Lowenberg_Paper Biodiversity Informatics, 13, 2018, pp. 11-26 A METRIC TO QUANTIFY ANALOGOUS CONDITIONS AND RANK ENVIRONMENTAL LAYERS PETER LÖWENBERG-NETO Instituto de Ciências da Vida e da Natureza, UNILA, Av. Tarquínio Joslin dos Santos, 1000 CEP 85870-901, Foz do Iguacu, Parana, Brasil. peter.lowenberg@unila.edu.br Abstract.—Analogous conditions in environmental variables are expected because environments are spatially autocorrelated and often present similar combinations over geographic space. That similar environmental combinations may be found at different localities provides a crucial basis for correlative species distribution modeling. An absolutely analogous variable is constant, while a non-analogous variable has no-repeating values, yet no current method allows researchers to quantify intermediate degrees of analogous conditions and rank environmental layers. I approached this issue from the perspective of dual-space correspondence, in which (a) variable range and modal frequency have a theoretical inverse relationship (y ∝ x-1), and (b) modal values of frequency are limited by the number of pixels in a given raster layer. For two geographic extents and two resolutions (2.5’ and 10’), I obtained range and modal frequency of 19 bioclimatic variables and 5 reference variables. Then, I measured Euclidean distances from candidate variables to the non-analogous variable as a metric for degree of analogous conditions, which were used to rank variables. Bioclimatic layers were plotted in log-log scatterplots of range vs. modal frequency; variables were located inside the upper-right triangle (except for one set), and no layer fit the inverse model. Temperature variables presented higher degrees of analogous conditions than precipitation for South America and the Araucaria Moist Forests ecoregion. Geographic extent and pixel resolution changed the degree of analogous conditions of derived variables (quarterly and monthly); however, a pattern of change was not observed, which suggested ad hoc hypotheses on geographic and temporal idiosyncrasies. Variables with high contribution in previous SDM/ENM studies (e.g., temperature seasonality and annual precipitation) showed low degree of analogous conditions. It is expected that heterogeneous layers would generate better correlational geographic distributional predictions than analogous variables, even though this hypothesis remains untested. Ranking layers can provide grounds for selecting variables in distribution and niche modeling, particularly as regards interpreting spatial projection and transferability. Alternatively, ranking can be used to compare degrees of analogous conditions of the same layer in different time spans. Key words.—analog conditions, bioclimatic variables, environmental space, geographic space An environmental digital layer is a common object of geographic information systems, in which values of a continuous variable are stored in a spatially referenced matrix (Chang 2017). In biogeography and macroecology, environmental layers include temperature, precipitation, humidity, radiation, soil, and human occupation. They can be used as a background in illustrating maps and as predictors in statistical analysis (Williams et al. 2012). Environmental layers are normally raster format objects, which implies some level of discretization of continuous space (Hijmans and Elith 2017). Environmental variables are not distributed heterogeneously across space. Variables like temperature are spatially autocorrelated, and show repeated or similar values over space (Legendre 1993). Additionally, combinations of variables can replicate more complex circumstances and cause localities to represent analogous conditions, for example, a monsoon- like climate in South America caused by heating and circulation regimes associated with topography (Zhou and Lau 1998). In the literature, ‘analogous’ or ‘analog’ has frequently been employed to designate equivalent climatic conditions through time. Several studies have inferred how populations and communities responded to past (Overpeck et al. 1992, Jackson and Overpeck 2000) and current climate change (Garcia et al. 2014) by comparing to modern climates against analogous past and/or future climate scenarios. They have also estimated new and disappearing climatic combinations in scenarios of change (Ohlemüller et al. 2006, Williams et al. 2007, Ackerly et al. 2010). 11 LÖWENBERG-NETO – ANALOGOUS CONDITIONS Contemporary analogous conditions, such as pixels with equal or similar values, provide grounds for species distribution modeling (Guisan and Zimmermann 2000). In a correlative approach, occurrences of species and digital environmental layers are used to estimate existing or realized niches of species (Peterson et al. 2011). Then, a given algorithm may search (Elith and Graham 2009) across the geographic extent for equivalent conditions (Elith and Leathwick 2009). An interesting topic relevant to correlational modeling is projection over space and the problem of non-analog climates (Fitzpatrick and Hargrove 2009). It is possible that native-range geographic range climate conditions are distinct from conditions in the invaded region (Fitzpatrick et al. 2006, Soberón and Peterson 2011). In these cases, challenging the equilibrium assumption, including invasion stages (Gallien et al. 2012), and controlling for possible niche shifts (Guisan et al. 2014) in spatial projections may overcome the problem. Correlational modeling procedures are based on the dual-space correspondence (Peterson et al. 2011, Soberón et al. 2017). Dual-space correspondence, or Hutchinson’s duality (Colwell and Rangel 2009), occurs when points from an input space (e.g. biotope space, geographic space) are plotted in a feature space (e.g. climatic space), where variables are axes and measures are coordinates (Hutchinson 1978, Colwell and Rangel 2009). Each input point g from geographic space (G) has a vector eg composed of measures of v variables, such as annual mean temperature, annual precipitation, and so on. The vector eg with v elements represents the coordinates of the point in a v- dimensional space, the environmental space (E) (Peterson et al. 2011). When enough variables are used at sufficient precision, points from G generally correspond one to one to points in E (Apinall and Lees 1994). However, this situation is not necessarily the case, and the same or very similar (analogous) climatic combinations may occur in separated geographic localities. In any case, the cloud of points can be interpreted as a particular realization of environmental conditions that occur across a given geographic extent at a particular time (i.e., the realized environment) (Jackson and Overpeck 2000). Two additional features from E are of particular interest: (i) empty environmental spaces denote combinations of conditions that are missing, such as warm (>30°C) and wet (>3000 mm) climates in California (Ackerly et al. 2010), and (ii) closely- located points that are environmentally analogous localities (de Oliveira et al. 2014). For a climatic variable, when many localities have the same value, their corresponding points pile up in a kernel in E, and represent high frequency regions (Figure 1). In this case, when more than one locality has the same environmental-variable value, if a single point was mapped back, it would represent all pixels in G in an asymmetric relationship (one-to-many, a partial reciprocity, Colwell and Rangel 2009). On the other hand, a non-analogous (non- repeating) variable in G has corresponding points (1:1) spread over E, with a maximum frequency of one. Let n = |G| the number of points in the rasterization of a variable; y is the range of values of the variables (y = ymax –ymin) and x is the modal frequency of variable y. If the extent and resolution of the discretization of G do not change, then an inverse relationship (y ∝ x-1) between the range of a variable and its modal frequency is to be expected. In other words, if the range of a variable is small, most cells in the raster have values in that small range. If the range is broad, the distribution of frequencies will tend to be flat (Figure 1). This effect occurs because (a) each variable’s range determines the span of its axis in E, and (b) the number of pixels is constant for a given extent and resolution, which provides a zero-sum scenario. Hence, when the span is low, density will be concentrated in a small region, and kernel modal frequency will be high; when the breadth is high, density is spread over the axis and maximum frequency is low. This point is important because of Hutchinson’s Duality: the same niche breadths in regions of contrasting spans of values of environmental variables may predict contrasting sizes of areas of distribution. Statistical selection of variables for species distribution modeling frequently aims at controlling variable collinearity and ranking variables (Negrão and Löwenberg-Neto in prep.). Current metrics for analogy of conditions 12 LÖWENBERG-NETO – ANALOGOUS CONDITIONS Figure 1. Inverse relationship between variable range in G (geographic space) and modal frequency in E (environmental space). Top row represents an absolutely analogous variable, zero-ranged, with modal frequency equal to number of pixels, and bipartite network asymmetric. Middle row represents a layer with intermediate degrees. Bottom row is a non-analogous layer: range equals the number of pixels minus one, modal frequency equals one; bipartite network is symmetric. Figure 2. Log-transformed scatterplots of variable range versus maximum frequency for two extents (South America, SA; Araucaria Moist Forests, AR), and two pixel resolutions (2.5’, 10’). Bioclimatic variables are labeled following Hijmans et al. (2006), and referential variables as CO = constant, SC = semi-constant and wide-range, RN = random normal, HT = heterogeneous, and NAN = non-analogous. 13 LÖWENBERG-NETO – ANALOGOUS CONDITIONS are available only in temporal frameworks and only for comparing pixels within layers (Garcia et al. 2014), which does not allow comparison among layers. In the present paper, I present a layer-scoped metric that quantifies overall degree of analogy of environmental layers under Hutchinson’s duality. I have then used the measurements to rank variables, and discuss the importance of these tools and ideas in the broader field of distributional ecology. METHODS I obtained 19 bioclimatic variables (Hijmans et al. 2005), and calculated their ranges and modal values. Measurements were developed for variables at two geographic extents: all of South America (SA) and the Araucaria Moist Forests ecoregion (AR) in southern Brazil (Olson et al. 2002). For both extents, I analyzed bioclimatic variables at two resolutions: 2.5’ and 10’ (Guisan and Thuiller 2005). For each combination of extent and resolution, I created 5 reference variables: (1) Constant variable (CO) is a homogeneous variable, with a modal frequency equal to the number of pixels and zero for range. For the log- transformed distance (see below), I assigned variable range to one. (2) Semi-constant, wide range (SC), is the second most homogeneous variable has a single, high modal frequency and a wide range with low frequencies. This variable is important variable because it controls for variables that are very homogeneous but that may mislead interpretation or metric quantification owing to its wide range of values. (3) Random normal (RN) is a heterogeneous variable drawn from a normal distribution; its range is similar to the bioclimatic variable with the broadest range in all combinations, annual precipitation. (4) Heterogeneous (HT) is the most heterogeneous variable, with a range similar to that of annual precipitation; it has repeating values with the lowest maximum frequency. Finally, (5) Non-analogous (NAN) is the absolutely heterogeneous variable, with no repeating values, range equal to the number of pixels, and a modal frequency of one. I compiled variable ranges and modal frequencies into data matrices. For each dataset, I measured the Euclidean distance matrix between rows using dist command and method = “euclidean” sqrt(sum((xi - yi)^2)) in R version 3.5.0. The distance between a given variable to the non-analogous variable (NAN) was used to quantify variable’s degree of analogous conditions; therefore, longer distances to NAN denoted more homogeneous (analogous) variables. The same measurement procedure was done for a log-transformed data matrix (log distance). Pearson’s correlations were calculated among distance, log distance, and secondary metrics, which included range, maximum frequency, and the Shannon-Weaver diversity index (Shannon 2001). The last metric was calculated using the command diversity in the ‘vegan’ R package (Oksanen et al. 2007). RESULTS Variable histograms used to measure range and modal frequency are presented in Appendix A; measurements are in Appendix B. For each variable, range and modal frequency were log- transformed and plotted (Figure 2). Correlation analyses showed that distance and log distance were strongly positively correlated; log distance was negatively correlated with variable range (Appendix C). For each combination of geographic extent and resolution, distances to the NAN variable were used to compare variables. Ranking showed that reference variables CO and SC had the highest degree of analogous conditions while RN, HT and NAN the least (Figure 3A). Statistical variables showed disparate degrees of analogous conditions (Figure 3B): mean diurnal range, temperature annual range, and precipitation seasonality had their degree of analogy of conditions affected by geographic extents, whereas isothermality and temperature seasonality, which are standardized variables, were less affected. Temperature variables presented higher degrees of analogy of conditions than precipitation in both regions (Figure 3C and 3D). DISCUSSION A metric that quantifies overall degree of analogy of conditions for individual environmental layers was presented. By creating a non-analogous variable in which the range equals the number of pixels and modal frequency equals one, it was possible to plot and measure 14 LÖWENBERG-NETO – ANALOGOUS CONDITIONS Figure 3. Ranking environmental variables by their Euclidean distance to the non-analogous (NAN) variable in a line graph for four combinations of extent and resolution (South America at 2.5’, South America at 10’, Araucaria Moist Forests at 2.5’, Araucaria Moist Forests at 10’). Variables were displayed in four subsets: (a) reference, (b) statistical, (c) temperature, and (d) precipitation. Bioclimatic variables were labeled following Hijmans et al. (2006), and reference variables are as follows: CO = constant, SC = semi-constant and wide-range, RN = random normal, HT = heterogeneous, and NAN = non-analogous. 15 LÖWENBERG-NETO – ANALOGOUS CONDITIONS the Euclidean distance to the candidate variable as an index of dissimilarity to the non-analogous variable. In this sense, higher distances to NAN denote that a variable has a high degree of analogous conditions. Log-transformed plots provided a better visualization of the variables and their spatial positions in the kernel, and allow visualization of expected upper limits, which are based on the number of pixels the expected inverse relationship between variable range and modal frequency (Figure 2). Bioclimatic layers were located inside the right-angled triangle, and no layer fit the inverse model, which was expected for climatic variables. Reference variables, especially SC and HT, showed consistent positions in all plots, providing internal references for the bioclimatic variables; conversely, RN was very close to realistic variables. Three variables were placed outside the triangle envelope, beyond the vertical axis. This effect occurred because variables had a wider range than the numbers of pixels available. The AR10 treatment comprised 684 pixels, and ranges were above 1000 for temperature seasonality, annual precipitation, and RN. Secondary metrics were not tested formally because I intended to provide a metric based on the duality ontology. Nevertheless, their correlations with Euclidean distance provided some information. For example, variable range was strongly negatively correlated with distance, which supports an exploratory approximation to analogous degree with no need for developing scatterplot. Modal frequencies were less correlated with distance in treatments with finer pixels; diversity index showed a weak relation to distance; the index did not discern the semi- constant wide-range variable well. Ranking bioclimatic layers showed that geographic extent and pixel resolution both affect the degree of analogy of conditions. It was expected that a change in grain size would not severely affect the degree of analogy of conditions (Guisan et al. 2007). For the same extent, ranking showed that pixel resolution modified a variable’s ranking position, even though it showed no pattern of increasing analogy of conditions with decreasing resolution or vice-versa. Regarding geographic extent, it was expected that extents limited the ranges of environmental variables (Thuiller et al. 2004, Randin et al. 2006), and that the smaller geographic extent would present higher degrees of analogy of conditions (Anderson and Raza 2010). This effect was observed only for a few variables, including statistical ones constructed by consideration of ranges of values (temperature diurnal and annual ranges) and coefficients of variation (precipitation seasonality). A third expectation was that variables arranged in temporal slices (quarterly and monthly) would show increasing degrees of analogy of conditions when compared to annual variables. In fact, annual mean temperature showed consistent ranking positions (8, 8, 8, 6, Fig. 3C), whereas time-sliced variables showed changeable positions (e.g., mean temperature of wettest quarter, ranks 5, 6, 14, 13). The same effect was observed for precipitation variables (e.g., annual precipitation, ranks 21, 20, 21, 21; precipitation of wettest quarter, ranks 18, 21, 12, 14); however, I did not observe any general trend of increasing degree by decreasing temporal slice size. In sum, geographic extent and pixel resolution changed the degree of analogous conditions of derived variables whereas annual variables tended to maintain their rankings. No consistent trend of change between extension/resolution and increasing degree of analogous conditions was recognized, which suggested ad hoc hypotheses for geographic and temporal idiosyncrasies. For the purpose of species distribution modeling (SDM), variables showing high degrees of analogy of conditions tend to estimate broad geographic ranges and few values are frequent across geographic space (Peterson et al. 2011). For a given range or niche-breadth, SDM models using variables close to the upper-left side of the triangle would predict larger geographic expanses than those in the lower- right part of the triangle. This observation thus offers a cautionary note for studies relating niche breadth to distributional area without controlling for variable degrees of analogy of conditions (Slatyer et al. 2013). In fact, this statement depends on each band of the variable histogram having a correlation with occurrences of species (Guisan and Thuiller 16 LÖWENBERG-NETO – ANALOGOUS CONDITIONS 2005). In any case, a study that summarized environmental variables that most contributed to estimating species’ geographic distributions showed that, for the WorldClim dataset, temperature seasonality, annual precipitation, and precipitation of the driest month had highest mean contributions (Bradie and Leung 2017). Interestingly, in this study, the former two variables were consistently ranked as showing low degree analogy for both extents and resolution; precipitation of the driest month showed an atypical trend, with high analogy for South America and low analogy for Araucaria Moist Forests (Fig. 3D), perhaps owing to odd contrasts in this variable in homogeneity across the two regions (Appendix A). Further, it is common in processing raster layers to transform decimal values into integers by multiplying by 10, 100, or 1000 (Hijmans et al. 2005), which produces increasing variable heterogeneity. It is also common to use raster layers arranged into categorical, nominal, or ordinal classes (Peterson et al. 2011), which dramatically homogenizes variables. Increasing or decreasing numbers of bins on the environmental axis affects modal frequencies in the E-space kernel and therefore the degree of analogy of conditions of environmental layers. In this paper, I have focused on quantification of analogy of conditions within the same variable layer as ‘contemporary’ analogous conditions. This quantification approach can be used to compare degrees of analogy of conditions in different time spans (Garcia et al. 2014). For a given variable with constant extent and resolution, the variable can be plotted for different spans, and degrees of analogous conditions in temporal scenarios of change can be ranked. By analyzing how analogous conditions were coded in G and E, I used two parameters to characterize degree of analogy of conditions. The Euclidean distance between the candidate layer and the non-analogous layer provided a metric of dissimilarity used to rank and compare variables by their degree of analogy. The resulting information may be used to select layers and interpret results in species distribution and ecological niche modeling. 1 DOI: 10.1002/9781118786352.wbieg0152. ACKNOWLEDGMENTS I am grateful to L.R.R. Faria Jr., T. Vasconcelos, J. Soberón, and Town Peterson, for suggestions that improved the manuscript. This research was conducted in the Biogeography and Macroecology Lab (ILACVN/UNILA/Brazil). REFERENCES Ackerly, D.D., S.R. Loarie, W.K. Cornwell, S.B. Weiss, H. Hamilton, R. Branciforte, and N.J.B. Kraft. 2010. The geography of climate change: Implications for conservation biogeography. Diversity and Distributions, 16:476-487. Aspinall, R. and B.G. Lees. 1994. Sampling and analysis of spatial environmental data. Proceedings of the Sixth International Symposium on Spatial Data Handling, vol 2. Edinburgh. Anderson, R. P., and Raza, A. 2010. The effect of the extent of the study region on GIS models of species geographic distributions and estimates of niche evolution: Preliminary tests with montane rodents (genus Nephelomys) in Venezuela. Journal of Biogeography, 37:1378-1393. Bradie, J., and B. Leung. 2017. A quantitative synthesis of the importance of variables used in MaxEnt species distribution models. Journal of Biogeography, 44:1344-1361. Chang, K.-T. 2017. Geographic Information System. The International Encyclopedia of Geography.1 Colwell, R.K. and T.F. Rangel. 2009. Hutchinson's duality: The once and future niche. Proceedings of the National Academy of Sciences USA 106:19651-19658. Elith, J., and Graham, C. H. 2009. Do they? How do they? WHY do they differ? On finding reasons for differing performances of species distribution models. Ecography 32:66-77. Elith, J., and J.R. Leathwick. 2009. Species distribution models: Ecological explanation and prediction across space and time. Annual Review of Ecology, Evolution, and Systematics 40:677- 697. Fitzpatrick, M.C., and W.W. Hargrove. 2009. The projection of species distribution models and the problem of non-analog climate. Biodiversity and Conservation 18:2255-2261. Fitzpatrick, M.C., J.F. Weltzin, N.J. Sanders, and R.R. Dunn. 2007. The biogeography of prediction error: Why does the introduced range of the fire ant over-predict its native range? Global Ecology and Biogeography 16:24-33. Gallien, L., R. Douzet, S. Pratte, N.E. Zimmermann, and W. Thuiller. 2012. Invasive species 17 LÖWENBERG-NETO – ANALOGOUS CONDITIONS distribution models—how violating the equilibrium assumption can create new insights. Global Ecology and Biogeography 21:1126-1136. Garcia, R. A., M. Cabeza, C. Rahbek, and M.B. Araújo. 2014. Multiple dimensions of climate change and their implications for biodiversity. Science 344:1247579. Guisan, A., C.H. Graham, J. Elith, and F. Huettmann. 2007. Sensitivity of predictive species distribution models to change in grain size. Diversity and Distributions 13:332-340. Guisan, A., B. Petitpierre, O. Broennimann, C. Daehler, and C. Kueffer. 2014. Unifying niche shift studies: Insights from biological invasions. Trends in Ecology and Evolution 29:260-269. Guisan, A., and N.E. Zimmermann. 2000. Predictive habitat distribution models in ecology. Ecological Modelling 135:147-186. Guisan, A., and W. Thuiller. 2005. Predicting species distribution: Offering more than simple habitat models. Ecology Letters 8:993-1009. Hijmans, R.J., S.E. Cameron, J.L. Parra, P.G. Jones, and A. Jarvis. 2005. Very high resolution interpolated climate surfaces for global land areas. International Journal of Climatology 25:1965- 1978. Hijmans, R. and J. Elith. 2017. Species Distribution Modeling with R.2 Hutchinson, G.E. 1978. An introduction to population ecology. Yale University Press, New Haven. Jackson, S.T. and J.T. Overpeck. 2000. Responses of plant populations and communities to environmental changes of the late Quaternary. Paleobiology 26:194-220. Legendre, P. 1993. Spatial autocorrelation: Trouble or new paradigm? Ecology 74:1659-1673. de Oliveira, G., T.F. Rangel, M.S. Lima-Ribeiro, L.C. Terribile, and J.A.F. Diniz-Filho. 2014. Evaluating, partitioning, and mapping the spatial autocorrelation component in ecological niche modeling: A new approach based on environmentally equidistant records. Ecography 37:637-647. Oksanen, J., F.G. Blanchet, M. Friendly, P. Kindt, P. Legendre, D. McGlinn, P.R. Minchin, R.B. O'Hara, G.L. Simpson, P. Solymos, M.H.H. Stevens, E. Szoecs, and H. Wagner. 2007. Community Ecology Package: Ordination, Diversity and Dissimilarities, the vegan package for R, version 2.4.3 Olson, D.M., E. Dinerstein, E.D. Wikramanayake, N.D. Burgess, G.V. Powell, E.C. Underwood, J.A. D’Amico, I. Itoua, H.E. Strand, J.C. Morrison, C.J. Loucks, T.F. Allnutt, T.H. Ricketts, J.F. 2 https://cran.r-project.org/web/packages/dismo/vignettes/sdm.pdf. Lamoreux, W.W. Wettengel, P. Hedao, and K.R. Kassem. 2001. Terrestrial ecoregions of the world: a new map of life on Earth. BioScience 51:933- 938. Overpeck, J.T., R.S. Webb, and T. Webb. 1992. Mapping eastern North American vegetation change of the past 18 ka: No-analogs and the future. Geology 20:1071-1074. Peterson, A.T., J. Soberón, R.G. Pearson, R.P. Anderson, E. Martínez-Meyer, M. Nakamura, and M.B. Araújo. 2011. Ecological niches and geographic distributions. Princeton University Press, Princeton. Randin, C.F., T. Dirnböck, S. Dullinger, N.E. Zimmermann, M. Zappa, and A. Guisan. 2006. Are niche-based species distribution models transferable in space? Journal of Biogeography 33:1689-1703. Shannon, C. E. 2001. A mathematical theory of communication. ACM SIGMOBILE Mobile Computing and Communications Review 5:3-55. Slatyer, R.A., M. Hirst, and J.P. Sexton. 2013. Niche breadth predicts geographical range size: A general ecological pattern. Ecology Letters 16:1104-1114. Soberón, J. and M. Nakamura. 2009. Niches and distributional areas: Concepts, methods, and assumptions. Proceedings of the National Academy of Sciences USA 106:19644-19650. Soberón, J., L. Osorio-Olvera, and A.T. Peterson. 2017. Diferencias conceptuales entre modelación de nichos y modelación de áreas de distribución. Revista Mexicana de Biodiversidad 88:437-441. Soberón, J., and A.T. Peterson. 2011. Ecological niche shifts and environmental space anisotropy: A cautionary note. Revista Mexicana de Biodiversidad 82:1348-1355. Thuiller, W., L. Brotons, M.B. Araújo, and S. Lavorel. 2004. Effects of restricting environmental range of data to project current and future species distributions. Ecography 27:165- 172. Williams, K.J., L. Belbin, L., M.P. Austin, J.L. Stein, and S. Ferrier. 2012. Which environmental variables should I use in my biodiversity model? International Journal of Geographical Information Science 26:2009-2047. Zhou, J. and K.-M. Lau. 1998. Does a monsoon climate exist over South America? Journal of Climate 11:1021-1040. 3 https://cran.r-project.org/web/packages/vegan/vegan.pdf. 18 LÖWENBERG-NETO – ANALOGOUS CONDITIONS APPENDIX 1: HISTOGRAMS OF BIOCLIMATIC VARIABLES. Figure A.1 Bioclimatic variables and reference variables presented in histograms for the extent of all of South America at 2.5’ resolution, with 883,760 pixels. 19 LÖWENBERG-NETO – ANALOGOUS CONDITIONS Figure A.2 Bioclimatic variables and reference variables presented in histograms for the extent of all of South America at 10’ resolution, with 55,377 pixels. 20 LÖWENBERG-NETO – ANALOGOUS CONDITIONS Figure A.3 Bioclimatic variables and reference variables presented in histograms for the extent of the Araucaria Moist Forest ecoregion at 2.5’ resolution, with 11,033 pixels. 21 LÖWENBERG-NETO – ANALOGOUS CONDITIONS Figure A.4 Bioclimatic variables and reference variables presented in histograms for the extent of the Araucaria Moist Forest ecoregion at 10’ resolution, with 684 pixels. 22 LÖWENBERG-NETO – ANALOGOUS CONDITIONS APPENDIX 2: VARIABLE MEASUREMENTS. Table B.1. Measurements for the extent of all of South America at a 2.5’ spatial resolution. Bioclimatic and reference variables: CO = constant, SC = semi-constant and wide range, RN = random normal, HT = heterogeneous, NAN = non-analogous. Variable Range Modal frequency Diversity Distance to NAN Log distance BIO1 399 87311 13.6680 1087163.5 7.3081 BIO2 202 33077 13.6732 1082891.1 7.1080 BIO3 80 37357 13.6732 1083249.1 7.4754 BIO4 6283 51005 13.3247 1076499.4 6.3375 BIO5 355 88750 13.6736 1087391.9 7.3503 BIO6 458 47793 13.6567 1083401.9 7.0026 BIO7 286 48816 13.6397 1083680.7 7.1585 BIO8 434 118219 13.6706 1091494.6 7.4176 BIO9 380 55925 13.6552 1084081.0 7.1281 BIO10 371 105921 13.6729 1089675.5 7.4149 BIO11 435 79219 13.6589 1086189.6 7.2395 BIO12 10577 42407 13.5000 1070686.8 6.1368 BIO13 1197 34742 13.5276 1081751.6 6.5779 BIO14 697 182027 13.0964 1104264.7 7.4798 BIO15 259 26691 13.5697 1082557.0 6.9363 BIO16 3450 59923 13.5211 1080650.0 6.5528 BIO17 2319 149545 13.1747 1094966.9 7.0824 BIO18 2574 38118 13.4865 1080237.2 6.4129 BIO19 2962 198539 13.1451 1105818.1 7.1615 CO 1 883760 13.6919 1530715.5 10.2994 SC 10600 873260 13.6896 1512443.2 7.6473 RN 17787 39148 13.4992 1061679.1 5.9963 HT 10600 8400 13.4977 1069447.7 5.3512 NAN 883760 1 13.4988 0.0 0.0000 Table B.2. Measurements for the extent of all of South America at a 10’ spatial resolution. Bioclimatic and reference variables: CO = constant, SC = semi-constant and wide range, RN = random normal, HT = heterogeneous, NAN = non-analogous. Variable Range Modal frequency Diversity Distance to NAN Log distance BIO1 355 5460 10.86252 67718.76 5.3070 BIO2 167 2083 10.90318 67666.22 5.1040 BIO3 55 2352 10.90312 67816.49 5.5296 BIO4 6214 3221 10.55476 60341.14 4.4511 BIO5 303 5531 10.90393 67790.77 5.3560 23 LÖWENBERG-NETO – ANALOGOUS CONDITIONS BIO6 414 2999 10.75527 67415.71 4.9916 BIO7 280 1243 10.86969 67496.91 4.7193 BIO8 395 7300 10.85650 67929.69 5.4130 BIO9 340 3527 10.84392 67544.47 5.1200 BIO10 318 6600 10.88888 67915.83 5.4236 BIO11 394 5013 10.80168 67619.34 5.2398 BIO12 9916 2665 10.73032 55773.64 4.2942 BIO13 980 2199 10.75779 66676.81 4.6217 BIO14 652 11355 10.32861 68451.50 5.4999 BIO15 232 1657 10.79936 67569.00 4.9018 BIO16 2787 1592 10.75131 64438.80 4.2316 BIO17 2159 9357 10.40633 66178.05 5.1607 BIO18 2427 2418 10.71755 64917.76 4.4653 BIO19 2787 5381 10.37613 64745.49 4.8381 CO 1 55377 10.92192 95914.04 8.2157 SC 9915 36308 10.85571 71256.49 5.6593 RN 23713 2443 10.72992 38895.48 4.1738 HT 9915 600 10.71872 55684.18 3.5234 NAN 55377 1 10.72878 0.00 0.0000 Table B.3. Measurements for the extent of the Araucaria Moist Forest ecoregion at a 2.5’ spatial resolution. Bioclimatic and reference variables: CO = constant, SC = semi-constant and wide range, RN = random normal, HT = heterogeneous, NAN = non-analogous. Variable Range Modal frequency Diversity Distance to NAN Log distance BIO1 93 426 9.3029 13372.10 4.1008 BIO2 70 464 9.3002 13402.14 4.2311 BIO3 19 1397 9.3044 13560.81 5.1271 BIO4 1092 302 9.3039 12144.04 3.2765 BIO5 102 312 9.3044 13356.38 3.9409 BIO6 85 365 9.2870 13379.19 4.0669 BIO7 83 351 9.3035 13381.08 4.0589 BIO8 125 284 9.2944 13327.28 3.8341 BIO9 133 237 9.2950 13316.11 3.7384 BIO10 98 409 9.3039 13365.19 4.0666 BIO11 86 387 9.3014 13378.89 4.0870 BIO12 1176 313 9.2993 12041.63 3.2797 BIO13 139 259 9.3007 13309.38 3.7606 BIO14 124 261 9.2798 13327.80 3.8017 BIO15 40 1023 9.2325 13485.09 4.7450 BIO16 397 587 9.3020 13009.46 3.8236 BIO17 406 385 9.2853 12987.14 3.6204 24 LÖWENBERG-NETO – ANALOGOUS CONDITIONS BIO18 347 590 9.3013 13070.80 3.8596 BIO19 402 317 9.2873 12989.29 3.5329 CO 1 11003 9.3059 19056.02 7.0001 SC 1200 9813 9.2849 16987.09 5.0290 RN 1200 453 9.1950 12018.93 3.4600 HT 1200 100 9.1034 12006.79 2.7183 NAN 11003 1 9.1128 0.00 0.0000 Table B.4. Measurements for the extent of the Araucaria Moist Forest ecoregion at a 10’ spatial resolution. Bioclimatic and reference variables: CO = constant, SC = semi-constant and wide range, RN = random normal, HT = heterogeneous, NAN = non-analogous. Variable Range Modal frequency Diversity Distance to NAN Log distance BIO1 72 30 6.5251 750.38 2.1695 BIO2 65 36 6.5223 759.33 2.2804 BIO3 18 86 6.5265 822.30 3.0589 BIO4 1002 22 6.5260 390.32 1.6566 BIO5 76 27 6.5265 745.33 2.1069 BIO6 73 28 6.5100 749.05 2.1349 BIO7 73 35 6.5255 749.48 2.2344 BIO8 108 22 6.5167 705.92 1.9150 BIO9 92 21 6.5170 725.46 1.9393 BIO10 75 30 6.5261 746.71 2.1576 BIO11 67 30 6.5237 756.50 2.1909 BIO12 1062 21 6.5215 463.74 1.6362 BIO13 104 20 6.5229 710.73 1.8822 BIO14 117 18 6.5021 694.74 1.8016 BIO15 39 67 6.4550 794.09 2.7061 BIO16 307 46 6.5242 465.01 2.0806 BIO17 379 28 6.5075 375.01 1.8000 BIO18 274 19 6.5234 502.63 1.6400 BIO19 377 26 6.5096 377.24 1.7617 CO 1 684 6.5280 1182.99 4.9105 SC 230 456 6.4635 787.22 3.3077 RN 1062 19 6.3412 463.48 1.5835 HT 230 6 6.3369 556.07 1.1155 NAN 684 1 6.3355 0.00 0.0000 25 LÖWENBERG-NETO – ANALOGOUS CONDITIONS APPENDIX 3: PEARSON CORRELATION COEFFICIENTS. Table C.1. Correlation coefficients for the parameters of two extents (South America SA, and Araucaria Moist Forests ecoregion AR) and two spatial resolutions (2.5’, 10’). NAN = non-analogous variable. SA2.5 Modal frequency Diversity Distance to NAN Log distance Range -0.1226 -0.0497 -0.8786 -0.8591 Modal frequency 1 0.1363 0.5759 0.4805 Diversity 0.1363 1 0.1488 0.1713 Distance to NAN 0.5759 0.1488 1 0.9259 SA10 Range -0.1218 -0.0818 -0.9207 -0.8089 Modal frequency 1 0.1520 0.4938 0.6196 Diversity 0.1520 1 0.1674 0.1884 Distance to NAN 0.4938 0.1674 1 0.9445 AR2.5 Range -0.0782 -0.6920 -0.7546 -0.8806 Modal frequency 1 0.1317 0.6270 0.5536 Diversity 0.1317 1 0.6466 0.6317 Distance to NAN 0.6270 0.6466 1 0.9278 AR10 Range -0.204 -0.1892 -0.6936 -0.4737 Modal frequency 1 0.0608 0.5563 0.8129 Diversity 0.0608 1 0.4542 0.4736 Distance to NAN 0.5563 0.4542 1 0.8559 26