Retrieval of among-stand variances from one observation per stand

https://doi.org/10.17221/141/2019-JFSCitation:Magnussen S., Breidenbach J. (2020): Retrieval of among-stand variances from one observation per stand. J. For. Sci., 66: 133-149.
download PDF

Forest inventories provide predictions of stand means on a routine basis from models with auxiliary variables from remote sensing as predictors and response variables from field data. Many forest inventory sampling designs do not afford a direct estimation of the among-stand variance. As consequence, the confidence interval for a model-based prediction of a stand mean is typically too narrow. We propose a new method to compute (from empirical regression residuals) an among-stand variance under sample designs that stratify sample selections by an auxiliary variable, but otherwise do not allow a direct estimation of this variance. We test the method in simulated sampling from a complex artificial population with an age class structure. Two sampling designs are used (one-per-stratum, and quasi systematic), neither recognize stands. Among-stand estimates of variance obtained with the proposed method underestimated the actual variance by 30-50%, yet 95% confidence intervals for a stand mean achieved  a coverage that was either slightly better or at par with the coverage achieved with empirical linear best unbiased estimates obtained under less efficient two-stage designs.

References:
Anderson T.W., Darling D.A. (1952): Asymptotic theory of certain „goodness of fit“ criteria based on stochastic processes. Annals of Mathematical Statistics, 23: 193–212. https://doi.org/10.1214/aoms/1177729437
 
Bailey R.L., Dell T.R. (1973): Quantifying diameter distributions with the Weibull function. Forest Science, 19: 97–104.
 
Bolker B.M., Brooks M.E., Clark C.J., Geange S.W., Poulsen J.R., Stevens M.H.H., White J.S.S. (2009): Generalized linear mixed models: a practical guide for ecology and evolution. Trends in Ecology and Evolution, 24: 127–135. https://doi.org/10.1016/j.tree.2008.10.008
 
Breidenbach J., Magnussen S., Rahlf J., Astrup R. (2018): Unit-level and area-level small area estimation under heteroscedasticity using digital aerial photogrammetry data. Remote Sensing of Environment, 212: 199–211. https://doi.org/10.1016/j.rse.2018.04.028
 
Breidenbach J., McRoberts R.E., Astrup R. (2016): Empirical coverage of model-based variance estimators for remote sensing assisted estimation of stand-level timber volume. Remote Sensing of Environment, 173: 274–281. https://doi.org/10.1016/j.rse.2015.07.026
 
Breidt F.J., Opsomer J.D., Sanchez-Borrego I. (2016): Nonparametric Variance Estimation Under Fine Stratification: An Alternative to Collapsed Strata. Journal of the American Statistical Association, 111: 822–833. https://doi.org/10.1080/01621459.2015.1058264
 
Cao Q.V. (2004): Predicting parameters of a Weibull function for modeling diameter distribution. Forest Science, 50: 682–685.
 
Casella G., Berger R.L. (2002): Statistical Inference. Pacific Grove, Duxbury Press: 660.
 
Cochran W.G. (1977): Sampling Techniques. New York, Wiley: 380.
 
Cordy C.B. (1993): An extension of the Horvitz-Thompson theorem to point sampling from a continuous universe. Statistics and Probability Letters, 18: 353–362. https://doi.org/10.1016/0167-7152(93)90028-H
 
Corona P. (2016): Consolidating new paradigms in large-scale monitoring and assessment of forest ecosystems. Environmental research, 144: 8–14. https://doi.org/10.1016/j.envres.2015.10.017
 
Corona P., Fattorini L. (2008): Area-based lidar-assisted estimation of forest standing volume. Canadian Journal of Forest Research, 38: 2911–2916. https://doi.org/10.1139/X08-122
 
Corona P., Fattorini L., Franceschi S., Chirici G., Maselli F., Secondi L. (2014): Mapping by spatial predictors exploiting remotely sensed and ground data: A comparative design-based perspective. Remote Sensing of Environment, 152: 29–37. https://doi.org/10.1016/j.rse.2014.05.011
 
Dahlke M., Breidt F.J., Opsomer J.D., Van Keilegom I. (2013): Nonparametric endogenous post-stratification estimation. Statistica Sinica, 23: 189–211. https://doi.org/10.5705/ss.2011.272
 
Dalenius T., Hodges J.L.J. (1959): Minimum variance stratification. Journal of the American Statistical Association, 54: 88–101. https://doi.org/10.1080/01621459.1959.10501501
 
Darroch N., Ratcliff D. (1971): A characterization of the Dirichlet distribution. Journal of the American Statistical Association, 66: 641–643. https://doi.org/10.1080/01621459.1971.10482324
 
Duane M.V., Cohen W.B., Campbell J.L., Hudiberg T., Turner D.P., Weyermann D.L. (2010): Implications of alternative field-sampling designs on Landsat-based mapping of stand age and carbon stocks in Oregon forests. Forest Science, 56: 405–416.
 
Dubey S.D. (1970): Compound gamma, beta and F distributions. Metrika, 16: 27–31. https://doi.org/10.1007/BF02613934
 
Duplat P., Perrotte G. (1981): Inventory and Estimation of the Growth of Forest Stands. Paris, Office National des Forêts: 432.
 
Eerikäinen K. (2009): A multivariate linear mixed-effects model for the generalization of sample tree heights and crown ratios in the Finnish National Forest Inventory. Forest Science, 55: 480–493.
 
Ene L.T., Gobakken T., Andersen H.E., Næsset E., Cook B.D., Morton D.C., Babcock C., Nelson R. (2018): Large-area hybrid estimation of aboveground biomass in interior Alaska using airborne laser scanning data. Remote Sensing of Environment, 204: 741–755. https://doi.org/10.1016/j.rse.2017.09.027
 
Evert F. (1983): An equation for estimating total volume of both stands and single trees of black spruce. The Forestry Chronicle, 59: 26–29. https://doi.org/10.5558/tfc59026-1
 
Fattorini L. (2015): Design-based methodological advances to support national forest inventories: a review of recent proposals. iForest - Biogeosciences and Forestry, 8: 6–11. https://doi.org/10.3832/ifor1239-007
 
Fattorini L., Franceschi S., Pisani C. (2009): A two-phase sampling strategy for large-scale forest carbon budgets. Journal of Statistical Planning and Inference, 139: 1045–1055. https://doi.org/10.1016/j.jspi.2008.06.014
 
Fazar W. (1959): Program evaluation and review technique. The American Statistician, 13: 10–16.
 
Fischer M. (2010): Multivariate Copulae. In: Kurowicka D., Joe H. (eds): Dependence Modeling. Singapore, World Scientific: 19–36.
 
Fraley C., Raftery A.E. (1998): How many clusters? Which clustering method? Answers via model-based cluster analysis. Computer Journal, 41: 314–327. https://doi.org/10.1093/comjnl/41.8.578
 
Goerndt M.E., Monleon V.J., Temesgen H. (2011): A comparison of small-area estimation techniques to estimate selected stand attributes using LiDAR-derived auxiliary variables. Canadian Journal of Forest Research, 41: 1189–1201. https://doi.org/10.1139/x11-033
 
Grafström A., Ringvall A.H. (2013): Improving forest field inventories by using remote sensing data in novel sampling designs. Canadian Journal of Forest Research, 43: 1015–1022. https://doi.org/10.1139/cjfr-2013-0123
 
Grafström A., Saarela S., Ene L.T. (2014): Efficient sampling strategies for forest inventories by spreading the sample in auxiliary space. Canadian Journal of Forest Research, 44: 1156–1164. https://doi.org/10.1139/cjfr-2014-0202
 
Grafström A., Schnell S., Saarela S., Hubbell S.P., Condit R. (2017): The continuous population approach to forest inventories and use of information in the design. Environmetrics, 28: e2480-n/a. https://doi.org/10.1002/env.2480
 
Harvey A.C. (1976): Estimating regression models with multiplicative heteroscedasticity. Econometrica, 44: 461–465. https://doi.org/10.2307/1913974
 
Holmström H., Fransson J.E.S. (2003): Combining remotely sensed optical and radar data in kNN-estimation of forest variables. Forest Science, 49: 409–418.
 
Izenman A.J. (1991): Recent development in nonparametric density estimation. Journal of the American Statistical Association, 86: 205–224. https://doi.org/10.2307/2289732
 
Jiang J., Lahiri P. (2006): Mixed Model Prediction and Small Area Estimation. TEST, 15: 1–96. https://doi.org/10.1007/BF02595419
 
Junttila V., Karuanne T., Leppänen V. (2010): Estimation of forest stand parameters from airborne laser scanning using calibrated plot databases. Forest Science, 56: 257–270.
 
Kangas A., Astrup R., Breidenbach J., Fridman J., Gobakken T., Korhonen K.T., Maltamo M., Nilsson M., Nord-Larsen T., Næsset E. (2018): Remote sensing and forest inventories in Nordic countries – roadmap for the future. Scandinavian Journal of Forest Research, 33: 397–412. https://doi.org/10.1080/02827581.2017.1416666
 
Katila M., Tomppo E. (2002): Stratification by ancillary data in multisource forest inventories employing k-nearest-neighbour estimation. Canadian Journal of Forest Research, 32: 1548–1561. https://doi.org/10.1139/x02-047
 
Kleinn C. (1994): Comparison of the performance of line sampling to other forms of cluster sampling. Forest Ecology and Management, 68: 365–373. https://doi.org/10.1016/0378-1127(94)90057-4
 
Koch B. (2011): Status and future of laser scanning, synthetic aperture radar and hyperspectral remote sensing data for forest biomass assessment. ISPRS Journal of Photogrammetry and Remote Sensing, 65: 581–590. https://doi.org/10.1016/j.isprsjprs.2010.09.001
 
Levene H. (1960): Robust tests for equality of variances. In: Olkin I. (Ed.): Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling. Standford, Standford University Press: 278–292.
 
Li F., Zhang L., Davis C.J. (2002): Modeling the joint distribution of tree diameters and heights by bivariate generalized beta distribution. Forest Science, 48: 47–58.
 
Liu G., Liang K.Y. (1997): Sample Size Calculations for Studies with Correlated Observations. Biometrics, 53: 937–947. https://doi.org/10.2307/2533554
 
Maas C.J.M., Hox J.J. (2005): Sufficient Sample Sizes for Multilevel Modeling. Methodology, 1: 86–92. https://doi.org/10.1027/1614-2241.1.3.86
 
MacLeod D. (1978): The Forest Management Institute Tree Data Bank. Ottawa, Forest Management Institute in Ottawa. Information Report FMR-X-112: 16.
 
Magnussen S. (1986): Diameter distributions in Picea abies described by the Weibull model. Scandinavian Journal of Forest Research, 1: 493–502. https://doi.org/10.1080/02827588609382440
 
Magnussen S. (2016): A new mean squared error estimator for a synthetic domain mean. Forest Science, 63: 1–9. https://doi.org/10.5849/forsci.16-056
 
Magnussen S. (2018): An estimation strategy to protect against over-estimating precision in a LiDAR-based prediction of a stand mean. Journal of Forest Science, 64: 497–505. https://doi.org/10.17221/120/2018-JFS
 
Magnussen S., Breidenbach J. (2017): Model-dependent forest stand-level inference with and without estimates of stand-effects. Forestry: An International Journal of Forest Research, 90: 675–685. https://doi.org/10.1093/forestry/cpx023
 
Magnussen S., Frazer G., Penner M. (2016): Alternative mean-squared error estimators for synthetic estimators of domain means. Journal of Applied Statistics, 43: 2550–2573. https://doi.org/10.1080/02664763.2016.1142942
 
Magnussen S., Mauro F., Breidenbach J., Lanz A., Kändler G. (2017): Area-level analysis of forest inventory variables. European Journal of Forest Research, 136: 839–855. https://doi.org/10.1007/s10342-017-1074-z
 
Magnussen S., Næsset E., Gobakken T. (2015): LiDAR supported estimation of change in forest biomass with time invariant regression models. Canadian Journal of Forest Research, 45: 1514–1523. https://doi.org/10.1139/cjfr-2015-0084
 
Mäkelä H., Pekkarinen A. (2004): Estimation of forest stand volumes by Landsat TM imagery and stand-level field-inventory data. Forest Ecology and Management, 196: 245–255. https://doi.org/10.1016/j.foreco.2004.02.049
 
Mauro F., Molina I., García-Abril A., Valbuena R., Ayuga-Téllez E. (2016): Remote sensing estimates and measures of uncertainty for forest variables at different aggregation levels. Environmetrics, 27: 225–238. https://doi.org/10.1002/env.2387
 
Mbachu H., Nduka E., Nja M. (2012): Designing a Pseudo R-Squared Goodness-of-Fit Measure in Generalized Linear Models. Journal of Mathematics Research, 4: 148. https://doi.org/10.5539/jmr.v4n2p148
 
Melville G., Stone C., Turner R. (2015): Application of LiDAR data to maximise the efficiency of inventory plots in softwood plantations. New Zealand Journal of Forestry Science, 45: 1–9. https://doi.org/10.1186/s40490-015-0038-7
 
Mostafa S.A., Ahmad I.A. (2017): Recent developments in systematic sampling: A review. Journal of Statistical Theory and Practice, 12: 1–21.
 
Muukkonen P., Heiskanen J. (2007): Biomass estimation over a large area based on standwise forest inventory data and ASTER and MODIS satellite data: A possibility to verify carbon inventories. Remote Sensing of Environment, 107: 617–624. https://doi.org/10.1016/j.rse.2006.10.011
 
Næsset E. (2014): Area-based inventory in Norway – from innovation to an operational reality. In: Maltamo M., Naesset E., Vauhkonen J. (eds) :Forestry Applications of Airborne Laser Scanning. Dordrecht, Springer: 215–240.
 
Newton P., Amponsah I. (2007): Comparative evaluation of five height–diameter models developed for black spruce and jack pine stand-types in terms of goodness-of-fit, lack-of-fit and predictive ability. Forest Ecology and Management, 247: 149–166. https://doi.org/10.1016/j.foreco.2007.04.029
 
Nothdurft A., Saborowski J., Breidenbach J. (2009): Spatial prediction of forest stand variables. European Journal of Forest Research, 128: 241–251. https://doi.org/10.1007/s10342-009-0260-z
 
Pagliarella M.C., Corona P., Fattorini L. (2018): Spatially-balanced sampling versus unbalanced stratified sampling for assessing forest change: evidences in favour of spatial balance. Environmental and Ecological Statistics, 25: 111–123. https://doi.org/10.1007/s10651-017-0378-y
 
Payandeh B. (1991): Plonski’s (metric) yield tables formulated. The Forestry Chronicle, 67: 545–546. https://doi.org/10.5558/tfc67545-5
 
Pinheiro J.C., Bates D.M. (2000): Mixed-effects models in S and S-plus. New York, Springer: 1–528.
 
Plonski W. (1960): Normal Yield Tables for Black Spruce, Jack Pine, Aspen, White Birch, Tolerant Hardwoods, White Pine, and Red Pine for Ontario. Ontario, Ontario Department od Land and Forests. Silvicultural Series Bulletin 2: 39.
 
Puliti S., Ene L.T., Gobakken T., Næsset E. (2017): Use of partial-coverage UAV data in sampling for large scale forest inventories. Remote Sensing of Environment, 194: 115–126. https://doi.org/10.1016/j.rse.2017.03.019
 
Räty M., Heikkinen J., Kangas A. (2018): Assessment of sampling strategies utilizing auxiliary information in large-scale forest inventory. Canadian Journal of Forest Research, 48: 749–757. https://doi.org/10.1139/cjfr-2017-0414
 
Rennolls K., Wang M. (2005): A new parameterization of Johnson’s SB distribution with application to fitting forest tree diameter data. Canadian Journal of Forest Research, 35: 575–579. https://doi.org/10.1139/x05-006
 
Saarela S., Grafström A., Ståhl G., Kangas A., Holopainen M., Tuominen S., Nordkvist K., Hyyppä J. (2015): Model-assisted estimation of growing stock volume using different combinations of LiDAR and Landsat data as auxiliary information. Remote Sensing of Environment, 158: 431–440. https://doi.org/10.1016/j.rse.2014.11.020
 
Saarela S., Holm S., Grafström A., Schnell S., Næsset E., Gregoire T.G., Nelson R.F., Ståhl G. (2016): Hierarchical model-based inference for forest inventory utilizing three sources of information. Annals of Forest Science, 73: 895–910. https://doi.org/10.1007/s13595-016-0590-1
 
Särndal C.E., Swensson B., Wretman J. (1992): Model Assisted Survey Sampling. New York, Springer: 694.
 
Self S.G., Mauritsen R.H. (1988): Power/Sample Size Calculations for Generalized Linear Models. Biometrics, 44: 79–86. https://doi.org/10.2307/2531897
 
Sexton J.O., Bax T., Siqueira P., Swenson J.J., Hensley S. (2009): A comparison of lidar, radar, and field measurements of canopy height in pine and hardwood forests of southeastern North America. Forest Ecology and Management, 257: 1136–1147. https://doi.org/10.1016/j.foreco.2008.11.022
 
Snedecor G.W., Cochran W.G. (1971): Statistical Methods. Iowa, Iowa State University Press: 593.
 
Snijders T.A. (2005): Power and Sample Size in Multilevel Linear Models. In: Everitt B.S., Howell D.C. (eds): Encyclopedia of Statistics in Behavioral Science, 3: 1570–1573.
 
Spurr S.H. (1952): Forest Inventory. New York, Ronald Press: 476.
 
Stevens D.L., Olsen A.R. (2004): Spatially balanced sampling of natural resources. Journal of the American Statistical Association, 99: 262–278. https://doi.org/10.1198/016214504000000250
 
Tam S.M. (1995): Optimal and robust strategies for cluster sampling. Journal of the American Statistical Association, 90: 379–382. https://doi.org/10.1080/01621459.1995.10476523
 
Tomppo E., Malimbwi R., Katila M., Mäkisara K., Henttonen H.M., Chamuya N., Zahabu E., Otieno J. (2014): A sampling design for a large area forest inventory: Case Tanzania. Canadian Journal of Forest Research, 44: 931–948. https://doi.org/10.1139/cjfr-2013-0490
 
von Lüpke N., Hansen J., Saborowski J. (2012): A three-phase sampling procedure for continuous forest inventory with partial re-measurement and updating of terrestrial sample plots. European Journal of Forest Research, 131: 1979–1990. https://doi.org/10.1007/s10342-012-0648-z
 
Wilhelm M., Tillé Y., Qualité L. (2017): Quasi-systematic sampling from a continuous population. Computational Statistics & Data Analysis, 105: 11–23.
 
download PDF

© 2020 Czech Academy of Agricultural Sciences