12th European Meeting of Statisticians. Absracts. Колектив

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  •  18-4-2017
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ISBN: Тегло (гр.): Формат: 140 / 200 Състояние: Мн. добро


THE STATISTICS OF SHAPE AUTHOR(®) David    G. Kendall (Cambrldge, Eng land)

In these lectures I shall be concerned with various aspects of the mathematics and the statistics of the shape of a configuration of к points in euclidean space (normally of 2 dlmensions). In the first lecture I shall present an account of work recently carrled out in the UK, by S. R. Broadbent who initiated this parti-cular statistical enquiry, and then by W. S. Kendall and myself in collaboration. Related work by D. M. Behrend and by C. G. Small ought also to be mentioned. Say we are given к points in the plane (they might be locations of к archaeological sites) and suppose that we are Interested in the number N(e) of the (^) triads of points for which the associated triangle has а largest angle which is not less than л - с. We may be asked, is N(e) too large to be compatible with а random "arrangement”? (If the answer is "yes", then the implication may be that the sites were deliberately arranged in a pattern involving many collinearities). This problem with e fixed (and set, say, to 40 minutes of are), and with the assumption that on the null hypothesis the sites are IID within a rectangle, with а uniform distribution therein for each site, was the one studied by Broadbent. Our development of his work consists (i) in avoiding so specific an assumption about the null-hypotheal» dletribution, and (ii) in allowing the parameter e to be an unknown nuisance-parameter
constrained only to lie in a stated range (TQ,Ti). We give a test which is largêly model-free, and which depends on the introduction of a plot which we call the pontoqram, and on the concept of simulations based on "Xateral perturbations of the data." The technique will be illustrated by using an archaeological data-set provided by Broadbent. In the second lecture I shall discuss further and more theoretical work on problems of this type.

Notice that "bluntness" (i.e. maximum angle not less than

и - e) is a shape-property of a triad of points.

Suppose now that we have a set of к points in two

dimensions, and want to consider all possible shape-

properties. It turns out that the right context for

this is provided by viewing the k-ad modulo the

similarity group as a point on the complex projective

k-2 2 manifold CP    (which reduces to the sphere S

when к = 3). The procrustean procedures of data-

analysis define a metric on this space which turns out

to be the classical Fubini-Study complex projective

metric, while any assumption (not necessarily IID) about


the к points induces a shape-distribution on CP

which turns out to be the unique invariant distribution

when the parent distribution is IID Gaussian with

circular symmetry. Specific shape-functlons (such as

the maximum angle of the triangle in the case к = 3)


define shape-contours on the space CP. Thus if we

Категория:     Математика
Издателство:     ---
Година:     1979
Cтраници:     289
Забележка:     Неизползвана книга.
Налични бройки:     1
Език:     Английски
Град на издаване:     Varna
Корици:     меки
Размери:      140/200/0 мм
Ключови думи:      статистика, книги за статистици
Категория › Математика

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