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An Invitation to Coupling and Copulas: with Applications to Multisensory Modeling Hans Colonius University of Oldenburg, Oldenburg, Germany H I G H L I G H T S

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An Invitation to Coupling and Copulas: with Applications to Multisensory Modeling Hans Colonius University of Oldenburg, Oldenburg, Germany H I G H L I G H T S Journal of Mathematical Psychology ( ) – Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp An invitation to coupling and copulas: With applications to multisensory modeling Hans Colonius University of Oldenburg, Oldenburg, Germany h i g h l i g h t s • Gives an introduction to the stochastic concepts of coupling and copula. • Demonstrates their role in modeling reaction times. • Presents an example from modeling multisensory integration. • Gives pointers to the advanced literature on coupling and copula and to the relevant multisensory literature. article info a b s t r a c t Article history: This paper presents an introduction to the stochastic concepts of coupling and copula. Coupling means Available online xxxx the construction of a joint distribution of two or more random variables that need not be defined on one and the same probability space, whereas a copula is a function that joins a multivariate distribution to its Keywords: one-dimensional margins. Their role in stochastic modeling is illustrated by examples from multisensory Coupling perception. Pointers to more advanced and recent treatments are provided. Copula ' 2016 Elsevier Inc. All rights reserved. Multisensory Race model Time window of integration model 1. Introduction derive measures of the amount of multisensory integration occur- ring in a given context. The concepts of coupling and copula refer to two related areas of In psychology, a need for the theory of coupling emerges from probability and statistics whose importance for mathematical psy- the following observations. Empirical data are typically collected chology has arguably been ignored so far. This paper gives an el- under various experimental conditions, e.g. speed vs. accuracy ementary, non-rigorous introduction to both concepts. Moreover, instructions in a reaction time (RT) experiment or different applications of both concepts to modeling in a multisensory con- numbers of targets or nontargets in a visual search paradigm. Data text are presented. Briefly, coupling means the construction of a collected under a particular experimental condition are considered joint distribution of two or more random variables that need not realizations of some random variable with respect to an underlying be defined on one and the same probability space, whereas a cop- probability (sample) space. For example, let Tk be the random ula is a function that joins a multivariate distribution to its one- variable denoting the time to detect a target in the presence of k dimensional margins. nontargets (distractors). Now consider random variable TkC1, the The theory of copulas has stirred a lot of interest in recent time to detect a target in the presence of k C 1 targets. Whenever years in several areas of statistics, including finance, mainly for response times Tk and TkC1 are collected in different trials (or the following reasons (see e.g., Joe, 2015): it allows one (i) to blocks of trials), there is no empirical coupling scheme, so these study the structure of stochastic dependency in a ``scale-free'' man- two random variables are ``stochastically unrelated'', i.e., they do ner, i.e., independent of the specific marginal distributions, and not refer to outcomes defined on one and the same underlying (ii) to construct families of multivariate distributions with speci- sample space. Alternatively, if the target is presented first under k and then fied properties. We will demonstrate in the final section how cop- under k C 1 nontargets, one immediately following the other, and ulas can be used to test models of multisensory integration and to responses are recorded in pairs at any given trial n, this imposes a coupling on Tk and TkC1, that is the existence of a bivariate distribution with marginal distributions equal to the distributions E-mail address: [email protected]. of Tk and TkC1. As explicated in the theory of ``contextuality- URL: http://www.uni-oldenburg.de/en/hans-colonius/. by-default'' being developed by E. N. Dzhafarov and J. Kujala http://dx.doi.org/10.1016/j.jmp.2016.02.004 0022-2496/' 2016 Elsevier Inc. All rights reserved. 2 H. Colonius / Journal of Mathematical Psychology ( ) – Table 1 Nomenclature Joint distribution of two Bernoulli random variables. O Xq Set of real numbers R 0 1 Ω Set of elementary outcomes ! 0 1 − q q − p 1−p F Sigma-algebra XO p 1 0 p p Probability measure on .Ω; F / P − B Borel sigma algebra 1 q q 1 P Probability on .R; B/ d −∞ C1 ≤ Dominated in distribution For every < x < and 0 < u < 1, we have a.s. Almost surely F.x/ ≥ u if; and only if; Q .u/ ≤ x: X ≤ YX .!/ ≤ Y .!/ for all ! (X; Y random variables) D D a ≤ b ai ≤ bi for i D 1;:::; n Thus, if there exists x with F.x/ u, then F.Q .u// u and Q .u/ is RanF Range (image) of function F the smallest value of x satisfying F.x/ D u. If F.x/ is continuous and FN Survival distribution corresponding to distribution strictly increasing, Q .u/ is the unique value x such that F.x/ D u. F Moreover, if U is a standard uniform random variable (i.e., de- CQ Dual of copula C fined on T0; 1U), then X D Q .U/ has its distribution function as C∗ Co-copula of copula C F.x/; thus, any distribution function can be conceived as arising δC Diagonal section of copula C from the uniform distribution transformed by Q .u/. ρ Spearman's rho τ Kendall's tau 2.1. Definition and examples r Pearson's linear correlation d D Equal in distribution Note to the reader: We enumerate Definitions, Theorems, Ex- Cov Covariance amples sequentially, so Definition 1 is followed by Example 2, E Expected value and so on. Definition 1. A coupling of a collection of random variables fXi; i 2 (see Dzhafarov and Kujala, 2016, in this issue) such a coupling Ig, with I denoting some index set, is a family of jointly distributed represents an empirically defined meaning of ``togetherness'' of the random variables responses. O O d Note that even if no ``natural'' coupling exists because responses .Xi V i 2 I/ such that Xi D Xi; i 2 I: cannot co-occur, a coupling satisfying some desirable properties Note that the joint distribution of the XO need not be the same as can always be constructed. This will be illustrated in Section4 with i that of X ; in fact, the X may not even have a joint distribution an example from multisensory modeling, coupling unisensory i i because they need not be defined on a common probability space. reaction times to visual and auditory stimuli presented in separate O blocks of trials. Moreover, stochastic unrelatedness does not However, the family .Xi V i 2 I/ has a joint distribution with the rule out numerically comparing averaged data under separate property that its marginals are equal to the distributions of the O conditions, or even the entire distributions functions. Specifically, individual Xi variables. The individual Xi is also called a copy of Xi. as shown in the next section, one can turn a statement about an ordering of two stochastically unrelated distributions into a Example 2 (Coupling Two Bernoulli Random Variables). Let Xp; Xq be statement about point-wise ordering of the corresponding random Bernoulli random variables, i.e., variables on a common probability space. P.Xp D 1/ D p and P.Xp D 0/ D 1 − p; 2. Coupling and Xq defined analogously. Assume p < q; we can couple Xp and Xq as follows: We begin with a few common definitions.1 Let X and Y be ran- Let U be a uniform random variable on T0; 1U, i.e., for 0 ≤ a < dom variables defined on a probability space .Ω; F ; P/, with val- b ≤ 1, ues in .R; B; P/. ``Equality'' of random variables can be interpreted P a U ≤ b D b − a in different ways. Random variables X and Y are equal in distribu- . < / : tion iff they are governed by the same probability measure: Define d D () 2 D 2 2 1; if 0 < U ≤ pI 1; if 0 < U ≤ qI X Y P.X A/ P.Y A/; for all A B: XO D XO D p 0; if p < U ≤ 1; q 0; if q < U ≤ 1: X and Y are point-wise equal iff they agree for almost all elementary events2: O Then U serves as a common source of randomness for both Xp and aD:s: () f j D g D O O d O d O O X Y P. ! X.!/ Y .!/ / 1: Xq. Moreover, Xp D Xp and Xq D Xq, and Cov.Xp; Xq/ D p.1 − q/. O O Let X be a real-valued random variable with distribution function The joint distribution of .Xp; Xq/ is presented in Table 1 and F.x/. Then the quantile function of X is defined as illustrated in Fig. 1. − Q .u/ D F 1.u/ D inffx j F.x/ ≥ ug; 0 ≤ u ≤ 1: (1) A simple, though somewhat fundamental, coupling is the fol- lowing: Example 3 (Quantile Coupling). Let X be a random variable with 1 If not indicated otherwise, most of the material on coupling in this section is distribution function F, that is, taken from the monograph by Thorisson(2000). 2 Here, a.s. is for ``almost surely''. P.X ≤ x/ D F.x/; x 2 R: H. Colonius / Journal of Mathematical Psychology ( ) – 3 Assume all the Xi take values in a finite or countable set E with P.Xi D x/ D pi.x/, for x 2 E.
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