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APPENDIX A Glossary of Symbols Used This section summarizes the mathematical notation used throughout this book.|A| The cardinality of set A. AC The complement of set A. Â An estimate of a set A. f ˆ An estimate of a function f. Ŷ An estimate or prediction of a variable Y. E{X} Expected value of random variable X. P(A) Probability of event A. P(A|B) Conditional probability of event A given B. The following list summarizes specific symbols used to express concepts in this book. a, b, c, d Entries of confusion matrix. ai,l Per-unit time probability of finding resource l. A Set of cells in geographic space where intrinsic growth rates are positive (abiotically suitable). b(g) Probability of a collector visiting cell g, or sampling bias if heterogeneous. bi,g Bionomic parameter. B Number of bootstrap samples in bootstrapping scheme. B Set of cells in G where biotic conditions are favorable for presence of a species. ĉ(X) A continuous output estimate of function f based on the result of an algorithm μ(Gdata, E). ci,g Mean field biotic interaction parameter d Number of excluded (deleted) datapoints in jackknife scheme. D Binary random variable to denote detection of species by collector. D(g) Conditional probability of detection of species by collector. 262 APPENDIX A di(e → g) Environmentally determined death rate of species i at environment e → g. e → g ⫽ (e1, e2, ... , ev)g Vector of ν environmental variables at cell g. A generic element of E. E-space Multidimensional space of scenopoetic variables. Mathematically, it is E. E Environmental space of scenopoetic environmental variables. Colloquially, it is referred to as “E-space.” E′ A generic subset of E obtained from the mapping function η. EA Scenopoetic existing fundamental niche, defined by η(GA). EI Invadable niche space, defined by η(GI). EO Occupied niche, defined by η(GO). EP Biotically reduced niche, defined by η(GP). Eval An estimate of expected loss, or average validation error, prediction error, or testing error. Ever An estimate of verification error, or calibration error, or training error. EK val An estimate of expected loss, using K-fold cross validation. Eval boot An estimate of expected loss, using bootstrap samples. η Function that maps geography into environment. η(A) For A ⊆ G, the direct image of set A, or the set {η(g)|g ∈ A}. η–1(A) For A ⊆ E, the inverse image of set A, or the set {g ∈ G|η(g) ∈ A}. f(X,Z) Nature’s response mechanism relating variables X, Y to response Y. f(X) Idealized approximation of nature’s response mechanism relating variables X to response Y, disregarding effects of Z. f ˆ(X) A binary estimate of function f based on the result of an algorithm μ(Gdata, E). f ˆ u Threshold-dependent binary model obtained by thresholding at value u. f ˆ –i Model calibrated by setting aside ith observation. f ˆ*i(X) Bootstrap replicates of fitted models. g A generic element of G (i.e., a single grid cell). G-space A set area of geography. Mathematically, it is G. GLOSSARY OF SYMBOLS USED 263 G Geographic space composed of cells or pixels, generally two-dimensional. Colloquially, it is “G-space.” G′ A generic spatial subset of space G. GA The abiotically suitable area, defined by η–1(EA). GB Biotically suitable area; generally referred to simply as B. GM Accessible area, based on the species’ present and historical movements; generally referred to simply as M. GI The invadable distributional area, defined as A ∩ B ∩ MC. GO The occupied distributional area, defined as A ∩ B ∩ M. GP The potential distributional area, defined as GO ∪ GI. Gdata Data; set of observations (presences, and, if existing, true absences). G⫹ Occurrence data documenting presences of species. ΔGO Change in occupied distributional area before and after some change event. h Number of cells that compose G, or |G|. I Binary random variable to denote species’ access to a site. J Binary random variable to denote abiotic suitability. K Binary random variable to denote biotic suitability. K Number of equal-sized pools for K-fold cross-validation. k Number of subsets in data-splitting scheme. k Negative binomial parameter in abundance estimation. k(i) Index notation for K-fold cross-validation. L(Y, Ŷ) Loss function for quantifying error committed when predicting Y with an estimate Ŷ. m Number of data points used for validating a model. M Movement set of geographic cells that have been accessible to a species within a given time span. μ(Gdata, E) Result...


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