Simulation study for the proposed spatial sampling schemes

Spatially balanced two-stage equal probability sampling

4.4 Sampling from populations with negative spatial autocorrelation

4.4.2 Simulation study for the proposed spatial sampling schemes

In this simulation study, three spatial frames: highly clustered, clustered and sparse, are considered. For each spatial frame, nine variables (or spatial populations) are simulated each of size N = 400: six with positive (z₁, ..., z₆) and three with negative (z₇, ..., z₉) spatial autocorrelation. Variables with positive spatial autocorrelation are simulated in the similar manner as in simulation study from Section4.2.2with spatial trend. In order to simulate variables with negative spatial autocorrelation, first a variable z was simulated from normal distribution with mean 5 and unity standard deviation. Then, spatially balanced samples of size 40,120 and 200 were selected from each frame. Values of the z variable for selected units were replaced by a value 10, which gave z₇, z₈, z₉ for selected values 40,120 and 200 respectively. One realization for z₇, z₈, z₉ for each of three spatial frames is show in Figure 4.2.

From each spatial population, 5000 equal probability samples are selected with respect to sampling fractions given byf = (0.05,0.11) i.e. n= (20,44). AMSE’s of HT-estimator of population totals for each of nine study variables are computed in the same manner as in the simulation study from Section4.2.2. Relative values of AMSE’s under spatial sampling design with respect to SRS are shown in Tables 4.10,4.11 and 4.12for the spatial frames with highly clustered, clustered and sparse spatial configuration of population units.

Results from the Tables 4.10, 4.11 and 4.12 show that proposed sampling schemes tend to be more efficient than spatial balanced sampling designs for populations with negative spatial autocorrelation, but also compromise some efficiency for populations with positive spatial autocorrelation. By increasing sample size, efficiency of the proposed sampling schemes increases; compromise of efficiency tend to decrease for spatial population with linear spatial trend, but it increase for populations with no spatial trend.

4.4. POPULATIONS WITH NEGATIVE SPATIAL AUTOCORRELATION

Figure 4.2: Populations with negative spatial autocorrelations: reading from left to right, top layer showz₇, z₈, z₉ for sparse spatial frame, middle layer shows z₇, z₈, z₉ for clustered spatial frame and bottom layer show z₇, z₈, z₉ for highly clustered spatial frame

One issue with the proposed sampling schemes which might not be acceptable that they can be less efficient than SRS for populations with no spatial trend having low and medium spatial autocorrelations. In this regard, two sampling schemes based on local cube method seems to be less problematic as compared to those based on cube method. On other hand, sampling schemes based on local cube method achieve lesser efficiency for the populations with negative spatial autocorrelation.

A direct comparison with the spatial sampling method (denoted by SPI) from Altieri and Cocchi (2021) may not be possible, as its implementation in R software is not available until now, to our best knowledge. An indirect comparison might be possible by computing the values of MSE’s relative to SRS from the simulation study conducted in the original article, see Table 4.13. A brief description of the simulation already provided earlier in this section. The results in Table4.13 shows that SPI method is always better than SRS

4.4. POPULATIONS WITH NEGATIVE SPATIAL AUTOCORRELATION

and spatially balanced sampling methods for populations with negative spatial autocorre- lation. However, it compromise more in terms of efficiency for populations with positive spatial autocorrelation as compared to proposed sampling strategies. For instance, per- cent gain in efficiency as compared to spatially balanced sampling is computed, using Eq.

(4.4) for SPI spatial sampling method (where MSE_sp denotes minimum MSE achieved by spatially balanced designs) and proposed sampling schemes using Eq. (4.5). Results are shown in Table4.14. For populations with negative spatial spatial autocorrelation, maxi- mum gain under SPI (as compared to spatially balanced sampling) is 48% and maximum loss is 1449% (for population with positive spatial autocorrelation). For the proposed sampling schemes, maximum gain is 26% while maximum loss is 362%.

Percent gain in efficiency = (

1− MSE_{SP I} MSEsp

)

×100 (4.4)

Percent gain in efficiency = (

1− AMSE_proposed AMSE_{P W D}

)

×100 (4.5)

The four proposed strategies are based on a heuristic idea of using eigenvector for spatial sampling, a further exploration about them may provide better results. For instance, selection of appropriate eigenvectors; usually a larger set of eigenvectors characterise neg- ative spatial autocorrelation as compared to those which characterise positive spatial autocorrelation. This was also evident from the simulation studies presented here. An adjustment in the threshold |λ₁/λ_N|>0.25, or selection of particular eigenvectors based on some known information about spatial pattern of the population units might be useful.

4.4. POPULATIONS WITH NEGATIVE SPATIAL AUTOCORRELATION

Table 4.10: Relative AMSE’s for propose sampling strategies and spatial balanced design with respect to SRS for highly clustered spatial populations

n z₁ z₂ z₃ z₄ z₅ z₆ z₇ z₈ z₉

20

SRS 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 GRTS 0.999 0.951 0.607 0.375 0.364 0.275 1.023 1.030 1.032 BBS 1.000 0.994 0.907 0.277 0.276 0.247 1.004 1.004 1.000 LPM1 0.999 0.929 0.463 0.303 0.286 0.167 1.037 1.043 1.043 LPM2 0.999 0.934 0.487 0.302 0.286 0.171 1.033 1.043 1.042 SCPS 0.999 0.935 0.484 0.299 0.284 0.168 1.035 1.042 1.038 DBSS 0.999 0.931 0.462 0.288 0.271 0.151 1.036 1.043 1.043 PWD 0.994 0.848 0.317 0.283 0.248 0.115 1.042 1.047 1.045 LCUBE_(sp,E) 1.008 0.988 0.595 0.287 0.282 0.179 1.025 1.016 0.985 LCUBE_(E,sp) 1.011 0.998 0.614 0.373 0.370 0.270 1.025 1.010 0.973 CUBE_(sp,E) 1.010 1.041 0.957 0.285 0.293 0.264 0.998 0.981 0.946 CUBE_(E,sp) 1.012 1.052 0.976 0.332 0.342 0.315 0.997 0.975 0.935

44

SRS.1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 GRTS.1 0.998 0.914 0.496 0.308 0.288 0.183 1.037 1.057 1.067 BBS.1 1.000 0.994 0.899 0.255 0.253 0.223 1.004 1.005 1.001 LPM1.1 0.999 0.864 0.364 0.273 0.241 0.117 1.066 1.096 1.102 LPM2.1 0.998 0.877 0.385 0.271 0.242 0.120 1.058 1.091 1.100 SCPS.1 0.998 0.879 0.381 0.269 0.240 0.116 1.058 1.089 1.091 DBSS.1 0.998 0.868 0.351 0.258 0.227 0.099 1.064 1.095 1.101 PWD.1 0.992 0.787 0.263 0.247 0.198 0.069 1.073 1.108 1.113 LCUBE_(sp,E) 1.009 0.945 0.457 0.262 0.246 0.124 1.049 1.053 1.023 LCUBE_(E,sp) 1.012 0.953 0.468 0.310 0.296 0.175 1.048 1.050 1.015 CUBE_(sp,E) 1.011 1.045 0.943 0.262 0.270 0.238 1.000 0.979 0.940 CUBE_(E,sp) 1.013 1.052 0.950 0.291 0.300 0.269 0.998 0.977 0.936

4.4. POPULATIONS WITH NEGATIVE SPATIAL AUTOCORRELATION

Table 4.11: Relative AMSE’s for propose sampling strategies and spatial balanced design with respect to SRS for clustered spatial populations

n z₁ z₂ z₃ z₄ z₅ z₆ z₇ z₈ z₉

20

SRS 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 GRTS 1.000 0.974 0.683 0.351 0.345 0.274 1.024 1.032 1.035 BBS 1.000 0.998 0.925 0.263 0.263 0.240 1.003 1.003 1.003 LPM1 0.999 0.965 0.563 0.301 0.293 0.197 1.037 1.044 1.046 LPM2 0.999 0.966 0.586 0.300 0.292 0.200 1.035 1.043 1.045 SCPS 1.000 0.969 0.576 0.291 0.283 0.188 1.035 1.043 1.045 DBSS 1.000 0.965 0.559 0.275 0.267 0.169 1.037 1.044 1.046 PWD 1.000 0.922 0.428 0.246 0.228 0.109 1.045 1.047 1.048 LCUBE_(sp,E) 1.002 1.090 0.861 0.275 0.295 0.235 1.003 0.945 0.863 LCUBE_(E,sp) 1.002 1.109 0.934 0.526 0.551 0.505 0.997 0.930 0.837 CUBE_(sp,E) 1.002 1.095 1.081 0.272 0.294 0.284 0.985 0.932 0.860 CUBE_(E,sp) 1.003 1.111 1.135 0.439 0.464 0.463 0.981 0.919 0.836

44

SRS 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 GRTS 1.000 0.951 0.570 0.288 0.277 0.186 1.040 1.061 1.070 BBS 1.000 0.998 0.914 0.245 0.244 0.220 1.004 1.005 1.003 LPM1 1.000 0.926 0.446 0.269 0.251 0.138 1.067 1.099 1.104 LPM2 1.000 0.932 0.466 0.265 0.250 0.140 1.061 1.094 1.101 SCPS 1.000 0.937 0.454 0.264 0.250 0.135 1.063 1.092 1.097 DBSS 1.000 0.929 0.434 0.252 0.236 0.119 1.065 1.097 1.103 PWD 1.000 0.877 0.343 0.239 0.211 0.086 1.079 1.111 1.114 LCUBE_(sp,E) 1.003 1.091 0.712 0.251 0.271 0.178 1.022 0.959 0.857 LCUBE_(E,sp) 1.003 1.099 0.746 0.389 0.411 0.324 1.020 0.953 0.844 CUBE_(sp,E) 1.004 1.106 1.073 0.249 0.273 0.259 0.985 0.924 0.837 CUBE_(E,sp) 1.003 1.114 1.101 0.353 0.378 0.369 0.983 0.919 0.827

4.4. POPULATIONS WITH NEGATIVE SPATIAL AUTOCORRELATION

Table 4.12: Relative AMSE’s for proposed spatial sampling schemes and spatial balanced design with respect to SRS for sparse spatial populations.

n z₁ z₂ z₃ z₄ z₅ z₆ z₇ z₈ z₉

20

SRS 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 GRTS 1.000 0.990 0.754 0.391 0.388 0.331 1.021 1.030 1.035 BBS 1.000 0.999 0.938 0.273 0.272 0.255 1.003 1.004 1.004 LPM1 1.000 0.985 0.632 0.326 0.322 0.237 1.037 1.044 1.047 LPM2 1.000 0.985 0.645 0.318 0.315 0.232 1.035 1.043 1.046 SCPS 0.999 0.987 0.618 0.298 0.295 0.205 1.038 1.043 1.044 DBSS 0.999 0.986 0.624 0.289 0.285 0.197 1.036 1.044 1.046 PWD 0.999 0.972 0.530 0.258 0.251 0.144 1.042 1.048 1.048 LCUBE_(sp,E) 1.000 1.012 0.699 0.280 0.282 0.204 1.027 1.023 1.005 LCUBE_(E,sp) 1.001 1.019 0.711 0.333 0.337 0.261 1.026 1.018 0.995 CUBE_(sp,E) 1.002 1.027 0.967 0.273 0.279 0.262 0.999 0.985 0.959 CUBE_(E,sp) 1.001 1.031 0.978 0.305 0.311 0.295 0.998 0.983 0.954

44

SRS 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 GRTS 0.999 0.979 0.647 0.323 0.318 0.238 1.038 1.060 1.073 BBS 1.000 0.999 0.929 0.255 0.254 0.235 1.005 1.005 1.005 LPM1 1.000 0.967 0.502 0.282 0.274 0.162 1.069 1.099 1.109 LPM2 1.000 0.968 0.516 0.280 0.272 0.163 1.065 1.095 1.106 SCPS 0.999 0.970 0.490 0.270 0.263 0.146 1.071 1.097 1.103 DBSS 0.999 0.968 0.489 0.262 0.254 0.138 1.066 1.097 1.106 PWD 0.997 0.939 0.397 0.248 0.234 0.103 1.080 1.110 1.116 LCUBE_(sp,E) 1.000 1.009 0.581 0.259 0.261 0.156 1.051 1.059 1.043 LCUBE_(E,sp) 1.001 1.015 0.593 0.295 0.297 0.194 1.049 1.056 1.035 CUBE_(sp,E) 1.001 1.033 0.962 0.257 0.264 0.244 1.001 0.984 0.953 CUBE_(E,sp) 1.002 1.036 0.967 0.280 0.287 0.268 0.999 0.982 0.948

4.4. POPULATIONS WITH NEGATIVE SPATIAL AUTOCORRELATION

Table 4.13: Values of MSE from simulation ofAltieri and Cocchi(2021); relative values of MSE with respect to SRS; and percent gain with respect to spatially balanced sampling.

Value of MSE MSE/MSE_srs

p n SRS SBS SPI SBS SPI Percent gain

Compact 30963 1944 10035 0.063 0.324 -416

Multicluster 31792 18390 25615 0.578 0.806 -39

50 Regular 29924 31112 26939 1.040 0.900 13

Random 30040 31016 24363 1.032 0.811 21

Compact 12026 448 3446 0.037 0.287 -669

Multicluster 11745 4734 4832 0.403 0.411 -2

0.5 125 Regular 11756 12042 7723 1.024 0.657 36

Random 12196 11657 9550 0.956 0.783 18

Compact 5659 139 1135 0.025 0.201 -717

Multicluster 5609 1623 2992 0.289 0.533 -84

250 Regular 5657 5439 2830 0.961 0.500 48

Random 5762 5592 4463 0.970 0.775 20

Compact 22969 1985 8075 0.086 0.352 -307

Multicluster 23459 12228 20761 0.521 0.885 -70

50 Regular 22312 23621 20801 1.059 0.932 12

Random 23028 21939 19939 0.953 0.866 9

0.25 Compact 8864 443 3416 0.050 0.385 -671

Multicluster 8670 3079 6903 0.355 0.796 -124

125 Regular 8693 9200 7234 1.058 0.832 21

Random 9087 8796 8131 0.968 0.895 8

Compact 4288 147 2277 0.034 0.531 -1449

Multicluster 4117 1083 2806 0.263 0.682 -159

250 Regular 4332 4468 4263 1.031 0.984 5

Random 4035 4080 3361 1.011 0.833 18