Characterizing prediction errors of a new tree height model for cut-to-length Pinus radiata stems through the Burr Type XII distribution

doi:10.1007/s11676-023-01632-3

Abstract

Abstract:

Unlike height-diameter equations for standing trees commonly used in forest resources modelling, tree height models for cut-to-length (CTL) stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed. This feature was merely noticed in previous studies but never thoroughly investigated. This study characterized the prediction error distribution of a newly developed such tree height model for Pinus radiata (D. Don) through the three-parameter Burr Type XII (BXII) distribution. The model’s prediction errors (

ε

) exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent. Structured serial correlations were also present in the data. A total of 14 candidate weighting functions were compared to select the best two for weighting

ε

in order to reduce its conditional heteroskedasticity. The weighted prediction errors (

ε w

) were shifted by a constant to the positive range supported by the BXII distribution. Then the distribution of weighted and shifted prediction errors (

ε w +

) was characterized by the BXII distribution using maximum likelihood estimation through 1000 times of repeated random sampling, fitting and goodness-of-fit testing, each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences. The nonparametric two sample Kolmogorov–Smirnov (KS) goodness-of-fit test and its closely related Kuiper’s (KU) test showed the fitted BXII distributions provided a good fit to the highly leptokurtic and heavy-tailed distribution of

ε

. Random samples generated from the fitted BXII distributions of

ε w +

derived from using the best two weighting functions, when back-shifted and unweighted, exhibited distributions that were, in about 97 and 95% of the 1000 cases respectively, not statistically different from the distribution of

ε

. Our results for cut-to-length P. radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution. The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P. radiata plantations, particularly when uncertainty assessments, statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.

Key words: Conditional heteroskedasticity, Leptokurtic error distribution, Skedactic function, Nonlinear quantile regression, Weighted prediction errors, Serial correlation, Random sampling and fitting, Nonparametric goodness-of-fit tests

Xinyu Cao, Huiquan Bi, Duncan Watt, Yun Li. Characterizing prediction errors of a new tree height model for cut-to-length Pinus radiata stems through the Burr Type XII distribution[J]. JOURNAL OF FORESTRY RESEARCH, 2023, 34(6): 1899-1914.

Xinyu Cao, Huiquan Bi, Duncan Watt, Yun Li. [J]. 林业研究(英文版), 2023, 34(6): 1899-1914.

References 97

1	Amemiya T. Advanced econometrics, 1985 Cambridge USA. p Harvard University Press 536
2	Arias-Rodil M, Diéguez-Aranda U, Burkhart HE. Effects of measurement error in total tree height and upper-stem diameter on stem volume prediction. For Sci, 2017, 63(3): 250-260, DOI
3	Barth A, Holmgren J. Stem taper estimates based on airborne laser scanning and cut-to-length harvester measurements for preharvest planning. Int J for Eng, 2013, 24(3): 161-169
4	Barth A, Möller JJ, Wilhelmsson L, Arlinger J, Hedberg R, Söderman U. A Swedish case study on the prediction of detailed product recovery from individual stem profiles based on airborne laser scanning. Ann for Sci, 2015, 72(1): 47-56, DOI
5	Bassett G, Koenker R. An empirical quantile function for linear models with iid errors. J Am Stat Assoc, 1982, 77(378): 407-415
6	Berger A, Gschwantner T, McRoberts RE, Schadauer K. Effects of measurement errors on individual tree stem volume estimates for the austrian national forest inventory. For Sci, 2014, 60(1): 14-24, DOI
7	Bi H, Hamilton F. Stem volume equations for native tree species in southern New South Wales and Victoria. Aust for, 1998, 61(4): 275-286, DOI
8	Bi H, Long Y. Flexible taper equation for site-specific management of Pinus radiata in New South Wales. Aust for Ecol Manag, 2001, 148(1): 79-91
9	Bi H, Jurskis V, O'Gara J. Improving height prediction of regrowth eucalypts by incorporating the of site trees in a modified Chapman-Richards equation. Aust for, 2000, 63(4): 257-266, DOI
10	Bi H, Fox JC, Li Y, Lei Y, Pang Y. Evaluation of nonlinear equations for predicting diameter from tree height. Can J for Res, 2012, 42(4): 789-806, DOI
11	Bi H, Murphy S, Volkova L, Weston C, Fairman T, Li Y, Law R, Norris J, Lei X, Caccamo G. Additive biomass equations based on complete weighing of sample trees for open eucalypt forest species in south-eastern Australia. For Ecol Manage, 2015, 349: 106-121, DOI
12	Biging GS, Dobbertin M. Evaluation of competition indices in individual tree growth models. For Sci, 1995, 41(2): 360-377
13	Bryson MC. Heavy-tailed distributions: properties and tests. Technometrics, 1974, 16(1): 61-68, DOI
14	Burkhart HE, Tomé M. Modeling forest trees and stands, 2012 New York London. p Springer, Dordrecht Heidelberg 457, DOI
15	Burr IW. Cumulative frequency functions. Ann Math Stat, 1942, 13(2): 215-232, DOI
16	Caccamo G, Iqbal IA, Osborn J, Bi H, Arkley K, Melville G, Aurik D, Stone C. Comparing yield estimates derived from LiDAR and aerial photogrammetric point-cloud data with cut-to-length harvester data in a Pinus radiata plantation in Tasmania. Aust for, 2018, 81(3): 131-141, DOI
17	Carey BP, Murphy GE. Mechanised versus manual log-making in two Chilean Pinus radiata stands. NZ J for Sci, 2005, 35(1): 25-34
18	Ciceu A, Garcia-Duro J, Seceleanu I, Badea O. A generalized nonlinear mixed-effects height–diameter model for Norway spruce in mixed-uneven aged stands. For Ecol Manage, 2020, 477: 118507, DOI
19	Davidian M, Carroll RJ. Variance function estimation. J Am Stat Assoc, 1987, 82(400): 1079-1091, DOI
20	Davidson R, MacKinnon JG (1993) Estimation and inference in econometrics (Vol. 63). Oxford University Press, New York Oxford. p 875
21	Davino C, Furno M, Vistocco D. Quantile regression: theory and applications, 2013 New York John Wiley & Sons 288
22	Delmaire M, Labelle ER. Use of harvester data to estimate the amount of merchantable non-utilized woody material remaining after mechanized cut-to-length forest operations. Forests, 2022, 13(6): 945, DOI
23	Dorado FC, Diéguez-Aranda U, Anta MB, Rodríguez MS, von Gadow K. A generalized height–diameter model including random components for radiata pine plantations in northwestern Spain. For Ecol Manage, 2006, 229(1–3): 202-213, DOI
24	Dutcă I, McRoberts RE, Næsset E, Blujdea VN. Accommodating heteroscedasticity in allometric biomass models. For Ecol Manage, 2022, 505: 119865, DOI
25	Fortin M, DeBlois J. A statistical estimator to propagate height prediction errors into a general volume model. Can J for Res, 2010, 40(10): 1930-1939, DOI
26	Foss S, Korshunov D, Zachary S (2011) An introduction to heavy-tailed and subexponential distributions (Vol 6, pp 0090−6778). 1st ed. New York: Springer. p 123
27	Goodman AC, Thibodeau TG. Age-related heteroskedasticity in hedonic house price equations. J Hous Res, 1995, 6: 25-42
28	Greene W (1999) Econometric analysis. 4th ed. Prentice Hall, Upper Saddle River, NJ. p 1040
29	Gujarati DN, Porter DC. Essentials of econometrics, 2010 4 Boston. p McGraw-Hill 554
30	Hao L, Naiman DQ. Quantile regression SAGE, quantitative applications in the social sciences, 2007 Calif. p Sage Publications, Thousand Oaks 136
31	Hauglin M, Hansen E, Sørngård E, Næsset E, Gobakken T. Utilizing accurately positioned harvester data: modelling forest volume with airborne laser scanning. Can J for Res, 2018, 48(999): 1-10
32	Holmgren J, Barth A, Larsson H, Olsson H. Prediction of stem attributes by combining airborne laser scanning and measurements from harvesters. Silva Fennica, 2012, 46(2): 227-239, DOI
33	Holopainen M, Vastaranta M, Rasinmäki J, Kalliovirta J, Mäkinen A, Haapanen R, Melkas T, Yu X, Hyyppä J. Uncertainty in timber assortment estimates predicted from forest inventory data. Eur J Forest Res, 2010, 129(6): 1131-1142, DOI
34	Howell SR, Song GZM, Chao KJ, Doley D. Functional evaluation of height–diameter relationships and tree development in an Australian subtropical rainforest. Aust J Bot, 2022, 70(2): 158-173, DOI
35	Huang S. Ecoregion-based individual tree height-diameter models for lodgepole pine in Alberta. West J Appl for, 1999, 14(4): 186-193, DOI
36	Huang S, Titus SJ, Wiens DP. Comparison of nonlinear height–diameter functions for major Alberta tree species. Can J for Res, 1992, 22(9): 1297-1304, DOI
37	Judge GG, Hill RC, Griffiths WE, Lütkepohl H, Lee TC. Introduction to the theory and practice of econometrics, 1988 2 New York John Wiley and Sons 1064
38	Kangas A. On the bias and variance in tree volume predictions due to model and measurement errors. Scand J for Res, 1996, 11(11–14): 281-290, DOI
39	Keefe RF, Zimbelman EG, Picchi G. Use of individual tree and product level data to improve operational forestry. Curr for Rep, 2022, 8: 148-165, DOI
40	Kemmerer J, Labelle ER. Using harvester data from on-board computers: a review of key findings, opportunities and challenges. Eur J Forest Res, 2021, 140(1): 1-17, DOI
41	Klebanov LB. Heavy tailed distributions, 2003 Prague. p Matfyz-press 208
42	Koenker R. Quantile regression, 2005 United Kingdom. p Cambridge University Press 349, DOI
43	Koenker R. Quantile regression: 40 years on. Ann Rev Econ, 2017, 9: 155-176, DOI
44	Koenker R, Bassett G. Regression quantiles. Econometrica, 1978, 46: 33-50, DOI
45	Koenker R, Bassett G. Robust tests for heteroscedasticity based on regression quantiles. Econ J Econ Soc, 1982, 50(1): 43-61
46	Koenker R (2022) Package ‘quantreg’ quantile regression. R package version 5.95. https://cran.r-project.org/web/packages/quantreg/quantreg.pdf
47	Lanzante JR. Testing for differences between two distributions in the presence of serial correlation using the Kolmogorov-Smirnov and Kuiper’s tests. Int J Climatol, 2021, 41: 6314-6323, DOI
48	Li W, Bi H, Watt D, Li Y, Ghaffariyan MR, Ximenes F. Estimation and spatial mapping of residue biomass following CTL harvesting in Pinus radiata plantations: an application of harvester data analytics. Forests, 2022, 13(3): 428, DOI
49	Lin F, Xie L, Hao Y, Miao Z, Dong L. Comparison of modeling approaches for the height–diameter relationship: an example with planted mongolian pine (Pinus sylvestris var. mongolica) trees in northeast China. Forests, 2022, 13(8): 1168, DOI
50	Lu K, Bi H, Watt D, Strandgard M, Li Y. Reconstructing the size of individual trees using log data from cut-to-length harvesters in Pinus radiata plantations: a case study in NSW. Aust J for Res, 2018, 29(1): 13-33, DOI
51	Magnussen S, Kleinn C, Fehrmann L. Wood volume errors from measured and predicted heights. Eur J Forest Res, 2020, 139(2): 169-178, DOI
52	Massey FJ. The Kolmogorov-Smirnov test for goodness of fit. J Am Stat Assoc, 1951, 46(253): 68-78, DOI
53	McRoberts RE, Westfall JA. Propagating uncertainty through individual tree volume model predictions to large-area volume estimates. Ann for Sci, 2016, 73: 625-633, DOI
54	Mead DJ (2013) Sustainable management of Pinus radiata plantations. Food and Agriculture Organization of the United Nations: Italy, Rome. p 246
55	Mehtätalo L, de Miguel S, Gregoire TG. Modeling height-diameter curves for prediction. Can J for Res, 2015, 45(7): 826-837, DOI
56	Molto Q, Rossi V, Blanc L. Error propagation in biomass estimation in tropical forests. Methods Ecol Evol, 2013, 4(2): 175-183, DOI
57	Müller F, Hanewinkel JDM. Digitization in wood supply – A review on howIndustry 4.0 will change the forest value chain. Comput Electron Agric, 2019, 162: 206-218, DOI
58	Murphy G, Wilson I, Barr B. Developing methods for pre-harvest inventories which use a harvester as the sampling tool. Aust for, 2006, 69(1): 9-15, DOI
59	Ogana FN, Ercanli I. Modelling height-diameter relationships in complex tropical rain forest ecosystems using deep learning algorithm. J for Res, 2022, 33(3): 883-898, DOI
60	Okasha MK, Matter MY. On the three-parameter Burr type XII distribution and its application to heavy tailed lifetime data. J Adv Math, 2015, 10(4): 3429-3442
61	Palander T, Vesa L, Tokola T, Pihlaja P, Ovaskainen H. Modelling the stump biomass of stands for energy production using a harvester data management system. Biosys Eng, 2009, 102(1): 69-74, DOI
62	Parresol BR. Additivity of nonlinear biomass equations. Can J for Res, 2001, 31(5): 865-878, DOI
63	Patrício MS, Dias CR, Nunes L. Mixed-effects generalized height-diameter model: a tool for forestry management of young sweet chestnut stands. For Ecol Manage, 2022, 514: 120209, DOI
64	Persson HJ, Olofsson K, Holmgren J. Two-phase forest inventory using very-high-resolution laser scanning. Remote Sens Environ, 2022, 271: 112909, DOI
65	Peuhkurinen J, Maltamo M, Malinen J. Estimating species-specific diameter distributions and saw log recoveries of boreal forests from airborne laser scanning data and aerial photographs: a distribution-based approach. Silva Fennica, 2008, 42(4): 625-641, DOI
66	Polosin VG, Mitroshin AN, Gerashchenko SI. Burr Type XII distribution in traffic control systems. Trans Res Procedia, 2023, 68: 433-440, DOI
67	Pretzsch H. Forest dynamics, growth, and yield, 2009 Heidelberg. p Springer-Verlag, Berlin 664, DOI
68	Priddle J (2005) Computer-controlled optimisation in cut-to-length harvesting systems and associated data flows. Available at: https://gottsteintrust.org/reports/
69	Rasinmäki J, Melkas T. A method for estimating tree composition and volume using harvester data. Scand J for Res, 2005, 20(1): 85-95, DOI
70	Rodrigues CK, Lopes ES, Figueiredo A, Pelissari AL, Silva MK. Modeling residual biomass from mechanized wood harvesting with data measured by forest harvester. An Acad Bras Ciênc, 2019, DOI
71	Rodriguez RN. A guide to the Burr type XII distributions. Biometrika, 1977, 64(1): 129-134, DOI
72	Romano JP, Wolf M. Resurrecting weighted least squares. J Econ, 2017, 197(1): 1-19, DOI
73	Rossit DA, Olivera A, Céspedes VV, Broz D. A big data approach to forestry harvesting productivity. Comput Electron Agric, 2019, 161: 29-52, DOI
74	Sánchez CAL, Varela JG, Dorado FC, Alboreca AR, Soalleiro RR, González JGÁ, Rodríguez FS. A height-diameter model for Pinus radiata D. Don in Galicia (Northwest Spain). Ann for Sci, 2003, 60(3): 237-245, DOI
75	Saunders MR, Wagner RG. Height-diameter models with random coefficients and site variables for tree species of Central Maine. Ann for Sci, 2008, 65(2): 1-10, DOI
76	Sellén D (2016) Big Data analytics for the forest industry: a proof-of-concept built on cloud technologies. Dissertation, Mid Sweden University, Ostersund, Sweden, p 80
77	Shan C, Bi H, Watt D, Li Y, Strandgard M, Ghaffariyan MR. A new model for predicting the total tree height for stems cut-to-length by harvesters in Pinus radiata plantations. J for Res, 2021, 32(1): 21-41, DOI
78	Shao Q. Notes on maximum likelihood estimation for the three-parameter Burr XII distribution. Comput Stat Data Anal, 2004, 45: 675-687, DOI
79	Siipilehto J, Lindeman H, Vastaranta M, Yu X, Uusitalo J. Reliability of the predicted stand structure for clear-cut stands using optional methods: airborne laser scanning-based methods, smartphone-based forest inventory application Trestima and pre-harvest measurement tool EMO. Silva Fenn, 2016, 50(3): 1568, DOI
80	Singer JD, Willett JB, Willett JB. Applied longitudinal data analysis: Modeling change and event occurrence, 2003 New York. p Oxford University Press 644, DOI
81	Söderberg J (2015) A method for using harvester data in airborne laser prediction of forest variables in mature coniferous stands. Dissertation, Swedish University of Agricultural Science, Uppsala, Sweden. p 31
82	Stendahl J, Dahlin B. Possibilities for harvester-based forest inventory in thinnings. Scand J for Res, 2002, 17(6): 548-555, DOI
83	Stephens MA. The goodness-of-fit statistic V_N: distribution and significance points. Biometrika, 1965, 52(3/4): 309-321, DOI
84	Tadikamalla PR. A Look at the Burr and related distributions. Int Stat Rev, 1980, 48(3): 337-344, DOI
85	Thupeng WM. Use of the Three-parameter Burr XII Distribution for modelling ambient daily maximum nitrogen dioxide concentrations in the Gaborone fire brigade. Am Sci Res J Eng Technol Sci, 2016, 26(2): 18-32
86	Tian D, Jiang L, Shahzad MK, He P, Wang J, Yan Y. Climate-sensitive tree height-diameter models for mixed forests in Northeastern China. Agric for Meteorol, 2022, 326: 109182, DOI
87	Uusitalo J (2017) Big data is transforming forestry. Available at: www.luke.fi/en/big-data-transforming-forestry
88	Varjo J (1995) Latvan hukkaosan pituusmallit männylle, kuuselle ja koivulle metsurimittausta varten. In: Puutavaran mittauksen kehittämistutkimuksia 1989–93 (Verkasalo E ed), Finnish Forest Research Institute, Research Papers 558, pp 21−23 (in Finnish)
89	Vermeesch P. Dissimilarity measures in detrital geochronology. Earth Sci Rev, 2018, 178: 310-321, DOI
90	Vesa L, Palander T. Modeling stump biomass of stands using harvester measurements for adaptive energy wood procurement systems. Energy, 2010, 35(9): 3717-3721, DOI
91	Watkins AJ. An algorithm for maximum likelihood estimation in the three parameter Burr XII distribution. Comput Stat Data Anal, 1999, 32(1): 19-27, DOI
92	Weiskittel AR, Hann DW, Kershaw JA, Vanclay JK. Forest growth and yield modeling, 2011 New York John Wiley and Sons 415, DOI
93	Woo H, Acuna M, Choi B, Han SK. FIELD: a software tool that integrates harvester data and allometric equations for a dynamic estimation of forest harvesting residues. Forests, 2021, 12(7): 834, DOI
94	Wooldridge JM, Wadud M, Lye J (2016) Introductory econometrics: Asia pacific edition with online study tools 12 months. Cengage Australia. p 455
95	Xie L, Widagdo FRA, Dong L, Li F. Modeling height–diameter relationships for mixed-species plantations of Fraxinus mandshurica Rupr and Larix olgensis Henry in northeastern China. Forests, 2020, 11(6): 610, DOI
96	Xin H, Zhu J, Tsai TR. Parameter estimation for the three-parameter Burr-XII distribution under accelerated life testing with type I censoring using particle swarm optimization algorithm. Int J Inn Comput Inf ConTrol, 2018, 14(5): 1959-1968
97	Zimmer WJ, Keats JB, Wang FK. The Burr XII distribution in reliability analysis. J Qual Technol, 1998, 30(4): 386-394, DOI