A fable of four functions.
نویسندگان
چکیده
The review by Blows (Blows, 2007) highlights a perspective with which we whole-heartedly agree: greater biological insight is obtained by focusing on fundamental structures of the two matrices G and c than by analysing the individual entries of those matrices. Here, we advocate extending this point of view further for any ‘function-valued trait’, i.e. a trait that is inherently a mathematical function, z(t). The function z(t) can be viewed as an extension of a multivariate trait vector z (Kirkpatrick & Heckman, 1989). We will introduce three more functions G(t,s), c(t,s) and b(t) that are, respectively, extensions of the matrices G and c, and the vector b discussed in the review. Function-valued traits arise naturally in many contexts. For example, Blows discusses a study of stabilizing selection on male song in the cricket Teleogryllus commodus, characterizing the song as a vector of five song attributes (chip number, interpulse interval, trill number, intercall interval and dominant frequency). However, the song itself is energy (or power) as a function of time and is thus a function-valued trait, z(t). Reducing this function-valued trait to a vector of five measurements allows the ‘song’ to be analysed in a familiar quantitative genetics framework for multivariate traits, but doing so has important drawbacks (Kirkpatrick & Heckman, 1989; Ramsay & Silverman, 1997). First and foremost, information is necessarily lost in the data reduction. (Consider the futility of grasping Debussy’s Clair de Lune from just its dominant frequency, duration, average inter-note interval, and a dozen other such elements!) Second, treating such data as functional ab initio leads to data reduction methods that retain functional information more efficiently (like Clair de Lune’s sheet music or its digital representation on a CD) than do ad hoc collections of attributes (Ramsay & Silverman, 1997). There are many other types of function-valued traits, including growth trajectories (size as a function of age), gene expression profiles (product as a function of time), reaction norms (phenotype as a function of environment) and morphological shapes (radial distance as a function of angle). Typically, these traits are measured at a finite number of values of the independent variable and the finite collection of trait measurements is treated as a multivariate trait. There are two problems with this. First, there may be more effective ways to combine the measurements taken (or more effective ways to take the finite set of measurements). Second, if measurements are taken at different index values on different individuals (e.g. sizes measured at different sets of ages for different individuals) then multivariate approaches would require some sort of binning procedure, usually improvised. By treating the trait at the outset as a function, however, these issues can be avoided altogether (Kirkpatrick et al., 1994). A function-centric quantitative genetics framework has been developed for describing function-valued traits, including their genetic variability, selection, and evolution (Kirkpatrick & Heckman, 1989; Kirkpatrick et al., 1990; Gomulkiewicz & Kirkpatrick, 1992; Kirkpatrick & Lofsvold, 1992; Kirkpatrick et al., 1994; Meyer, 1998; Pletcher & Geyer, 1999; Kingsolver et al., 2001). In this framework, which is a direct extension of multivariate quantitative genetics, a trait function z(t) is usually represented mathematically by an ‘orthogonal function series expansion’. These expansions can describe practically any biologically plausible function shape, including nonsmooth and discontinuous functions that cannot be written in terms of the perhaps more familiar Taylor series expansion. Orthogonal function expansions can be used with least-squares or likelihood statistical methods to develop estimates of z(t) based on the data collected rather than some preconceived notion about the form of the function. These estimates provide natural interpolations of the original data. The extension of the genetic variance–covariance matrix G is the genetic variance–covariance function G(t,s). When t „ s, the G-function gives the additive genetic covariance of the trait expressed at index values t and s (like the off-diagonal elements of G) and when s 1⁄4 t it equals the additive genetic variance of the trait expressed at index value t (like the diagonal elements of G). What does a G-function look like? Consider the example of a growth trajectory, size as a function of age, and suppose all individuals in a study were measured at the same five ages. Given an appropriate breeding design, one could estimate a five-dimensional genetic covariance matrix G, the elements of which can be visualized as a three-dimensional histogram. The corresponding G-function would be a smoothed surface enveloping the tops of the histogram. Figures 1 and 2 of Kirkpatrick et al. (1990) show these visualizations of a genetic covariance matrix G and the corresponding G-function. Like G, the G-function can be ‘diagonalized’ and understood in terms of its eigenstructure. The G-function can be decomposed into a series of eigenvalues, each of which is the variance associated with a principal component. Principal components are represented by loading functions (eigenfunctions) rather than loading vectors Correspondence: Joel Kingsolver, Department of Biology, CB-3280, University of North Carolina, Chapel Hill, NC 27599, USA. Tel.: 919 843 6291; fax: 919 962 1625; e-mail: [email protected]
منابع مشابه
The animal fable and Greek iambus: ainoi and half-ainoi in Archilochus
Animal fable is often discussed as a genre in its own right, associated with moralising, criticism, and popular ideology. 1 Yet any instantiation of a fable in a literary text intersects with the conventions of the host-genre. While fable is found in various ancient genres, it has a particular association with Greek iambus, and Hellenistic and Roman imitators of archaic iambus turn regularly to...
متن کاملThe Imaginary Audience and the Personal Fable: A Test of Elkind’s Theory of Adolescent Egocentrism
The aim of this research was to test empirically Elkind’s (1967, 1970, 1978) Piagetian theoretical formulation for the developmental nature of adolescent egocentrism. The contribution of this study is threefold because it includes: 1) Pubertal development (with a distinction between status and timing), which has been systematically ignored by other investigators; 2) a broad age range (11 18 yea...
متن کاملPersonal fable - Wikipedia, the free encyclopedia
Adolescent egocentrism can be divided into two separate forms: the imaginary audience and the personal fable. The first relates to the adolescent living life believing he is constantly being watched and judged by others, and that others are as concerned with his appearance and behaviour as he himself is.[1] The personal fable results in the adolescent perceiving himself as special and unique, b...
متن کاملFables: ways of knowing and understanding meaning in nursing.
Fables are literary short stories intended to convey a moral maxim for some aspect of human living. Within the discipline of nursing, a fable is a medium by which stories and themes of humanly lived phenomena may be viewed in light of a particular philosophical lens of understanding. The fable offers possibilities for instruction through illuminating fictitious characters and offering notions f...
متن کاملRooks Use Stones to Raise the Water Level to Reach a Floating Worm
In Aesop's fable "The Crow and the Pitcher," a thirsty crow uses stones to raise the level of water in a pitcher and quench its thirst. A number of corvids have been found to use tools in the wild, and New Caledonian crows appear to understand the functional properties of tools and solve complex physical problems via causal and analogical reasoning. The rook, another member of the corvid family...
متن کاملMedieval anxieties: translation and authorial self-representation in the vernacular beast fable
My dissertation examines the concept of vernacular translation in the Middle Ages, particularly examining French and Middle English texts. It focuses on a specific genre of literature popular in the Middle Ages but relatively ignored in contemporary literary scholarship: the beast fable. My argument is that some of the principal writers of vernacular fables from the twelfth through the fifteent...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Journal of evolutionary biology
دوره 20 1 شماره
صفحات -
تاریخ انتشار 2007