The TAOS Project: Statistical Analysis of Multi-Telescope Time Series Data

Lehner, M. J.; Coehlo, N. K.; Zhang, Z.-W.; Bianco, Federica; Wang, J.-H.; Rice, John A.; Protopapas, Pavlos; Alcock, C.; Axelrod, T. S.; Byun, Yong Ik; Chen, W. P.; Cook, K. H.; Pater, Imke de; Kim, Dae Won; King, S. K.; Lee, T.; Marshall, Stuart; Schwamb, Megan E.; Wang, S.-Y.; Wen, C. Y.

doi:10.1086/655443

Cited by 11 publications

(16 citation statements)

References 20 publications

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“…If

rp ⩽ n,

there are no divisors larger than n . In that case

s = 0,

and the expression reduces to the formula offered by Lehner et al [6].…”

Section: Exact Distribution Of Rank Productsmentioning

confidence: 91%

“…If the P ‐values

r_{j} / (n + 1)

are uniform and continuous on the unit interval [0,1], the probability distribution of

w_{j} = ‐ \log (r_{j} / [n + 1])

is given by the exponential distribution

p (w_{j}) = e^{‐ w_{j}}

with scale parameter 1, denoted as Exp(1). Given that

w_{j}

is distributed as Exp(1), the sum of

w_{j}

over

k

independent replicates has a gamma

(k, 1)

distribution, i.e.,

p (z) = Γ (k)^{‐ 1} z^{k ‐ 1} e^{‐ z},

where

z = \sum_{j = 1}^{k} w_{j}

[see 2,6,8]. Koziol…”

Section: Gamma Approximation For Rank Productsmentioning

confidence: 99%

“…In their discussion of rank statistics, Lehner et al [6] have shown that we can enumerate the number of ordered k ‐tuples such that their product equals

rp,

using

H false(italicrp; k, n false) = true \prod_{t = 1}^{m} (\begin{matrix} a_{t} + k ‐ 1 \\ k ‐ 1 \end{matrix}) if italicrp ⩽ n .

…”

Section: Exact Distribution Of Rank Productsmentioning

confidence: 99%

“…The computation of

H (rp; k, n)

is an application of the so‐called Piltz divisor function [11, see also Sloane's (A007425) at http://oeis.org/A007425], and intimately related to the study of ordered factorizations of integers [12]. For a proof see Nathanson [10], Theorem 7.5, and Lehner et al [6]. The above expression for

H (rp; k, n)

is a valid method for counting the representations of rp as long as the rank product is less than or equal to the number of genes.…”

Section: Exact Distribution Of Rank Productsmentioning

confidence: 99%

“…This note provides a combinatorial exact expression for calculating the probability mass function of the rank product statistic and the exact P ‐values based on the fundamental theorem of arithmetic. The underlying method has previously been suggested by Lehner et al [6] in a different research area, but their expression is exact only for the restricted case that the rank product is not larger than the number of genes in the array. In this paper, we remove this restriction, making the resulting counting method generally applicable to the analysis of microarray and other data.…”

Section: Introductionmentioning

confidence: 99%

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The exact probability distribution of the rank product statistics for replicated experiments

2013

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Edited by Paul Bertone Keywords:Rank product method Microarray Permutational inference Gamma approximation Exact inference a b s t r a c tThe rank product method is a widely accepted technique for detecting differentially regulated genes in replicated microarray experiments. To approximate the sampling distribution of the rank product statistic, the original publication proposed a permutation approach, whereas recently an alternative approximation based on the continuous gamma distribution was suggested. However, both approximations are imperfect for estimating small tail probabilities. In this paper we relate the rank product statistic to number theory and provide a derivation of its exact probability distribution and the true tail probabilities.

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“…If

rp ⩽ n,

there are no divisors larger than n . In that case

s = 0,

and the expression reduces to the formula offered by Lehner et al [6].…”

Section: Exact Distribution Of Rank Productsmentioning

confidence: 91%

“…If the P ‐values

r_{j} / (n + 1)

are uniform and continuous on the unit interval [0,1], the probability distribution of

w_{j} = ‐ \log (r_{j} / [n + 1])

is given by the exponential distribution

p (w_{j}) = e^{‐ w_{j}}

with scale parameter 1, denoted as Exp(1). Given that

w_{j}

is distributed as Exp(1), the sum of

w_{j}

over

k

independent replicates has a gamma

(k, 1)

distribution, i.e.,

p (z) = Γ (k)^{‐ 1} z^{k ‐ 1} e^{‐ z},

where

z = \sum_{j = 1}^{k} w_{j}

[see 2,6,8]. Koziol…”

Section: Gamma Approximation For Rank Productsmentioning

confidence: 99%

“…In their discussion of rank statistics, Lehner et al [6] have shown that we can enumerate the number of ordered k ‐tuples such that their product equals

rp,

using

H false(italicrp; k, n false) = true \prod_{t = 1}^{m} (\begin{matrix} a_{t} + k ‐ 1 \\ k ‐ 1 \end{matrix}) if italicrp ⩽ n .

…”

Section: Exact Distribution Of Rank Productsmentioning

confidence: 99%

“…The computation of

H (rp; k, n)

H (rp; k, n)

is a valid method for counting the representations of rp as long as the rank product is less than or equal to the number of genes.…”

Section: Exact Distribution Of Rank Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The exact probability distribution of the rank product statistics for replicated experiments

2013

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A kilometre-sized Kuiper belt object discovered by stellar occultation using amateur telescopes

et al. 2019

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The Taos Project: Results From Seven Years of Survey Data

Zhang

Lehner

Wang

et al. 2013

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The Taiwanese-American Occultation Survey (TAOS) aims to detect serendipitous occultations of stars by small (∼1 km diameter) objects in the Kuiper Belt and beyond. Such events are very rare (< 10 −3 events per star per year) and short in duration (∼200 ms), so many stars must be monitored at a high readout cadence. TAOS monitors typically ∼500 stars simultaneously at a 5 Hz readout cadence with four telescopes located at Lulin Observatory in central Taiwan. In this paper, we report the results of the search for small Kuiper Belt Objects (KBOs) in seven years of data. No occultation events were found, resulting in a 95% c.l. upper limit on the slope of the faint end of the KBO size distribution of q = 3.34 to 3.82, depending on the surface density at the break in the size distribution at a diameter of about 90 km.

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The TAOS Project: Statistical Analysis of Multi-Telescope Time Series Data

Cited by 11 publications

References 20 publications

The exact probability distribution of the rank product statistics for replicated experiments

The exact probability distribution of the rank product statistics for replicated experiments

A kilometre-sized Kuiper belt object discovered by stellar occultation using amateur telescopes

The Taos Project: Results From Seven Years of Survey Data

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