First we’ll load up the package and some data:
library(forestmangr)
data(exfm1)
data(exfm2)
data(exfm3)
data(exfm4)
data(exfm5)
data_acs_pilot <- as.data.frame(exfm3)
data_acs_def <- as.data.frame(exfm4)
data_ace_pilot <- as.data.frame(exfm1)
data_ace_def <- as.data.frame(exfm2)
data_as <- as.data.frame(exfm5)The objective of this example is to survey an area of 46.8 ha using the simple random sampling method. The aimed error is 20%. 10 plots of 3000 m² each were measured for a pilot inventory. The data collected is shown below:
data_acs_pilot
#> TOTAL_AREA PLOT_AREA VWB
#> 1 46.8 3000 41
#> 2 46.8 3000 33
#> 3 46.8 3000 24
#> 4 46.8 3000 31
#> 5 46.8 3000 10
#> 6 46.8 3000 32
#> 7 46.8 3000 62
#> 8 46.8 3000 16
#> 9 46.8 3000 66
#> 10 46.8 3000 25Now we’ll calculate the inventory variables for a 20% error, considering a finite population with the sprs function. Area values must be inserted in square meters, and total area values must be in hectares:
sprs(data_acs_pilot, "VWB", 3000, 46.8,error = 20, pop = "fin")
#> Variables Values
#> 1 Total number of sampled plots (n) 10.0000
#> 2 Number of maximum plots (N) 156.0000
#> 3 Variance Quoeficient (VC) 53.2670
#> 4 t-student 2.2622
#> 5 recalculated t-student 2.0452
#> 6 Number of samples regarding the admited error 25.0000
#> 7 Variance (S2) 328.0000
#> 8 Standard deviation (s) 18.1108
#> 9 Mean (Y) 34.0000
#> 10 Standard error of the mean (Sy) 5.5405
#> 11 Absolute Error 12.5335
#> 12 Relative Error (%) 36.8634
#> 13 Estimated Total Value (Yhat) 5304.0000
#> 14 Total Error 1955.2326
#> 15 Inferior Confidence Interval (m3) 21.4665
#> 16 Superior Confidence Interval (m3) 46.5335
#> 17 Inferior Confidence Interval (m3/ha) 71.5549
#> 18 Superior Confidence Interval (m3/ha) 155.1118
#> 19 inferior Total Confidence Interval (m3) 3348.7674
#> 20 Superior Total Confidence Interval (m3) 7259.2326With these results, we can see that in order to meet the desired error, we’ll need 15 more samples. After a new survey was done, this are the new data:
data_acs_def
#> TOTAL_AREA PLOT_AREA VWB
#> 1 46.8 3000 41
#> 2 46.8 3000 33
#> 3 46.8 3000 24
#> 4 46.8 3000 31
#> 5 46.8 3000 10
#> 6 46.8 3000 32
#> 7 46.8 3000 62
#> 8 46.8 3000 16
#> 9 46.8 3000 66
#> 10 46.8 3000 25
#> 11 46.8 3000 44
#> 12 46.8 3000 7
#> 13 46.8 3000 57
#> 14 46.8 3000 22
#> 15 46.8 3000 31
#> 16 46.8 3000 40
#> 17 46.8 3000 43
#> 18 46.8 3000 27
#> 19 46.8 3000 17
#> 20 46.8 3000 50
#> 21 46.8 3000 38
#> 22 46.8 3000 20
#> 23 46.8 3000 35
#> 24 46.8 3000 31
#> 25 46.8 3000 26Now the definitive inventory can be done:
sprs(data_acs_def, "VWB", 3000, 46.8, error = 20, pop = "fin")
#> Variables Values
#> 1 Total number of sampled plots (n) 25.0000
#> 2 Number of maximum plots (N) 156.0000
#> 3 Variance Quoeficient (VC) 45.4600
#> 4 t-student 2.0639
#> 5 recalculated t-student 2.0930
#> 6 Number of samples regarding the admited error 20.0000
#> 7 Variance (S2) 226.6933
#> 8 Standard deviation (s) 15.0563
#> 9 Mean (Y) 33.1200
#> 10 Standard error of the mean (Sy) 2.7595
#> 11 Absolute Error 5.6952
#> 12 Relative Error (%) 17.1957
#> 13 Estimated Total Value (Yhat) 5166.7200
#> 14 Total Error 888.4555
#> 15 Inferior Confidence Interval (m3) 27.4248
#> 16 Superior Confidence Interval (m3) 38.8152
#> 17 Inferior Confidence Interval (m3/ha) 91.4159
#> 18 Superior Confidence Interval (m3/ha) 129.3841
#> 19 inferior Total Confidence Interval (m3) 4278.2645
#> 20 Superior Total Confidence Interval (m3) 6055.1755The desired error was met.
The area values can also be inserted as variables:
sprs(data_acs_def, "VWB", "PLOT_AREA", "TOTAL_AREA",
error = 20, pop = "fin")
#> Variables Values
#> 1 Total number of sampled plots (n) 25.0000
#> 2 Number of maximum plots (N) 156.0000
#> 3 Variance Quoeficient (VC) 45.4600
#> 4 t-student 2.0639
#> 5 recalculated t-student 2.0930
#> 6 Number of samples regarding the admited error 20.0000
#> 7 Variance (S2) 226.6933
#> 8 Standard deviation (s) 15.0563
#> 9 Mean (Y) 33.1200
#> 10 Standard error of the mean (Sy) 2.7595
#> 11 Absolute Error 5.6952
#> 12 Relative Error (%) 17.1957
#> 13 Estimated Total Value (Yhat) 5166.7200
#> 14 Total Error 888.4555
#> 15 Inferior Confidence Interval (m3) 27.4248
#> 16 Superior Confidence Interval (m3) 38.8152
#> 17 Inferior Confidence Interval (m3/ha) 91.4159
#> 18 Superior Confidence Interval (m3/ha) 129.3841
#> 19 inferior Total Confidence Interval (m3) 4278.2645
#> 20 Superior Total Confidence Interval (m3) 6055.1755It’s also possible to run multiple simple random sampling inventories. To demonstrate this, we’ll use the example dataset for stratified sampling, but running simple random statistics. We’ll still use the sprs function, but use the .groups argument to run a simple random sampling inventory for each stratum:
sprs(data_ace_def, "VWB", "PLOT_AREA", "STRATA_AREA",
.groups = "STRATA" ,error = 20, pop = "fin")
#> Variables STRATA1 STRATA2 STRATA3
#> 1 Total number of sampled plots (n) 14.0000 20.0000 23.0000
#> 2 Number of maximum plots (N) 144.0000 164.0000 142.0000
#> 3 Variance Quoeficient (VC) 24.4785 15.8269 16.7813
#> 4 t-student 2.1604 2.0930 2.0739
#> 5 recalculated t-student 2.4469 4.3027 4.3027
#> 6 Number of samples regarding the admited error 9.0000 11.0000 12.0000
#> 7 Variance (S2) 2.1829 3.6161 5.3192
#> 8 Standard deviation (s) 1.4774 1.9016 2.3063
#> 9 Mean (Y) 6.0357 12.0150 13.7435
#> 10 Standard error of the mean (Sy) 0.3752 0.3984 0.4402
#> 11 Absolute Error 0.8105 0.8339 0.9130
#> 12 Relative Error (%) 13.4288 6.9409 6.6431
#> 13 Estimated Total Value (Yhat) 869.1429 1970.4600 1951.5739
#> 14 Total Error 116.7157 136.7670 129.6455
#> 15 Inferior Confidence Interval (m3) 5.2252 11.1811 12.8305
#> 16 Superior Confidence Interval (m3) 6.8462 12.8489 14.6565
#> 17 Inferior Confidence Interval (m3/ha) 52.2519 111.8105 128.3048
#> 18 Superior Confidence Interval (m3/ha) 68.4624 128.4895 146.5647
#> 19 inferior Total Confidence Interval (m3) 752.4271 1833.6930 1821.9284
#> 20 Superior Total Confidence Interval (m3) 985.8586 2107.2270 2081.2194The objective of this example is to survey an area using the stratified random sampling method. The area was divided into 3 strata: one with 14.4 ha and 7 plots, another with 16.4 ha and 8 plots, and another with 14.2 ha and 7 plots. The plots have an area of 1000 square meters. In total, 22 plots were sampled for the pilot inventory. The data is shown below:
data_ace_pilot
#> STRATA STRATA_AREA PLOT_AREA VWB
#> 1 1 14.4 1000 7.90
#> 2 1 14.4 1000 3.80
#> 3 1 14.4 1000 4.40
#> 4 1 14.4 1000 6.25
#> 5 1 14.4 1000 5.55
#> 6 1 14.4 1000 8.10
#> 7 1 14.4 1000 6.10
#> 8 2 16.4 1000 10.20
#> 9 2 16.4 1000 15.25
#> 10 2 16.4 1000 13.40
#> 11 2 16.4 1000 13.60
#> 12 2 16.4 1000 14.20
#> 13 2 16.4 1000 9.85
#> 14 2 16.4 1000 10.20
#> 15 2 16.4 1000 11.55
#> 16 3 14.2 1000 10.65
#> 17 3 14.2 1000 12.15
#> 18 3 14.2 1000 14.60
#> 19 3 14.2 1000 10.90
#> 20 3 14.2 1000 16.55
#> 21 3 14.2 1000 17.90
#> 22 3 14.2 1000 13.35We’ll calculate the statistics with an aimed error of 5%, considering a finite population using the strs function. Area values can be inserted as a numeric vector, or as a variable. The plot area must be inserted in square meters, and strata area must be in hectares:
strs(data_ace_pilot, "VWB", 3000, c(14.4, 16.4, 14.2),
strata = "STRATA", error = 5, pop = "fin")
#> $Table1
#> Variables STRATA 1 STRATA 2
#> 1 Plot Area 3000.0000 3000.0000
#> 2 Number of sampled plots per stratum (nj) 7.0000 8.0000
#> 3 Total number of sampled plots (n) 22.0000 22.0000
#> 4 Number of maximum plots per stratum (Nj) 48.0000 54.6667
#> 5 Number of maximum plots (N) 150.0000 150.0000
#> 6 Nj/N Ratio (Pj) 0.3200 0.3644
#> 7 Stratum sum (Eyj) 42.1000 98.2500
#> 8 Stratum quadratic sum (Eyj2) 268.8950 1237.2275
#> 9 Mean of Yi per stratum (Yj) 6.0143 12.2812
#> 10 PjSj2 0.8370 1.5929
#> 11 PjSj 0.5175 0.7619
#> 12 PjYj 1.9246 4.4758
#> 13 t-student 2.0796 2.0796
#> 14 recalculated t-student 2.0129 2.0129
#> 15 Number of samples regarding the admited error 45.0000 45.0000
#> 16 Optimal number of samples per stratum (nj optimal) 11.0000 16.0000
#> 17 Optimal number of samples (n optimal) 46.0000 46.0000
#> 18 Total value of Y per stratum (Yhatj) 288.6857 671.3750
#> STRATA 3
#> 1 3000.0000
#> 2 7.0000
#> 3 22.0000
#> 4 47.3333
#> 5 150.0000
#> 6 0.3156
#> 7 96.1000
#> 8 1365.5500
#> 9 13.7286
#> 10 2.4316
#> 11 0.8760
#> 12 4.3321
#> 13 2.0796
#> 14 2.0129
#> 15 45.0000
#> 16 19.0000
#> 17 46.0000
#> 18 649.8190
#>
#> $Table2
#> Variables value
#> 1 t-student 2.0796
#> 2 Standard error of the mean (Sy) 0.4228
#> 3 Stratified Variance 4.8614
#> 4 Stratified Standard Deviation 2.1554
#> 5 Variance Quoeficient (VC) 20.0829
#> 6 Stratified Mean (Y) 10.7325
#> 7 Absolute Error 0.8793
#> 8 Relative Error (%) 8.1925
#> 9 Estimated Total Value (Yhat) 1609.8798
#> 10 Total Error 131.8894
#> 11 Inferior Confidence Interval (m3) 9.8533
#> 12 Superior Confidence Interval (m3) 11.6118
#> 13 Inferior Confidence Interval (m3/ha) 32.8442
#> 14 Superior Confidence Interval (m3/ha) 38.7060
#> 15 inferior Total Confidence Interval (m3) 1477.9904
#> 16 Superior Total Confidence Interval (m3) 1741.7691Analyzing the first table, we can see that in order to achieve the desired error, we must sample 24 additional plots. 4 in stratum 1, 8 in stratum 2 and 12 in stratum 3.
After a new survey, the new data is shown below:
data_ace_def
#> STRATA STRATA_AREA PLOT_AREA VWB
#> 1 1 14.4 1000 7.90
#> 2 1 14.4 1000 3.80
#> 3 1 14.4 1000 4.40
#> 4 1 14.4 1000 6.25
#> 5 1 14.4 1000 5.55
#> 6 1 14.4 1000 8.10
#> 7 1 14.4 1000 6.10
#> 8 1 14.4 1000 6.60
#> 9 1 14.4 1000 7.40
#> 10 1 14.4 1000 5.35
#> 11 1 14.4 1000 5.90
#> 12 1 14.4 1000 4.65
#> 13 1 14.4 1000 4.25
#> 14 1 14.4 1000 8.25
#> 15 2 16.4 1000 10.20
#> 16 2 16.4 1000 15.25
#> 17 2 16.4 1000 13.40
#> 18 2 16.4 1000 13.60
#> 19 2 16.4 1000 14.20
#> 20 2 16.4 1000 9.85
#> 21 2 16.4 1000 10.20
#> 22 2 16.4 1000 11.55
#> 23 2 16.4 1000 9.25
#> 24 2 16.4 1000 11.30
#> 25 2 16.4 1000 13.95
#> 26 2 16.4 1000 12.70
#> 27 2 16.4 1000 10.15
#> 28 2 16.4 1000 14.90
#> 29 2 16.4 1000 10.80
#> 30 2 16.4 1000 11.55
#> 31 2 16.4 1000 13.90
#> 32 2 16.4 1000 9.20
#> 33 2 16.4 1000 12.45
#> 34 2 16.4 1000 11.90
#> 35 3 14.2 1000 10.65
#> 36 3 14.2 1000 12.15
#> 37 3 14.2 1000 14.60
#> 38 3 14.2 1000 10.90
#> 39 3 14.2 1000 16.55
#> 40 3 14.2 1000 17.90
#> 41 3 14.2 1000 13.35
#> 42 3 14.2 1000 14.90
#> 43 3 14.2 1000 9.70
#> 44 3 14.2 1000 15.20
#> 45 3 14.2 1000 13.45
#> 46 3 14.2 1000 12.40
#> 47 3 14.2 1000 14.45
#> 48 3 14.2 1000 13.55
#> 49 3 14.2 1000 12.30
#> 50 3 14.2 1000 15.65
#> 51 3 14.2 1000 14.20
#> 52 3 14.2 1000 17.80
#> 53 3 14.2 1000 14.80
#> 54 3 14.2 1000 9.35
#> 55 3 14.2 1000 12.60
#> 56 3 14.2 1000 13.80
#> 57 3 14.2 1000 15.85Now we’ll run the inventory again, this time with the definitive data:
strs(data_ace_def, "VWB", "PLOT_AREA", "STRATA_AREA",
strata = "STRATA", error = 5, pop = "fin")
#> $Table1
#> Variables STRATA 1 STRATA 2
#> 1 Plot Area 1000.0000 1000.0000
#> 2 Number of sampled plots per stratum (nj) 14.0000 20.0000
#> 3 Total number of sampled plots (n) 57.0000 57.0000
#> 4 Number of maximum plots per stratum (Nj) 144.0000 164.0000
#> 5 Number of maximum plots (N) 450.0000 450.0000
#> 6 Nj/N Ratio (Pj) 0.3200 0.3644
#> 7 Stratum sum (Eyj) 84.5000 240.3000
#> 8 Stratum quadratic sum (Eyj2) 538.3950 2955.9100
#> 9 Mean of Yi per stratum (Yj) 6.0357 12.0150
#> 10 PjSj2 0.6985 1.3179
#> 11 PjSj 0.4728 0.6930
#> 12 PjYj 1.9314 4.3788
#> 13 t-student 2.0032 2.0032
#> 14 recalculated t-student 2.0141 2.0141
#> 15 Number of samples regarding the admited error 46.0000 46.0000
#> 16 Optimal number of samples per stratum (nj optimal) 12.0000 17.0000
#> 17 Optimal number of samples (n optimal) 47.0000 47.0000
#> 18 Total value of Y per stratum (Yhatj) 869.1429 1970.4600
#> STRATA 3
#> 1 1000.0000
#> 2 23.0000
#> 3 57.0000
#> 4 142.0000
#> 5 450.0000
#> 6 0.3156
#> 7 316.1000
#> 8 4461.3350
#> 9 13.7435
#> 10 1.6785
#> 11 0.7278
#> 12 4.3368
#> 13 2.0032
#> 14 2.0141
#> 15 46.0000
#> 16 18.0000
#> 17 47.0000
#> 18 1951.5739
#>
#> $Table2
#> Variables value
#> 1 t-student 2.0032
#> 2 Standard error of the mean (Sy) 0.2339
#> 3 Stratified Variance 3.6949
#> 4 Stratified Standard Deviation 1.8936
#> 5 Variance Quoeficient (VC) 17.7851
#> 6 Stratified Mean (Y) 10.6471
#> 7 Absolute Error 0.4685
#> 8 Relative Error (%) 4.4003
#> 9 Estimated Total Value (Yhat) 4791.1768
#> 10 Total Error 210.8250
#> 11 Inferior Confidence Interval (m3) 10.1786
#> 12 Superior Confidence Interval (m3) 11.1156
#> 13 Inferior Confidence Interval (m3/ha) 101.7856
#> 14 Superior Confidence Interval (m3/ha) 111.1556
#> 15 inferior Total Confidence Interval (m3) 4580.3518
#> 16 Superior Total Confidence Interval (m3) 5002.0018The desired error was met.
Now we’ll survey an area of 18 hectares in which 18 plots of 200 m² each were systematically sampled:
data_as
#> TOTAL_AREA PLOT_AREA VWB
#> 1 18 200 6
#> 2 18 200 8
#> 3 18 200 9
#> 4 18 200 10
#> 5 18 200 13
#> 6 18 200 12
#> 7 18 200 18
#> 8 18 200 19
#> 9 18 200 20
#> 10 18 200 20
#> 11 18 200 24
#> 12 18 200 23
#> 13 18 200 26
#> 14 18 200 30
#> 15 18 200 31
#> 16 18 200 31
#> 17 18 200 33
#> 18 18 200 32First, let’s see what error we would get, if we used the simple random sampling method:
sprs(data_as, "VWB", 200, 18)
#> Variables Values
#> 1 Total number of sampled plots (n) 18.0000
#> 2 Number of maximum plots (N) 900.0000
#> 3 Variance Quoeficient (VC) 44.6505
#> 4 t-student 2.1098
#> 5 recalculated t-student 1.9873
#> 6 Number of samples regarding the admited error 79.0000
#> 7 Variance (S2) 81.9771
#> 8 Standard deviation (s) 9.0541
#> 9 Mean (Y) 20.2778
#> 10 Standard error of the mean (Sy) 2.1341
#> 11 Absolute Error 4.5025
#> 12 Relative Error (%) 22.2042
#> 13 Estimated Total Value (Yhat) 18250.0000
#> 14 Total Error 4052.2580
#> 15 Inferior Confidence Interval (m3) 15.7753
#> 16 Superior Confidence Interval (m3) 24.7803
#> 17 Inferior Confidence Interval (m3/ha) 788.7634
#> 18 Superior Confidence Interval (m3/ha) 1239.0143
#> 19 inferior Total Confidence Interval (m3) 14197.7420
#> 20 Superior Total Confidence Interval (m3) 22302.2580We got a 22.2% error. Now, let’s calculate the sampling error using the method of successive differences, with the ss_diffs function. To use this function, the data must be set in the measured order, the plot area must be in square meters, and the total area value must be in hectares.
ss_diffs(data_as, "VWB", 200, 18)
#> Variables Values
#> 1 Total number of sampled plots (n) 18.0000
#> 2 Number of maximum plots (N) 900.0000
#> 3 Variance Quoeficient (VC) 44.6505
#> 4 t-student 2.1098
#> 5 recalculated t-student 1.9873
#> 6 Number of samples regarding the admited error 79.0000
#> 7 Variance (S2) 81.9771
#> 8 Standard deviation (S) 9.0541
#> 9 Mean (Y) 20.2778
#> 10 Standard error of the mean (Sy) 0.4041
#> 11 Absolute Error 0.8527
#> 12 Relative Error (%) 4.2050
#> 13 Estimated Total Value (Yhat) 18250.0000
#> 14 Total Error 767.4046
#> 15 Inferior Confidence Interval (m3) 19.4251
#> 16 Superior Confidence Interval (m3) 21.1304
#> 17 Inferior Confidence Interval (m3/ha) 971.2553
#> 18 Superior Confidence Interval (m3/ha) 1056.5225
#> 19 inferior Total Confidence Interval (m3) 17482.5954
#> 20 Superior Total Confidence Interval (m3) 19017.4046We got a 4.2% error, which is significantly lower than before.