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## Agrociencia

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*On-line version* ISSN 2521-9766*Print version* ISSN 1405-3195

### Agrociencia vol.52 n.7 México Oct./Nov. 2018

Natural Renewable Resources

Density funcions: an application for delimiting optimal intervals of climate and physiography for forest species

^{1} División de Estudios de Postgrado-Instituto de Estudios Ambientales, Universidad de la Sierra Juárez, Avenida Universidad S/N, Ixtlán de Juárez, 68725 Oaxaca, México.

^{2} Instituto Tecnológico del Valle del Guadiana. Carretera Durango-México Km 22.5 Villa Montemorelos, C.P. 34371, Durango, México.

The space a species occupies in a natural system can be delimited by the physical-geographic medium or by the environmental conditions that define it. The objective of this study was to delimit climate intervals in which the maximum presence rate occurs of three tree species native to the Sierra Norte of Oaxaca, Mexico (*Pinus pseudostrobus Lindl* (var. Apulcensis), *Pinus patula* Schl. et Cham, and *Quercus macdougallii* Martínez), in function of nine environmental variables using the Weibull density function and the finite Gaussian mixture model. To this end, we used data from 634 plots measuring 1,000 m^{2}, which were established systematically in the study area. The results showed that high dispersion of the two pines species is related to mean precipitation from April to September. In contrast, the scarce presence of *Quercus magdougalli*, an endemic species, seems to be related to the reduced intervals of winter precipitation and to altitude. The two density functions tested allowed definition of optimal environmental intervals for each species. The finite mixture model was more flexible than the Weibull function when identifying bimodal distributions, particularly for the two pines species, whose observed dispersion pattern was more heterogeneous than that of *Quercus*. The results obtained will serve to prioritize areas for purposes of conservation or commercialization.

**Key words: **Gaussian mixture model; Weibull function; *Quercus magdougalli*; Sierra Norte of Oaxaca; temperate forest

El espacio que ocupa una especie en el sistema natural puede delimitarse por el medio físico-geográfico o por las condiciones ambientales que lo definen. El objetivo de este estudio fue delimitar intervalos climáticos en los que ocurre la tasa de presencia máxima de tres especies arbóreas (*Pinus pseudostrobus Lindl* (var. Apulcensis)*, Pinus patula* Schl. et Cham *y Quercus macdougallii* Martínez) nativas de la Sierra Norte de Oaxaca, México, en función de nueve variables ambientales usando la función de densidad de Weibull y el modelo de Gauss de mezclas finitas. Para lo anterior, se usaron datos de 634 parcelas de 1,000 m^{2} las cuales se establecieron sistemáticamente en el área de estudio. Los resultados mostraron que la alta dispersión de dos de las especies estudiadas (ambas de pino) está relacionada con la precipitación media de abril a septiembre; en contraste, la escasa presencia de *Quercus magdougalli* (especie endémica) parece estar relacionada con los intervalos reducidos de la precipitación en el invierno y la altitud. Las dos funciones de densidad probadas permitieron definir los intervalos ambientales óptimos para cada especie. El modelo de mezclas finitas fue más flexible que la función de Weibull al identificar distribuciones bimodales, en particular para las dos especies de pino cuyo patrón de dispersión observado fue más heterogéneo que el de *Quercus*. Los resultados obtenidos podrían servir para priorizar áreas con fines de conservación y comercialización.

**Palabras clave: **Modelo de Gauss de mezclas finitas; función de Weibull; *Quercus magdougalli*; Sierra Norte de Oaxaca; bosque templado

Introduction

Several analytical tools were explored for studying distribution and abundance of live organisms, mainly statistical models or these in combination with geographic information systems, to characterize species’ habitats (^{Austin, 1987}; ^{Segurado and Araujo, 2004}; ^{Elith et al., 2006}) or to evaluate the response of a specific species in function of change in environmental variables that define the bioclimate niche (^{Antúnez et al., 2017a}). Models based on maximum entropy algorithms are also used to predict the potential distribution of organisms (^{Brotons et al., 2004}; ^{Phillips et al., 2009}; ^{Franklin, 2010}).

The space a species occupies can be delimited by the physical-geographic medium or by environmental conditions (^{Pearman et al., 2008}; ^{Elith y Leathwick, 2009}). Maps are a valuable tool for illustrating the physical-geographic space where a species can find ideal conditions. But delimiting and representing only the environmental space is not easy because each variable of the natural system is a variable in magnitude and intensity (^{Martínez-Antúnez 2013}; ^{Antúnez et al., 2017a}). This task could be facilitated if we knew the optimal intervals of those variables whose effect can limit or potentiate the abundance of a species in a location. That is, it is easier to delimit the optimal space of a species in function of the most relevant variables that delimit the space defined by all the variables (multi-dimensional space) (^{Hutchinson, 1957}; ^{Austin and Smith, 1990}).

Statistical likelihood functions are used to describe the relationship between living organisms and the environment parting from observed patterns, to explain the relationship between the species and their area of greatest abundance, or to determine the spatial pattern and identify optimal climate values (^{Borda-de-Água et al., 2002}; ^{Magurran, 2004}; ^{Gowda, 2011}; ^{Verberk, 2012}; ^{Martínez-Antúnez, 2015}).

Given that it is possible to model the largest concentration of data in a probabilistic space, it is also possible to use the same principle to define an interval of any environmental variable based on maximum likelihood of a density function (^{Antúnez et al., 2017a}). In this sense, a probability density function can be a useful tool in defining climate values in which the maximum probability that an abundance of a species would occur.

The objective of this study was to determine the environmental intervals where the maximum abundance of individuals of three forest species native to the Sierra Norte of Oaxaca, Mexico, would occur using the Weibull density function and the finite Gaussian mixture model. The hypothesis was that these functions permit defining the width of the partial niche with each of the climate variables whose effects are significant to the distribution and abundance of forest species.

Materials and Methods

Study area

Santiago Comaltepec is located in the Sierra Norte region of Oaxaca (17°33’35” N and -99°26’32” W) southeast of Mexico City and has an area of approximately 26.5 km^{2} (Figure 1). Altitude varies between 1700 and 3000 m. Mean annual high temperature is 13.4 ºC, the mean annual low temperature is 4.7 ºC and summer rainfall is 600 to 1200 mm (^{CNA, 2017}; ^{INEGI, 2015}).

Sampling and studied variables

In the study of live organisms, the most conventional abundance indicators are dominance, frequency and density (^{Schweik, 2017}). In our study, relative density of each plot was used as the indicator of abundance, which is defined as the relationship between the number of individuals of each species registered in each plot and the total of individuals of the same species in all the plots. Systematic sampling was used to establish the sampling plots, and each sampling unit had an area of 1000 m^{2}, where individuals with a diameter at breast height larger than or equal to 7.5 cm were counted. In the study area, 634 plots were established.

The tree species studied were *Pinus pseudostrobus* Lindl (var. Apulcensis), *Pinus patula* Schl. et Cham and *Quercus macdougallii* Martínez. The first species is often used (in the study region) to reforest areas with degraded soils or sites without vegetation because it is a fast-growing species. The second is of high demand in sawmills, furniture factories, and cellulose and paper industries (^{Muñoz et al., 2011}). *Quercus macdougallii* is a species endemic to the Sierra Juárez and was registered in 33 of the 634 plots; it has no commercial use and is in the red list of endangered species in the category of “vulnerable” of the International Union for Conservation of Nature (^{IUCN, 2017}).

The variables selected for the study were altitude above sea level of each site (ALT, m), dominant slope of each plot (PEN, %), geographic exposure (EXP: zenithal (1), north (2), northeast (3), east (4), southeast (5), south (6), southwest (7), west (8), and northwest (9), mean winter precipitation (Nov+Dec+Jan+Feb) (WINP, mm), Julian date of the last freezing date of spring (SDAY, day), balance of precipitation summer/spring (Jul+Aug)/Apr/May) (SMRSPRPB), precipitation from April to September (GSP, mm), annual aridity index (BHH) whose value was estimated using the square root of the sum of degree-days above 5 ºC divided by mean annual precipitation (^{Rehfeldt et al., 2006}; ^{Sáenz-Romero et al., 2012}), and mean summer precipitation (Jul+Aug) (SMRP, mm). These variables were selected by a multivariate correlation analysis using the bootstrapping method (^{Yoder et al., 2004}), selecting the variables that had the highest coefficients (< 0.8 with at least one species) of a total of 22 available variables that include temperature measurements (high, low, average), precipitations in specific periods and frosts (^{Rehfeldt et al., 2006}). Physiographic variables were recorded in the field with a GPS receiver (global positioning system) for altitude and a Suunto^{®} clinometer for exposure and slope. The other variables were obtained with the ANUSPLIN^{®} modeler of the Forest Service of the US Department of Agriculture (^{Rehfeldt et al., 2006}; ^{Crookston et al., 2008}; ^{Sáenz-Romero et al., 2010}), whose algorithms are based on historical climate information from more than 4,000 weather stations in Mexico, southern US, Guatemala, Belize and Cuba, from 1961 to 1990. These variables were used in similar studies because of their importance for forest species (^{Tchebakova et al., 2005}; ^{Martínez-Antúnez et al., 2015}; ^{Rehfeldt et al., 2015}).

Data analysis

To estimate the value of an environmental variable in which the maximum abundance rate of a species occurs, two probability density functions were tested: 1) the two-parameter Weibull function (W2p), and 2) the finite Gaussian mixture model, using the density of each species expressed in relative terms as the variable of interest. The Weibull function and the finite Gaussian mixture model generate robust, flexible models. The Weibull function allows analytical expression of the value of the integral using the functions of accumulated distribution (^{Torres, 2005}). The Gaussian model offers satisfactory results because of the contributions of each Gaussian mixture in terms of likelihood (e.g. ^{Bilmes, 1998}; ^{Yang and Ahuja, 1998}; ^{Paalanen et al., 2006}).

The two-parameter Weibull likelihood density function is expressed as follows:

And its accumulated function is:

where ^{Marsaglia et al., 2003}). Moreover, to obtain consistent, asymptotically efficient estimators, the final estimation of the Weibull parameters was done with the maximum likelihood method (MLE) (^{Zarnoch and Dell, 1985}; ^{Borders et al., 1987}; ^{Seguro and Lambert, 2000}).

The finite Gaussian mixture model is expressed as follows:

where *W*_{ik} are the contributions of each
Gaussian mixture in terms of probability from the
*k*^{th} mixture to *M* total
Gaussian distributions, whose sum ^{Paalanen
et al., 2006}). The initial expressions of
the non-singular multivariate normal distribution of a random variable with
D-dimensions and its expression to describe the probability density function
of a random vector, as well as their derivations of the original expression
using the normal distribution can be consulted in ^{Bilmes (1998)}, ^{Xuan
et al. (2001)} and ^{Paalanen et al. (2006)}.

Probability densities of the finite mixture model ^{Dempster et al., 1977}), according to the methodology of ^{Fraley et al. (2012)} with the *mclust* package in R (^{R Core Team, 2017}).

The optimal abundance interval for each species was delimited using a probabilistic cluster
defined by the density of the finite mixture model whose space can be
classified into tau^{th} probabilities (^{Chen et al., 2006}; ^{Fraley et al., 2012};
^{Fraley et al.,
2017}). In our study, tau is a standardized measure of probability
and takes any possible value (elementary successions) of the probabilistic
space (between 1 and 100), the zone near the centroid of the cluster being
that of greatest probability. A tau of 0.35 is used because 98 % of the
maximum probabilities defined by both models were distributed between the
limits of this probabilistic region (from the center outward). For the two
pines species, the Gaussian models with two mixed components were used
(^{Chen et al.,
2006}) in order to identify distributions with multi-modal
tendencies, and for *Q. macdougallii* a Gaussian model with a
single component was adjusted because a smaller number of individuals was
recorded in the study area.

Results and Discussion

The density curves projected by the two functions used revealed the environmental values in which the maximum likelihood of abundance of each species occurred. For example, the optimal abundance rate of *P. pseudostrobus* occurred when the balance of summer/spring precipitation (SMRSPRPB) has a value near 6.6 (Figure 2A), and the optimal abundance rate of *Q. macdougallii* occurs near 2,775 m altitude (Figure 2B).

The distance between the upper and lower limits, referred to in our study as the optimal abundance interval (IOA), varied for each species, although they grow in the same eco-graphic region (Table 1). For example, *P. patula* had an IOA at sites whose slopes fluctuated between 8 and 80%, with a broader interval in relation to slope, followed by *Q. macdougallii* (50) and *P. pseudostrobus* (34). Regarding altitude, *Q. macdougallii* had a narrower IOA than the other two species with a width of only 550 m. In contrast, *P. patula* had a broader IOA with limits at 2200 to 2900 m (a width of 700 m) (Table 1).

Especies | WINP (mm) | PEND (%) | SDAY (días) | |||||||||

LI | MAX | LS | AI | LI | MAX | LS | AI | LI | MAX | LS | IOA | |

Pinus patula |
150 | 185.4 | 447 | 297 | 18 | 50.6 | 70 | 52 | 8 | 12.9 | 80 | 72 |

Pinus pseudostrobus |
180 | 252.1 | 447 | 267 | 28 | 52.5 | 62 | 34 | 18 | 56 | 68 | 50 |

Quercus macdougallii |
165 | 336.4 | 425 | 260 | 10 | 15.5 | 60 | 50 | 10 | 55.6 | 79 | 69 |

BHH | AI | SMRSPRPB | AI | ALT(msnm) | AI | |||||||

Pinus patula |
0.02 | 0.032 | 0.046 | 0.026 | 5.3 | 5.4 | 5.8 | 0.5 | 2200 | 2263 | 2900 | 700 |

Pinus pseudostrobus |
0.017 | 0.026 | 0.034 | 0.017 | 5.4 | 5.5 | 5.8 | 0.4 | 2300 | 2613 | 2890 | 590 |

Quercus macdougallii |
0.01 | 0.019 | 0.035 | 0.025 | 5.3 | 5.6 | 5.8 | 0.5 | 2350 | 2775 | 2900 | 550 |

GSP(mm) | AI | SMRP (mm) | AI | EXP | ||||||||

Pinus patula |
1100 | 1118.7 | 2200 | 1100 | 450 | 489.6 | 1000 | 550 | oeste y noroeste | |||

Pinus pseudostrobus |
1150 | 1385.3 | 2100 | 950 | 500 | 615.1 | 980 | 480 | noreste, noroeste | |||

Quercus macdougallii |
1150 | 1782.2 | 2100 | 950 | 500 | 807.5 | 950 | 450 | suroeste, noroeste, noreste (predomina suroeste) |

WINP: Winter precipitation; PEN: dominant slope; SDAY: Julian date of the last freezing date of spring; BHH: annual aridity index; SMRSPRPB: summer/spring precipitation balance (Jul+Aug)/(Abr+May); ALT: altitude over sea level; GSP: precipitation from April to September; SMRP: summer precipitation; EXP: geographic exposure; LI: lower limit of the optimal abundance interval; LS: upper limit of the optimal abundance interval; MAX: value of the respective variable where the maximum abundance rate occurs; and IOA: optimal abundance interval.

Similar observations can be made regarding other variables such as precipitation recorded in specific periods, annual aridity index and day of the last frost in spring. For example, the optimal amount of precipitation for *P. patula* from April to September (GSP) was 1,100 to 2,200 mm, the optimal in summer was 450 to 1,000 mm, and in winter 150 to 447 mm, but *Q. macdougallii* had narrower IOA in the same periods: 1,150 to 2,100 mm from April to September (a width of 950 mm), 500 to 950 mm in summer and 165 to 425 in winter. The narrowest IOA of the aridity index was observed in *P. pseudostrobus*, 0.017 to 0.034, followed by optimal intervals of *Q. macdougallii* and *P. patula* whose limits were 0.01 to 0.035 and 0.02 to 0.046, respectively (Table 1).

During data collection we observed variations in density of individuals that depended on the predominating physiographic variables, particularly the exposures of each unit of sampling. Thus, the maximum abundance rate of *P. patula* was observed in sites with west and northwest exposures, for *P. pseudostrobus* in northeast and northwest exposures, and *Q. macdougalli* with greater presence in southwest, northwest and northeast exposures (Table 1).

The two probability density functions used robustly modeled maximum abundance of the three species studied, with greater sensitivity in the projections generated with the finite Gaussian mixture models. This model detected bimodal trends of several species in the face of variation in an environmental variable, such as the case of *P. pseudostrobus* in function of summer precipitation whose values of maximum probability were observed when precipitation was 615.1 mm (Table 1) and 872.8 mm (Figure 3A and 3B). Likewise, *Q. macdougalli* behaved in a similar way with altitude above sea level (Figure 2B) when a second smaller scale vertex of the density curve was observed around 2200 masl, as a response to the concentration of sample data between 2000 and 2500 m altitude (Figure 2B). The greatest plasticity of the finite Gaussian mixture model could correspond to the larger number of parameters in its structure and, above all, to the individual contribution of each Gaussian mixture (^{Bilmes, 1998}; ^{Xuan et al., 2001}; ^{Paalanen et al., 2006}). However, despite the reduced number of parameters of the Weibull function (Equations 1 and 2), this function also projected a maximum abundance probability similar to the mixed model (Figures 2A and 2B).

The value of a variable at which the probability of maximum abundance of a species occurs does not always remain in the center of the distribution, given that in most cases they do not follow normal distributions (Figures 2B and 3A). Moreover, abundance does not follow a unique pattern of distribution because of changing environmental variables and, when more environmental variables are added, the resulting space will not have a geometric form, nor could it be modeled with the Gaussian standard normal function (^{Antúnez et al., 2017b}).

The results of our study suggest that the scarce distribution of *Q. macdougallii* in the study area could be related to narrow optimal intervals (IOAs) of summer and winter precipitation and altitude, whose intervals were small compared with those of *Pinus patula* and *P. pseudostrobus* (Table 1). The broad distribution of the latter seems to correspond to broad IOA of the number of frost and rainfall events from April to September, variable whose effect is significant on several conifers and latifoliate species in northwestern Mexico such as *Abies durangensis*, *Pinus maximinoi*, *Quercus resinosa*, *Q. acutifolia* and *Q. urbanii* (^{Martínez-Antúnez et al., 2013}).

In our study, optimal intervals of the species were not identified in function of the annual aridity index, due to the small values of this variable. However, like precipitation from April to September and degree-days above 5 ºC, this index has a significant effect on forest species diversity (^{Silva-Flores et al., 2014}) and on their distribution and abundance, according to ^{Sáenz-Romero et al. (2010)} and ^{Sáenz-Romero et al. (2012)}.

When field data were being collected, evidence of forest fire was observed on adult tree trunks and, particularly, in areas where more *Q. macdougallii* were present. The fire could have altered the density of this species, and abundance of plants is affected by other factors not considered in our study, such as edaphological characteristics or human activity (^{Clark et al., 1998}; ^{Rajakaruna, 2004}). It should also be taken into account that the absence of a species in a given location is not necessarily due to scarcity of resources or absence of optimal environmental conditions, but that the species has not explored that location (^{Soberón and Peterson, 2005}; ^{Soberón and Miller, 2009}).

Because the optimal intervals of abundance delimited with density functions are not similar to any geometric figure (Figure 3A), particularly when two or more mixed components are included, our study could be complemented with other analytical tools that would allow study of the undefined shapes that species IOA take on, for example, using tools from differential geometry.

Conclusions

The density functions tested in our study allowed definition of the optimal interval of a relevant environmental variable for a species. In this interval, the highest probability of abundance of the entire spectrum of values of any variable occurs, for example, to establish plantations of these or other species of ecological interest in the face of a climatic contingency or one caused by different types of biological factors or agents.

Literatura Citada

Antúnez, P., C. Wehenkel, C. A. López-Sánchez, and J. C. Hernández-Díaz. 2017. The role of climatic variables for estimating probability of abundance of tree species. Pol. J. Ecol. 65: 324-338. [ Links ]

Antúnez, P., J. C. Hernández-Díaz, C. Wehenkel, and R. Clark-Tapia. 2017b. Generalized models: an application to identify environmental variables that significantly affect the abundance of three tree species. Forests 8: 2-14. [ Links ]

Austin, M. P., and T. M. Smith. 1990. A new model for the continuum concept. *In*: Progress in Theoretical Vegetation Science. Springer Netherlands. pp: 35-47.
[ Links ]

Austin, M. P. 1987. Models for the analysis of species' response to environmental gradients. Vegetation 69: 35-45. [ Links ]

Bilmes, J. A. 1998. A gentle tutorial of the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models. Int. Comp. Scien. Inst. 4: 126. [ Links ]

Borda-de-Água, L., S. P. Hubbell, and M. McAllister. 2002. Species-area curves, diversity indices, and species abundance distributions: a multifractal analysis. The Am. Nat. 159: 138-155. [ Links ]

Borders, B. E., R. A. Souter, R. L Bailey, and K. D. Ware. 1987. Percentile-based distributions characterize forest stand tables. Forest Sci. 33: 570-576. [ Links ]

Brotons, L., W. Thuiller, M. B. Araújo, and A. H. Hirzel. 2004. Presence-absence versus presence-only modelling methods for predicting bird habitat suitability. Ecography 27: 437-448. [ Links ]

Chen, T., J. Morris, and E. Martin. 2006. Probability density estimation via an infinite Gaussian mixture model: application to statistical process monitoring. J. Roy. Stat. Soc. C-App 55: 699-715. [ Links ]

Clark, D. B., D. A. Clark, and J. M. Read. 1998. Edaphic variation and the mesoscale distribution of tree species in a neotropical rain forest. J. Ecol. 86: 101-112. [ Links ]

CNA. 2017. Comisión Nacional del Agua. Servicio Meteorológico Nacional. https://smn.cna.gob.mx (Consulta: enero 2017). [ Links ]

Crookston, N. L., G. E. Rehfeldt., D. E. Ferguson, and M. Warwell. 2008. FVS and global Warming: A prospectus for future development. *In*: Havis, R. N., and N. L. Crookston (comps). Third Forest Vegetation Simulator Conference. Department of Agriculture, Forest Service, Rocky Mountain Research Station: U.S. pp: 7-16.
[ Links ]

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. Maximum likelihood from incomplete data via the EM algorithm. J. Royal Stat. Soc. 39: 1-38. [ Links ]

Elith, J., and J. R. Leathwick. 2009. Species distribution models: ecological explanation and prediction across space and time. Annu. Rev. Ecol. Evol. Syst. 40: 677-697. [ Links ]

Elith, J., C. H. Graham, R. P Anderson, M. Dudık, S. Ferrier, A. Guisan, and J. Li. 2006. Novel methods improve prediction of species’ distributions from occurrence data. Ecography 29: 129-151. [ Links ]

Fraley, C., A. E. Raftery, T. B. Murphy, L. Scrucca. 2012. Mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation - http://www.stat.cmu.edu/~rnugent/PCMI2016/papers/fraleymclust.pdf . (Consulta: junio 2015). [ Links ]

Fraley, C., A. Raftery, L. Scrucca, T. B. Murphy, M. Fop, and M. L. Scrucca. 2017. Gaussian Mixture Modelling for Model-Based Clustering, Classification, and Density Estimation. ftp://193.1.193.66/pub/cran.r-project.org/web/packages/mclust/mclust.pdf (Consulta: noviembre 2017). [ Links ]

Franklin, J. 2010. Mapping Species Distributions: Spatial Inference and Prediction - Cambridge University Press. Cambridge CB2 8RU, UK. 319 p. [ Links ]

Gowda, D. M. 2011. Probability models to study the spatial pattern, abundance and diversity of tree species. Annual Conference on Applied Statistics in Agriculture. http://newprairiepress.org/cgi/viewcontent.cgi?article=1048&context=agstatconference . (Consulta: enero 2017). [ Links ]

Hutchinson, G., E. 1957. Concluding remarks. Cold Spring Harbor symposia on quantitative biology. 22: 415-427. [ Links ]

INEGI (Instituto Nacional de Estadística y Geografía). 2015. Datos vectoriales de uso de suelo y tipos de vegetación serie V. escala 1:250000. México. [ Links ]

IUCN (International Union for Conservation of Nature and Natural Resources). 2017. Red List of Threatened Species. Version 2017-1. Gland, Suiza y Cambridge, Reino Unido. [ Links ]

Magurran, A. E. 2004. Measuring biological diversity. John Wiley y Sons. United Kingdom. http://www2.ib.unicamp.br/profs/thomas/NE002_2011/maio10/Magurran%202004%20c2-4.pdf (Consulta: marzo 2017). [ Links ]

Marsaglia, G., W. W. Tsang, and J. Wang. 2003. Evaluating Kolmogorov's distribution. J. Stat. Soft. 8: 1-14. [ Links ]

Martínez-Antúnez, P., C. Wehenkel, J. C. Hernández-Díaz, and J. J. Corral-Rivas. 2015. Use of the weibull function to model maximum probability of abundance of tree species in northwest México. Ann. For. Sci. 72: 243-251. [ Links ]

Martínez-Antúnez, P., C. Wehenkel, J. C. Hernández-Díaz, M. González-Elizondo, J. J. Corral-Rivas, and A. Pinedo-Álvarez. 2013. Effect of climate and physiography on the density of trees and shrubs species in Northwest México. Pol. J. Ecol. 61: 295-307. [ Links ]

Muñoz, F., H. J., J. Sáenz-Reyes, J. J. García-Sánchez, E. Hernández-Máximo y J. Anguiano-Contreras. 2011. Áreas potenciales para establecer plantaciones forestales comerciales de *Pinus pseudostrobus* Lindl. y *Pinus greggii* Engelm. en Michoacán. Rev. Mex. Cienc. Forest. 2: 29-44.
[ Links ]

Paalanen, P., J. K. Kamarainen, J. Ilonen, and H. Kälviäinen. 2006. Feature representation and discrimination based on Gaussian mixture model probability densities-practices and algorithms. Pattern. Recognit. 39: 1346-1358. [ Links ]

PCRM (Planeación comunitaria para el manejo de los recursos naturales). Santiago Comaltepec, Oaxaca. 1992. http://era-mx.org/biblio/ERPCOM.pdf . (Consulta: marzo 2017). [ Links ]

Pearman, P. B., A. Guisan, O. Broennimann, and C.F. Randin. 2008. Niche dynamics in space and time. Trends Ecol. Evol. 23: 149-158. [ Links ]

Phillips, S. J., M. Dudík, J. Elith, C. H. Graham, A. Lehmann, J. Leathwick, and S. Ferrier. 2009. Sample selection bias and presence-only distribution models: implications for background and pseudo-absence data. Ecol. Appl. 19:181-197. [ Links ]

R Core Team. 2017 - R: a language and environment for statistical computing - R Foundation for Statistical Computing. Vienna, Austria, 3551 p. https://cran.r-project.org/doc/manuals/r-release/fullrefman.pdf . (Consulta: mayo 2017). [ Links ]

Rajakaruna, N. 2004. The edaphic factor in the origin of plant species. Int. Geol. Rev. 46: 471-478. [ Links ]

Rehfeldt, G. E., J. J. Worrall, S. B. Marchetti, and N. L. Crookston. 2015. Adapting forest management to climate change using bioclimate models with topographic drivers. Forestry 88: 528-539. [ Links ]

Rehfeldt, G. E., N. L. Crookston, M. V. Warwell, and J. S. Evans. 2006. Empirical analyses of plants climate relationships for the western United States. Int. J. Plant Sci. 167: 1123-1150. [ Links ]

Sáenz-Romero, C., A. Martínez-Palacios, J. M. Gómez-Sierra, N. Pérez-Nasser, y N. M. Sánchez-Vargas. 2012. Estimación de la disociación de *Agave cupreata* a su hábitat idóneo debido al cambio climático. Rev, Chapingo. Serie Cienc. For. Ambiente 18: 291-301.
[ Links ]

Sáenz-Romero, C., G. E. Rehfeldt, N. L. Crookston, P. Duval, R. St-Amant, J. Beaulieu, and B. A. Richardson. 2010. Spline models of contemporary, 2030, 2060 and 2090 climates for Mexico and their use in understanding climate-change impacts on the vegetation. Clim. Chang. 102: 595-623. [ Links ]

Schweik, C. M. 2017. Social norms and human foraging: An investigation into the spatial distribution of *Shorea robusta* in Nepal. Forest, trees and people programme.
http://www.treesforlife.info/fao/Docs/P/X2104E/X2104E06.htm
. (Consulta: febrero 2017).
[ Links ]

Segurado, P., and M. B. Araujo. 2004. An evaluation of methods for modelling species distributions. J. Biogeogr. 31: 1555-1568. [ Links ]

Seguro, J. V., and T. W. Lambert. 2000. Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis. J. Wind Eng. Ind. Aerodyn. 85: 75-84. [ Links ]

Silva-Flores, R., G. Pérez-Verdín, and C. Wehenkel. 2014. Patterns of tree species diversity in relation to cimatic factors on the Sierra Madre Occidental, Mexico - PLoS ONE 9(8): 105034. doi:10.1371/journal.pone.0105034. [ Links ]

Soberón, J., and A. T. Peterson. 2005. Interpretation of models of fundamental ecological niches and species’ distributional areas. Biodiversity Informatics. 2: 1-10. [ Links ]

Soberón, J., y C. P. Miller. 2009. Evolución de los nichos ecológicos. Misc. Mat. 49: 83-99. [ Links ]

Tchebakova, N. M., Rehfeldt, G. E., Parfenova, E. I. 2005. Impacts of climate change on the distribution of Larix spp. and *Pinus sylvestris* and their climatypes in Siberia. Mitig. Adapt. Strateg. Glob. Chang. 11: 861-882.
[ Links ]

Torres, R., y J. M. 2005. Predicción de distribuciones diamétricas multimodales a través de mezclas de distribuciones Weibull. Agrociencia 39: 211-220. [ Links ]

Verberk, W. C. E. P. 2012. Explaining general patterns in species abundance and distributions. Nat. Edu. Know. 3: 38. [ Links ]

Xuan, G., W. Zhang, and P. Chai. 2001. EM algorithms of Gaussian mixture model and hidden Markov model. http://grxuan.org/httpdocs/english/(ICIP2001)EM%20Algorithm%20of%20Gaussian%20Mixture%20Model%20and%20Hidden%20Markov%20Model.pdf . (Consulta: mayo 2016). [ Links ]

Yang, M. H., and N. Ahuja. 1998. Gaussian mixture model for human skin color and its applications in image and video databases. *In*: Electronic Imaging'99. International Society for Optics and Photonics. pp: 458-466
[ Links ]

Yoder, P. J., J. U. Blackford, N. G. Waller, and G. Kim. 2004. Enhancing power while controlling family-wise error: an illustration of the issues using electrocortical studies. J. Clin. Exp. Neuropsyc. 26: 320-331. [ Links ]

Zarnoch, S. J., and T. R. Dell. 1985. An evaluation of percentile and maximum likelihood estimators of Weibull parameters. For. Sci. 31: 260-268. [ Links ]

Received: July 2017; Accepted: January 2018