I 63/2| C£L ■ 2, bi > 2, n > 2, and the accompanying stochastic model, based on the following assumptions: (V i e W1)(ai ~ N(0, aj)) ; (V(i, j) eW x W2)(Pij ~ N(0,a2)) ; (V(i, j, k) e W1 x W2 x W3) (s ijk ~ N(0, a2)) ; (2a) (V(i, j, k) e W x W2 x W3) (cov(ai, pj) = cov(ai, sijk) = cov(pj., sijk) = 0) ; (V(i, j,k),(p,q,r) e W1 x W2 x W3)((i * p v j * q v k * r) ^ cov(sijk,spqr) = 0) , and, consequently: (V(i, j,k) e W x W2 x W3)(Ajk ~ N(0,al)), (2b) where the following notations were introduced: Ajk — the true error of the fixed double-difference L0 solution for the relative coordinate e, n or u, for an a.. - the joint random effect oftropospheric and ionospheric refraction (the nesting factor); ft. — the random effect of "far-field" multipath (factor nested within ai); and s..k — the purely random error (nested within ftj). In order to make a connection with the presentation below, the next notations are introduced here: N, =1 ^ nj = n,.. 5 = £ a^h = b • N = £ a=11 % "j- <3> Based on (1) and (2a-b), one can write: >2 +a2), (4) where u2a, aft and a2 are the variance components to be estimated. A graphical presentation of the true error statistical distribution for the coordinates e, n and u is given by frequency histograms in Figure 1, Figure 2 and Figure 3 (January 2009 is chosen as an example). It is obvious that the validity of the assumptions given by (2a-b) exists. (V( i, j,k) e W xW2 x W3)(D{A#} = all = aj +a2 Figure 1: Frequency histogram for the true errors of the coordinate e (January 2009) - baseline Bar-Podgorica. | 262 | ANOVA ESTIMATES | 260-271 | GEODETSKI VESTNIK | 63/2 | Figure 2: Frequency histogram for the true errors of the coordinate n (January 2009) - baseline Bar-Podgorica. Figure 3: Frequency histogram for the true errors of the coordinate u (January 2009) - baseline Bar-Podgorica. In this paper, separately for each coordinate, the author divided the set of all true errors for the four-year period (2008-2011) by months (not by days as in his previous research) and then, within each subset established, he carried out the same procedure as described in Andic (2016), with exceptions of its parts regarding the step 8 (related to hypothesis testing of influence of the nesting factor a) and the step 10 (related to outlier detection method used). As for the step 8, i.e. testing for statistical significance of the influence of the nesting factor (a), the following test statistic is used herein (following Sahai and Ojeda, 2005): Fa H0,a = MSN MSB H0,a ~ F(fMSN'B — a)• (5) The sums MSN and MSB, with the corresponding degrees of freedom, are calculated as follows: MSN = (u / u3 )MSA + (1 - u / u3 )MSE, fMSN = MSN (U / U3)2 2 (1 — Ui / u3)2 2 V 1 3 MSA2 +--1-MSE2 i — 1 MSB = with Ya Vbl n-(n—1Ynj ^-u— N—1Ybl Ynj A-L^i=1Lu j=1 nj (nj L^k=1 jk Zuj=1Z^ k=1 j k ijk ' B — a N — B fMSB = B — a d.f, d.f, (5a) (5b) N — k 12 and U3 = k12 — k3 B — a a — 1 where a, B and N are from (1) and (3). Calculation of k12 and k3 was given in Andic (2016). IANC ANOVA ESTIMATES | 260-271 | (5c) I 263 | | 63/2 | GEODETSKI VESTNIK ^ When Fa > F095 (fMSN, B - a), there is a statistically significant influence of the nesting factor. ¡j^ On the other side, as for the step 10, i.e. outlier detection, beside the ''three-sigma'' criterion, the ¡53 PEROBHIK2S method was also used. Thus, a ''fine'' elimination of gross errors was also provided. For ä details regarding the related mathematical tool, see Perovic (2015, 2017). cc cc 2.2 ANOVAestimatesandestimatesoftheirvariancesandcovariances The ANOVA estimates are calculated as follows (see also Andic (2016)): ^ a a 2 a ny A \2 g ^2 2 L<=1 L j=1 L k=1 A 'jk ~ Li=1 L j=1 n9 (L k=1A'jk ) CT., = ms =-----; (6) £ s s N - 5 CC l a=1 l b=1 n-- '(l n=1 a ijk )2 -l a=1 n^li l - ajk )2 - (5 - a m 2 2 '<=1^ j=1 'J v^k=1 >Jk' i—ti=1 ' v=: ct2 = m =-N=k (7) 12 L L NT'( L^ L n= Aijk )2 - N-1(La=1 Lb=1 L j Aijk )2 - (k12 - ¿3^ - (a - 1)m2 (8) N - k1 with the correspondent degrees of freedom: f = N - B (6a) » (MSB - MSE)2 N^ . n , fa =-2-t = /p ; (7a) P MSB2 MSE2 P B - a N - B « = [MSA - (o3/ ot )MSB + (o3/ ot - 1)MSE]2 N(8a) a MSA2 , , ,2 MSB2 , , MSE2 a' -+ (u3 / u,)2-+ (u3 / u, -1)2- a -1 3 B - a 3 1 N - B whereby the latter two, as signified, may be regarded as theoretical ones in the case of a large number of data. The estimates of the ANOVA estimate variances and covariances are calculated as follows (see also Andic (2016)): 2 2ms m 2 =-— ms N - 5 (9) 2(k7 + Nk3 - 2k5 )mß + 4(N - k12 +—(-—-) ' 3 5' p 12'tp MB ,, m22 =-2-N-5--(10) mß (N - k12)2 2 = 2(^1ma +^2mp +^3ms + 2^4mamß + 2^5mqms + 2^6mpms ) (11) m = (N - k1 )2(N -k12)2 (k12 - k3)(5 - a) - (a -1) m m n - k1 ¿22 =--m22 (12) Darko An® | SEZONSKA SPREMENLJIVOST ČASOVNIH VRST GPS-KOORDINAT NA PODLAGI ANOVA-VARIANC | SEASONAL PATTERN IN TIME SERIES OF VARIANCES OF GPS RESIDUAL ERRORS | 264 | ANOVA ESTIMATES | 260-271 | GEODETSKI VESTNIK I 63/2 | K 2 2 m B - a N - k m22 12 K 2( k5 - k7 + kg - k4 4 2(a -1)(B - a) 4 N LH4 + N - B - (N - ki2)(ki2 - k3)mm (N -k{)(N -k12) and then also the correlation coefficient estimates: (13) (14) S K 2 2 _ m ,ma r 2 2 —- ms ,ma m 2 m K 22 ms ,mR r 2 2 =- ms m m 2 m 2 and K 22 mR ,ma r 2 2 =- mP ,ma m 2 m 2 (15) ^ Remark: During the calculation process, the adopted interval-variant B from Andic (2016) was used. 2.3 Testing homogeneity of variances within the entire four-year period considered Bartlett's test (Bartlett, 1937) is used to test if k ANOVA estimates obtained are statistically equal. Namely, for each coordinate (e, n and u), the test is performed particularly for nesting (a) and nested (P) factor, as well as for the pure random effect (e). For that purpose, the following test statistic is used: 3(k i -1)[f t Inm2tk^,(fc,t,p Inm2tp)] 55 Xli I Ho =-—k----- I H0 ~ X2kci -1 (16) 3( kc,i -1) +S ^(1/fc,i, - ) - 1/fci C" where (c, i,p) e {e, n, u} x {e, P, a} x {1, 2, „., kc}, and whereby the following variance with the corresponding degrees of freedom is previously calculated: = fcj Zptl fc,i,pmh,p ' with fc,i =ZP=i fc,i,p d-f> (17) One should note that a good approximation to %2— distribution is achieved when degrees of freedom are greater than or equal to 4 (Bolsev and Smirnov, 1968), and that is largely fulfilled in this study, because one states fcapf > 74, with (c, p) e {e, n, u} x {1, 2, ..., kca} (degrees of freedom for the nesting factor, comparing to those for the nested factor and the purely random effect, always take minimal values). So, if %2; — %1-a.k ._1, then H0 is accepted (all variances are equal). Otherwise, H0 is rejected (there is at least one variance that differs from the others). 3 RESULTS The results obtained by applying the procedure described in Chapter 2 are presented numerically and graphically in the continuation. At first, extreme values of square roots of ANOVA estimates, calculated by months for the coordinates e, n and u, are shown in Table 1. Time series graphs of those square roots are presented in Figure 4. Maximums and minimums of the estimates of the ANOVA estimate variances are presented in Table 2, and there is also Figure 5 representing the corresponding time series. Darko Andič | SEZONSKA SPREMENLJIVOST ČASOVNIH VRST GPS-KOORDINAT NA PODLAGI ANOVA-VARIANC | SEASONAL PATTERN IN TIME SERIES OF VARIANCES OF GPS RESIDUAL ERRORS ANOVA ESTIMATES | 260-265 | | 271 | | 63/2 | GEODETSKI VESTNIK Table 1: Maximums and minimums of the square roots of the ANOVA estimates (in mm). Square Roots Year 2008 Year 2009 Year 2010 Year 2011 of ANOVA Estimates min max min max min max min max mz 3.3 4.6 3.7 4.6 4.1 4.5 3.8 4.5 e m 3.5 5.6 4.0 5.6 4.0 5.7 3.7 5.7 ma 2.4 4.5 2.1 5.0 2.3 5.1 2.1 5.2 m 4.3 6.1 4.9 6.1 5.4 6.2 4.9 5.7 n m 4.9 7.0 5.0 6.7 5.1 7.0 4.6 6.5 m 3.0 7.0 3.1 7.6 3.0 7.1 2.4 7.0 mz 8.3 13.1 8.8 12.4 10.4 12.7 9.4 10.9 u m 9.9 15.4 11.0 15.3 10.8 17.2 10.6 15.6 m 16.6 29.7 15.6 31.1 13.5 34.5 16.4 36.5 Table 2: Maximums and minimums of the estimates of the ANOVA estimate variances (in mm4). Estimates Year 2008 Year 2009 Year 2010 Year 2011 of ANOVA Estimate Variances min max min max min max min max 2 m 2 mt 0.0035 0.0185 0.0069 0.0150 0.0092 0.0133 0.0061 0.0369 e 2 m 2 m, 0.0324 0.2462 0.0626 0.1944 0.0573 0.2162 0.0428 0.2203 2 m 2 ma 0.2038 2.8503 0.1358 3.9016 0.1978 4.3530 0.1548 4.6704 2 m 2 m2 0.0106 0.0544 0.0226 0.0481 0.0279 0.0476 0.0165 0.0950 n 2 m 2 m 0.1157 0.6128 0.1348 0.4349 0.1591 0.5221 0.1157 0.3772 2 m 2 ma 0.5399 18.6456 0.8347 21.1847 0.6035 16.4889 0.4170 16.2761 2 m 2 m2 0.1487 1.1675 0.2367 0.8474 0.4195 0.8608 0.2396 2.0631 u 2 m 2 m. 1.8719 14.0572 3.0859 12.0416 2.9764 16.8701 2.7603 16.4249 2 m 2 ma 482.43 5995.81 344.02 5815.81 208.91 8562.71 460.38 11001.9 Figure 4 provides the information that a seasonal pattern in the time series of (the square roots of) the ANOVA estimates is present. It can be concluded that maximums and minimums are present in the summer and winter period, respectively. This ascertainment is also valid for the time series of the estimates of the ANOVA estimate variances (see Figure 5). The highest values exist for the coordinate u. It is also obvious that values related to the coordinate e are lower than the correspondent ones, which were obtained for the coordinate n. Darko An® | SEZONSKA SPREMENLJIVOST ČASOVNIH VRST GPS-KOORDINAT NA PODLAGI ANOVA-VARIANC | SEASONAL PATTERN IN TIME SERIES OF VARIANCES OF GPS RESIDUAL ERRORS | 266 | ANOVA ESTIMATES | 260-271 | GEODETSKI VESTNIK I 63/2 | Figure 4: Time series of the square roots of the ANOVA estimates (solid - ms; dash - dot - ma). In Figure 5, one can also spot that convincingly highest values of the estimates of the ANOVA estimate variances are related to the coordinate u, especially when considering the nesting factor. On the other hand, the estimates for the coordinate e are with the lowest values, no matter if the purely random effect, the nested or the nesting factor is observed, but very close to the correspondent ones for the coordinate n when considering the nested factor. With the aim to avoid potential doubts regarding the peaks in the diagrams in the first row of Figure 5, connected to January 2011, the author want to emphasize that the correspondent values were obtained on the basis of a significantly decreased number of the true errors due to a larger percentage of rejected data in a ''fine'' (i.e. additional) elimination of gross errors, by using the aforementioned PEROBHIK2S method (45.6%, 44.6% and 65.8% in the case of the coordinates e, n and u, respectively), which led to a decrease in denominator in equation (9), causing, in that way, an increase of the related estimate of the ANOVA estimate variance for the purely random error. The values related to the peaks weren't intentionally omitted in the diagrams to testify how a smaller input dataset affects the estimates. ma 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 | r 2 2 | m ,ma 0.1 0.3 0.1 0.3 0.1 0.4 0.1 0.2 The Bartlett's test was applied within the entire four-year period (2008-2011), primarily for testing all ANOVA estimates (Approach 1), then ANOVA estimates obtained for the same months in different years (January 2008-2011, ... , December 2008-2011) (Approach 2), and, after that, for testing the ANOVA estimates obtained for the winter (January-February-March 2008-2011), spring (months: April-May-June 2008-2011), summer (months: July-August-September 2008-2011) and autumn (months: October-November-December 2008-2011) period separately (Approach 3). The significance level of 0.05 was previously adopted for all approaches, and as a homogeneity indicator, Test Statistic-to -Quantile Ratio, %2 / j21_ak_1 , was chosen. It turned out that for any effect (related to e, P and a) and for any coordinate (n, e and u), there is no equality in Approach 1, and the results for Approach 2 and Approach 3 are presented in Table 4. Darko An® | SEZONSKA SPREMENLJIVOST ČASOVNIH VRST GPS-KOORDINAT NA PODLAGI ANOVA-VARIANC | SEASONAL PATTERN IN TIME SERIES OF VARIANCES OF GPS RESIDUAL ERRORS | 268 | ANOVA ESTIMATES | 260-271 | GEODETSKI VESTNIK I 63/2 | Table 4: Summarized results of Bartlett's test applied in Approach 2 and Approach 3. Test Statistic-to- Approach 2 Approach 3 Quantile Ratio (Extreme and Mean Values) S P a S P a min(X2/ Xa95kJ 33.67 2.75 0.75 102.75 35.94 3.10 e max(x2 / Xo.95;k—1) 688.86 58.93 5.04 387.90 62.21 11.66 (x2 / x2 ) VA A0.95k-1' mean 166.01 20.26 2.65 220.20 49.54 6.58 Homogeneity no no for january no no no min(X° / X°.95kJ 20.59 3.24 0.52 112.85 31.13 4.65 n max(X2 / Xo.95;k—1) 629.15 68.24 9.31 438.64 62.39 12.41 (X2 / X2 ) VA A0.95k_1' mean 193.72 22.04 2.52 253.87 48.37 7.74 Homogeneity no no for march and june no no no min(X2 / X°.95kJ 104.33 1.20 0.56 350.59 35.26 4.61 u max(X2 / X°.95;k—1) 751.91 45.39 7.75 750.22 52.09 13.04 (X2 / X2 ) VA A0.95;k-^ mean 406.93 15.79 3.46 529.15 47.19 8.58 Homogeneity no no for april no no no The values in Table 4 show that the inhomogeneity of the residual error variances is also present between the same months from different years (with an irrelevant exception of only one or two months, listed in Table 4 for the nesting factor, Approach 2), as well as within each of the four seasons. The results of testing in Approach 1 and Approach 2 were somewise expected, but it is not the case when it comes to the results obtained applying Approach 3. 4 DISCUSSION On the basis of the results obtained through the study, it is obvious that the highest values of the ANOVA estimates of the ''far-field'' multipath and the combined tropospheric and ionospheric residuals are present in the summer period and the lowest ones in the winter period, regardless of whether a calendar year with minimal or increased solar activity is considered. Besides, the applied statistical test provided the results proving the inhomogeneity of the residual error variances. It turned out that the seasonal pattern in the time series of the estimates of the ANOVA estimate variances is also present, indicating extreme values that exist in the summer and winter period. In addition, there is no correlation between the ANOVA estimates for the purely random and the combined tropospheric and ionospheric effect at all, and, practically, there is no other correlation. This is stated because the absolute values of the related correlation coefficients do not exceed 9.3%. For a GPS baseline of 40 km in length, located in the mid-latitude region, and four-year period, involving minimal and an increased solar activity, it turned out that, for the coordinates e, n and u, the square roots of the ANOVA estimates for the residuals arising due to the combined tropospheric and ionospheric effects are within intervals 2.1-5.2 mm, 2.4-7.6 mm and 13.5-36.5 mm, respectively. Following the same order in presenting values, for the ''far-field'' residuals we have intervals 3.5-5.7 mm, 4.6-7.0 mm and 9.9-17.2 mm. Darko Andič | SEZONSKA SPREMENLJIVOST ČASOVNIH VRST GPS-KOORDINAT NA PODLAGI ANOVA-VARIANC | SEASONAL PATTERN IN TIME SERIES OF VARIANCES OF GPS RESIDUAL ERRORS ANOVA ESTIMATES | 260-269 | | 271 | I 63/2| GEODETSKI VESTNIK ^ In addition, one should be noted that the part of these results obtained for the joint influence of tropo- LJ_J Cj spheric and ionospheric residuals during the year 2008 (see corresponding values in Table 1), in the case ^ of the coordinates e and n, is in a good agreement with the results from the first stage od the research 3s project presented (see comments in the paragraph below Table 5 in Andic (2016)), where those were obtained by unifying ANOVA estimates, calculated using daily subsets of data, and where intervals 2S 1.8—3.9 mm (for the coordinate e) and 2.2—6.5 mm (for the coordinate n) were established. As one Cl_ ^ can see, practically, there is no significant difference between the results of the ANOVA approach based 3 on the daily and that based on the monthly level when it is about coordinates in the horizontal plane. However, this is not the case with, always critical, coordinate u. Herein we have interval 16.6-29.7 mm, which is not in so good agreement with that established in the previous research, and it is 9.4-20.2 mm. LJ_I ^ That's what was actually expected because, even if we consider a year with the lowest solar activity, there are different atmospheric conditions between different days within such a year and this had a significant impact on unifying ANOVA estimates, since a large percentage, i. e. 35% of those, was rejected through Bartlett's test (see comments above Table 5 in Andic (2016)). As for the multipath residual effect, comparing the results for the year 2008 presented in this paper with those given in the previous author's research, one can be said a large discrepancy exists, especially when it is about the square roots of the ANOVA estimate maximums. Namely, it turned out that the maximums ware increased 67.9%, 70.0% and 93.5% for the coordinates e, n and u, respectively, and, following the same order of presentation, the minimums were decreased 11.4%, 14.3% and 17.2% (to get these values, compare the related values in Table 1 to the corresponding ones for the Variant B in Table 1, Table 2 and Table 3 presented in Andic (2016)). So, in this case, we can conclude that there is a difference between the results obtained in considering daily and monthly data subsets. 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Real Estate Administration of Montenegro Ul. Bracana Bracanoviča b.b, Podgorica, Montenegro e-mail: andjic.darko@gmail.com Darko Andič | SEZONSKA SPREMENLJIVOST ČASOVNIH VRST GPS-KOORDINAT NA PODLAGI ANOVA-VARIANC | SEASONAL PATTERN IN TIME SERIES OF VARIANCES OF GPS RESIDUAL ERRORS ANOVA ESTIMATES | 260-271 | | 271 |