Indirect approach for estimation of forest degradation in non-intact dry forest: modelling biomass loss with Tweedie distributions
- Klaus Dons^{1}Email author,
- Sushma Bhattarai^{3},
- Henrik Meilby^{2},
- Carsten Smith-Hall^{2} and
- Toke Emil Panduro^{2}
DOI: 10.1186/s13021-016-0051-z
© The Author(s) 2016
Received: 10 November 2015
Accepted: 31 May 2016
Published: 29 June 2016
Abstract
Background
Implementation of REDD+ requires measurement and monitoring of carbon emissions from forest degradation in developing countries. Dry forests cover about 40 % of the total tropical forest area, are home to large populations, and hence often display high disturbance levels. They are susceptible to gradual but persistent degradation and monitoring needs to be low cost due to the low potential benefit from carbon accumulation per unit area. Indirect remote sensing approaches may provide estimates of subsistence wood extraction, but sampling of biomass loss produces zero-inflated continuous data that challenges conventional statistical approaches. We introduce the use of Tweedie Compound Poisson distributions from the exponential dispersion family with Generalized Linear Models (CPGLM) to predict biomass loss as a function of distance to nearest settlement in two forest areas in Tanzania.
Results
We found that distance to nearest settlement is a valid proxy variable for prediction of biomass loss from fuelwood collection (p < 0.001) and total subsistence wood extraction (p < 0.01). Biomass loss from commercial charcoal production did not follow a spatial pattern related to settlements.
Conclusions
Distance to nearest settlement seems promising as proxy variable for estimation of subsistence wood extraction in dry forests in Tanzania. Tweedie GLM provided valid parameters from the over-dispersed continuous biomass loss data with exact zeroes, and observations with zero biomass loss were successfully included in the model parameters.
Keywords
Compound Poisson distribution Spatial analysis REDD+ Forest monitoring TanzaniaBackground
Measuring forest degradation using remote sensing (RS) is generally more challenging than measuring deforestation [1] and, combined with scarce funds and insufficient technical capacity in many developing countries, lack of estimates of emissions from forest degradation is limiting implementation of reduced emissions from deforestation and degradation (REDD+) [2]. The indirect RS approach that maps the infrastructure facilitating degradation, such as settlements and roads, offers a way to delineate intact forest from non-intact forest and to model and estimate emissions from forest degradation in non-intact forest [3]. This approach may also be used to estimate past carbon emissions from present measurements of degradation activities and establishing the historic baseline necessary to demonstrate additionality [4]. Most forest degradation studies on carbon stock changes focus on degradation in undisturbed humid forests with high levels of carbon stock per hectare [3, 5, 6]. However, it has been suggested that degradation in dry forests may be more widespread on a global scale [7] and that such degradation needs to be quantified [8, 9]. Degradation in dry forests is likely to imply a slow reduction in carbon stock over time [8], including subsistence extraction due to underlying drivers such as population growth and poverty [10] which further limits the use of RS for estimation of emission levels. The potential carbon benefit per unit area in dry forests is low, but countries may access benefits from REDD+ funding if they can quantitatively monitor significant degradation activities over large forest areas likely to be degraded [4]. Forest degradation can be defined at national or sub-national levels, as can the specific degradation activities targeted for monitoring [7]. In general, degradation drivers can be categorized as either subsistence wood extraction activities, commercial activities, or uncontrolled wild fires [4, 11]. The first two categories are compatible with von Thünen’s model of agricultural land use, saying that harvesters are willing to travel longer for high value products [12]—low value subsistence products are extracted in proximity to population centres, whereas higher value commercial products can be harvested in remote forest areas. In consequence, we would expect that the level of subsistence degradation decreases with increasing distance to populated areas [13, 14].
In this study, we use an indirect RS approach to estimate biomass loss, and hence carbon emissions, in a tropical dry forest. It is applied to estimate biomass loss from both commercial and subsistence activities. A common problem in inventories is that sampling of biomass loss produces continuous data including a high number of true zeroes. We define true zeroes as those that represent actual response outcomes, i.e. ‘no disturbance’ in a particular sampling unit and not zeroes created by missing data. Statistical data analysis of such zero-inflated continuous data is challenging as ordinary distributions, such as normal or gamma, fit the data poorly and log-transformations or similar are not possible due to the zeroes [15]. A common solution in studies of biological resources, such as in marine biology and fisheries, has been to sub-set or transform the data with a constant to remove the zeroes and then use standard models such as generalized linear models (GLM) with log-normal or gamma distributions on the remaining continuous data [16, 17]. However, the assumptions of these approaches are associated with statistical problems [18, 19] and have been found to overestimate the quantity of biomass from fisheries [20]. Another solution is to use a separate model for the zeroes, e.g. delta-type models by use of logit, or probit to estimate the rate of zeroes, followed by a model of the continuous positive data [21]. Such two-step models allow different estimates for the two components and while this has been found useful in econometric studies [15] it is less applicable to biomass data because of limitations to application in a multiplicative structure [19]. Hence, when modelling forest degradation, the zeroes are an inherent part of the data and should be actively included for correct parameter estimates of biomass loss. Here we apply the exponential dispersion model (EDM) family of distributions [22, 23]. EDM distributions are response distributions for GLM and include the Tweedie family of distributions, which has proven especially useful for modelling positive continuous data with a proportion of exact zeroes [24, 25]. Tweedie GLM implements a multiplicative structure on the dependent variable [19] by combining the discrete and continuous probabilities and thereby provides valid estimates where true zeroes are included in the estimate of the continuous response variable. The implementation in the established GLM framework makes model results comprehensible to readers. EDMs have been available for decades but have a density function which is analytically intractable and thus not included in commercial statistical software until recently [26].
This paper assesses the accuracy of an indirect RS approach to estimate commercial and subsistence wood extraction in dry tropical forests. We use Tweedie Compound Poisson distributions from the exponential dispersion family with GLM (CPGLM) to predict biomass loss as a function of distance to nearest settlement. Furthermore, we demonstrate a simple GIS approach to establish area-based CPGLM predictions of biomass loss from subsistence degradation activities for potential application to REDD+ monitoring systems.
Results
Observed levels of wood extraction
Biomass loss (Mg ha^{−1}) | Number of stumps ha^{−1} | |||||
---|---|---|---|---|---|---|
IDO | KIW | Total | IDO | KIW | Total | |
Fuelwood | 1.226 | 1.144 | 1.182 | 43 | 28 | 35 |
Total subsistence (incl. fuelwood) | 3.425 | 2.678 | 3.027 | 97 | 56 | 75 |
Commercial charcoal | 2.102 | 5.508 | 3.915 | 9 | 30 | 21 |
Total | 5.527 | 8.187 | 6.942 | 106 | 86 | 96 |
Parameters for the models of biomass loss
Dependent variable | P | ϕ | Intercept α | Location effect γ (KIW = 1) | Spatial trend (slope β) |
---|---|---|---|---|---|
Fuelwood | 1.5579 | 23.1135 | 6.48168 (0.61441)*** | 1.11205 (0.53172)* | −0.00098 (0.00022)*** |
Fuelwood (global) | 1.5627 | 24.0375 | 6.45821 (0.65354)*** | – | −0.00074 (0.00022)** |
Total subsistence ext | 1.5471 | 22.7754 | 6.48894 (0.47176)*** | – | −0.00038 (0.00014)** |
Charcoal | 1.4475 | 142.0440 | 6.38165 (0.75172)*** | – | −0.00024 (0.00022) ns |
Application of CPGLM for spatial prediction of biomass loss
Discussion
We applied a CPGLM model for estimation of biomass loss by subsistence wood extraction in non-intact forest areas as a function of distance to nearest settlement. Model parameters demonstrated significant decrease in biomass loss with increasing distance to nearest settlement. The result was valid for fuelwood (p < 0.001) and for total subsistence wood harvest (p < 0.01). The per plot levels of wood extraction for charcoal production were higher than that of subsistence harvest. All charcoal extraction sites but one were situated within 4500 m from nearest settlements, but otherwise charcoal extraction does not seem to follow a specific spatial pattern with regard to settlements. This was expected considering von Thünen’s theory of locational rent. From field studies we experienced that charcoal producers’ choice of extraction site follows the availability of a combination of tree species and ground surface conditions. The high value of the product makes the transportation cost worthwhile and we suggest that commercial degradation activities are perhaps best detected with direct RS approaches focusing on canopy cover changes. Because of the likely acceptance of longer travel distances [27] indirect indicators, such as infrastructure and concentration of population, may not apply for estimation of charcoal extraction.
In the case of sampling for biomass loss, most models applied are not able to facilitate a multiplicative structure allowing for the zeroes to take part in the final prediction of biomass. CPGLM seems to provide good fit for this type of data, so why has CPGLM not already been widely applied for estimation of forest degradation? One reason may be that Tweedie models in general have been seen as intractable and perhaps inaccurate as most common optimization procedures previously available are inappropriate [26]. In later years, improvements in optimization routines and statistical software [28] have increased the use of the models [26]. In recent ecological studies, CPGLM has been found particularly useful for estimating biomass from fisheries with no-catch occasions and the CPGLM seems to provide more accurate estimates and better fit to these data than, e.g. delta approaches [18, 19, 29, 30]. The Tweedie CPGLM has also been applied with success for near-optimal modelling of monthly rainfall [24]. A few studies on terrestrial ecology has also applied CPGLM to model for example reproductive capacity of moss [31] and the area of forest fires [32].
We also found good fits with the CPGLM and it was possible to provide highly significant predictions of biomass loss from subsistence wood harvesting as a function of increasing distance to nearest settlement. Model performance was comparable with delta log-normal models in terms of AIC and quantile residuals, but an advantage of CPGLM is that the response variable is maintained at its original scale, thereby providing higher variance stability compared to models that need back-transformation of the dependent variable in order to derive predicted values at original scale [20]. The predictions were used to produce maps of forest degradation in order to provide full area-based predictions of subsistence biomass loss. The use of concentric buffer distances with average extraction levels predicted by the CPGLM model for different distance intervals is simple and easy to implement at various spatial resolutions. It requires a forest vector, e.g. from Landsat, and settlements from remote sensing or existing GIS. This indirect remote sensing based prediction of subsistence wood extraction can be applied across large geographical areas using default values for extraction-distance relationships or be locally calibrated depending on the chosen accuracy level for measurement of subsistence extraction under REDD+.
In spite of the optimism regarding the model fit by CPGLM, this study includes a number of limitations. Biomass loss is estimated through two models. A stem shape model for conversion of stump DM to DBH and an allometric model for DBH to biomass (kg per tree). Although stumps represent true evidence of wood extraction, the use of two models introduces unquantifiable uncertainty in the biomass loss estimates. The spatial analysis tool illustrated here includes only few variables. Proxy variables such as population pressure, access roads, cost-distance and forest productivity (normalized difference vegetation index) were not included. It is expected that a larger study with more field sample plots would gain improved model fit by including more relevant proxy variables in a multiplicative way, e.g. by establishing various spatial predictions individually for a multiplicative raster surface overlay. Another improvement to this study would be to investigate the possible effects of spatial autocorrelation as villages and plots follow similar spatial distributions. This could lead to similarities in the data that may be confounded with the wood harvest. Finally, in this study we have assumed that harvested wood equals biomass loss. In doing so we have not included the potentially increased productivity in remaining trees that may result from reduced competition.
CPGLM offers potential for REDD+ monitoring approaches as we may expect better model fit for zero-inflated continuous data, thereby reducing the chance of overestimating biomass loss. Monitoring of forest degradation with remote sensing has been viewed as more complex and challenging than for deforestation. The introduction of degradation into REDD+ at COP 13 in Bali [33] was recently described as the end of the idea of a simple forest-based climate change mitigation system [34]. However, in this study we suggest a way to simplify quantification of subsistence wood extraction which has been identified as the most complex activity to estimate by remote sensing [4, 27].
If integrated in a GIS environment and locally calibrated, accurate estimates of subsistence wood harvest levels may be expected from non-intact dry forest areas using CPGLM, although further studies are needed to demonstrate applicability outside of the study areas assessed here.
Conclusions
We assessed the accuracy of an indirect remote sensing approach to estimate commercial and subsistence wood extraction in dry tropical forests in Tanzania. We used Tweedie Compound Poisson distributions from the exponential dispersion family with GLM (CPGLM) to predict biomass loss as a function of distance to nearest settlement. We found that levels of fuelwood extraction as well as total subsistence wood harvest decrease significantly with increasing distance to nearest settlement. The level of biomass loss associated with commercial charcoal production does not follow a systematic spatial pattern related to settlements in the study areas. CPGLM offers potential for REDD+ monitoring approaches as we expect better model fit for continuous data with high numbers of true zeroes, thereby reducing the chance of overestimating biomass loss. Based on the present results we suggest a low cost GIS approach to establish area based CPGLM predictions of biomass loss from subsistence wood extraction using distance to nearest settlement as proxy variable. Further studies are needed to demonstrate if the approach is valid on a regional level for implementation in REDD+ monitoring systems.
Methods
Study area
Data
We established a forest vector by supervised classification of a Landsat TM scene of 6 December 2009. A majority filter cleaned up minor forest patches outside the forest and minor gaps inside. The scene was captured at the beginning of the rainy season where there is highest separability between ground herbs and foliage of shrubs and trees. We used a very high resolution (VHR) QuickBird (QB) image for ground truthing and obtained an overall accuracy of 77.4 %, with Kappa coefficient 0.56. We compared available local GIS data, the National Reconnaissance Level Land Use and Natural Resources Mapping Project of 1997 [38], and found poor agreement with settlements in the QB image. Settlements were therefore digitized on single house level directly on VHR images from 2003 with Google Earth^{TM}. The quality was visually approved with QB in KIW. We established 25 transects, each 4 km, at regular distances perpendicular to the forest edge in each site. In KID, a major vehicle road across the western part of the forest was considered as forest edge. In order to maintain the direction perpendicular to the forest edge, while respecting the systematic random sample design, all transects face south-east in area 1 and north in area 2. As the purpose of this study is to quantify forest degradation and not deforestation, all transects start at the forest edge. Above ground biomass loss was recorded in circles with radius 15 m, established at distances 500, 2000, and 3500 m along the transect. Diameter (DM) of all stumps >5 cm remaining after wood extraction was recorded. Stump DM was measured at 20 cm above ground or at height of the cut. In order to list local uses of the forest and reasons for wood extraction a qualitative pre-study was carried out. This pre-study was based on interviews with different groups of forest users and key informants including charcoal makers, fuelwood collectors, natural resource committee members, and pastoralists. Forest degradation activities in the areas include fuelwood extraction for domestic and commercial use, commercial charcoal production, poles and logs for local construction, animal grazing, sub-canopy fires and clearance for agriculture in small patches. In the field, local forest users with experience in a variety of wood harvest activities and knowledge of species participated in classification of stumps to commercial charcoal production or subsistence uses. They mainly used species, type of cut, stump DM, and location of the stumps in relation to other signs of degradation activities to decide the type of wood product associated with each stump.
Estimation of biomass loss
Analysis of variance
Source | DF | Sum of squares | Mean square | Approximate F value | Pr > F |
---|---|---|---|---|---|
Model | 2 | 289,095 | 144,548 | 8775.58 | <0.0001 |
Error | 532 | 8762.9 | 16.716 | ||
Uncorrected total | 534 | 297,858 |
Parameter estimates of the stem shape model
Approx. parameter estimate | Std error | Approximate 95 % confidence limits | ||
---|---|---|---|---|
b | −0.0452 | 0.0173 | −0.0792 | −0.0113 |
c | −0.00108 | 0.000396 | −0.00185 | −0.00030 |
Data analysis
Prediction models with CPGLM and confidence intervals by bootstrapping were implemented using the R environment for statistical computation and graphics. The tweedie models were computed with four different R packages; Statmod [41], Tweedie [42], CPLM [26], and fishmod [43] and results were comparable. Here we present the results using the CPLM and the fishmod packages.
Declarations
Authors’ contributions
All authors contributed to the study design and to writing the manuscript. SB conducted data collection. KD, HM, and TEP performed data analysis. All authors read and approved the final manuscript.
Acknowledgements
The authors wish to thank Filemon Elisante from Sokoine University of Agriculture and people from villages in Idodi and Kiwele, Tanzania, for their valuable assistance during field work. We are particularly grateful that Dunn and Smyth (2005) and Zhang (2013) chose to produce R programs for implementation of the Tweedie family of model distributions. This study was financially supported by the O.H.F and A.J.-E Heilmanns Foundation, Oticon, The AD scholarship of University of Copenhagen, and the Ministry of Foreign Affairs, Denmark (Grant No. LIFE 10-068).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Böttcher H, Eisbrenner K, Fritz S, Kindermann G, Kraxner F, McCallum I, et al. An assessment of monitoring requirements and costs of reduced emissions from deforestation and degradation. Carbon Balance Manag. 2009;4:7.View ArticleGoogle Scholar
- Herold M. An assement of national forest capabilities in tropical non-Annex 1 countries: recommandations for capacity building: GOFC-GOLD Land Cover Project Office, Friedrich Schiller University Jena, prepared for The Prince’s rainforest project and the government of Norway; 2009.
- Mollicone D, Achard F, Federici S, Eva HD, Grassi G, Belward A, et al. An incentive mechanism for reducing emissions from conversion of intact to non-intact forests. Clim Change. 2007;83(4):477–93.View ArticleGoogle Scholar
- Herold M, Roman-Cuesta RM, Mollicone D, Hirata Y, Van Laake P, Asner GP, et al. Options for monitoring and estimating historical carbon emissions from forest degradation in the context of REDD+. Carbon Balance Manag. 2011;6(1):13.View ArticleGoogle Scholar
- Souza CM Jr, Roberts DA, Cochrane MA. Combining spectral and spatial information to map canopy damage from selective logging and forest fires. Remote Sens Environ. 2005;(98):329–43.View ArticleGoogle Scholar
- Asner GP, Knapp DE, Broadbent EN, Oliveira PJC, Keller M, Silva JN. Selective logging in the Brazilian Amazon. Science. 2005;310:480–2.View ArticleGoogle Scholar
- Herold M, Skutsch M. Monitoring, reporting and verification for national REDD plus programmes: two proposals. Environ Res Lett. 2011;6(1):014002–10.View ArticleGoogle Scholar
- Grainger A. Constraints on modelling the deforestation and degradation of tropical open woodlands. Glob Ecol Biogeogr. 1999;8(3–4):179–90.View ArticleGoogle Scholar
- Chidumayo EN. Forest degradation and recovery in a miombo woodland landscape in Zambia: 22 years of observations on permanent sample plots. For Ecol Manag. 2013;291:154–61.View ArticleGoogle Scholar
- le Polain de Waroux Y, Lambin EF. Monitoring degradation in arid and semi-arid forests and woodlands: the case of the argan woodlands (Morocco). Appl Geogr. 2012;32(2):777–86. View ArticleGoogle Scholar
- GOFC-GOLD. A sourcebook of methods and procedures for monitoring and reporting anthropogenic greenhouse gas emissions and removals caused by deforestation, gains and losses of carbon stocks in forests remaining forests and forestation. Report version COP19-1. Alberta, Canada: Natural Resources Canada; 2013.
- Hall P, editor. Von Thünen's Isolated State (English translation by Carla M. Wartenberg, with an introduction by the editor). Pergamon Press; 1966.
- Shackleton CM, Griffin NJ, Banks DI, Mavrandonis JM, Shackleton SE. Community structure and species composition along a disturbance gradient in a communally managed South African savanna. Vegetatio. 1994;115(2):157–67. Google Scholar
- Albers HJ, Robinson EJZ. A review of the spatial economics of non-timber forest product extraction: implications for policy. Ecol Econ. 2013;92:87–95.View ArticleGoogle Scholar
- Min J, Agresti A. Modeling nonnegative data with clumping at zero: a survey. JIRSS. 2002;1(1–2):33.Google Scholar
- Brynjarsdóttir J, Stefánsson G. Analysis of cod catch data from Icelandic groundfish surveys using generalized linear models. Fish Res. 2004;70(2–3):195–208.View ArticleGoogle Scholar
- Ortiz M, Arocha F. Alternative error distribution models for standardization of catch rates of non-target species from a pelagic longline fishery: billfish species in the Venezuelan tuna longline fishery. Fish Res. 2004;70(2–3):275–97.View ArticleGoogle Scholar
- Shono H. Application of the Tweedie distribution to zero-catch data in CPUE analysis. Fish Res. 2008;93(1–2):154–62.View ArticleGoogle Scholar
- Foster S, Bravington M. A Poisson-Gamma model for analysis of ecological non-negative continuous data. Environ Ecol Stat. 2013;20(4):533–52.View ArticleGoogle Scholar
- Tascheri R, Saavedra-Nievas JC, Roa-Ureta R. Statistical models to standardize catch rates in the multi-species trawl fishery for Patagonian grenadier (Macruronus magellanicus) off Southern Chile. Fish Res. 2010;105(3):200–14. View ArticleGoogle Scholar
- Duan N, Manning WG Jr, Morris CN, Newhouse JP. A comparison of alternative models for the demand for medical care. J Bus Econ Stat. 1983;1(2):115–26. Google Scholar
- Jorgensen B. Exponential dispersion models. J Roy Stat Soc Ser B (Methodol). 1987;49(2):127–62. Google Scholar
- Jorgensen B. The theory of dispersion models. Abingdon: Taylor & Francis; 1997.Google Scholar
- Hasan MM, Dunn PK. Two Tweedie distributions that are near-optimal for modelling monthly rainfall in Australia. Int J Climatol. 2011;31(9):1389–97. doi:10.1002/joc.2162.View ArticleGoogle Scholar
- Smyth GK. Regression modelling of quantity data with exact zeros. In: Wilson RJ, Osaki S, Murthy DP, editors. Proceedings of the second Australia-Japan workshop on stochastic models in engineering, technology and management. Technology management centre. St Lucia: University of Queensland; 1996. p. 572–80.Google Scholar
- Zhang Y. Likelihood-based and Bayesian methods for Tweedie compound Poisson linear mixed models. Stat Comput. 2013;23(6):743–57. View ArticleGoogle Scholar
- Peres CA, Barlow J, Laurance WF. Detecting anthropogenic disturbance in tropical forests. Trends Ecol Evol. 2006;21(5):227–9. View ArticleGoogle Scholar
- Dunn P, Smyth G. Series evaluation of Tweedie exponential dispersion model densities. Stat Comput. 2005;15(4):267–80. View ArticleGoogle Scholar
- Candy SG. Modelling catch and effort data using generalised linear models, the tweedie distribution, random vessel effects and random stratum-by-year effects. Ccamlr Sci. 2004;11:59–80.Google Scholar
- Ancelet S, Etienne MP, Benoît H, Parent E. Modelling spatial zero-inflated continuous data with an exponentially compound Poisson process. Environ Ecol Stat. 2010;17(3):347–76.View ArticleGoogle Scholar
- Stark LR, Brinda JC, McLetchie DN. An experimental demonstration of the cost of sex and a potential resource limitation on reproduction in the moss Pterygoneurum (Pottiaceae). Am J Bot. 2009;96(9):1712–21.View ArticleGoogle Scholar
- Podur JJ, Martell DL, Stanford D. A compound Poisson model for the annual area burned by forest fires in the province of Ontario. Environmetrics. 2010;21(5):457–69. doi:10.1002/env.996.Google Scholar
- United Nations Framework Convention of Climate Change. Reporting on global observing systems for climate. FCCC/CP/2007/6/Add.2; 2007
- Nguon P, Kulakowski D. Natural forest disturbances and the design of REDD+ initiatives. Environ Sci Policy. 2013;33:332–45. View ArticleGoogle Scholar
- Munishi P, Mringi S, Shirima D, Linda S. The role of the miombo woodlands of the southern highlands of Tanzania as carbon sinks. J Ecol Nat Env. 2010;2(12):261–2.
- Skutsch M, McCall M, Trines E. Reference scenarios for degradation under REDD. Kyoto: Think Global, Act Local project. KTGAL Policy Paper 5; 2009.
- Lund JF, Treue T. Are we getting there? Evidence of decentralized forest management from the tanzanian miombo woodlands. World Dev. 2008;36(12):2780–800. View ArticleGoogle Scholar
- Hunting technical services, cartographer National reconnaissance level land use and natural resources Mapping Project. Ministry of natural resources and tourism, Tanzania;1997.
- Chamshama SAO, Mugasha AG, Zahabu E. Stand biomass and volume estimation for Miombo woodlands at Kitulangalo, Morogoro Tanzania. South Afr For J. 2004;200:59–70.Google Scholar
- Malimbwi RE, Solberg B, Luoga EJ. Estimation of biomass and volume in miombo woodland at Kitulanghalo Forest Reserve, Tanzania. J Trop For Sci. 1994;7:230–42.Google Scholar
- Smyth GK. Statistical modeling—package ‘statmod’’. CRAN. version 1.4.17: 2013.
- Dunn PK. Tweedie exponential family models—package ‘tweedie’. CRAN.version 2.1.7 2013.
- Foster SD. Fits Poisson-sum-of-Gammas GLMs, Tweedie GLMs, and delta log-normal models. Package ‘fishmod’. CRAN. version 0.25. 2015.