject Reject Chi-squareChi -square Reject Reject Anderson-DarlingKolmogorovKolmogorov-Smirnov-Smirnov No reject rejectNo reject KolmogorovAnderson-DarlingKolmogorov-Smirnov-Smirnov RejectReject Reject   Dispersion measures DispersionChi-square: The latitudemeasures error: The hasReject latitude a dispersion error Chi-squarehas (s )a of dispersion about one ( sand)Reject of aabout half one and a half times higher thantimes longitude higher thanerror longitude with a similar error range with aKolmogorov-value. similar range value. Kolmogorov-Smirnov No reject Reject  Skewness: The Skewness:latitude and The longitude latitude errorsand longitude exhibit aerrors verySmirnov weakexhibit asymmetry, a very weak thus asymmetry, thus both distributionsboth can distributions be considered can to be be considered symmetrical. to be symmetrical. Sensors 2021, Sensors21, x FOR 2021Sensors PEER, 21, REVIEWx2021 FOR, Sensors21 PEER, x FOR REVIEW2021 PEER, 21, REVIEWx FOR PEER REVIEW 13 of 21 13 of 21 13 of 21 13 of 21  Kurtosis: The latitudeKurtosis: and The longitude latitude errorsand longitude are leptokurtic errors are(Kurt leptokurtic > 0), which (Kurt means > 0), which means Table 9. Analysis of ﬁt between empiricalthat they data are of moreϕthatand concentrated theyλ errors are and more distributions around concentrated the other mean around than value the the normalthan mean the distribution normalvalue than distribution for the the normal distribution DGPS system in 2014. Table 9. AnalysisTable of9.Table Analysisfit between 9. Analysis ofTable fitempirical wouldbetween of9. Analysisfit data between empiricalsuggest. of ofφ fitempiricaland data betweenwould λThe errorsof φ datavery empiricaland andsuggest. of λ distributionshigh φerrors and data λconcentrationand The errorsof distributionsφ other andvery and λ thandistributions errorshigh the other ofand normalconcentration 1D thandistributions other position distributionthe than normal the other errors distribution normaloffor than 1D distributionthepositionrepresented normalfor distribution forerrors by representedthe for by the the DGPS systemthe DGPS in the 2014. system DGPS insystemthe 2014. kurtosisDGPS in 2014. system value in 2014. [kurtosisKurt () =value 6.204 [ Kurtand (Kurt) = (6.)204 = 12.956] and Kurt needs() = emphasis. 12.956] needs Compared emphasis. to Compared to Best Fit Distribution for ϕ Error Best Fit Distribution for λ Error Best Fit DistributionBest Fit BestDistribution forthe Fit φ Error 2006DistributionBest for measurement Fit φ ErrorDistribution forthe φ Error 2006 for campaign, measurementφ ErrorBest Fit Distribution theBest campaign, system Fit BestDistribution for Fit λ has Error Distribution the Best significantly for systemFit λ Error Distribution for λ hasError increased significantlyfor λ Error this increasedpa- this parameter, which rameter,results in which an increase results in in 2D an positionincrease accuracy.in 2D position accuracy.  Statistical  testing: Statistical The Anderson testing: The-Darling Anderson and Kolmogorov-Darling and-Smirnov Kolmogorov tests- Smirnov have tests have shown that φ errorshown fits thatthe normalφ error distribution.fits the normal However, distribution. all tests However, were rejected all tests for were λ rejected for λ error. error.  Fit: The statistical Fit: distribution The statistical that distribution fits empirical that data fits bestempirical is the datalogistic best distribution is the logistic distribution

(φ and λ errors).(φ and λ errors). Statistical analysisStatisticalStatistical of analysis φ and analysisStatisticalλ of errors φ and ofpresented analysis λφ errorsand λ of presentedinerrors φTables and presented λ 8 in errorsand Tables 9 presentedinallows 8Tables and for 9 8 inallows theand Tables fol- 9 forallows 8 theand forfol- 9 allowsthe fol- for the fol- lowingSimilar conclusions:lowing analyses lowingconclusions: conclusions:Similar werelowing carriedanalyses conclusions: out were with carried respect out to withthe 2D respect position to the error. 2D Theirposition results error. Their results Table 10. Statisticalare analysis Centralpresented of tendency distributionCentral in are Tables Centralmeasures tendency presented measures 10tendency: Central measuresTheand inmean of11 measuresTables tendency 2D.: Thevalues position mean 10: measuresofThe and φ values error meanand 11 λ: using.ofThe valueserrors φ meanandthe ofare λφ DGPS valuesveryerrors and closeλ ofare systemerrors φ veryto and ze- are incloseλ veryerrors 2014. to close ze- are veryto ze- close to zero, which isro, indicative whichro, iswhich ofindicative a symmetrical isro, indicative which of a symmetrical is distributionofindicative a symmetrical distributionof of a 1Dsymmetrical distribution position of 1D errors distribution position of 1D in positionthe errors N of- S1D in errors positionthe N in-S the errors N-S in the N-S and E-W directions.and E-Wand directions. E-W directions.andProbability E-W directions. Density Function 2D Position Error DistributionTable Measure 10. StatisticalTable analysis 2D 10. Position Statistical of distribution Error analysis measures of distribution of 2D position measures error of 2D using position the DGPS error systemusing the in 2014.DGPS system in 2014.  Dispersion Dispersion measures Dispersion: measuresThe latitudeDispersion measures: Thefor error latitude 2D: measuresThe has Position latitudea error dispersion: The has error Error latitudea dispersion( shas) of a abouterror dispersion ( shas one) of a aboutand dispersion(s) aof Distributiononehalf about and (s one) aof half aboutand a onehalf and a half Distribution MeasureDistribution 2D Position Measure Errortimes 2D higher PositiontimesProbability than higher timesErrorlongitude Densitythanhigher timeserlongitudeProbability rorthan Function wihigher longitudeth era similarror than forDensity wi 2D erlongitudeth rorrange Position a Function similarwi thvalue. era Error similarrorrange for wi 2D thvalue. range aPosition similar 2Dvalue. Position rangeError value. Error Distribution2D Position Error Distribution SampleSample size size Sample900 size’000 Skewness: 900’000 Skewness:900The ’000 latitudeSkewness: The and latitude Skewness:longitudeThe latitude and errorslongitudeThe and latitude exhibitlongitude errors and a very exhibiterrorslongitude weak exhibita very asymmetry,errors weaka very exhibit asymmetry, weak thus a very asymmetry, weak thus asymmetry, thus thus ArithmeticArithmetic mean mean Arithmetic0.367 mean m both 0.367 distributions mboth0.367 distributions mboth can bedistributions consideredboth can bedistributions consideredcan to be be symmetrical. considered can to be be symmetrical. considered to be symmetrical. to be symmetrical. MedianMedian Median0.330  m Kurtosis: 0.330 The mKurtosis:0.330 latitude mKurtosis: The and latitude longitudeTheKurtosis: latitude and errorslongitudeThe and latitude arelongitude leptokurticerrors and areerrorslongitude leptokurtic (Kurt are >leptokurticerrors 0), (whichKurt are >leptokurtic means(0),Kurt which > 0), means(whichKurt > means0), which means RangeRange Range5.076 m that 5.076 they are mthat5.076 more they mthat concentrated are they more arethat concentrated more around they concentrated are the more around mean concentrated valuearound the mean than the value aroundthe mean normal than value the the meandistribution than normal value the normaldistribution than the distribution normal distribution would suggest. The very high concentration of 1D position errors represented by the VarianceVariance Variance0.064 m would 0.064 suggest. m0.064 Themwould very suggest.highwould concentration The suggest. very high Theof 1D concentrationvery position high concentrationerrors of 1D represented position of 1D errors by position the represented errors represented by the by the kurtosis value [Kurt() = 6.204  and Kurt () = 12.956] needs emphasis. Compared to Standard deviationStandard deviation0.252 m kurtosis value0.252 [ Kurtmkurtosis ( ) = 6. value204kurtosis and [Kurt Kurt value( )( =) [6.=Kurt 20412.956]( and) = needsKurt6.204( andemphasis.) = 12.956] Kurt( )Comparedneeds = 12.956] emphasis. needsto emphasis.Compared Compared to to Standard deviationthe 0.252 2006 measurement mthe 2006the measurement 2006 campaign, measurementthe 2006 campaign, the measurement system campaign, the has system significantly campaign, the system has significantly the increasedhas system significantly thisincreasedhas significantlypa- increased this pa- thisincreased pa- this pa- Skewness Skewness4.513 4.513 Skewnessrameter, 4.513 whichrameter, resultsrameter, which in an resultswhich rameter,increase inresults an inwhich increase 2D in positionan results increase in 2D in accuracy. positionan in increase 2D position accuracy. in 2D accuracy. position accuracy. Kurtosis Kurtosis54.192 54.192 Kurtosis Statistical 54.192 Statistical testing: Statistical The testing: Anderson Statistical testing: The - AndersonDarling The testing: Anderson and-Darling The Kolmogorov - AndersonDarling and Kolmogorov and--SmirnovDarling Kolmogorov and- testsSmirnov Kolmogorov have-Smirnov tests have- testsSmirnov have tests have 2DRMS(2D) 2DRMS(2D)0.885 m 0.885 m 2DRMS(2D)shown 0.885 that mshown φ error thatshown fits φ the error that normalshown φfits error the distribution. that fitsnormal φ the error normaldistribution. However,fits the distribution. normal allHowever, tests distribution. However, were all rejectedtests allHowever, were tests for rejected λ were all rejectedtests for λ were for rejected λ for λ R95(2D) R95(2D)0.748 m error.0.748 m error. R95(2D)error. 0.748 m error.  Fit: The statistical distribution that fits empirical data best is the logistic distribution  Fit: The statistical Fit:distribution The statisticalFit: that The fits distributionstatistical empirical distribution thatdata fits best empirical isthat the fits logistic dataempirical bestdistribution is data the bestlogistic is the distribution logistic distribution (φ and λ errors).(φ and (λφ errors). and λ errors). (φ and λ errors). Similar analysesSimilar were Similaranalyses carried analyses wereSimilar out carried with were analyses respect carriedout with were to out therespect carried with 2D position respectto out the with 2D to error. positiontherespect 2D Their position to error. resultsthe 2D Their error. position results Their error. results Their results are presentedare inpresented Tablesare presented 10 in and Tablesare 11 inpresented. 10Tables and 1110 in. andTables 11. 10 and 11.

Table 10. StatisticalTable 10. Tableanalysis Statistical 10. of Statistical distributionTableanalysis 10. analysis of Statistical distribution measures of distributionanalysis of measures 2D positionof distribution measures of 2Derror position ofusing measures 2D position theerror DGPS ofusing 2Derror system position the using DGPS in theerror 2014. system DGPS using insystem the 2014. DGPS in 2014. system in 2014.

Distribution DistributionMeasureDistribution 2D Measure Position Distribution Measure Error2D Position 2D MeasureProbability PositionError Error2D DensityProbability Position FunctionProbability Error Density for Function2DDensityProbability Position Function for Error2DDensity Position for Function2D PositionError2D Position for Error2D PositionError2D Position Distribution Error2D PositionError Distribution Error2D Position Distribution Error Distribution Sample size Sample sizeSample 900’000 size Sample900’000 size 900’000 900’000 Arithmetic meanArithmetic Arithmetic mean0.367 mmeanArithmetic 0.367 mmean 0.367 m 0.367 m Median Median Median0.330 m Median0.330 m 0.330 m 0.330 m Range Range Range5.076 m Range5.076 m 5.076 m 5.076 m Variance Variance Variance0.064 m Variance0.064 m 0.064 m 0.064 m Standard deviationStandard Standard deviation0.252 deviation mStandard 0.252 deviation m 0.252 m 0.252 m Skewness SkewnessSkewness 4.513 Skewness4.513 4.513 4.513 Kurtosis Kurtosis Kurtosis54.192 Kurtosis54.192 54.192 54.192 2DRMS(2D) 2DRMS(2D)0.885 m 0.885 m 2DRMS(2D) 2DRMS(2D)0.885 m 0.885 m R95(2D) R95(2D) R95(2D)0.748 m R95(2D)0.748 m 0.748 m 0.748 m

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Table 11. Analysis of ﬁt between empirical data of 2D position error and distributions other than the normal distributionTable 11. Analysis for the DGPS of fit systembetween in empirical 2014. data of 2D position error and distributions other than thethe normalnormal distributiondistribution forfor thethe DGPSDGPS systemsystem inin 20142014.. Best Fit Distribution For 2D Position Error Best Fit Distribution For 2D Position Error

Tables 10 and 11 allows for the following conclusions: Statistical Please analysis note of ϕ thatand thereλ errors are presented no outliers in Tables in the8 measurementsand9 allows for theunder following analysis, which conclusions: indicatesindicates thethe highhigh qualityquality ofof thethe positioningpositioning servicesservices providedprovided byby thethe DGPSDGPS system.system. • Central tendencyThe values measures: of 2DRMS The and mean R95 values are similar. of ϕ and Moreover,λ errors arethe veryvalues close of 2DRMS to zero, and R95 which is indicativeare below of1 m, a symmetrical which proves distribution the very good of 1D accuracy position of errors the system. in theN-S and E-Wdirections. The figure of the 2D position error distribution may suggest that the empirical dis- • Dispersiontributiontribution measures: hashas Theaa “linearlinear latitude trendtrend error”.. However,However, has a dispersion thisthis isis notnot (s )thethe of aboutcase, because one and less a half than 0.17% times higher(1496(1496 than fixes)fixes) longitude ofof thethe studiedstudied error with populationpopulation a similar rangehavehave anan value. errorerror greatergreater thanthan oror equalequal toto 22 m.m. • Skewness:Therefore, The latitude they andcan be longitude considered errors as outliers. exhibita very weak asymmetry, thus both distributions The distribution can be consideredof 2D position to be error symmetrical. is, by its nature, asymmetrical, hence the best fit • Kurtosis:distr Theibutionsibutions latitude include:include: and longitude beta,beta, gamma,gamma, errorsare lognormal,lognormal, leptokurtic RayleighRayleigh (Kurt >andand 0), WeibullWeibull which means distributions.distributions. that they are more concentrated around the mean value than the normal distribution would3.3. suggest. EGNOS The2006 very Results high concentration of 1D position errors represented by the kurtosis valueTwo EGNOS [Kurt(ϕ) measurement = 6.204 and Kurt campaigns(λ) = 12.956] were needs tested emphasis. from two Compared different to implementa- the 2006tiontion measurement phases.phases. TheThe campaign, 20062006 campaigncampaign the system datesdates has fromfrom significantly thethe periodperiod increased whenwhen thethe this systemsystem parameter, waswas notnot fullfully whichoperational results in [[ anitit waswas increase thenthenin inin 2D thethe position initialinitial operationaloperational accuracy. capabilitycapability (IOC)(IOC) phasephase]].. ThisThis cancan bebe in-in- • Statisticalterpretedterpreted testing: asas aaThe periodperiod Anderson-Darling inin whichwhich thethe systemsystem and Kolmogorov-Smirnov waswas notnot fullyfully stablestable testsandand therethere have shownmaymay havehave beenbeen thatsomeϕ error position fits the errors normal classified distribution. as gross. However, However, all tests the were 2014 rejected campaign for λwaserror. made in the • Fit: TheFull statisticalOperational distribution Capability that (FOC). fits empiricalTable 12 presents data best the is theresults logistic of statistical distribution analysis and (ϕ andteststestsλ errors).ofof φ and λ errors determined using the EGNOS system in 2006. These include the Similarevaluation analyses of were selected carried distribution out with measures respect to and the 2Dthe positionresults of error. testing Their the results statistical fit of are presentedφ and in λTables errors 10with and the 11 normal. distribution. Tables 10 and 11 allows for the following conclusions: Table 12. Statistical• analysisPlease of distribution note that theremeasures are and no statistical outliers intests the of measurementsφ and λ errors using under the analysis,EGNOS system which in 2006. Distribution Measure φ Errorindicates λ Error the high qualityProbability of Density the positioning Function forfor services φ Error providedProbability by the Density DGPS Function system. forfor λ Error Sample size • The900 values’’000 of 2DRMS and R95 are similar. Moreover, the values of 2DRMS and R95 Arithmetic mean −0.297are m below 0.026 1 m,m which proves the very good accuracy of the system. Median • 0.194The m figure0.111 of m the 2D position error distribution may suggest that the empirical dis- Range 321.739tribution m 161.565 has m a “linear trend”. However, this is not the case, because less than 0.17% Variance 18.051 m 7.509 m (1496 fixes) of the studied population have an error greater than or equal to 2 m. Standard deviation 4.249 m 2.740 m Skewness −0.567Therefore, 1.410 they can be considered as outliers. Kurtosis • 44.819The distribution68.963 of 2D position error is, by its nature, asymmetrical, hence the best fit distributions include: beta,Anderson gamma,--Darling lognormal, Reject Rayleigh andAnderson Weibull-- distributions.Darling Reject Chi--squaresquare Reject Chi--squaresquare Reject 3.3. EGNOS 2006 Results Kolmogorov--Smirnov Reject Kolmogorov--Smirnov Reject Two EGNOSStudies measurement have shown campaigns a very wide were range tested of from both two φ (321(321 different..739 m) implementation and λ (161(161..565 m) er- phases. Therors. 2006 Therefore, campaign data dates represented from the by period the kurtosis when the for system φ and was λ errors not fully were opera- very concen- tional [ittrated, wastrated, then althoughalthough in the initial thethe averageaverage operational valuesvalues capability ofof bothboth (IOC) variablesvariables phase]. areare This closeclose can toto be zero.zero. interpreted Skewness calcu- as a periodlatedlated in which forfor λ error the system also reached was not a fully high stable value and (1.410). there may have been some position errors classifiedInIn asorderorder gross. toto detdet However,ermine thethe 2014numerical campaign scale was of outlier made inmeasurements the Full Operational that caused this Capabilityanomaly, (FOC). Figure Table 12 4 presents histograms the results ofof statisticalφ and λ errors analysis to make and teststhe number of ϕ and of outliers λ errors determinedvisible. using the EGNOS system in 2006. These include the evaluation of

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Table 11.Table Analysis 11. Analysis of fit between of fit between empirical empirical data of data 2D positionof 2D position error and error distributions and distributions other than other than the normalthe normal distribution distribution for the for DGPS the DGPSsystem system in 2014 in. 2014. Best FitBest Distribution Fit Distribution For 2D For Position 2D Position Error Error

TablesTables 10 and 10 11 and allows 11 allows for the for following the following conclusions: conclusions:  Please Please note that note there that thereare no are outliers no outliers in the in measurements the measurements under under analysis, analysis, which which indicatesindicates the high the quality high quality of the ofpositioning the positioning services services provided provided by the by DGPS the DGPS system. system.  The valuesThe values of 2DRMS of 2DRMS and R95 and are R95 similar. are similar. Moreover, Moreover, the values the values of 2DRMS of 2DRMS and R95 and R95 are beloware below 1 m, which 1 m, which proves proves the very the good very accuracygood accuracy of the ofsystem. the system.  The figureThe figure of the of2D the position 2D position error distributionerror distribution may suggest may suggest that the that empirical the empirical dis- dis- tributiontribution has a has“linear a “ lineartrend” trend. However,”. However, this is this not isthe not case, the becausecase, because less than less 0.17%than 0.17% (1496 (1496fixes) fixes)of the of studied the studied population population have anhave error an greatererror greater than orthan equal or equalto 2 m. to 2 m. Therefore,Therefore, they can they be can considered be considered as outliers. as outliers.  The distributionThe distribution of 2D ofposition 2D position error is,error by itsis, bynature, its nature, asymmetrical, asymmetrical, hence hencethe best the fit best fit distributionsdistributions include: include: beta, gamma,beta, gamma, lognormal, lognormal, Rayleigh Rayleigh and Weibull and Weibull distributions. distributions. Sensors 2021, 21, 31 16 of 23 3.3. EGNOS3.3. EGNOS 2006 R 2006esults R esults Two EGNOSTwo EGNOS measurement measurement campaigns campaigns were testedwere tested from twofrom different two different implementa- implementa- tion phases.tion phases. The 2006 The campaign2006 campaign dates datesfrom thefrom period the period when whenthe system the system was not was full noty fully operationaloperational [it was [it then was in then the ininitial the initialoperational operational capability capability (IOC) (IOC)phase phase]. This] .can This be can in- be in- selected distributionterpretedterpreted as measuresa period as a period in which and in thewhich the resultssystem the system was of testingnot was fully notthe stablefully statistical stableand there and ﬁtmaythere of have mayϕ and beenhaveλ errorsbeen with the normalsome someposition distribution. position errors errors classified classified as gross. as gross. However, However, the 2014 the campaign2014 campaign was mad wase madin thee in the Full OperationalFull Operational Capability Capability (FOC). (FOC). Table Table12 presents 12 presents the results the results of statistical of statistical analysis analysis and and Table 12. Statistical analysis of distributiontests measures oftests φ and of φ andλ and errors statistical λ errors determined determined tests ofusingϕ and usingtheλ EGNOS errorsthe EGNOS usingsystem system the in 2006. EGNOS in These2006. system Theseinclude ininclude 2006.the the evaluationevaluation of selected of selected distribution distribution measures measures and the and results the results of testi ofng testi theng statistical the statistical fit of fit of Distribution φ andφ λ anderrors Probabilityλ errors with the with normal the Density normal distribution. distribution. Function Probability Density Function ϕ Error λ Error Measure for ϕ Error for λ Error Table 12.Table Statistical 12. Statistical analysis analysis of distribution of distribution measures measures and statistical and statistical tests of tests φ and of φλ errorsand λ errorsusing theusing EGNOS the EGNOS system system in 2006. in 2006.

Sample sizeDistributionDistribution Measure 900’000 Measure φ Error φ Errorλ Error λ ErrorProbability Probability Density Density Function Function for φ Error for φ ErrorProbability Probability Density Density Function Function for λ Error for λ Error Arithmetic mean Sample−0.297Sample size msize 0.026900’000 m900 ’000 MedianArithmetic 0.194Arithmetic mean m mean− 0.297 0.111 m−0.297 m 0.026m m0.026 m MedianMedian 0.194 m0.194 m0.111 m0.111 m Range 321.739Range Range m 321.739 161.565321.739 m 161.565 mm 161.565 m m VarianceVariance 18.051Variance m 18.051 7.509 m18.051 m 7.509m m7.509 m StandardStandard deviation deviation 4.249 m4.249 m2.740 m2.740 m Standard deviationSkewness 4.249Skewness m −0.567 2.740 −0.567 m 1.410 1.410 Skewness Kurtosis−0.567Kurtosis 44.819 1.410 44.81968.963 68.963 Kurtosis 44.819 68.963 AndersonAnderson-Darling-Darling Reject Reject AndersonAnderson-Darling-Darling Reject Reject Chi-squareChi- square Reject Reject Chi-squareChi- square Reject Reject Anderson-DarlingKolmogorovKolmogorov-Smirnov-Smirnov RejectReject Reject KolmogorovAnderson-DarlingKolmogorov-Smirnov-Smirnov Reject Reject Chi-square Reject Chi-square Reject StudiesStudies have shownhave shown a very a widevery rangewide rangeof both of φboth (321 φ.739 (321 m).739 and m) λ and (161 λ.565 (161 m).565 er- m) errors. Therefore,rors. Kolmogorov-SmirnovTherefore, data representeddata represented by the by kurtosisReject the kurtosis for φKolmogorov-Smirnov forand φ λ and errors λ errors were verywere concen-very Rejectconcen- trated,trated, although although the average the average values values of both of variablesboth variables are close are toclose zero. to Skewnesszero. Skewness calcu- calcu- lated forlated λ errorfor λ alsoerror reached also reached a high a value high value(1.410). (1.410). Studies haveIn order shownIn orderto det a toermine very determine wide the numerical rangethe numerical of scale both ofscale ϕoutlier(321.739 of outlier measurements m)measurements and λthat(161.565 caused that caused this m) errors. this Therefore,anomaly, dataanomaly, represented Figure Figure 4 presents by4 presents the histograms kurtosis histograms of φ for and ofϕ φ λ andand errors λλ errors toerrors make to makethe were number the very number ofconcentrated, outliers of outliers although thevisible. averagevisible. values of both variables are close to zero. Skewness calculated for λ error also reached a high value (1.410). In order to determine the numerical scale of outlier measurements that caused this Sensors 2021, 21, x FOR PEER REVIEWanomaly, Figure4 presents histograms of ϕ and λ errors to make the number15 of of 21 outliers

visible.

(a) (b)

Figure 4.FigureHistograms 4. Histograms of ϕ (ofa )φ and (a) andλ ( bλ )(b errors) errors using using the EGNOS EGNOS system system in 2006. in 2006. The chart The was chart cut was to show cut tothe shownumber the of number Commented [M3]: 1. please add the caption for each sub- outliers. figures. of outliers. 2. Should the abscissa in the figure b be  errors？ Figure 4 shows that both for  and  errors outliers even by −60–60 m were quite pleaseconfirm. Figure4 shows that both for ϕ and λ errors outliers even by −60–60 m were quite frequent (more than 10 fixes). There is no doubt that the assessment of the statistical dis- Commented [MS4]: Thank You very much. It was corrected. frequenttribution (more of position than 10 errors fixes). from There the EGNOS is no doubt 2006 measurement that the assessment campaign cannot of the be statistical distributionconsidered of representative position errors and no from general the EGNOSconclusions 2006 canmeasurement be drawn from it. campaign Predictably, cannot be consideredstatistical representative testing of the fit andbetween no generalempirical conclusions data of φ and can  errors be drawn with the from normal it. Predictably,dis- statisticaltribution testing showed of a lack the of fit fit. between empirical data of ϕ and λ errors with the normal distributionDespite showed the anomalies a lack of identified fit. in this campaign resulting from the status of the systemDespite (IOC), the the anomalies EGNOS identifiedsystem was intested this in campaign the same resultingway as thefrom DGPS the system. status The of the sys- results are presented in Tables 13–15. tem (IOC), the EGNOS system was tested in the same way as the DGPS system. The results Table 13. Analysis of fit betweenare presented empirical indata Tables of φ and 13 λ– 15errors. and distributions other than the normal distribution for the EGNOS system in 2006.

Best Fit Distribution for φ Error Best Fit Distribution for λ Error

Results presented in Table 13 indicate that the Cauchy distribution is the best fit for  and  errors. Similar to the DGPS system, the analysis was carried out in relation to the 2D position error. The results are presented in Tables 14 and 15.

Table 14. Statistical analysis of distribution measures of 2D position error using the EGNOS system in 2006.

Distribution Measure 2D Position Error Probability Density Function for 2D Position Error 2D Position Error Distribution Sample size 900’000 Commented [M5]: is it correct? is it should be 90,0000 Arithmetic mean 3.033 m Median 1.823 m Commented [MS6]: It is correct. Several days ago, the Range 187.687 m Assistant Editor from Sensors recommended such a notation to me. Please see: Variance 16.447 m https://www.mdpi.com/1424-8220/20/24/7144 Standard deviation 4.056 m Skewness 5.633 Kurtosis 72.045 2DRMS(2D) 8.390 m R95(2D) 9.984 m

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(a) (a) (b) (b) Figure 4. HistogramsFigure of 4. φ Histograms and λ errors of using φ and the λ errorsEGNOS using system the EGNOSin 2006. Thesystem chart in was2006. cut The to chartshow was the cutnumber to show of out- the number of outliers. liers.

Figure 4 showsFigure that both4 shows for that and both  errors for  outliers and  errorseven byoutliers −60–60 even m were by − 60quite–60 m were quite frequent (morefrequent than 10 (fixes).more thanThere 10 is fixes). no doubt There that is theno doubtassessment that the of theassessment statistical of dis- the statistical dis-

tribution of positiontribution errors of position from the errors EGNOS from 2006 the measurementEGNOS 2006 measurementcampaign cannot campaign be cannot be (a) (a) (a) (a) (b) (b) (b) (b) considered representativeconsidered representativeand no general and conclusions no general can conclusions be drawn fromcan be it. drawn Predictably, from it. Predictably, Figure 4. Histograms of φ and λ errors using the EGNOS system in 2006. The chart was cut to show the number of out- Figure 4. Histograms Figureof φ and 4. λHistograms errorsFigure using 4. ofHistograms theφ and EGNOS λ errors of systemφ andusing λin errorsthe 2006. EGNOS usingThe chart systemthe EGNOSwas in cut 2006. systemto showThe inchart the 2006. number was The cut chart of to out- show was thecut numberto show ofthe out- number of outliers. liers. liers. statisticalliers. testingstatistical of the fit testing between of the empirical fit between data empiricalof φ and data errors of φwith and the errors normal with dis- the normal distribution showedtribution a lack showedof fit. a lack of fit. FigureDespite 4 showsFigure the that 4 Figure anomalies showsbothDespite for 4that shows Figure bothand theidentified that  for 4anomalieserrors shows both and outliersfor thatin errors this andbothidentified even  outlierscampaignfor errors by  −and60 evenoutliers in– 60 errorsthis mbyresulting wereeven− 60campaign outliers– 60quiteby m −from60 wereeven–60 resulting mquitebythe were − 60 status– 60quite mfrom wereof the thequite status of the frequentfrequent (more than (morefrequent 10 fixes).than (10 moreThere fixes). than is Thereno 10 doubt fixes). is no that Theredoubt the isassessmentthat no thedoubt assessment thatof the the statistical assessmentof the statistical dis- of the dis- statistical dis- frequentsystem (more (IOC), than 10systemthe fixes). EGNOS There (IOC), issystem no the doubt EGNOS was that thetested system assessment in thewas of same thetested statistical way in theas dis- the same DGPS way system. as the DGPSThe system. The tribution oftribution positiontribution of errors position from oftribution position errors the EGNOS from of errors position the 2006 from EGNOS errors measurement the EGNOS 2006 from measurement the 2006 campaign EGNOS measurement 2006cannot campaign measurement be campaign cannot be cannot campaign be cannot be consideredresults considered representativeare presentedconsideredresults representative and in areconsiderednorepresentative Tables general presented and noconclusions13representative general– and15. in no Tables conclusionsgeneral can and be 13 drawnconclusionsno– general15.can from be drawnconclusions canit. Predictably, be from drawn it.can Predictably,from be drawn it. Predictably, from it. Predictably, Sensors 2021, 21, 31 17 of 23 statistical testingstatistical of statisticalthe testing fit between of testing statisticalthe fit empirical of between the testing fit databetween empirical of ofthe φ fit empiricaland data between  errorsof φ data empiricaland with of  φerrors the and data normal with errorsof φ thedis- and withnormal  errorsthe dis-normal with thedis- normal dis- Table 13. AnalysisTable of fit 13. betweentribution Analysis showedempirical tributionof fit betweena tributionlack showeddata of offit. empirical showed a φ lacktribution and of λ afit. lackerrorsdata showed of of fit.and φa lackand distributions ofλ fit.errors and other distributions than the normal other thandistribution the normal for distribution for Despite the anomalies identified in this campaign resulting from the status of the the EGNOS systemthe inEGNOS 2006. systemDespite inthe 2006. anomalies Despite identified theDespite anomalies in this the campaign identifiedanomalies resulting in identified this campaign from in thethis statusresultingcampaign of thefrom resulting the status from ofthe the status of the system (IOC),system the EGNOS(IOC),system the system(IOC), EGNOSsystem thewas EGNOSsystem(IOC), tested thewas insystem theEGNOS tested same was in system waytested the assame inwasthe the way DGPStested same as system.in theway the DGPS assame Thethe system. wayDGPS as Thesystem. the DGPS The system. The Table 13. AnalysisBest ofFit ﬁt Distribution betweenresultsBest are empirical Fit presentedforresults Distribution φ Errorare dataresults in presented Tables of areϕ for and presentedresults13 in –φ15. Tablesλ Error errors are in presented13 Tables– and15. distributions13 in–15. Tables Best 13 Fit–15. other Distribution thanBest the normalFit for Distribution λ Error distribution for forλ Error the EGNOS system in 2006. Table 13. AnalysisTable 13.of fit TableAnalysis between 13. ofAnalysis empirical fitTable between of13. datafit Analysis empiricalbetween of φ and of empirical datafit λ betweenerrors of φ data andand empirical λofdistributions errors φ and dataand λ errors distributionsof other φ and than λdistributions errors the other normal and than distributions other distribution the normal than the otherdistributionfor normal than distribution the for normal fordistribution for the EGNOS systemthe EGNOS inthe 2006. system EGNOS in system the2006. EGNOS in 2006. system in 2006. Best Fit Distribution for ϕ Error Best Fit Distribution for λ Error Best Fit DistributionBest Fit Distribution Bestfor φ Fit Error Distribution forBest φ Fit Error Distribution for φ Error for φ ErrorBest Fit DistributionBest Fit Distribution Bestfor λ Fit Error Distribution forBest λ FitError Distribution for λ Error for λ Error

Results presentedResults in Table presented 13 indicate in Table that 13 the indicate Cauchy that distribution the Cauchy is distributionthe best fit for is the best fit for

   and  errors. Resultsand presentederrors.Results in presentedResults Table 13presented inindicateResults Table in 13thatpresented Table indicate the 13Cauchy inindicatethat Table the distribution 13Cauchythat indicate the distributionCauchy is thatthe bestthe distribution Cauchy fitis thefor best distribution is thefit for best fit is forthe best fit for  and  errors.Similar and  toerrors. and the  DGPSSimilarerrors. and system, to errors. the DGPSthe analysis system, was the carried analysis out was in relationcarried outto the in relation2D posi- to the 2D posi- Table 14. Statistical analysistionSimilar error. of to distribution the SimilarThe DGPS tionresults toSimilar system,the error. measures DGPS are to the theThe Similarpresented system, analysis DGPS of results 2D to thesystem, position thewas analysis inareDGPS carried Tables the presented error system, analysiswas out 14 usingcarriedin theand relationwas in analysis theout carried15.Tables EGNOStoin therelation was out 142D carriedin systemposi-and torelation the 15.out 2D in toin 2006.posi- therelation 2D posi- to the 2D position error. Thetion results error.tion Theare error. presentedresults Thetion are results error. in presented Tables Theare presentedresults14 in and Tables are15. in presented14 Tables and 15. 14 in and Tables 15. 14 and 15. Table 14. Statistical analysis of distributionProbability measures Density of Function2D position error using2D Positionthe EGNOS Error system in 2006. DistributionTableTable Measure 14. 14. StatisticalTable 14. analysis Table Statisticalanalysis 2D 14. Positionof distributionStatistical analysis ofTable distribution 14.of Erroranalysis distributionmeasures Statistical of distribution measuresof analysis measures2D position of measuresdistributionof error 2D2D position positionusing of 2Dmeasures the error position EGNOS error using of error2D usingsystemthe position usingEGNOS inthe the 2006.error EGNOSsystem EGNOS using in system the2006. system EGNOS in 2006. in system 2006. in 2006. for 2D Position Error Distribution DistributionDistribution Measure MeasureDistributionDistribution Distribution2D2D Measure Position Position Measure MeasureErrorDistribution2D ErrorPosition 2D2DProbability ErrorMeasure PositionPosition Probability DensityErrorProbability2D Error Position Function DensityProbability DensityError forProbability 2DFunction Density PositionProbability for Function DensityErrorfor 2D Density 2DPosition for Position Function 2D Function 2DError Position Position Error for for Error 2DError 2D 2D Position Position DistribuPosition2D 2DError Errortion Position Position Error Distribu Error 2D Errortion Position Distribu 2DDistribu Errortion Position Distribution Error tion Distribution SampleSample size size SampleSample sizeSample900 900 size’000 ’size 000 Sample900 900’000’000 size 900 900’000’000 900’000 Arithmetic meanArithmetic Arithmetic mean3.033 m meanArithmetic 3.033 m mean3.033 m 3.033 m Arithmetic meanArithmetic mean 3.033 m3.033 m ArithmeticMedian mean Median 1.823Median3.033 m m 1.823Median m 1.823 m 1.823 m MedianMedianRange RangeMedian 187.687Range1.823 m m 187.687 1.823Range m 187.687 m1.823 m m 187.687 m RangeRangeVariance VarianceRange 16.447Variance187.687 m m16.447 187.687Variance m 16.447187.687 m m m 16.447 m Standard deviationStandard deviationStandard4.056 deviationm 4.056 m 4.056 m StandardVarianceVariance deviation Variance4.05616.447 m m 16.44716.447 m m Skewness Skewness Skewness5.633 Skewness5.633 5.633 5.633 StandardStandardKurtosis deviation deviation Standard Kurtosis 72.045Kurtosisdeviation4.056 m 72.045Kurtosis 4.056 m72.0454.056 m 72.045 SkewnessSkewness2DRMS(2D) 2DRMS(2D)Skewness2DRMS(2D)8.390 5.633 m 2DRMS(2D)8.390 5.633 m 8.390 5.633 m 8.390 m R95(2D) 9.984R95(2D) m 9.984 m KurtosisKurtosisR95(2D) Kurtosis9.984R95(2D)72.045 m 72.0459.98472.045 m 2DRMS(2D) 2DRMS(2D)8.390 m 8.390 m8.390 m R95(2D)SensorsSensorsR95(2D) 20212021 ,, 2121,, xx R95(2D) FORFOR PEER9.984PEER m REVIEWREVIEW 9.984 m9.984 m 1616 ofof 2121

Table 15. Analysis of ﬁt between empirical data of 2D position error and distributions other than the Table 15. Analysis of fit between empirical data of 2D position error and distributions other than normal distributionTable 15. Analysis for the EGNOS of fit between system empirical in 2006. data of 2D position error and distributions other than thethe normalnormal distributiondistribution forfor thethe EGNOSEGNOS systemsystem inin 20062006.. Best Fit Distribution for 2D Position Error BestBest FitFit DistributionDistribution forfor 2D2D PositionPosition ErrorError

FromFrom TablesTables 1414 andand 15,15, itit followsfollows thatthat aa considerableconsiderable numbernumber ofof outliersoutliers andand thethe lacklack Resultsofof fitfit presented betweenbetween in thethe Table errorserrors 13 indicateandand thethe thatnormalnormal the Cauchydistributiondistribution distribution causedcaused isthethe the valuesvalues best fit ofof for 2DRMS2DRMSϕ andand and λ errors.R95R95 toto differdiffer significantly.significantly. TheThe distributionsdistributions beingbeing thethe bestbest fitfit forfor thethe EGNOSEGNOS 20062006 2D2D Similarpositionposition to the errorerror DGPS are:are: system, beta,beta, exponential,exponential, the analysis wasgamma,gamma, carried lognormallognormal out in relation andand WeibullWeibull to the 2D distributions.distributions. position error. The results are presented in Tables 14 and 15. From3.4.3.4. Tables EGNOSEGNOS 14 and 20142014 15 RR,esultsesults it follows that a considerable number of outliers and the lack of fit betweenTheThe the errorsmeasurementsmeasurements and the normal ofof thethe EGNOSEGNOS distribution systemsystem caused carriedcarried the valuesoutout inin 2014,2014, of 2DRMS whichwhich and areare R95 analysedanalysed inin to differthis significantly.this subsubsection,section, The shouldshould distributions bebe consideredconsidered being the fullyfully best representative,representative, fit for the EGNOS sincesince 2006 inin 2014 2D2014 position thethe systemsystem op-op- error are:eratederated beta, exponential,inin FOCFOC mode.mode. gamma, TablesTables lognormal 1616––1919 presentpresent and the Weibullthe resultsresults distributions. ofof statisticalstatistical analyses.analyses. TheThe methodmethod usedused waswas identicalidentical asas forfor thethe DGPSDGPS 20062006 andand 20142014 studies.studies. 3.4. EGNOS 2014 Results TableTable 16.16. StatisticalStatistical analysisanalysisThe ofof measurements distributiondistribution measuresmeasures of the andand EGNOS statisticalstatistical system teststests of carriedof φφ andand outλλ errorserrors in 2014, usingusing which thethe EGNOSEGNOS are analysed systemsystem inin 2014.2014. in this subsection, should be considered fully representative, since in 2014 the system DistributionDistribution MeasureMeasure φφ ErrorError λλ ErrorError ProbabilityProbability DensityDensity FunctionFunction forfor φφ ErrorError ProbabilityProbability DensityDensity FunctionFunction forfor λλ ErrorError operated in FOC mode. Tables 16–19 present the results of statistical analyses. The method SampleSample sizesize 900900’’000000 ArithmeticArithmetic meanmean used0.0050.005 was mm identical−−0.0040.004 mm as for the DGPS 2006 and 2014 studies. MedianMedian −−0.0570.057 mm −−0.0830.083 mm RangeRange 4.6894.689 mm 2.3602.360 mm VarianceVariance 0.1510.151 mm 0.0550.055 mm StandardStandard deviationdeviation 0.3880.388 mm 0.2350.235 mm SkewnessSkewness 1.5791.579 −−0.2040.204 KurtosisKurtosis 4.3754.375 0.9830.983 AndersonAnderson--DarlingDarling RejectReject AndersonAnderson--DarlingDarling NoNo rejectreject ChiChi--squaresquare RejectReject ChiChi--squaresquare RejectReject KolmogorovKolmogorov--SmirnovSmirnov RejectReject KolmogorovKolmogorov--SmirnovSmirnov NoNo rejectreject

TableTable 17.17. AnalysisAnalysis ofof fitfit betweenbetween empiricalempirical datadata ofof φφ andand λλ errorserrors andand distributionsdistributions otherother thanthan thethe normalnormal distributiondistribution forfor thethe EGNOSEGNOS systemsystem inin 2014.2014. BestBest FitFit DistributionDistribution forfor φφ ErrorError BestBest FitFit DistributionDistribution forfor λλ ErrorError

StatisticalStatistical analysisanalysis ofof φφ andand λλ errorserrors presentedpresented inin TablesTables 1616 andand 1717 allowsallows forfor thethe followingfollowing conclusions:conclusions:  CentralCentral tendencytendency measuresmeasures:: TheThe meanmean valuesvalues ofof φφ andand λλ errorserrors areare veryvery closeclose toto ze-zero,ro, whichwhich isis indicativeindicative ofof aa symmetricalsymmetrical distributiondistribution ofof 1D1D positionposition errorserrors inin thethe NN--SS andand EE--WW directions.directions.  DispersionDispersion measuresmeasures:: TheThe dispersiondispersion ofof φφ errorerror ((ss)) andand thethe rangerange valuevalue areare almostalmost twicetwice thethe valuevalue forfor λλ error.error.  Skewness:Skewness: TheThe latitudelatitude errorerror exhibitsexhibits significantsignificant skewnessskewness toto thethe right,right, whereaswhereas thethe longitudelongitude errorerror exhibitsexhibits slightslight skewnessskewness toto thethe left.left.

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TableTable 15. Analysis 15. Analysis of fit betweenof fit between empirical empirical data ofdata 2D of position 2D position error errorand distributions and distributions other otherthan than the normalthe normal distribution distribution for the for EGNOS the EGNOS system system in 2006 in. 2006. Best BestFit Distribution Fit Distribution for 2D for Position 2D Position Error Error

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Table 15. AnalysisTable 15.Table of Analysis fit 15.between Analysis ofTable fit empirical between 15.of fit Analysis between dataempirical of of empirical2D fit data betweenposition of data2D errorempirical position of 2Dand position distributionserrordata ofand error2D distributions position andother distributions than error other and than distributions other than other than the normalthe distribution FromnormaltheFrom normaldistributionTables for the Tables distributionthe EGNOS14 normalfor and the14 system EGNOS15,distributionforand the it in 15, EGNOSfollows system2006 it for. follows systeminthe 2006that EGNOS .in athat 2006 considerable system. a considerable in 2006. number number of outliers of outliers and theand lack the lack of fitof between fit betweenBest the Fit errors theDistributionBest errors Fit andBest Distribution Fittheand for Distribution 2Dnormal Bestthe Position for normalFit 2D Distributiondistribution forPositionError 2Ddistribution Position Error for caused 2D Error Position caused the Error values the values of 2DRMS of 2DRMS and and R95 toR95 differ to differ significantly. significantly. The distributionsThe distributions being being the bestthe bestfit for fit the for EGNOSthe EGNOS 2006 20062D 2D positionposition error error are: beta,are: beta,exponential, exponential, gamma, gamma, lognormal lognormal and Weibulland Weibull distributions. distributions.

3.4. EGNOS3.4. EGNOS 2014 2014Results Results Sensors 2021, 21, 31 18 of 23

The measurementsThe measurements of the of EGNOS the EGNOS system system carried carried out in out 2014, in 2014, which which are analysed are analysed in in Fromthis TablessubthisFromsection, sub14 Tables Fromandsection, 15, shouldTables14 it and follows Fromshould 1415, beand itTables that followsconsidered 15, be a it considerable14 consideredfollows thatand a15, considerable thatfully it followsnumber a considerable fullyrepresentative, that of numberrepresentative, outliers a considerable number of and outliers sincethe of outliersnumberlack and since in the 2014 andof lackin outliers the 2014 the lack systemandthe the system lack op- op- of fit betweeneratedof fiterated inbetweentheof FOC fiterrors in between FOCthemode. andof errors fit themode.the betweenTables normalanderrors Tablesthe and the16distribution normal– errorsthe19 16 normalpresent –distribution 19and caused present thedistribution the normal the caused results thevalues distribution caused resultsthe of values 2DRMSstatistical the of caused values ofstatistical and2DRMS theofanalyses. 2DRMS valuesand analyses. ofand The 2DRMS methodThe and method Table 16. Statistical analysisR95 of to distributionused differR95used was tosignificantly. R95differ identicalwas measuresto significantly.differ identical R95The significantly.as and todistributions for differ statistical asThe the for significantly.distributions DGPS Thethe being tests distributionsDGPS 2006 ofthe Theϕbeing best2006and distributions fitthe beingλ 2014and errorsfor best the 2014studies. fit usingEGNOS best beingfor studies. thefit the the for2006EGNOS EGNOS bestthe 2D EGNOSfit 2006 systemfor the2D 2006 EGNOS in 2D 2014. 2006 2D position errorposition are:position error beta, are:exponential, errorposition beta, are: exponential, beta, error gamma, exponential, are: lognormalbeta,gamma, exponential, gamma, lognormal and Weibulllognormal gamma, and distributions.Weibull andlognormal Weibull distributions. and distributions. Weibull distributions. Distribution Probability Density Function Probability Density Function for TableTable 16. Statistical 16. Statistical analysisϕ Error analysis of distribution ofλ distributionError measures measures and statistical and statistical tests oftests φ and of φ λ and errors λ errors using using the EGNOS the EGNOS system system in 2014. in 2014. Measure 3.4. EGNOS3.4. 2014 EGNOS3.4. Results EGNOS 2014 R3.4. esults2014 EGNOS R esults 2014for ϕ ResultsError λ Error DistributionDistribution Measure Measure φ ErrorφThe Error measurementsλ ErrorTheλ ErrormeasurementsTheProbability ofmeasurements theProbability EGNOSThe of Density themeasurements system EGNOSof Density the Function EGNOScarried system Function of forthe outsystem carried φ EGNOS in forError 2014, carried φout Error systemwhich in 2014,out Probability are carriedin which Probability2014,analysed out whichareDensity in analysedin 2014, Densityare Function analysed which in Function are for in analysed λ Errorfor λ Error in Sample size 900’000 SampleSample size size this 900sub’section,000this900 sub’000 thisshouldsection, sub section,be should thisconsidered subshould besection, considered fully be consideredshould representative, fully be representative, consideredfully sincerepresentative, infully 2014since representative, the in since 2014system in the 2014 op- systemsince the in system op- 2014 theop- system op- ArithmeticArithmeticArithmetic mean mean mean 0.005 0.005erated mm0.005 in erated −FOCm0.004− 0.004 mode. −erated inm0.004 FOC m Tables min mode. FOCerated 16 mode.Tables–19 in present FOC Tables16– 19mode. the present16 –results19 Tables present the of 16results statistical– the19 present resultsof statistical analyses. ofthe statistical results analyses. The methodof analyses. statistical The method The analyses. method The method used was identical as for the DGPS 2006 and 2014 studies. MedianMedianMedian −0.057−0.057used− m 0.057was −identical m0.083− 0.083−used m0.083 as mwas for m theidenticalused DGPS was as 2006 identical for andthe DGPS2014 as for studies. 2006 the DGPSand 2014 2006 studies. and 2014 studies. RangeRangeRange 4.6894.689 mm4.689 m2.360 2.360 m2.360 m m Table 16. StatisticalTable 16.Table Statisticalanalysis 16. Statistical of Tableanalysis distribution 16. analysis of Statistical distribution measures of distribution analysis measuresand statisticalof distributionmeasures and testsstatistical and ofmeasures statisticalφ andtests λ of anderrors testsφ andstatistical ofusing λ φ errors and the tests λ EGNOSusing errors of φ the andusing system EGNOS λ errorsthe in EGNOS system2014. using systeminthe 2014. EGNOS in 2014. system in 2014. VarianceVarianceVariance 0.1510.151 mm0.151 m0.055 0.055 m0.055 m m StandardDistributionStandard deviationStandardDistribution deviationMeasureDistribution deviation Measure φ Error 0.3880.388 MeasureDistribution φ mλm0.388 ErrorError φm0.235 Measure Error 0.235λProbability Error m0.235 λ m φError m ProbabilityError Density Probability Functionλ ErrorDensity for DensityFunctionProbability φ Error Function for φDensity ErrorProbability for φ Function ErrorProbability Density for Probabilityφ Function Error Density for DensityFunctionProbability λ Error Function for λDensity Error for λ Function Error for λ Error Sample size 900’000 SampleSkewness sizeSkewness Sample 900 size1.579’000 Sample 1.579 size −0.204 900− ’0000.204 900’000 SkewnessArithmetic Arithmeticmean Arithmetic mean0.005 1.579m meanArithmetic0.005−0.004 m m 0.005 mean−−0.004 m0.204 m− 0.0040.005 m m −0.004 m KurtosisKurtosis 4.375 4.375 0.983 0.983 KurtosisMedian Median− 0.057Median 4.375 m −−0.0570.083Median m m− 0.057− 0.083 0.983 m m− 0.083−0.057 m m −0.083 m Range Range 4.689Range m 4.6892.360Range m m 4.689 2.360 m m 2.3604.689 m m AndersonAnderson-Darling2.360Anderson m -Darling-Darling Reject Reject Anderson-DarlingAndersonAnderson-Darling-Darling NoNo reject rejectNo reject Variance Variance0.151Variance m 0.1510.055Variance m m 0.1510.055 m m 0.0550.151 m m 0.055Chi m-square Chi-square Reject Reject Chi-squareChi-square Reject Reject Standard deviation Standard0.388 m deviation0.235 m 0.388 m 0.235Chi-square m Reject Chi-square Reject Standard deviation Standard0.388 deviation m 0.235 m 0.388 m 0.235 m KolmogorovKolmogorov-Smirnov-Smirnov Reject Reject KolmogorovKolmogorov-Smirnov-Smirnov No rejectNo reject Skewness SkewnessSkewness1.579 1.579−0.204Skewness 1.579 −0.204 −0.2041.579 −0.204 Sensors 2021, 21, x FORSensors PEER 2021 REVIEW, 21, x FOR PEER REVIEW Kolmogorov-Smirnov Reject Kolmogorov-Smirnov No reject17 of 21 17 of 21 Kurtosis KurtosisKurtosis 4.375 4.3750.983Kurtosis 4.375 0.983 0.9834.375 0.983 TableTable 17. Analysis 17. Analysis of fit betweenof fit between empiricalAnderson empirical data-DarlingAnderson ofdata φ Anderson and -ofDarling φ λ and -errorsRejectDarlingAnderson λ errors andReject- Darlingdistributions andAnderson Reject distributions -DarlingAnderson Rejectother Anderson-Darling otherthanNo -Darling therejectthanAnderson normal theNo reject- Darlingnormal distributionNo reject distribution No forreject for Sensors 2021, 21 , x FOR PEER REVIEW Chi-square Chi-squareChi -squareReject Chi-Rejectsquare RejectChi-square ChiReject-square Chi -squareReject Chi-Rejectsquare Reject Reject 17 of 21 theSensors EGNOSthe 2021 EGNOS ,system 21, x systemFOR in PEER2014. in REVIEW2014. 17 of 21 Table 17. Analysis of ﬁt between empiricalKolmogorov dataKolmogorov of-Smirnovϕ andKolmogorov -λSmirnoverrorsReject-Smirnov Kolmogorov and distributionsReject-KolmogorovSmirnov Reject Kolmogorov-Smirnov otherRejectKolmogorov than- Smirnov theNo-Smirnov Kolmogorovreject normal No reject distribution-SmirnovNo reject forNo reject the EGNOS systemBest inFitBest 2014. Distribution Fit DistributionKurtosis for φ :for ErrorThe φ ErrorlatitudeKurtosis and: The longitude latitude errorsandBest longitude are FitBest leptokurticDistribution Fit errorsDistribution are( forKurt leptokurtic λ forError> 0), λ Errorwhich ( Kurt means > 0), which means Table 17. AnalysisTable 17.Table of Analysis fit 17.between Analysis ofTable fit empirical between 17.of fit Analysis between dataempirical of of φempirical fit anddata between λ of errors φdata andempirical andof λ φ errors distributionsand data λ and errors of distributions φ and andother λdistributions errorsthan other the and normal than distributions other the distribution than normal the other normaldistribution for than distribution the normalfor fordistribution for the EGNOS system inthe 2014. EGNOSthat system they inare 2014. more that concentrated they are more around concentrated the mean around value the than mean the normalvalue than distribution the normal distribution the EGNOS systemthe in EGNOS 2014. system in 2014. Kurtosis: The latitude and longitude errors are leptokurtic (Kurt > 0), which means Best Fit Distribution forwouldϕ Error suggest.Kurtosiswould : The suggest.latitude and longitude Best Fit Distribution errors are leptokurtic for λ Error (Kurt > 0), which means Best Fit DistributionBest FitBest Distribution for Fit φDistribution ErrorBest for Fitφthat ErrorDistribution for φthey Error are for moreφ Error concentratedBest Fit DistributionBest FitBest Distributionaround for Fit λDistribution Error theBest for mean Fitλ Error Distribution for λvalue Error thanfor λ Error the normal distribution  Statisticalthat testing: theyStatistical are The more Anderson testing: concentrated The-Darling Anderson around and the - KolmogorovDarling mean value and-Smirnov than Kolmogorov the normaltests- Smirnov have distribution tests have would suggest. shown thatwould λ errorshown suggest. fits thatthe normalλ error fitsdistribution. the normal However, distribution. all tests However, were rejected all tests for were φ rejected for φ  Statistical testing: The Anderson-Darling and Kolmogorov-Smirnov tests have error. Statisticalerror. testing: The Anderson-Darling and Kolmogorov-Smirnov tests have shown that λ error fits the normal distribution. However, all tests were rejected for φ  Fit: Statisticalshown distributionsFit: that Statistical λ error fitsthat distributions the fit normalempirical distribution.that data fit empiricalbest However,are lognormal data bestall tests are(φ error)werelognormal rejected and (φ for error) φ and error. logistic (error.λ error) logistic distributions. (λ error) These distributions. distributions These exhibit distributions a much exhibit better fita much to em- better fit to em-  Fit: Statistical distributions that fit empirical data best are lognormal (φ error) and piricalStatistical dataStatistical than analysispirical theanalysis normal ofdata φ ofthanand distribution. φ λandthe errors normalλ errors presented distribution. presented in Tables in Tables 16 and 16 17and allows 17 allows for the for the Statisticallogistic analysis (λStatistical oferror) φ and distributions.analysis λ errors of presented φ and These λ inerrors Tables distributions presented 16 and 17in Tablesallowsexhibit 16for and athe much 17 allows better for thefit to em- Statisticalfollowingfollowing analysis conclusions:Statistical ofconclusions: φ and analysis λ errors of φpresented and λ errors in Tables presented 16 and in 17Tables allows 16 forand the 17 allows for the followingSimilarfollowing conclusions:following toconclusions:pirical the conclusions:DGPSfollowing Similardata thansystem, conclusions:to the normaltheDGPS analysis system, distribution. was the carried analysis out was in relationcarried outto the in 2Drelation posi- to the 2D posi- Table 18. Statistical analysistionCentral error. ofCentral tendency Central distribution The CentralCentral tionresultstendency measures  error. tendency measurestendency are measuresCentral: The measures Thepresented mean measures oftendencymeasuresresults: 2DThe values :positionmean The:in Thearemeasures ofTables: values mean meanTheφpresented and error :mean valuesof λThe18values usingerrorsφ and meanand ofvaluesin are theλ19.φ of Tables values errorsand very φ EGNOS ofandλ closeareoferrors φ18 φ λveryand and system toerrors are ze- closeλ 19.very errorserrors in areto close2014. ze- are very are tovery ze- veryclose close close to ze-ze- to zero, whichro,ro, is which indicativewhichro,Similarro, which whichisis indicative indicativeof to ais ro, symmetrical theindicativeindicative which ofDGPS a ofsymmetrical is of aindicative distribution system,symmetricalaof symmetrical a symmetrical distribution of the aof symmetrical 1D distributionanalysis distribution position of distribution 1D was distributionerrorsposition of 1D carriedof inposition 1Derrorsthe of of Nposition 1D-out Sinerrors theposition position in Nin relation-errors Sthe errors N errors-S in int othe the the in NN the-2DS-S Nposi--S Table 18. StatisticalTable analysisand 18. EStatistical- Wofandtion and directions.distribution E andE error.--W andanalysis directions.Edirections. E-W -TheW measures directions. of andresults distribution Probability E -W of directions.are 2D presented position measures Density error ofin Function2D Tablesusing position the 18 EGNOSand error 19. using system 2D the Position in EGNOS 2014. Error system in 2014. Distribution Measure 2D Position Dispersion Error measures Dispersion: The dispersion measures of: Theφ error dispersion (s) and ofthe φ rangeerror (values) and are the almost range value are almost Dispersion Dispersion measuresDispersionDispersion : measuresThe dispersionmeasuresmeasures: Thefor: Theof: 2DdispersionφThe dispersionerror Position dispersion (s) and of of Errorφthe φerror rangeoferror φ(s )value error and(s) andthe are (s range) almost theand Distributionvaluerange the rangeare value almost value are almost are almost Distribution MeasureDistributionTable 2D Position18. Measure Statisticaltwice Error the 2Danalysistwice value PositionProbability the fortwice of valueλ Errordistribution error.the for value Density twice λ error. forProbability the Function measuresλ valueerror. for forDensity ofλ 2D error.2D Position positionFunction Error errorfor 2D using Position2D the Position ErrorEGNOS Error system Distribution2D Position in 2014. Error Distribution SampleSample size size Sample900 size ’000 Skewness:  900’000twiceSkewness: The 900twice the latitudeSkewness:’000 value The the errorlatitude value TheforSkewness: exhibits λlatitude forerror error. λ Thesignificantexhibits error.error latitude exhibits significant skewness error significant exhibits skewness to the skewnesssignificant right, to the whereas right,to skewness the whereastheright, to whereas the the right, the whereas the ArithmeticArithmeticDistribution mean mean Arithmetic Measure0.363 mean 2Dmlongitude Position 0.363Skewness: longitude error Error m0.363Skewness: longitudeexhibits m error The Probabilityslight exhibits error latitudeThelongitude skewness exhibits latitude slight Density error skewness slightto exhibitserrorthe exhibits Function skewnessleft. toexhibits slight the significantfor left.to skewness 2Dthe significant Positionleft. to skewness the Error left. skewness to 2Dthe to Position right, the right, whereasError whereasDistribution the the MedianMedian Sample sizeMedian 0.304 m 900 0.304longitude’000 m0.304longitude m error error exhibits exhibits slight slight skewness skewness to the to left. the left. RangeRangeArithmetic meanRange 3.071 m 0.363 3.071 m m3.071 m VarianceVariance Median Variance 0.074 m 0.304 0.074 m m0.074 m StandardStandard deviation deviationRangeStandard deviation0.273 m 3.071 0.273 m m0.273 m Skewness Skewness VarianceSkewness 2.548 0.074 2.548 m 2.548 KurtosisStandard deviation 9.920 0.273 m KurtosisKurtosis 9.920 9.920 2DRMS(2D)Skewness 2DRMS(2D) 0.901 m 2.548 0.901 m 2DRMS(2D)Kurtosis 0.9019.920 m R95(2D) KurtosisR95(2D) 0.854 m 9.920 0.854 m R95(2D)2DRMS(2D) 0.901 0.854 m m R95(2D) 0.854 m Table 19. AnalysisTable of fit 19. between Analysis empirical of fit between data of empirical 2D position data error of 2D and position distributions error and other distributions than other than Tablethe normal 19. Analysis distributionthe of normal ﬁt betweenfor the distribution EGNOS empirical system for data the in ofEGNOS 2014 2D position. system error in 2014 and. distributions other than the normal distributionTable 19. Analysis for the EGNOS of fit between system empirical in 2014. data of 2D position error and distributions other than the normal distributionBest Fit for Distribution the EGNOSBest Fit systemfor Distribution 2D in Position 2014. for Error 2D Position Error Best Fit Distribution for 2D Position Error Best Fit Distribution for 2D Position Error

Tables 18 and 19 allows for the following conclusions: Tables 18 and 19 allows for the following conclusions:  Please note thatPlease there note are no that outliers there are in the no outliers measurements in the measurements under analysis, under which analysis, which Tables 18 and 19 allows for the following conclusions: indicates the highindicates quality the of high the positioningquality of the services positioning provided services by the provided EGNOS by sys- the EGNOS sys-  Please note that there are no outliers in the measurements under analysis, which tem. Please tem. note that there are no outliers in the measurements under analysis, which indicates the high quality of the positioning services provided by the EGNOS sys-  The valuesindicates of 2DRMSThe the values highand of R95quality 2DRMS are similar.of andthe R95positioning Moreover, are similar. servicesthe valuesMoreover, provided of 2DRMS the byvalues andthe ofEGNOSR95 2DRMS sys- and R95 tem. are belowtem. 1 m, arewhich below proves 1 m, the which very proves good accuracythe very goodof the accuracy system. of the system.  The values of 2DRMS and R95 are similar. Moreover, the values of 2DRMS and R95  The distributionThe valuesThe of 2Ddistribution of position2DRMS and errorof 2D R95 is, position byare its similar. nature, error Moreover,is, asymmetrical, by its nature, the values henceasymmetrical, ofthe 2DRMS best fithence and theR95 best fit are below 1 m, which proves the very good accuracy of the system. distributionsare below include:distributions 1 m, beta, which gamma, include: proves lognormal, beta, the very gamma, good Rayleigh lognormal, accuracy and Weibullof Rayleigh the system. distributions. and Weibull distributions.  The distribution of 2D position error is, by its nature, asymmetrical, hence the best fit 4. Discussiondistributions 4. Discussion include: beta, gamma, lognormal, Rayleigh and Weibull distributions. In order to assessIn orderwhich to of assess the statistical which of distributions the statistical are distributions the best fit arefor theempirical best fit for empirical 4. Discussion data of DGPS anddata EGNOS of DGPS systems, and EGNOS Table ssystems, 20 and 21 Table summarises 20 and the 21 summariseanalyses and the studies analyses and studies In order to assess which of the statistical distributions are the best fit for empirical carried out. PointsIncarried order (1– 10)toout. assess were Points assignedwhich (1–10) of towere the individual statistical assigned distributions distributionsto individual to aredistributions allow the thebest selection fit to for allow empirical the selection data of DGPS and EGNOS systems, Tables 20 and 21 summarise the analyses and studies of the bestdata fit. of Theof DGPS the distributions best and fit. EGNOS The being distributions systems, the best Table fitbeing weres 20 the andassigned best 21 fitsummarise 10were points. assigned the analyses 10 points. and studies carried out. Points (1–10) were assigned to individual distributions to allow the selection of the best fit. The distributions being the best fit were assigned 10 points.

Sensors 2021, 21, 31 19 of 23

Statistical analysis of ϕ and λ errors presented in Tables 16 and 17 allows for the following conclusions: • Central tendency measures: The mean values of ϕ and λ errors are very close to zero, which is indicative of a symmetrical distribution of 1D position errors in the N-S and E-W directions. • Dispersion measures: The dispersion of ϕ error (s) and the range value are almost twice the value for λ error. • Skewness: The latitude error exhibits significant skewness to the right, whereas the longitude error exhibits slight skewness to the left. • Kurtosis: The latitude and longitude errors are leptokurtic (Kurt > 0), which means that they are more concentrated around the mean value than the normal distribution would suggest. • Statistical testing: The Anderson-Darling and Kolmogorov-Smirnov tests have shown that λ error fits the normal distribution. However, all tests were rejected for ϕ error. • Fit: Statistical distributions that fit empirical data best are lognormal (ϕ error) and logistic (λ error) distributions. These distributions exhibit a much better fit to empirical data than the normal distribution. Similar to the DGPS system, the analysis was carried out in relation to the 2D position error. The results are presented in Tables 18 and 19. Tables 18 and 19 allows for the following conclusions: • Please note that there are no outliers in the measurements under analysis, which indicates the high quality of the positioning services provided by the EGNOS system. • The values of 2DRMS and R95 are similar. Moreover, the values of 2DRMS and R95 are below 1 m, which proves the very good accuracy of the system. • The distribution of 2D position error is, by its nature, asymmetrical, hence the best fit distributions include: beta, gamma, lognormal, Rayleigh and Weibull distributions.

4. Discussion In order to assess which of the statistical distributions are the best fit for empirical data of DGPS and EGNOS systems, Tables 20 and 21 summarise the analyses and studies carried out. Points (1–10) were assigned to individual distributions to allow the selection of the best fit. The distributions being the best fit were assigned 10 points. Sensors 2021, 21, 31 20 of 23

Table 20. Statistical distributions being the best ﬁt for empirical data from DGPS 2006 and 2014 measurement campaigns.

Ranking of the Best Fit Distributions DGPS 2006 DGPS 2014 ϕ Error λ Error 2D Position Error ϕ Error λ Error 2D Position Error 1. Beta 10 pt 1. Lognormal (3P) 10 pt 1. Weibull 10 pt 1. Logistic 10 pt 1. Logistic 10 pt 1. Weibull 10 pt 2. Weibull (3P) 9 pt 2. Logistic 9 pt 2. Beta 9 pt 2. Lognormal (3P) 9 pt 2. Beta 9 pt 2. Lognormal (3P) 9 pt 3. Normal 8 pt 3. Beta 8 pt 3. Gamma (3P) 8 pt 3. Normal 8 pt 3. Normal 8 pt 3. Beta 8 pt 4. Logistic 7 pt 4. Gamma (3P) 7 pt 4. Lognormal (3P) 7 pt 4. Beta 7 pt 4. Lognormal (3P) 7 pt 4. Gamma (3P) 7 pt 5. Lognormal (3P) 6 pt 5. Normal 6 pt 5. Gamma 6 pt 5. Gamma (3P) 6 pt 5. Gamma (3P) 6 pt 5. Rayleigh 6 pt 6. Gamma (3P) 5 pt 6. Weibull (3P) 5 pt 6. Weibull (3P) 5 pt 6. Laplace 5 pt 6. Weibull (3P) 5 pt 6. Weibull (3P) 5 pt 7. Laplace 4 pt 7. Laplace 4 pt 7. Rayleigh 4 pt 7. Cauchy 4 pt 7. Laplace 4 pt 7. Gamma 4 pt 8. Cauchy 3 pt 8. Cauchy 3 pt 8. Rayleigh (2P) 3 pt 8. Weibull (3P) 3 pt 8. Cauchy 3 pt 8. Lognormal 3 pt 9. Chi-square (2P) 2 pt 9. Rayleigh (2P) 2 pt 9. Normal 2 pt 9. Chi-square (2P) 2 pt 9. Rayleigh (2P) 2 pt 9. Rayleigh (2P) 2 pt 10. Rayleigh (2P) 1 pt 10. Expotential (2P) 1 pt 10. Lognormal 1 pt 10. Rayleigh (2P) 1 pt 10. Chi-square (2P) 1 pt 10. Logistic 1 pt

Table 21. Statistical distributions being the best ﬁt for empirical data from EGNOS 2006 and 2014 measurement campaigns.

Ranking of the Best Fit Distributions EGNOS 2006 EGNOS 2006 ϕ Error λ Error 2D Position Error ϕ Error λ Error 2D Position Error 1. Cauchy 10 pt 1. Cauchy 10 pt 1. Lognormal 10 pt 1. Lognormal (3P) 10 pt 1. Logistic 10 pt 1. Lognormal (3P) 10 pt 2. Laplace 9 pt 2. Student’s 9 pt 2. Lognormal (3P) 9 pt 2. Gamma (3P) 9 pt 2. Beta 9 pt 2. Lognormal 9 pt 3. Student’s 8 pt 3. Laplace 8 pt 3. Expotential 8 pt 3. Beta 8 pt 3. Normal 8 pt 3. Gamma (3P) 8 pt 4. Logistic 7 pt 4. Logistic 7 pt 4. Expotential (2P) 7 pt 4. Logistic 7 pt 4. Lognormal (3P) 7 pt 4. Beta 7 pt 5. Lognormal (3P) 6 pt 5. Normal 6 pt 5. Weibull (3P) 6 pt 5. Normal 6 pt 5. Gamma (3P) 6 pt 5. Weibull 6 pt 6. Gamma (3P) 5 pt 6. Gamma (3P) 5 pt 6. Weibull 5 pt 6. Cauchy 5 pt 6. Weibull (3P) 5 pt 6. Weibull (3P) 5 pt 7. Normal 4 pt 7. Beta 4 pt 7. Beta 4 pt 7. Laplace 4 pt 7. Laplace 4 pt 7. Gamma 4 pt 8. Beta 3 pt 8. Lognormal (3P) 3 pt 8. Gamma (3P) 3 pt 8. Weibull (3P) 3 pt 8. Cauchy 3 pt 8. Rayleigh 3 pt 9. Chi-square (2P) 2 pt 9. Chi-square (2P) 2 pt 9. Chi-square 2 pt 9. Rayleigh (2P) 2 pt 9. Chi-square (2P) 2 pt 9. Rayleigh (2P) 2 pt 10. Rayleigh (2P) 1 pt 10. Rayleigh (2P) 1 pt 10. Cauchy 1 pt 10. Chi-square (2P) 1 pt 10. Rayleigh (2P) 1 pt 10. Logistic 1 pt Sensors 2021, 21, 31 21 of 23

The use of distribution gradations (from the best to the worst fit) and assigning points to them made it possible to determine those distributions which present the best fit in three categories: (1) Universal distribution where all the results from Tables 20 and 21 were taken into account for evaluation. Both concerning 1D and 2D position errors. (2) Best fit 1D position error distribution where the fit results for 1D errors were analysed from the following measurement campaign: DGPS 2006, DGPS 2014 and EGNOS 2014. (3) Best fit 2D position error distribution where the fit results for 2D error were analysed from the following measurement campaign: DGPS 2006, DGPS 2014 and EGNOS 2014. In the analyses in points 2 and 3, the results of the EGNOS 2006 measurement campaign were omitted due to its low representativeness. Cumulative results are presented in Table 22.

Table 22. Statistical distributions which were the best ﬁt for the measurement campaigns analysed depending on the position dimension (1D, 2D or 1D and 2D).

Ranking of the Best Fit Distributions 1D Position Error 2D Position Error 1D+2D Position Errors 1. Logistic 53 pt 1. Lognormal (3P) 26 pt 1. Lognormal (3P) 93 pt 2. Beta 51 pt 2. Weibull 26 pt 2. Beta 86 pt 3. Lognormal (3P) 49 pt 3. Beta 24 pt 3. Gamma (3P) 75 pt 4. Normal 44 pt 4. Gamma (3P) 23 pt 4. Logistic 69 pt 5. Gamma (3P) 39 pt 5. Weibull (3P) 15 pt 5. Normal 56 pt 6. Weibull (3P) 30 pt 6. Gamma 14 pt 6. Weibull (3P) 51 pt 7. Laplace 25 pt 7. Rayleigh 13 pt 7. Laplace 42 pt 8. Cauchy 21 pt 8. Lognormal 13 pt 8. Cauchy 42 pt 9. Rayleigh (2P) 9 pt 9. Rayleigh (2P) 7 pt 9. Weibull 31 pt 10. Chi-square (2P) 8 pt 10. Normal 2 pt 10. Lognormal 23 pt

The results presented in Table 22 indicate that: • The universality of the lognormal distribution which approximates both 1D and 2D position errors. • Beta, gamma, logistic and Weibull distributions ﬁt almost as well as the lognormal distribution. • The normal distribution, commonly used for analysing navigation positioning system errors, is only suitable for 1D applications. • The chi-square distribution, which is often recommended for position error analysis (especially 2D), shows no signiﬁcant similarity to empirical data obtained from navigation positioning systems.

5. Conclusions The Gauss distribution is commonly used to present results of accuracy analyses for the position determination by navigation systems. Due to the simplicity of calculations, the special features of standard deviation, as well as the intuitive perception of randomness in statistics to which this distribution corresponds, it is commonly used in science. It should be noted however that the necessary conditions for a random variable to be normally distributed include the independence of measurements and identical conditions of their realisation, which is not the case in the iterative method of determining successive positions, the ﬁltration of coordinates or the dependence of the position error on meteorological conditions. The consistency of ϕ and λ errors was tested on DGPS and EGNOS systems. For each of the systems, the analyses used two measurement campaigns from 2006 and 2014. Sensors 2021, 21, 31 22 of 23

Studies of DGPS (2006 and 2014) and EGNOS (2014) systems confirmed that ϕ and λ errors alternately fit the normal distribution, but also showed that the normal distribution is not an optimal statistical distribution to describe the navigation positioning system errors. The distributions that describe positioning system errors more accurately include: beta, gamma, logistic, lognormal and Weibull distributions. The results of the EGNOS 2006 measurement campaign cannot be considered representative and no general conclusions can be drawn from it. This is due to the fact that the EGNOS system was then in the IOC phase, hence numerous position errors classified as gross have appeared during the measurements (Figure4). The research proved that in order to reliably determine navigation positioning system errors, statistical analyses should be performed using various distributions (by selecting the best one) for a representative sample size. This is the second article in a series of monothematic publications [26], the aim of which will be statistical distribution analysis of navigation positioning system errors. One of the next research issues that has not been studied in this article will be to determine the impact of GNSS errors (ionospheric and tropospheric effects, multipath, noise, etc.) that influence the consistency of the empirical distributions of navigation positioning system errors with theoretical distributions.

Funding: This research was funded from the statutory activities of Gdynia Maritime University, grant number WN/2020/PZ/05. Conﬂicts of Interest: The author declares no conﬂict of interest.

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