The bivariate Power-Normal and the bivariate ... - bibsys brage

The bivariate Power-Normal and the bivariate Johnson’s System bounded distribution in forestry, including height curves. Supplementary material.

Title:

The bivariate Power-Normal and the bivariate Johnson’s System bounded distribution in forestry, including height curves. Supplementary material.

Author: Erik Mønness Hedmark University College, ØLR P.O. Box 104, 2450 Rena, Norway Phone: +47 62430598 Fax: +47 62430500 e-mail: [email protected] Number: 9

Year: 2014

Pages: 80

ISBN: 978-82-7671-958-1 ISSN: 1501-8563

Financed by:

Keywords: Bi-normal, bivariate Johnson’s System bounded distribution, bivariate power-normal distribution, height curve, Box-Cox transformation.

Content The bivariate Power-Normal and the bivariate Johnson’s System bounded distribution in forestry, including height curves. Supplementary material. ................................................................................................................................... 1 The bivariate Power-Normal and the bivariate Johnson’s System bounded distribution in forestry, including height curves. Supplementary material. ................................................................................................................................... 2 Summary:............................................................................................................................................................................................................................................................................................................................................................. 3 Plots of all stands ................................................................................................................................................................................................................................................................................................................................................. 4 Stand characteristics.......................................................................................................................................................................................................................................................................................................................................... 75 SAS programs: Two-dimensional Kolmogorov-Smirnov in 4 versions............................................................................................................................................................................................................................................................... 78

Summary: This document contain supplementary material concerning two published papers: Mønness, E. 2011b. The Power-Normal Distribution: Application to forest stands. Canadian Journal of Forest Research 41(4): 707-714. doi: 10.1139/X10-246. Mønness, E. (2014). The bivariate power-normal distribution and the bivariate Johnson system bounded distribution in forestry, including height curves. Canadian Journal of Forest Research, 45(3), 307-313. doi: 10.1139/cjfr-2014-0333 Supplementary material to the above article; SAS programs and computing details are found in Mønness, E. 2011a. The Power-Normal Distribution and Johnsons System bounded distribution: Computing details and programs. Available from http://hdl.handle.net/11250/133513 3] . From published abstracts: The Power-Normal (PN) distribution, originated from the inverse Box-Cox transformation, is presented and some possibilities in forest research are explored. The Power-Normal achieve shapes, by a Skewness * Kurtosis value, common to diameters and heights of forest stands. The estimation of the parameters by maximum likelihood is straightforward with good numerical properties. The shapes achieved by PN are very diverse even with only three parameters: The Johnson System bounded distribution (SB), also used in forestry, can encounter numerical problems with maximum likelihood estimation. The PN distribution is seen to give good estimates of diameter and height distributions, judged by the Kolmogorov-Smirnov statistic and visual inspection. It seems to perform better than the SB, especially on heights. A bivariate diameter and height distribution yields a unified model of a forest stand. The bivariate Johnson’s System bounded distribution and the bivariate power-normal distribution are explored. The power-normal originates from the well-known Box-Cox transformation. As evaluated by the bivariate Kolmogorov-Smirnov distance, the bivariate power-normal distribution seems to be superior to the bivariate Johnson’s System bounded distribution. The conditional median height given the diameter is a possible height curve and is compared with a simple hyperbolic height curve. Evaluated by the height deviance, the hyperbolic function yields the best height prediction. A close second is the curve generated by a bivariate power-normal distribution. Johnson’s System bounded distributions suffer from the sigmoid shape of the association between height and diameter. The data is from Vestjordet, E. 1977. Precommercial thinning of young stands of Scots Pine and Norway Spruce I: Data stability, dimension distribution etc. Medd. Nor. inst. skogforsk 33(9): 1-436.

Plots of all stands PowerNormal Johnson System Bounded Observed histogram grouped in units of 10 Vestjordet2 height curve Individual trees with LOWESS regression

Histogram Diameter (mm) RED BLUE BLACK

Histogram Height (dm) RED BLUE BLACK

Plot Diameter (mm) * Height (dm) RED BLUE

BLACK GREEN The curves is a LOWESS based on point estimates of the density and may sometimes appear strange. The scales are fixed, equal in all figures.

The histograms are marginal diameter and height distributions. The plots show diameter/height relation with estimated height given diameter curve.

The scales vary dependent of actual data values.

Results (↓) for STAND = 1











































































































































Stand characteristics. STAND 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

No.Obs. Dmin Hmin Dmax Hmax Dg (mm) Hl (dm) Dskew Hskew Dkurtosis Hkurtosis 48 31 44 160 110 108.35 83.48 -0.16 -0.12 0.05 1.70 132 18 30 182 118 94.49 90.97 0.41 -0.67 -0.40 0.62 123 18 19 148 87 77.12 63.26 0.29 -0.26 0.30 -0.03 217 17 23 152 92 73.75 67.58 0.57 -0.13 0.51 -0.18 128 21 27 106 84 57.55 60.35 0.42 0.02 0.37 0.45 90 51 51 193 113 105.57 84.94 0.59 0.10 0.05 -0.21 101 36 40 149 119 85.08 79.68 0.57 0.21 0.13 0.96 80 27 37 164 98 87.78 78.33 0.14 -0.51 -0.14 -0.21 87 14 20 106 69 56.23 50.34 0.33 -0.20 0.07 -0.05 58 18 21 106 76 61.76 51.66 0.03 0.06 -0.57 0.12 50 10 17 96 61 58.47 48.89 -0.03 -0.22 -0.22 -0.02 84 4 15 101 73 54.82 50.84 0.24 0.23 -0.35 -0.63 93 17 20 112 60 53.45 47.67 0.39 -0.20 0.23 -0.78 51 6 15 93 54 46.62 41.96 0.51 0.15 -0.05 -0.56 100 88 71 180 128 128.69 108.55 0.21 -0.68 -0.17 1.20 189 55 54 153 131 101.71 104.40 0.15 -0.87 -0.41 1.22 50 107 85 212 124 153.82 109.41 -0.09 -0.49 0.38 -0.27 112 59 57 161 124 120.94 108.44 -0.40 -1.35 -0.31 2.87 136 54 60 159 122 109.08 102.96 0.07 -0.88 -0.48 0.69 208 40 64 148 118 96.07 101.47 0.18 -0.78 -0.44 0.99 63 86 84 208 126 146.55 110.91 0.05 -0.43 -0.10 -0.30 106 26 30 116 86 63.81 62.12 0.38 0.12 -0.05 -0.18 116 24 29 113 82 72.78 64.77 0.01 -0.72 0.01 0.89 116 23 23 102 69 53.79 53.33 0.69 0.17 -0.03 -0.82 86 16 22 117 76 63.13 53.76 0.27 -0.24 0.03 0.31 89 34 35 115 78 75.34 62.55 -0.05 -0.58 -0.70 0.12 74 24 26 113 75 63.62 57.60 0.46 -0.09 0.04 -0.48 74 26 27 154 100 76.84 77.84 0.46 -0.30 0.25 0.01 95 12 19 145 98 83.29 78.45 0.02 -0.90 -0.53 0.67 92 39 49 152 105 89.52 81.82 0.22 -0.25 -0.05 0.16 228 19 26 150 75 74.04 58.69 0.43 -0.20 -0.16 -0.09 124 19 23 131 71 79.79 56.73 0.02 -0.53 -0.15 0.39 79 55 43 169 90 113.56 65.42 0.26 0.05 -0.31 -0.08 198 10 14 156 78 75.19 60.03 0.21 -1.03 0.80 2.90 122 36 32 152 81 89.62 61.15 -0.01 -0.66 0.00 0.75 107 16 23 102 66 61.87 51.29 0.24 -0.13 -0.40 -0.58 106 13 26 130 79 89.01 64.76 -0.60 -0.72 -0.21 0.16 107 10 20 135 86 78.01 61.52 -0.03 -0.40 -0.77 0.12 110 12 22 156 72 81.37 58.02 0.24 -0.82 1.15 1.76 108 7 17 133 71 62.16 56.13 0.55 -0.19 -0.39 -0.54 108 52 36 136 75 85.19 58.14 0.21 -0.04 0.21 0.49 107 24 21 121 69 67.59 52.64 0.25 0.00 -0.49 -0.55 85 25 32 159 86 109.78 73.28 -0.75 -1.08 0.93 2.90 119 19 27 156 94 93.65 75.08 0.05 -0.50 -0.56 0.01

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

118 196 76 152 96 176 116 56 111 154 113 77 96 160 162 137 158 75 78 80 79 137 114 98 163 189 120 63 105 131 88 94 51 254 175 129 223 161 120 235 169 126 170 180 84 138

21 17 41 17 35 19 45 64 64 32 61 58 47 28 33 22 16 32 18 48 12 41 33 37 17 12 26 10 13 30 37 29 25 16 32 45 20 7 7 19 27 20 16 20 54 27

30 22 36 23 31 30 47 63 83 40 68 63 54 36 51 28 30 30 22 45 19 36 42 37 25 21 35 19 24 37 40 34 26 22 33 47 28 16 17 26 37 21 19 27 42 35

119 143 122 128 134 130 190 209 168 188 213 224 212 177 158 198 185 150 126 133 129 159 182 211 179 173 181 175 157 155 163 140 122 147 155 167 154 156 162 124 140 146 184 140 147 165

74 93 71 79 71 104 120 122 131 150 158 148 152 124 121 151 140 100 86 85 86 118 115 121 121 116 125 124 107 126 130 101 82 87 92 101 105 98 98 88 93 92 110 96 96 117

76.90 77.40 85.55 76.61 83.84 81.09 107.79 127.46 119.47 107.50 131.87 145.59 138.17 96.69 96.12 110.01 101.31 93.93 83.48 90.60 78.11 98.54 115.68 115.05 90.76 81.57 98.99 97.57 83.26 89.91 106.38 86.43 83.10 70.98 84.97 99.50 80.48 83.64 96.34 69.80 84.06 88.58 78.23 85.61 97.85 91.04

59.11 70.50 55.81 63.07 58.99 82.43 99.16 95.49 112.48 118.61 132.57 126.40 125.76 100.68 99.15 118.60 107.68 74.35 67.41 67.50 71.42 87.23 93.90 96.10 91.58 90.22 94.84 95.22 82.03 95.43 99.47 79.94 66.44 66.03 73.10 78.19 74.11 70.80 73.07 69.37 74.94 73.25 76.62 73.15 74.23 81.09

-0.03 0.15 -0.15 -0.03 0.14 -0.29 0.34 0.31 0.16 0.11 0.48 -0.10 -0.24 0.17 0.45 0.20 -0.06 0.24 -0.37 0.18 -0.14 0.24 -0.14 0.20 0.33 0.41 0.08 0.21 0.30 0.21 -0.03 0.03 -0.32 0.09 0.26 0.15 0.33 0.43 -0.36 0.37 -0.16 -0.45 0.85 -0.32 0.06 0.25

-0.13 -0.55 0.00 -0.26 -0.46 -0.75 -0.46 0.13 0.22 -0.98 -0.58 -1.03 -1.20 -1.13 -0.51 -0.81 -1.03 -0.50 -1.01 -0.06 -1.16 -0.63 -0.85 -0.71 -0.80 -0.27 -0.59 -0.58 -0.57 -0.63 -0.48 -0.96 -0.82 -0.78 -0.48 -0.18 -0.51 -0.32 -1.06 -0.40 -0.73 -1.00 -0.43 -0.84 -0.20 -0.47

-0.40 -0.59 -0.18 -0.47 0.20 -0.22 -0.47 0.66 -0.64 -0.34 -0.36 -0.73 -0.42 -0.15 -0.37 -0.70 -0.47 -0.35 0.38 -0.42 0.13 -0.60 -0.32 0.27 0.09 -0.24 0.21 0.18 0.29 -0.38 -0.99 -0.17 -0.04 0.50 0.59 0.17 0.15 0.59 0.77 -0.29 -0.06 0.25 1.12 -0.19 -0.42 -0.16

-0.53 -0.11 -0.47 -0.36 0.27 0.09 -0.61 0.22 -0.26 1.27 0.73 0.85 3.31 1.10 0.39 0.02 0.41 1.08 1.28 -0.47 2.14 0.73 1.19 0.94 0.49 -0.65 0.14 0.21 0.72 0.43 -0.03 0.91 0.34 0.75 0.56 -0.02 0.43 0.52 2.36 0.30 0.77 1.53 0.15 0.95 0.64 0.38

91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136

180 148 341 261 527 158 170 77 122 126 85 79 127 94 114 222 87 112 149 135 77 51 162 106 193 78 143 102 50 60 174 121 143 53 76 53 142 162 66 121 90 157 94 33 82 35

5 8 11 7 12 19 15 16 32 12 16 23 4 16 19 5 52 11 8 3 23 33 13 10 5 28 9 13 28 25 8 15 14 40 22 39 8 15 28 14 24 6 24 17 10 14

14 19 18 16 19 22 18 22 39 20 21 28 12 21 28 16 52 17 16 13 24 32 19 17 15 30 17 18 26 27 17 23 20 31 26 37 17 22 30 21 27 15 25 19 19 18

172 182 183 163 143 178 164 206 176 146 171 137 84 111 113 136 169 117 110 104 95 121 124 110 127 150 133 108 92 122 114 140 143 179 143 126 133 159 141 141 148 155 133 117 114 113

117 113 123 112 101 139 105 117 108 109 116 81 61 69 78 92 95 69 84 76 71 70 70 70 80 86 88 63 61 80 86 89 78 77 83 73 75 110 102 94 88 100 91 64 62 64

87.95 94.98 88.44 86.43 67.93 90.81 84.39 118.47 108.85 73.54 91.40 87.21 49.83 66.60 72.06 70.96 109.00 55.33 65.39 52.92 62.23 70.47 60.81 64.46 60.29 91.13 78.94 63.48 63.90 76.72 58.23 80.54 63.34 89.10 84.46 84.20 62.22 79.60 97.49 79.57 78.63 80.47 83.79 79.85 63.70 67.76

87.68 89.93 96.31 84.40 67.01 88.87 76.80 93.97 86.42 74.28 85.26 64.39 42.45 51.50 57.79 66.90 78.93 46.20 59.82 52.96 51.15 54.16 53.02 50.68 56.12 66.48 63.04 49.59 47.32 60.58 59.21 71.27 51.30 58.42 64.35 59.87 58.14 82.22 80.01 70.98 62.75 76.48 68.29 53.12 50.09 49.15

0.32 0.26 0.20 0.24 0.61 0.20 0.23 -0.18 -0.15 0.37 0.39 0.01 -0.15 -0.15 -0.17 0.30 0.33 0.29 -0.06 0.29 0.06 0.41 0.37 0.01 0.15 -0.15 -0.01 0.21 0.01 0.04 0.30 -0.18 0.77 0.91 -0.22 -0.04 0.24 0.31 -0.23 -0.06 0.44 0.10 0.01 -0.44 0.50 -0.12

-0.38 -0.63 -0.82 -0.38 0.09 -0.25 -0.73 -0.56 -0.71 -0.05 -0.31 -0.50 -0.12 -0.25 -0.21 -0.36 -0.21 0.16 -0.39 0.18 -0.08 0.35 -0.30 0.01 -0.49 -0.41 -0.12 -0.17 -0.01 0.00 -0.09 -0.84 0.26 0.10 -1.11 -0.14 -0.12 -0.26 -0.47 -0.51 0.05 -0.65 -0.38 -0.91 -0.19 -0.33

-0.67 -0.45 -0.34 -0.73 0.09 -0.53 0.02 -0.18 -0.79 -0.48 -0.41 -0.24 -0.34 -0.29 0.34 -0.18 0.75 0.50 -0.26 -0.64 -0.60 -0.31 -0.41 -0.68 -0.29 -0.03 -0.57 -0.21 -0.48 -0.66 -0.35 -0.01 1.73 2.35 -0.23 -0.39 -0.37 -0.31 -0.55 -0.59 -0.04 -0.51 -0.66 -0.23 0.51 0.20

-0.02 -0.08 0.13 -0.43 0.01 -0.10 1.06 0.81 0.47 -1.07 -0.56 0.01 0.17 -0.05 0.39 0.23 -0.08 -0.01 0.25 -0.76 -0.13 -0.72 -0.67 -0.49 -0.10 -0.16 -0.19 -0.27 0.41 -0.64 -0.29 0.53 -0.14 -0.63 1.26 -0.59 -0.88 0.01 1.55 -0.08 -0.60 0.09 0.13 0.84 -0.10 0.17

137 138 139

69 115 123

17 26 17

19 30 25

70 149 136

51 96 95

42.45 101.27 77.89

37.38 79.59 70.02

SAS programs: Two-dimensional Kolmogorov-Smirnov in 4 versions P (D ≤ d  H ≤ h)

/* 2014 juli Beregning Bivariat KolmogorovSmirnov H< og d< FIL=PN_JSB_KS-2D */ options ps=63; data hoydiam.dhKolmogorovSmirnov2D; merge hoydiam.dhhorisontal hoydiam.dhbox1 hoydiam.dhparjsbf; by flate; array dd{550} 3 dd1-dd550; /* enkelt-dataverdier */ array hh{550} 3 hh1-hh550; /* enkelt-dataverdier */ array ant{550,550} _temporary_; /* beregn for en flate */ drop dd1-dd550 hh1-hh550 k l I ld lh distpn distjsb diff;

/* if flate=1 then DO */ DO k=1 TO n; DO l=1 TO n; ant{k,l}=0; distjsb=0; distpn=0; DO i=1 TO n; IF ((dd{i}0) then KK=PROBBNRM((1+dlanda*dbmy)/(dlanda*dbsigma), (1+hlanda*hbmy)/(hlanda*hbsigma),ropn); else

else

else

if (dlanda0) AND (hlanda>0) then KK=PROBBNRM((1+dlanda*dbmy)/(dlanda*dbsigma), (1+hlanda*hbmy)/(hlanda*hbsigma),ropn); else if (dlanda=dd{k}) AND (hh{i}0) AND (hlanda>0) then KK=PROBBNRM((1+dlanda*dbmy)/(dlanda*dbsigma),

else

else

else

(1+hlanda*hbmy)/(hlanda*hbsigma),ropn); if (dlanda FIL=PN_JSB_KS-2D-IV */ options ps=63; data hoydiam.dhKolmogorovSmirnov2D_IV; merge hoydiam.dhhorisontal hoydiam.dhbox1 hoydiam.dhparjsbf; by flate; array dd{550} 3 dd1-dd550; /* enkelt-dataverdier */ array hh{550} 3 hh1-hh550; /* enkelt-dataverdier */ array ant{550,550} _temporary_; /* beregn for en flate */ drop dd1-dd550 hh1-hh550 k l I ld lh distpn distjsb diff;

/* if flate=1 then DO */ DO k=1 TO n; DO l=1 TO n; ant{k,l}=0; distjsb=0; distpn=0; DO i=1 TO n; IF ((dd{i}>=dd{k}) AND (hh{i}>=hh{l})) then ant{k,l}=ant{k,l}+1; End; ant{k,l}= ant{k,l}/n; /* Beregne Kolomgorov-Smirov for SJB */ ld=(log((dd{k}-dtau)/(dteta-dd{k}))-dmy)/dbeta; lh=(log((hh{l}-htau)/(hteta-hh{l}))-hmy)/hbeta; diff= ABS(1-(PROBBNRM(1000,lh,rojsb)+ PROBBNRM(ld, 1000,rojsb)PROBBNRM(ld, lh, rojsb))- ant{k,l});

IF (distjsb < diff) then distjsb = diff; /* Beregne Kolomgorov-Smirov for PN */ ld=(dd{k}**dlanda-(1+dlanda*dbmy))/(dlanda*dbsigma); lh=(hh{l}**hlanda-(1+hlanda*hbmy))/(hlanda*hbsigma); KK=1; if (dlanda>0) AND (hlanda>0) then KK=PROBBNRM((1+dlanda*dbmy)/(dlanda*dbsigma), (1+hlanda*hbmy)/(hlanda*hbsigma),ropn); else

else

else

if (dlanda