# Plot design optimization

The function metrics.variables used for the calculation of stand-level variables and metrics (see vignette “Stand-level”) requires arguments specifying the plot designs and sizes. If the optimal plot design and size for the calculation of stand-level variables is not known, the optimal plots design for the corresponding TLS data can be determined by two different approaches implemented in FORTLS. The approaches depend on whether field data for the sample plots is available or not.

## Estimating optimal plot size without field data

If no field data is available, the function estimation.plot.size can be applied to determine the optimal plot design and size. This function uses the data frame containing the list of detected trees (introduced in tree.tls) and estimates stand-level density ($$N$$, trees/ha) and basal area ($$G$$, m$$^2$$/ha) for many simulated differently-sized plots and the three plot designs (circular fixed area, k-tree and angle-count) by increasing continuously their sizes.

Thus, circular fixed area plots with increasing radius (increment of 0.1 m) to the maximum radius defined by radius.max in plot.parameters (by default set to 25, if radius is larger than furthest tree, the horizontal distance to this furthest tree is considered as maximum radius) will be simulated and for each plot, the variables (N and G) are estimated. Similarly, k-tree plots with tree numbers ($$k$$) ranging from 1 to k.max (specified in plot.parameters, default value set to 50 or total number of trees in the plot) and angle-count plots with increasing basal area factor (BAF, increments of 0.1 m$$^2$$/ha) to the maximum value specified by BAF.max in plot.parameters (set to 4 by default) are simulated and the respective stand-level variables are calculated. Optionally, the minimum diameter at breast height (dbh.min, in cm) to include the trees in the estimations can be defined. By default the minimum $$dbh$$ is set to 4 cm.

The function generates size-estimation charts i.e., plots showing the estimated stand-level density ($$N$$) and basal area ($$G$$) on the $$y$$ axes respective to the different plot sizes ($$x$$ axes). The estimations will be performed for simulated plots corresponding to all sample plots. By default the output graphs will contain one line for each sample plot. When average is set to TRUE, the average of all estimations (for all plots) as a continuous line and the standard deviation as grey shaded area will be drawn instead of multiple lines for each sample plot. One chart for each plot design is drawn by default. If all.plot.designs is set to TRUE, the line charts of all three plot design will be drawn in one graph with different colours for each plot design.

estimation.plot.size(tree.tls = tree.tls,
plot.parameters = data.frame(radius.max = 25, k.max = 50, BAF.max = 4),
dbh.min = 4,
average = TRUE, all.plot.designs = FALSE)

The continuous line represents the average over all sample plots (i.e. 16 plots in the example shown here) of the estimated density ($$N$$) on the left and the basal area ($$G$$) on the right. The dotted line indicates the number of sample plots. This figure helps to find suitable plot designs for the calculation of stand-level metrics and variables. The optimal plot design and size should be chosen within a range where the estimated values for $$N$$ and $$G$$ reach a stable level. A too small plot leads to high errors of estimation, since only few trees enter the plot and therefore the sample is too small. In the example above, the basal area estimated for fixed area plots with radius smaller than 5 m is much higher (around 40-50 m$$^2$$/ha) than the true value (around 20 m$$^2$$/ha). On the other hand, too large plots come along with systematic errors due to occlusion of trees. Therefore, the basal area in the same example of fixed area plots with radius bigger than 20 m is estimated lower than the true value. In order to avoid both types of errors, the figure helps to find a plot size range with stable values.

## Validation with field data and optimizing plot design

When data from field measurements are available for the same sample plots, the TLS-based estimates can be validated and the optimal plot designs can be found applying functions implemented in FORTLS. In the first step of the optimization process, the function simulations simulates plots with incremental size and computes the corresponding stand-level metrics and variables (similar to the function metrics.variables, see “Stand-level” vignette). Based on the simulated data, two different processes can be performed. First, the bias between TLS data and field data for each individual estimated variable can be assessed with the function relative.bias. Second, correlations between all estimated variables and metrics based on TLS-data (output data of the simulations function) and the variables estimated from field data can be calculated with the correlations function. This function calculates both the Pearson and Spearman correlation coefficients. To visualize the correlation coefficients, heat maps can be drawn with the optimize.plot.design function.

### Plot simultaion and estimation of metrics and variables

The simulations function is applied as follows:

simulations <- simulations(tree.tls = tree.tls, tree.ds = tree.ds, tree.field = tree.field,
plot.design = c("fixed.area", "k.tree", "angle.count"),
plot.parameters = data.frame(radius.max = 25, k.max = 50, BAF.max = 4),
scan.approach = "single", var.metr = list(tls = NULL, field = NULL),
dbh.min = 4, h.min = 1.3, max.dist = Inf,
dir.data = dir.data, save.result = FALSE, dir.result = NULL)

#### The input data frames

Both TLS and field data from the same sample plots are required to compute the function. The TLS data introduced in tree.tls should have the same format as the data frame returned from the tree.detection.single.scan and tree.detection.multi.scan functions. Thus, each row must correspond to a detected tree and it must contain at least the following columns: id, file, tree, x, y, phi.left, phi.right, horizontal.distance, dbh, num.points, num.points.hom, num.points.est, num.points.hom.est and partial.occlusion. The data frame containing the field data must be inserted in the argument tree.field. Similar to the TLS data table, each row must correspond to a tree (specified in the columns id and tree) and the values for horizontal.distance, dbh, total.height and an integer value indicating whether the tree is dead (1) or alive (NA, specified in dead) must be included in the data frame.

When the distance sampling method for correction of occlusion effects was applied (function distance.sample, see “Stand-level” vignette), a list with the results in the output data frames from the aforesaid function can be introduced in tree.ds. The list must contain at least the data frame tree with the detection probabilities (P.hn, P.hn.cov, P.hr and P.hr.cov) for each tree. By default tree.ds is set to NULL and as a result, the calculations of the variables based on occlusion correction will not be performed.

#### Specifying designs of simulated plots

A vector containing the names of the plot designs can specify the plot designs ("fixed.area", "k.tree", "angle.count") that are to be considered for the simulations in plot.design. By default, this argument is set to NULL and all three plots designs will be considered.

Furthermore the argument plot.parameters allows for manually specifying the design of the simulated plots. Many differently-sized plots of the plot designs specified in plot.design are simulated. The list introduced in plot.parameters can include the following elements to customize the generated plots. The elements radius.max, k.tree.max and BAF.max define the maximum radius (in m), the maximum number of trees and the maximum BAF (in m$$^2$$/ha) respectively to which the sizes of circular fixed area, k-tree and angle-count plots respectively should increase. By default the values are set to radius.max = 25, k.tree.max = 50 and BAF.max = 4. The increment by which the sizes of circular fixed area and angle-count plots sequentially increase can also be customized by specifying the elements radius.increment and BAF.increment respectively. The default settings are radius.increment = 0.1 (in m) and BAF.increment = 0.1 (in m$$^2$$/ha). An additional element of the list can be num.trees defining the number of dominant trees per hectare (trees/ha). This value is needed for the calculation of dominant diamters and heights and is set to 100 trees/ha by default.

Similar to the other functions, the scan approach can be specified in scan.approach and is by default set to "multi". Metrics and variables of interest can be defined as a vector in var.metr. Thus, only those metrics and variables named in the vector are calculated. If not specified, var.metr is set to NULL and all possible metrics and variables are computed. The arguments dbh.min, h.min and max.dist can optionally define the minimum $$dbh$$, height and maximum distance of a tree to be included in the calculations. The default values are dbh.min = 4 (in cm), h.min = 1.3 (in m) and max.dist = NULL (no maximal distance is considered).

The argument dir.data should specify the working directory of the .txt files with the normalized reduced point clouds (output of normalize function). If not specified, it is set to NULL and the current working directory is assigned to it. The output files will be saved by default, since the argument save.result is set to TRUE, to the directory path indicated in dir.result. For each plot design (circular fixed area, k-tree and angle-count plots), a .csv file is created using the write.csv function from the utils package.

#### Output of the simulations function

The simulations function generates a list with one element for each plot design. The elements are data frames containing the simulated plot. Each row represents a simulated plot defined by their respective plot identification number id and their size determined by either radius, k or BAF depending on the plot design. The columns N, G, V, V.user, W.user, d, dg, dgeom, dharm, h, hg, hgeom, hharm, d.0, dg.0, dgeom.0, dharm.0, h.0, hg.0, hgeom.0 and hharm.0 display the stand-level variables based on the field data. The remaining columns contain the stand-level variables and metrics estimated for each plot based on the TLS data. As an example the data frame for circular fixed area plots is shown below.

head(simulations$fixed.area) id radius N G V V.user W.user d dg dgeom dharm h hg hgeom hharm d.0 dg.0 dgeom.0 dharm.0 h.0 hg.0 hgeom.0 hharm.0 N.tls N.hn N.hr N.hn.cov N.hr.cov N.sh G.tls G.hn G.hr G.hn.cov G.hr.cov G.sh V.tls V.hn V.hr V.hn.cov V.hr.cov V.sh d.tls dg.tls dgeom.tls dharm.tls h.tls hg.tls hgeom.tls hharm.tls d.0.tls dg.0.tls dgeom.0.tls dharm.0.tls h.0.tls hg.0.tls hgeom.0.tls hharm.0.tls n.pts n.pts.est n.pts.red n.pts.red.est P01 P05 P10 P20 P25 P30 P40 P50 P60 P70 P75 P80 P90 P95 P99 mean.z mean.q.z mean.g.z mean.h.z median.z mode.z max.z min.z var.z sd.z CV.z D.z ID.z kurtosis.z skewness.z p.a.mean.z p.a.mode.z p.a.2m.z p.b.mean.z p.b.mode.z p.b.2m.z CRR.z L2.z L3.z L4.z L3.mu.z L4.mu.z L.CV.z median.a.d.z mode.a.d.z weibull_c.z weibull_b.z mean.rho mean.q.rho mean.g.rho mean.h.rho median.rho mode.rho max.rho min.rho var.rho sd.rho CV.rho D.rho ID.rho kurtosis.rho skewness.rho p.a.mean.rho p.a.mode.rho p.b.mean.rho p.b.mode.rho CRR.rho L2.rho L3.rho L4.rho L3.mu.rho L4.mu.rho L.CV.rho median.a.d.rho mode.a.d.rho weibull_c.rho weibull_b.rho mean.r mean.q.r mean.g.r mean.h.r median.r mode.r max.r min.r var.r sd.r CV.r D.r ID.r kurtosis.r skewness.r p.a.mean.r p.a.mode.r p.b.mean.r p.b.mode.r CRR.r L2.r L3.r L4.r L3.mu.r L4.mu.r L.CV.r median.a.d.r mode.a.d.r weibull_c.r weibull_b.r 4 2.5 509.2958 53.87560 485.1621 385.4172 175.3602 36.7 36.7 36.7 36.7 16.6 16.6 16.6 16.6 36.7 36.7 36.7 36.7 16.6 16.6 16.6 16.6 509.2958 535.4928 535.6553 553.4044 547.1048 509.2958 51.99859 54.67328 54.68987 56.50203 55.85885 51.99859 383.8610 403.6059 403.7284 417.1060 412.3579 383.8610 36.05502 36.05502 36.05502 36.05502 13.32373 13.32373 13.32373 13.32373 36.05502 36.05502 36.05502 36.05502 13.32373 13.32373 13.32373 13.32373 4752.000 324.3204 28.33333 31.89828 1.06937 8.61100 9.339 9.590 9.694 9.824 10.110 10.321 11.033 12.6206 13.646 13.767 14.188 15.06200 15.238 11.00676 11.34071 10.354835 7.979817 10.321 10.102 25.396 0.102 7.463079 2.731864 0.2481988 25.294 3.952 6.099774 -1.117082 40.05622 60.49542 98.11661 59.94378 39.41439 1.8818861 0.4334052 8556406 103592905 1293244862 -178939297 2951898465 1.3e-06 1.4267575 0.9047575 4.578460 12.04913 1.647650 1.719348 1.571360 1.493911 1.586860 0.5915731 2.499989 0.5915731 0.2414102 0.4913351 0.2982035 1.908416 0.8601007 1.810289 0.1171962 46.19790 99.99850 53.80210 0 0.6590631 196670.2 377892.9 765026.4 -594229.5 1477928 8.4e-06 0.4246912 1.056077 NA NA 11.18274 11.47030 10.77472 10.101851 10.47458 9.178085 25.47709 2.1692727 6.514317 2.552316 0.2282371 23.30782 3.878540 5.003789 -0.8179068 40.12085 94.35134 59.87915 5.647161 0.4389330 8753077 106671499 1339243442 -186975775 3135302332 1.3e-06 1.4446079 2.004653 5.019574 12.17664 16 2.5 509.2958 44.62240 395.1908 315.3808 142.6170 33.4 33.4 33.4 33.4 16.3 16.3 16.3 16.3 33.4 33.4 33.4 33.4 16.3 16.3 16.3 16.3 509.2958 535.4928 535.6553 543.4899 539.0934 509.2958 36.97431 38.87619 38.88798 39.45676 39.13759 36.97431 284.0880 298.7008 298.7915 303.1616 300.7093 284.0880 30.40325 30.40325 30.40325 30.40325 13.93300 13.93300 13.93300 13.93300 30.40325 30.40325 30.40325 30.40325 13.93300 13.93300 13.93300 13.93300 6507.667 708.8804 89.33333 50.63408 1.98046 8.69900 9.048 10.592 10.936 11.038 11.325 11.533 11.885 12.1700 12.368 12.459 13.043 13.39300 14.568 11.24222 11.42735 10.862746 9.367235 11.533 12.445 14.948 0.102 4.196767 2.048601 0.1822239 14.846 1.432 11.224370 -2.336350 63.75997 20.81752 98.98918 36.24003 79.10180 1.0108184 0.7520887 22982316 271444389 3247255172 -503669713 8468728684 5.0e-07 0.9057775 1.2027775 6.413071 12.07408 1.666723 1.754167 1.564441 1.451082 1.709399 0.4703062 2.499988 0.4703062 0.2991372 0.5469344 0.3281495 2.029682 0.9615330 1.808112 -0.1776052 51.89550 99.99943 48.10450 0 0.6666925 541557.5 1073007.2 2230061.9 -1634862.8 4102985 3.1e-06 0.4802624 1.196417 NA NA 11.40354 11.56120 11.15961 10.723566 11.66687 12.453883 14.99513 0.9449329 3.620746 1.902826 0.1668628 14.05020 1.431501 9.581083 -2.0602270 63.49917 24.14316 36.50083 75.856269 0.7604825 23523874 280290716 3381548744 -524472512 8950638614 5.0e-07 0.8801652 1.050345 7.052386 12.18588 4 2.6 470.8726 49.81102 448.5596 356.3398 162.1304 36.7 36.7 36.7 36.7 16.6 16.6 16.6 16.6 36.7 36.7 36.7 36.7 16.6 16.6 16.6 16.6 470.8726 495.0932 495.2434 511.6534 505.8291 470.8726 48.07562 50.54852 50.56386 52.23930 51.64464 48.07562 354.9010 373.1563 373.2696 385.6379 381.2481 354.9010 36.05502 36.05502 36.05502 36.05502 13.32373 13.32373 13.32373 13.32373 36.05502 36.05502 36.05502 36.05502 13.32373 13.32373 13.32373 13.32373 4752.000 324.3204 28.33333 31.89828 1.00941 3.52310 9.098 9.508 9.623 9.738 10.061 10.271 10.691 12.0940 13.240 13.727 14.125 15.04800 15.227 10.72964 11.13278 9.954041 7.537012 10.271 10.102 25.396 0.102 8.813723 2.968791 0.2766907 25.294 3.617 5.158607 -1.100687 39.57380 58.06012 97.61027 60.42620 41.85642 2.3869469 0.4224933 8910081 107128487 1330448258 -179674917 2887255543 1.2e-06 1.2446393 0.6276393 4.061632 11.82730 1.715051 1.794777 1.629195 1.541556 1.651641 0.5915731 2.599997 0.5915731 0.2798282 0.5289879 0.3084387 2.008424 0.9503215 1.749552 0.0754967 46.90851 99.99861 53.09149 0 0.6596356 231576.9 466972.4 992378.5 -724516.0 1875788 7.4e-06 0.4713072 1.123477 NA NA 10.93564 11.27652 10.44544 9.669257 10.41573 9.178085 25.47709 2.1692727 7.571908 2.751710 0.2516278 23.30782 3.648276 4.401105 -0.8483287 39.43887 90.90429 60.56113 9.094323 0.4292341 9141658 110604689 1381277002 -189302292 3102508288 1.2e-06 1.2825717 1.757553 4.509889 11.98177 16 2.6 470.8726 41.25592 365.3762 291.5873 131.8575 33.4 33.4 33.4 33.4 16.3 16.3 16.3 16.3 33.4 33.4 33.4 33.4 16.3 16.3 16.3 16.3 470.8726 495.0932 495.2434 502.4869 498.4222 470.8726 34.18483 35.94322 35.95413 36.47999 36.18490 34.18483 262.6553 276.1657 276.2495 280.2900 278.0226 262.6553 30.40325 30.40325 30.40325 30.40325 13.93300 13.93300 13.93300 13.93300 30.40325 30.40325 30.40325 30.40325 13.93300 13.93300 13.93300 13.93300 6507.667 708.8804 89.33333 50.63408 2.02500 8.58600 9.038 10.315 10.918 11.024 11.318 11.525 11.869 12.1570 12.358 12.456 13.037 13.39200 14.560 11.22232 11.40733 10.846357 9.363614 11.525 12.445 14.948 0.102 4.186814 2.046171 0.1823305 14.846 1.440 10.984884 -2.288059 64.27412 20.61485 99.01514 35.72588 79.30825 0.9848639 0.7507572 24364239 287347738 3433378621 -532919146 8945155375 5.0e-07 0.9336819 1.2226819 6.409016 12.05311 1.719710 1.811796 1.610972 1.489595 1.776552 0.4703062 2.599995 0.4703062 0.3252045 0.5702671 0.3316065 2.129689 1.0203854 1.808793 -0.2034102 52.55830 99.99947 47.44170 0 0.6614281 614615.3 1259319.9 2705262.3 -1911549.9 4948564 2.8e-06 0.5091441 1.249404 NA NA 11.39275 11.55032 11.15117 10.723123 11.66348 12.453883 14.99513 0.9449329 3.615002 1.901316 0.1668882 14.05020 1.442599 9.359456 -2.0148956 63.87088 24.08590 36.12912 75.913563 0.7597633 24978854 297407285 3586119564 -556323552 9485674908 5.0e-07 0.9038278 1.061130 7.051229 12.17445 4 2.7 436.6391 46.18964 415.9483 330.4331 150.3431 36.7 36.7 36.7 36.7 16.6 16.6 16.6 16.6 36.7 36.7 36.7 36.7 16.6 16.6 16.6 16.6 873.2782 918.1976 918.4762 941.7921 932.5193 872.8359 83.92237 88.23914 88.26591 90.50658 89.61547 83.87987 638.9710 671.8382 672.0420 689.1021 682.3173 638.6474 34.96277 34.97982 34.94570 34.92864 13.82336 13.83238 13.81432 13.80530 36.05502 36.05502 36.05502 36.05502 13.32373 13.32373 13.32373 13.32373 9720.333 628.9908 72.00000 61.86391 1.07525 3.14325 7.160 9.422 9.560 9.672 9.994 10.223 10.551 11.8730 12.787 13.691 14.071 15.03400 15.217 10.51156 10.95263 9.695039 7.393004 10.223 10.102 25.396 0.102 9.467294 3.076897 0.2927156 25.294 3.227 4.563186 -1.013085 40.66258 55.91363 97.68611 59.33742 44.00773 2.3113116 0.4139060 9305897 110969906 1369445141 -182486188 2872952321 1.1e-06 1.1575571 0.4095571 3.816339 11.62778 1.783608 1.870852 1.688341 1.590305 1.737114 0.5915731 2.699997 0.5915731 0.3188325 0.5646525 0.3165788 2.108424 1.0150070 1.703296 0.0258272 48.02578 99.99871 51.97422 0 0.6605962 271519.1 572872.7 1273189.1 -879967.0 2368671 6.6e-06 0.5077340 1.192035 NA NA 10.74166 11.11126 10.21645 9.416921 10.37449 9.178085 25.47709 2.1692727 8.076948 2.841997 0.2645771 23.30782 3.231135 3.988811 -0.7922456 39.60941 87.77957 60.39059 12.219143 0.4216203 9577417 114927230 1426118305 -193702332 3118482248 1.1e-06 1.2037969 1.563575 4.267396 11.80698 16 2.7 436.6391 38.25652 338.8125 270.3882 122.2711 33.4 33.4 33.4 33.4 16.3 16.3 16.3 16.3 33.4 33.4 33.4 33.4 16.3 16.3 16.3 16.3 436.6391 459.0988 459.2381 465.9549 462.1857 437.3046 31.69951 33.33006 33.34018 33.82781 33.55417 31.74783 243.5597 256.0878 256.1655 259.9122 257.8097 243.9309 30.40325 30.40325 30.40325 30.40325 13.93300 13.93300 13.93300 13.93300 30.40325 30.40325 30.40325 30.40325 13.93300 13.93300 13.93300 13.93300 6507.667 708.8804 89.33333 50.63408 2.02200 8.58500 9.041 10.242 10.913 11.023 11.315 11.527 11.867 12.1530 12.351 12.453 13.043 13.43685 14.553 11.22242 11.40653 10.845990 9.334158 11.527 12.445 14.948 0.102 4.166236 2.041136 0.1818801 14.846 1.438 11.041497 -2.286062 64.30236 20.46210 99.01174 35.69764 79.46481 0.9882597 0.7507642 25633166 302258980 3611012454 -560736947 9412526857 4.0e-07 0.9335763 1.2225763 6.426180 12.05149 1.765761 1.862154 1.651184 1.522645 1.836115 0.4703062 2.699995 0.4703062 0.3497093 0.5913622 0.3349051 2.229689 1.0605005 1.819670 -0.2151981 52.95336 99.99949 47.04664 0 0.6539866 683166.0 1440851.1 3186039.3 -2178061.5 5789524 2.6e-06 0.5317053 1.295455 NA NA 11.40089 11.55754 11.16129 10.737206 11.67122 12.453883 14.99513 0.9449329 3.596266 1.896382 0.1663363 14.05020 1.444901 9.360365 -2.0056493 63.67499 24.19485 36.32501 75.804642 0.7603062 26316332 313490162 3782025311 -586596001 10009343558 4.0e-07 0.9060391 1.052990 7.076416 12.18090 As described above, plots with increasing sizes are estimated and the corresponding variables an metrics are calculated. The plots are ordered from the smallest to the biggest. Plots with small radius can not be simulated for all sample plots (see table above), since trees not always enter, because the plots are too small. But plots with e.g. a radius of 20 m, can be simulated for all sample plots (see end of table below). tail(simulations$fixed.area)
id radius N G V V.user W.user d dg dgeom dharm h hg hgeom hharm d.0 dg.0 dgeom.0 dharm.0 h.0 hg.0 hgeom.0 hharm.0 N.tls N.hn N.hr N.hn.cov N.hr.cov N.sh G.tls G.hn G.hr G.hn.cov G.hr.cov G.sh V.tls V.hn V.hr V.hn.cov V.hr.cov V.sh d.tls dg.tls dgeom.tls dharm.tls h.tls hg.tls hgeom.tls hharm.tls d.0.tls dg.0.tls dgeom.0.tls dharm.0.tls h.0.tls hg.0.tls hgeom.0.tls hharm.0.tls n.pts n.pts.est n.pts.red n.pts.red.est P01 P05 P10 P20 P25 P30 P40 P50 P60 P70 P75 P80 P90 P95 P99 mean.z mean.q.z mean.g.z mean.h.z median.z mode.z max.z min.z var.z sd.z CV.z D.z ID.z kurtosis.z skewness.z p.a.mean.z p.a.mode.z p.a.2m.z p.b.mean.z p.b.mode.z p.b.2m.z CRR.z L2.z L3.z L4.z L3.mu.z L4.mu.z L.CV.z median.a.d.z mode.a.d.z weibull_c.z weibull_b.z mean.rho mean.q.rho mean.g.rho mean.h.rho median.rho mode.rho max.rho min.rho var.rho sd.rho CV.rho D.rho ID.rho kurtosis.rho skewness.rho p.a.mean.rho p.a.mode.rho p.b.mean.rho p.b.mode.rho CRR.rho L2.rho L3.rho L4.rho L3.mu.rho L4.mu.rho L.CV.rho median.a.d.rho mode.a.d.rho weibull_c.rho weibull_b.rho mean.r mean.q.r mean.g.r mean.h.r median.r mode.r max.r min.r var.r sd.r CV.r D.r ID.r kurtosis.r skewness.r p.a.mean.r p.a.mode.r p.b.mean.r p.b.mode.r CRR.r L2.r L3.r L4.r L3.mu.r L4.mu.r L.CV.r median.a.d.r mode.a.d.r weibull_c.r weibull_b.r
2559 11 20 286.4789 24.66506 211.9293 168.9124 76.25320 32.78889 33.10929 32.47470 32.16775 15.72500 15.77696 15.67285 15.62057 38.00919 38.13158 37.88871 37.77067 16.04577 16.07842 16.01360 15.98188 246.6902 259.3793 259.4580 260.5347 258.7484 268.1751 20.68138 21.74518 21.75178 21.84204 21.69229 22.48258 145.2360 152.7066 152.7530 153.3868 152.3352 157.8850 31.89665 32.67149 31.15759 30.44960 12.65043 12.77711 12.52558 12.40361 38.70450 39.11473 38.33744 38.01243 12.78380 12.93334 12.63747 12.49658 14256.333 9777.434 1500.667 1605.642 0.503 2.4010 4.6700 7.224 7.856 8.285 9.248 10.039 10.827 11.663 12.072 12.620 13.126 13.525 15.02800 9.498822 10.06134 8.423595 5.564067 10.012 12.934 17.773 0.101 11.00303 3.317082 0.3492098 17.672 4.182 3.269055 -0.8280228 57.05479 13.94600 95.73290 42.94521 86.01888 4.265055 0.5344524 143572397 1617370185 18927095193 -2473934204 35199802352 1e-07 2.177178 3.435178 3.136683 10.61561 8.882625 10.67035 5.742351 2.067638 8.794915 0.1000084 19.99997 0.1000084 34.95529 5.912300 0.6656028 19.89997 10.615460 1.781430 0.1237297 49.45539 99.99993 50.54461 0 0.4441318 161478966 2351353463 36674517035 -1951716342 29574870040 1e-07 5.341997 8.782616 1.532740 9.864254 14.14869 14.66584 13.60292 13.00865 13.24735 13.098382 24.85415 0.6629205 14.90152 3.860249 0.2728344 24.19123 5.473503 2.583600 0.2985818 42.06498 53.03285 57.93502 46.967080 0.5692686 305051362 4938483951 84413085236 -8009741985 171321681760 0 2.696722 1.050308 4.125092 15.58229
2560 12 20 310.3521 24.82251 213.2653 170.4284 76.73561 31.55128 31.91174 31.19873 30.85627 15.68462 15.70780 15.66138 15.63810 37.23372 37.36789 37.10049 36.96879 16.14607 16.16621 16.12614 16.10643 278.5212 292.8476 292.9365 296.5834 294.2724 306.3638 19.92268 20.94746 20.95382 21.21468 21.04937 21.91428 152.2515 160.0830 160.1316 162.1252 160.8618 167.4715 29.35498 30.17865 28.56768 27.78941 13.58869 13.78073 13.34545 13.02087 36.13364 36.69115 35.66487 35.27259 14.04560 14.20998 13.86409 13.66583 16950.000 20145.792 1529.000 1611.616 0.571 2.7250 5.4500 8.167 8.718 9.215 10.147 10.844 11.372 11.939 12.360 12.803 13.586 14.332 15.70900 10.147569 10.66770 9.111729 6.193954 10.847 11.377 17.591 0.101 10.82666 3.290390 0.3242540 17.490 3.677 3.910328 -1.0263343 59.70916 40.07179 96.22142 40.29084 59.89833 3.776970 0.5768614 163020972 1916656950 23302716336 -3046140794 46225758098 1e-07 1.887569 1.229431 3.406567 11.29426 9.876787 11.45792 7.051412 2.883396 10.152802 0.1000020 19.99997 0.1000020 33.73304 5.808015 0.5880470 19.89997 9.963760 1.849015 -0.0444131 51.63187 99.99993 48.36813 0 0.4938401 188067386 2799595313 44437689036 -2772907266 43910514235 1e-07 4.806447 9.776785 1.755560 11.091832 15.13936 15.65515 14.56380 13.87074 14.68149 11.392439 25.97687 0.6149849 15.88368 3.985434 0.2632499 25.36188 6.475083 2.525968 0.0540181 45.86457 80.26023 54.13543 19.739704 0.5828014 351088358 6009103904 107754837364 -9936643458 226675724189 0 3.239521 3.746917 4.291125 16.63549
2561 13 20 358.0986 26.99181 244.0854 196.0309 88.25248 30.65111 30.97916 30.32416 29.99877 16.56222 16.62685 16.49787 16.43399 36.28503 36.40635 36.16509 36.04694 17.45026 17.50284 17.39668 17.34230 246.6902 259.3793 259.4580 255.9155 254.3892 267.6363 15.99621 16.81901 16.82412 16.59441 16.49544 17.35443 123.7290 130.0934 130.1328 128.3561 127.5905 134.2347 28.09787 28.73344 27.44249 26.77323 13.80882 13.94154 13.67060 13.52743 34.37567 34.53079 34.22309 34.07339 14.52437 14.60481 14.44138 14.35609 13008.500 17940.994 1275.000 1394.355 0.583 2.5130 4.8500 7.940 8.712 9.317 10.587 11.266 11.796 12.335 12.651 12.989 14.029 14.874 16.40200 10.364617 10.95352 9.208809 6.160306 11.266 11.862 18.915 0.101 12.55426 3.543199 0.3418552 18.814 3.939 3.606888 -0.9508453 62.09272 38.57363 96.01505 37.90728 61.40006 3.983775 0.5479576 163722283 1994322322 25172528270 -3096431943 48018428978 1e-07 2.085617 1.497383 3.211962 11.57003 10.191565 11.55485 8.352277 6.224284 10.326048 0.5452360 19.99992 0.5452360 29.64660 5.444869 0.5342525 19.45468 9.254569 1.790222 0.0777802 50.49330 99.99993 49.50670 0 0.5095804 182191996 2698559726 42779529019 -2871903068 46312819801 1e-07 4.590674 9.646329 1.952156 11.494005 15.43861 15.92150 14.90807 14.29910 14.96905 11.396051 26.89557 1.4374041 15.14360 3.891478 0.2520615 25.45816 5.717303 2.620821 0.0727989 46.68826 86.39205 53.31174 13.607874 0.5740204 345914279 5984357012 108257760931 -10036937746 233390206831 0 2.851825 4.042555 4.501368 16.91736
2562 14 20 326.2676 26.33464 237.9450 190.6221 86.02561 31.82439 32.05765 31.58643 31.34362 16.50488 16.57180 16.43725 16.36921 36.37367 36.43150 36.31820 36.26505 17.35692 17.41033 17.30122 17.24326 294.4366 309.5818 309.6757 321.4368 317.4355 324.9851 21.44220 22.54513 22.55197 23.40847 23.11708 23.66687 164.5789 173.0444 173.0969 179.6709 177.4344 181.6543 30.00426 30.45045 29.56737 29.14418 13.59718 13.87527 12.99057 11.23805 36.11562 36.24826 35.98738 35.86391 14.12081 14.16125 14.07725 14.03035 19581.000 18593.551 1679.333 1730.414 0.664 2.9189 5.7390 8.515 9.120 9.706 10.623 11.250 11.907 12.563 12.916 13.336 14.203 15.009 16.13600 10.626730 11.15787 9.583030 6.692478 11.252 11.947 19.481 0.101 11.57074 3.401579 0.3200965 19.380 3.794 3.961588 -1.0479797 59.97503 39.17448 96.56767 40.02497 60.80192 3.430308 0.5454920 209490136 2570590219 32592792092 -4107992892 65267931931 1e-07 1.984270 1.320270 3.455819 11.81871 10.273074 11.72287 7.960387 4.273023 9.915276 0.1000310 20.00000 0.1000310 31.88965 5.647093 0.5496984 19.89996 9.347442 1.922395 0.0446746 47.52190 99.99994 52.47810 0 0.5136538 231243050 3491619887 56565658882 -3635109042 59513819528 0e+00 4.563232 10.173043 1.891395 11.575362 15.68101 16.18406 15.12666 14.47760 15.28735 8.591582 26.04291 1.6018739 16.02964 4.003703 0.2553217 24.44104 5.813319 2.588043 0.0760458 46.19425 97.16024 53.80575 2.839701 0.6021221 440733186 7765273419 143169937397 -12968148341 306342704434 0 2.911257 7.089431 4.438153 17.19710
2563 15 20 334.2254 26.67290 240.8075 192.9812 87.05307 31.64286 31.87649 31.40932 31.17674 16.51905 16.58523 16.44784 16.37013 36.27554 36.33279 36.22188 36.17161 17.47622 17.52059 17.43296 17.39087 238.7324 251.0123 251.0884 259.4727 256.5049 256.6323 16.59437 17.44794 17.45324 18.03603 17.82974 17.83859 131.9304 138.7166 138.7587 143.3921 141.7520 141.8224 29.36490 29.74950 28.99923 28.65520 14.37034 14.49912 14.23496 14.09320 34.15560 34.34168 33.97126 33.79001 14.78784 14.94781 14.61244 14.42169 9236.667 9075.978 1201.667 1241.396 0.626 2.7690 5.6178 8.942 9.568 10.088 11.015 11.820 12.512 13.182 13.440 13.741 14.614 15.136 16.25800 10.982258 11.54074 9.838005 6.621983 11.799 13.401 20.245 0.101 12.57862 3.546635 0.3229423 20.144 3.884 4.031515 -1.1552750 60.19676 25.80000 96.31534 39.80324 74.17410 3.682199 0.5424677 184063068 2332018798 30435513757 -3732263096 61191217784 1e-07 2.142742 2.418742 3.421984 12.22040 9.448153 11.09171 6.694392 2.933080 9.323490 0.1000019 20.00000 0.1000019 33.75845 5.810202 0.6149564 19.90000 10.312777 1.772333 0.0504223 49.34366 99.99993 50.65634 0 0.4724077 170018638 2501603522 39308118584 -2317481170 35828894184 1e-07 5.157570 9.348151 1.671244 10.576315 15.54378 16.00671 15.01593 14.34900 15.08682 13.174380 27.73252 2.1417932 14.60546 3.821709 0.2458674 25.59073 4.942452 3.094322 0.0077833 44.61860 75.01833 55.38140 24.981602 0.5604893 354081706 6131851416 110882537246 -10379448499 242930048830 0 2.434861 2.369403 4.626210 17.00557
2564 16 20 310.3521 23.85538 211.8378 169.8424 76.46164 30.97692 31.28390 30.67259 30.37392 16.18974 16.25163 16.12409 16.05316 36.18010 36.25185 36.10918 36.03926 16.93134 16.98705 16.87890 16.82953 318.3099 334.6830 334.7846 338.5946 336.0030 357.4331 21.02449 22.10594 22.11265 22.36431 22.19313 23.60860 162.7122 171.0817 171.1336 173.0812 171.7565 182.7110 28.29084 28.99965 27.55287 26.79226 13.70120 13.94374 13.38122 12.93075 36.21691 36.32707 36.10425 35.98921 15.02366 15.14870 14.90190 14.78418 22187.833 26385.098 1690.000 1884.641 0.632 2.7600 5.4520 8.768 9.430 9.909 10.720 11.258 11.803 12.294 12.580 12.916 13.831 14.564 16.19075 10.521718 11.03270 9.477378 6.514663 11.258 11.930 19.809 0.101 11.01390 3.318720 0.3154161 19.708 3.150 4.308202 -1.1747411 62.62202 37.11315 96.37187 37.37798 62.85850 3.626607 0.5311585 167849455 2026457926 25217643949 -3271733774 51422539428 1e-07 1.767282 1.408282 3.512850 11.69181 10.054045 11.67658 7.726990 5.116570 9.872622 0.4703062 19.99999 0.4703062 35.25883 5.937914 0.5905996 19.52968 10.917843 1.671939 0.0757414 48.78392 99.99993 51.21608 0 0.5027026 188013065 2889834451 47324710513 -2781038923 45137050481 1e-07 5.452152 9.583739 1.747202 11.287794 15.47980 16.06434 14.82711 14.05150 14.96039 12.453883 27.16916 0.9449329 18.43877 4.294039 0.2773962 26.22423 6.647488 2.493410 0.0942334 45.76022 72.88986 54.23978 27.110064 0.5697564 355862521 6306167488 117543510235 -10219870722 238710281311 0 3.276466 3.025920 4.050228 17.06615

### Calculation of relative bias

In order to estimate the goodness of the TLS-based estimation of the variables, the function relative.bias computes the relative bias between the variables estimated from field data and their respective TLS-based estimates. The relative bias is calculated for each sample plot and each simulated plot (i.e. different plot sizes and designs). Therefore, the input data for this function (introduced in simulations) must be a list of data frames containing the estimated variables (based on field and TLS data) for all the simulated plots. Thus, a similar list to the output of the simulations function (see above) is required, that has the same description and format.

Optionally, the variables for which the relative bias will be computed can be specified in a vector in variables. Only, the names of the field data based estimates can be introduced. If not otherwise specified, the argument will be set to c("N", "G", "V", "d", "dg", "d.0", "h", "h.0") by default. Other possible variables are dgeom, dharm, dg.0, dgeom.0, dharm.0, hg, hgeom, hharm, hg.0, hgeom.0 or hharm.0.

The arguments save.result and dir.result define whether and to which directory the output files should be saved. Two different output files are generated. First, the data frames for each plot design (as shown below for circular fixed areas) are saved as .csv files using the write.csv function from the utils package. Second, interactive line charts representing the relative biases are saved as .html files by means of the saveWidget function in the htmlwidgets package. An example of these interactive line charts is provided below.

bias <- relative.bias(simulations = Rioja.simulations,
variables = c("N", "G", "d", "dg", "d.0", "h", "h.0"),
save.result = FALSE, dir.result = NULL)
#> Computing relative bias for fixed area plots
#>  (0.31 secs)
#> Computing relative bias for k-tree plots
#>  (0.08 secs)
#> Computing relative bias for angle-count plots
#>  (0.04 secs)

The function calculates the relative bias between the field data estimates (specified in variables) and the counterpart variables that are estimated based on TLS data. The TLS counterparts for the density (N) are the variables N.tls, N.hn, N.hr, N.hn.cov, N.hr.cov and N.sh for circular fixed area and k-tree plots, and N.tls and N.pam for angle-count plots. The same pattern applies to the basal area (G) and the volume (V) where the corresponding TLS-based estimates are G.tls, G.hn, G.hr, G.hn.cov, G.hr.cov, G.sh and G.pam, and V.tls, V.hn, V.hr, V.hn.cov, V.hr.cov, V.sh and V.pam respectively. In case of mean and dominant diameters (d, dg, dgeom, dharm, d.0, dg.0, dgeom.0, and dharm.0) and heights (h, hg, hgeom, hharm, h.0, hg.0, hgeom.0 and hharm.0), for all three plot designs their respective counterpart variables are d.tls, dg.tls, dgeom.tls, dharm.tls, d.0.tls, dg.0.tls, dgeom.0.tls and dharm.0.tls (for the diameter), and h.tls, hg.tls, hgeom.tls, hharm.tls, h.0.tls, hg.0.tls, hgeom.0.tls, hharm.0.tls and in addition P99 (for the height). The relative bias are calculated as follows

$\frac{\frac{1}{n} \sum_{i=1}^{n}y_i - \frac{1}{n}\sum_{i=1}^{n}x_i}{\sum_{i=1}^{n}x_i}$

where $$x_i$$ is the value of the field estimate and $$y_i$$ the value of its TLS counterpart corresponding to plot $$i$$ of $$n$$ sample plots. For each plot size defined by the radius, k or BAF, the biases are calculated and stored as a data frame (shown below). Each row represents the a simulated plot of a certain size (here defined by radius) and the columns contain the calculate bias between the variables indicated in the column names. The two compared variables are joint with . as separation, e.g. N.N.tls means that the bias between N and N.tls was calculated.

head(bias$fixed.area) radius N.N.tls N.N.hn N.N.hr N.N.hn.cov N.N.hr.cov N.N.sh G.G.tls G.G.hn G.G.hr G.G.hn.cov G.G.hr.cov G.G.sh d.d.tls dg.dg.tls d.0.d.0.tls h.h.tls h.P99 h.0.h.0.tls h.0.P99 2.5 0.00000 5.143769 5.175673 7.687337 6.637258 0.0000000 -9.6703456 -5.023997 -4.995178 -2.577931 -3.554964 -9.6703456 -5.1950393 -5.1950393 -5.1950393 -17.15280 -9.404255 -17.15280 -9.404255 2.6 0.00000 5.143769 5.175673 7.687337 6.637258 0.0000000 -9.6703456 -5.023997 -4.995178 -2.577931 -3.554964 -9.6703456 -5.1950393 -5.1950393 -5.1950393 -17.15280 -9.462006 -17.15280 -9.462006 2.7 50.00000 57.715653 57.763509 61.202593 59.709142 50.0255646 36.9178682 43.960606 44.004289 47.235110 45.855820 36.9247528 -6.7531804 -6.7288477 -5.1950393 -15.63418 -9.513678 -17.15280 -9.513678 2.8 33.33333 40.191691 40.234230 43.781747 42.351883 33.4575777 33.2182993 40.070740 40.113242 43.792068 42.337512 33.3234726 -0.8541952 -0.8376348 0.2062484 -14.02461 -11.170292 -15.06550 -11.170292 2.9 25.00000 31.429711 31.469591 34.807426 33.467629 25.2969253 25.9246555 32.401928 32.442104 35.906474 34.537062 26.2018841 -0.2929137 -0.2804175 0.5072745 -14.43016 -14.105740 -15.18488 -14.105740 3.0 0.00000 5.143769 5.175673 7.378223 6.372637 0.3139353 0.5167802 5.687131 5.719200 8.226848 7.170623 0.8472341 -3.6238902 -3.6289551 -3.6911015 -14.53109 -13.990006 -15.31393 -14.150873 For better visualization, line charts are created that show the relative bias of a given variable and plot design. As an example, the interactive graphic showing the relative bias of basal area estimations (G) for fixed are plots can be seen when following the link (RB.G.fixed.area.html). ### Functions facilitating model-based or model-assisted sampling approaches To facilitate the application of model-based sampling, two additional functions are included in the FORTLS package. The function correlations computes the correlations between variables estimated from field data and those estimated from TLS data and calculates the respective Pearson and Spearman correlation coefficients. The results are saved as .csv files and represented as line charts and heat maps (when applying the function optimize.plot.design). #### Computing correlations The correlations function computes the correlations for all the plot designs that are introduces as elements of a list in simulations. The format and description must be the same as the output list of the simulations function. Also similar to the function relative.bias, the variables for which the correlations are to be calculated can be specified in variables. By default, this argument is set to variables = c("N", "G", "V", "d", "dg", "d.0", "h", "h.0"). If only one of the two above-mentioned correlation measures should be calculated, it can be specified in method. This argument is set to method = c("pearson", "spearman") by default and both correlation coefficients are computed. fixed.area.simulations <- list(fixed.area = Rioja.simulations$fixed.area[Rioja.simulations$fixed.area$radius < 7.5, ])
cor <- correlations(simulations = fixed.area.simulations,
variables = c("N", "G", "d", "dg", "d.0", "h", "h.0"),
method = c("pearson", "spearman"),
save.result = FALSE, dir.result = NULL)
#> Computing correlations for fixed area plots
#>  (30.99 secs)

In addition to the calculation of the correlation measures, the function also performs tests of association and returns the p-values. The output of this function is a list containing the following three elements correlations, correlations.pval and opt.correlations. Each of them is a list including, if not otherwise specified (in method), two elements pearson and spearman. These two elements are lists again that include separate data frames for each plot design (circular fixed area, k-tree and angle-count plots). The data frames contain the corresponding correlation coefficients (in correlations), the calculated p-values (in correlations.pval) and the optimal correlations for a given plot size and field data estimate (in opt.correlations).

cor$correlations$pearson$fixed.area[20:26,1:15] radius N.N.tls N.N.hn N.N.hr N.N.hn.cov N.N.hr.cov N.N.sh N.n.pts N.n.pts.est N.n.pts.red N.n.pts.red.est N.G.tls N.G.hn N.G.hr N.G.hn.cov 4.7 0.8427010 0.8427010 0.8427010 0.8372458 0.8383825 0.8416905 0.3827329 0.6163229 0.8403905 0.8064847 0.6918248 0.6918248 0.6918248 0.6733457 4.8 0.5341718 0.5341718 0.5341718 0.5474591 0.5450412 0.5385519 0.3803623 0.5561943 0.6830626 0.4886011 0.4470624 0.4470624 0.4470624 0.4507763 4.9 0.5341718 0.5341718 0.5341718 0.5474591 0.5450412 0.5385629 0.3803623 0.5561943 0.6830626 0.4886011 0.4470624 0.4470624 0.4470624 0.4507763 5.0 0.5341718 0.5341718 0.5341718 0.5474591 0.5450412 0.5384909 0.3803623 0.5561943 0.6830626 0.4886011 0.4470624 0.4470624 0.4470624 0.4507763 5.1 0.4267459 0.4267459 0.4267459 0.4406877 0.4382988 0.4300013 0.2931150 0.4148178 0.6415550 0.3953048 0.3313156 0.3313156 0.3313156 0.3375288 5.2 0.6374909 0.6374909 0.6374909 0.6407699 0.6404134 0.6395937 0.3792903 0.4078300 0.6505892 0.4761103 0.5101267 0.5101267 0.5101267 0.5014544 5.3 0.6064784 0.6064784 0.6064784 0.6012499 0.6020179 0.6052299 0.2815443 0.2962698 0.5317175 0.4570482 0.4429577 0.4429577 0.4429577 0.4322508 All mentioned data frames are divided into rows each of which represent a given plot size defined by radius (for circular fixed area plots), k (for k-tree plots) or BAF (for angle-count plots). The columns of the data frames in correlations (shown above) and correlations.pval (shown below) contain the calculated coefficients and p-values respectively for the corresponding correlation. The column names are composed of the two variables (e.g. N and N.tls) separated by . (giving N.N.tls) that were correlated as described for the relative.bias function. cor$correlations.pval$pearson$fixed.area[20:26,1:15]
radius N.N.tls N.N.hn N.N.hr N.N.hn.cov N.N.hr.cov N.N.sh N.n.pts N.n.pts.est N.n.pts.red N.n.pts.red.est N.G.tls N.G.hn N.G.hr N.G.hn.cov
4.7 0.0022053 0.0022053 0.0022053 0.0025099 0.0024441 0.0022596 0.2750097 0.0577525 0.0023308 0.0048223 0.0266593 0.0266593 0.0266593 0.0328198
4.8 0.0905190 0.0905190 0.0905190 0.0813023 0.0829309 0.0874083 0.2485167 0.0755965 0.0205199 0.1272621 0.1680119 0.1680119 0.1680119 0.1640786
4.9 0.0905190 0.0905190 0.0905190 0.0813023 0.0829309 0.0874006 0.2485167 0.0755965 0.0205199 0.1272621 0.1680119 0.1680119 0.1680119 0.1640786
5.0 0.0905190 0.0905190 0.0905190 0.0813023 0.0829309 0.0874511 0.2485167 0.0755965 0.0205199 0.1272621 0.1680119 0.1680119 0.1680119 0.1640786
5.1 0.1905500 0.1905500 0.1905500 0.1748975 0.1775217 0.1868221 0.3816974 0.2045907 0.0333605 0.2288530 0.3195963 0.3195963 0.3195963 0.3100452
5.2 0.0348622 0.0348622 0.0348622 0.0336470 0.0337777 0.0340795 0.2499634 0.2130948 0.0301856 0.1387719 0.1088955 0.1088955 0.1088955 0.1160739
5.3 0.0479069 0.0479069 0.0479069 0.0503955 0.0500245 0.0484933 0.4016223 0.3763530 0.0922936 0.1575664 0.1724261 0.1724261 0.1724261 0.1842722

The data frames of the opt.correlations list (example shown below) are also divided into rows that represent the different plot sizes. For a given plot size and variable (specified in the argument variables), the best correlating TLS-based estimate and the corresponding correlation coefficient is displayed in this table. The columns named <variable>.metric (with <variable> being here N, G, d, dg, d.0, h and h.0) contain the TLS-based variable or metric that yielded the best correlation with the respective field data-based variable of the column name for a certain plot radius. The columns <variable>.cor display the measures of the respective correlations. That means, in the example shown here, the TLS-based estimate that yielded the best correlation with the field data based variable density (N) for circular fixed area plots with a radius of 4.4 m is ID.rho. And the correlation coefficient is 0.7286.

cor$opt.correlations$pearson$fixed.area[20:26,] radius N.cor N.metric G.cor G.metric d.cor d.metric dg.cor dg.metric d.0.cor d.0.metric h.cor h.metric h.0.cor h.0.metric 20 4.7 0.8427010 N.hn 0.8673762 G.hr.cov 0.8661827 d.tls 0.8480509 dg.tls 0.8243517 d.0.tls 0.9542529 h.tls 0.8614859 hg.tls 21 4.8 0.7697247 p.b.mode.r 0.7355321 V.hn.cov 0.7636362 dg.tls 0.7533423 dg.tls 0.8279615 d.0.tls 0.8425797 P95 0.8115912 h.0.tls 22 4.9 -0.7663825 p.a.mode.r 0.7355321 V.hn.cov 0.7636362 dg.tls 0.7533423 dg.tls 0.8279615 d.0.tls 0.8443094 P95 0.8115912 h.0.tls 23 5.0 -0.7626689 p.a.mode.r 0.7355321 V.hn.cov 0.7636362 dg.tls 0.7533423 dg.tls 0.8279615 d.0.tls 0.8449040 P95 0.8115912 h.0.tls 24 5.1 -0.7979226 p.a.mode.r -0.6614491 p.a.mode.r 0.7653580 dg.tls 0.7572720 dg.tls 0.8279615 d.0.tls 0.8572419 P95 0.8115912 h.0.tls 25 5.2 -0.7586088 mean.h.z -0.6689808 P05 0.7948788 dg.tls 0.7840312 dg.tls 0.8279615 d.0.tls 0.9003149 hg.tls 0.8115912 h.0.tls 26 5.3 -0.7823391 mean.h.z -0.7073419 P05 0.8182592 dg.tls 0.8155215 dg.tls 0.8279615 d.0.tls 0.8918826 hg.tls 0.8115912 h.0.tls The correlations functions creates different files and saves them (if save.result = TRUE, default setting) to the directory indicated in dir.result. These files are, on the one hand, .csv files of the data frames in the lists correlations and opt.correlations created by means of the write.csv function from the utils package. These .csv files will be named as correlations.<plot design>.<method>.csv and opt.correlations.<plot design>.plot.<method>.csv with <plot design> being fixed.area.plot, k.tree.plot or angle.count.plot and <method> being pearson or spearman. On the other hand, interactive line charts representing the correlation coefficients will be created for each variable (selected in variables) as .html file using the saveWidget function in the htmlwidgets package. As an example, the interactive line chart for the variable height (h) and fixed area plots (pearson measure) is shown (correlations.h.fixed.area.pearson.html). #### Visualizing correlations In order to visualize the optimal correlations, the function optimize.plot.design creates heat maps and is applied as follows: optimize.plot.design(correlations = cor$opt.correlations,
variables = c("N", "G", "d", "dg", "d.0", "h", "h.0"),
dir.result = NULL)

The function creates the heat maps based on the optimal correlation list (opt.correlations from the output of the correlations function) introduced in correlations. The introduced list must have the same format and description as the opt.correlation list. Similar to the other functions described above, the variables of interest can be selected in variables (default setting: variables = c("N", "G", "V", "d", "dg", "d.0", "h", "h.0")). This function generates interactive heat maps with the saveWidget function of the htmlwidgets package and saves these graphics to the directory indicated in dir.result (or by default to the working directory). For each plot design and correlation measure a plot is generated and named as opt.correlations.<plot design>.<method>.html where <plot design> can be fixed.area.plot, k.tree.plot or angle.count.plot and <method> either pearson or spearman according to the plot design and correlation measure.

As an example, the heat map of the pearson correlation coefficient for fixed area plots and pearson measure is provided and can be seen when opening the link (opt.correlations.fixed.area.pearson.html).