A multivariate uniformity test for the case of unknown support

Berrendero, José R.; Cuevas, Antonio; Pateiro-López, Beatriz

doi:10.1007/s11222-010-9222-z

Cited by 17 publications

(25 citation statements)

References 24 publications

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“…Indeed note that d(x, ∂C r (Y n )) is relatively simple to calculate; this is done in Berrendero, Cuevas and Pateiro-López (2012) in the two-dimensional case although can be in fact used in any dimension. Observe first that ∂C r (Y n )) is included in a finite union of spheres of radius r, with centres in Z = {z 1 , .…”

Section: On the Estimation Of The Maximum Distance To The Boundarymentioning

confidence: 99%

Detection of low dimensionality and data denoising via set estimation techniques

Aaron¹,

Cholaquidis²,

Cuevas³

2017

Electron. J. Statist.

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This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set S ⊂ R d . The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of S, of geometric or topological character. The available information is just a random sample of points drawn on S. The term "to identify" means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is S full dimensional? Is S "close to a lower dimensional set" M? If so, can we estimate M or some functionals of M (in particular, the Minkowski content of M)? As an important auxiliary tool in the answers of these questions, a denoising procedure is proposed in order to partially remove the noise in the original data. The theoretical results are complemented with some simulations and graphical illustrations. arXiv:1702.05193v2 [math.ST] 3 Nov 2017 Estimation of some other relevant quantities in a manifold, Niyogi, Smale and Weinberger (2008), Chen and Müller (2012). Dimensionality reduction, Genovese et al. (2012a), Tenebaum et al. (2000).The problems under study. The contents of the paper. We are interested in getting some information (in particular, regarding dimensionality and Minkowski content) about a compact set M ⊂ R d . While the set M is typically unknown, we are supposed to have a random sample of points X 1 , . . . , X n whose distribution P X has a support "close to M". To be more specific, we consider two different models:The noiseless model : the support of P X is M itself; Aamari and Levrard (2015), Amenta et al. (2002), Cholaquidis et al. (2014), Cuevas and Fraiman (1997). The parallel (noisy) model : the support of P X is the parallel set S of points within a distance to M smaller than R 1 , for some R 1 > 0, where M is a d -dimensional set and d ≤ d; Berrendero et al. (2014). Note that other different models "with noise" are considered in Genovese et al. (2012a), Genovese et al. (2012b) and Genovese et al (2012c).In Section 3 we first develop, under the noiseless model, an algorithmic procedure to identify, eventually, almost surely (a.s.), whether or not M has an empty interior; this is achieved in Theorems 1 and 2 below. A positive answer would essentially entail (under some conditions, see the beginning of Section 3) that M has a dimension smaller than that of the ambient space.Then, assuming the noisy model andM = ∅ ( whereM denotes the interior of M) Theorems 3 (i) and 4 (i) provide two methods for the estimation of the maximum level of noise R 1 , giving also the corresponding convergence rates. If R 1 is known in advance, the remaining results in Theorems 3 and 4 allow us also to decide whether or not the "inside set" M has an empty interior.The identification methods are "algorithmic" in the sense that they are based on automatic procedures to perform them with arbitrary precision. This will requi...

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Section: On the Estimation Of The Maximum Distance To The Boundarymentioning

confidence: 99%

Detection of low dimensionality and data denoising via set estimation techniques

Aaron¹,

Cholaquidis²,

Cuevas³

2017

Electron. J. Statist.

Self Cite

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“…This paper is not concerned with giving an overview over the available literature (see e.g. [2], [6], [10], [11]), but to turn attention to the maximum spacings test studied in [1]. To be specific, let K be a bounded set in R d , d ≥ 1, where |K| = 1 and |∂K| = 0 and | · | denotes Lebesgue measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the case d = 1 and K = [0, 1], result (1.1) is due to L. Weiss, see [13]. This paper has been largely forgotten, since it is referenced neither in [9] nor in [1]. The latter paper suggests to use V n as a statistic for testing the hypothesis H 0 that X 1 has a uniform distribution over K. Using (1.1) and denoting by g 1−α the (1 − α)-quantile of G, an asymptotic level-α-test rejects H 0 if V n > c n,α , where c n,α = g 1−α + log n + (d − 1) log log n + β n .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the univariate case, this test has been proposed in [13], and [13] also proves its consistency against general alternatives. The authors of [1] prove the consistency of the test based on V n (see Theorem 1 of [1]). This proof, however, hinges on a heuristic argument (see line 4 of the left-hand column of page 270 of [1]), and the method of proof does not cover the case of testing for uniformity on the surface of a sphere or, more generally, on a lower-dimensional…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On the consistency of the spacings test for multivariate uniformity, including on manifolds

Henze¹

2018

J. Appl. Probab.

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We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ R d that is based on the maximal spacing generated by i.i.d. points X 1 , . . . , X n in K, i.e., the volume of the largest convex set of a given shape that is contained in K and avoids each of these points. Since asymptotic results for the case d > 1 are only availabe under uniformity, a key element of the proof is a suitable coupling.

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“…A variety of other well-documented and tested methods are also available (Petrie and Willemain, 2013) and could be used interchangeably with those we specifically mention in this paper. A few examples include those that perform a two-sample test on a subsample of points in a high-density region and a subsample in a low-density region (Jain et al, 2002), or those that consider the distribution of distances from points to the boundary of support, both in the case of known support (Berrendero et al, 2006) and unknown support (Berrendero et al, 2012).…”

Section: Discussionmentioning

confidence: 99%

A common goodness-of-fit framework for neural population models using marked point process time-rescaling

Long

Weber

Arai

et al. 2018

Preprint

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A critical component of any statistical modeling procedure is the ability to assess the goodness-of-fit between a model and observed data. For neural spike train models of individual neurons, many goodness-of-fit measures rely on the time-rescaling theorem to assess the statistical properties of rescaled spike times. Recently, there has been increasing interest in statistical models that describe the simultaneous spiking activity of neuron populations, either in a single brain region or across brain regions. Classically, such models have used spike sorted data to describe relationships between the identified neurons, but more recently clusterless modeling methods have been used to describe population activity using a single model. Here we develop a generalization of the time-rescaling theorem that enables comprehensive goodness-of-fit analysis for either of these classes of population models. We use the theory of marked point processes to model population spiking activity, and show that under the correct model, each spike can be rescaled individually to generate a uniformly distributed set of events in time and the space of spike marks. After rescaling, multiple well-established goodness-of-fit procedures and statistical tests are available. We demonstrate the application of these methods both to simulated data and real population spiking in rat hippocampus.

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A multivariate uniformity test for the case of unknown support

Cited by 17 publications

References 24 publications

Detection of low dimensionality and data denoising via set estimation techniques

Detection of low dimensionality and data denoising via set estimation techniques

On the consistency of the spacings test for multivariate uniformity, including on manifolds

A common goodness-of-fit framework for neural population models using marked point process time-rescaling

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