Correlated binomial models and correlation structures

Hisakado, Masato; Kitsukawa, K.; Mori, Shintaro

doi:10.1088/0305-4470/39/50/005

Cited by 54 publications

(53 citation statements)

References 5 publications

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“…In fact, N 1 /N obeys a beta distribution beta(1, a), and the correlation coefficient among M n is 1/(a + 2) in thus limit [41]. N) vs. p. We adopt the same set for the parameters, as in Figure 5.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Phase transition of social learning collectives and the echo chamber

Mori

Nakayama

Hisakado³

2016

Phys. Rev. E

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We study a simple model for social learning agents in a restless multi-armed bandit. There are N agents, and the bandit has M good arms that change to bad with the probability q c /N . If the agents do not know a good arm, they look for it by a random search (with the success probability q I ) or copy the information of other agents' good arms (with the success probability q O ) with probabilities 1 − p or p, respectively. The distribution of the agents in M good arms obeys the Yule distribution with the power-law exponent 1 + γ in the limit N, M → ∞ and γ = 1 +The system shows a phase transition at p c = q I q I +qo . For p < p c (> p c ), the variance of N 1 per agent is finite (diverges as ∝ N 2−γ with N ). There is a threshold value N s for the system size that scales as ln N s ∝ 1/(γ − 1). The expected value of the number of the agents with a good arm N 1

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Section: Acknowledgmentsmentioning

confidence: 99%

Phase transition of social learning collectives and the echo chamber

Mori

Nakayama

Hisakado³

2016

Phys. Rev. E

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“…Pólya urn is one of the simplest models for this process [9][10][11]. In this model, an urn consists of t balls, where the proportion of red balls is z(t) ∈ (0, 1) and the rest of the balls are blue.…”

Section: Introductionmentioning

confidence: 99%

Correlation function for generalized Pólya urns: Finite-size scaling analysis

Mori

Hisakado²

2015

Phys. Rev. E

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We describe a universality class for the transitions of a generalized Pólya urn by studying the asymptotic behavior of the normalized correlation function C(t) using finite-size scaling analysis. X(1), X(2), · · · are the successive additions of a red (blue) ball (X(t) = 1 (0)) at stage t and C(t) ≡ Cov(X(1), X(t + 1))/Var(X(1)). Furthermore, z(t) = t s=1 X(s)/t represents the successive proportions of red balls in an urn to which, at the t + 1-th stage, a red ball is added (X(t + 1) = 1) with probability q(z(t)) = (tanh[J(2z(t) − 1) + h] + 1)/2, J ≥ 0, and a blue ball is added (X(t + 1) = 0) with probability 1 − q(z(t)). A boundary (J c (h), h) exists in the (J, h) plane between a region with one stable fixed point and another region with two stable fixed points for q(z). C(t) ∼ c + c ′ · t l−1 with c = 0 (> 0) for J < J c (J > J c ), and l is the (larger) value of the slope(s) of q(z) at the stable fixed point(s). On the boundary J = J c (h), C(t) ≃ c + c ′ · (ln t) −α ′ and c = 0 (c > 0), α ′ = 1/2 (1) for h = 0 (h = 0). The system shows a continuous phase transition for h = 0 and C(t) behaves as C(t) ≃ (ln t) −α ′ g((1 − l) ln t) with a universal function g(x) and a length scale 1/(1 − l) with respect to ln t. β = ν || · α ′ holds with β = 1/2 and ν || = 1.

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“…This may lead to biased estimates in a model that assumes independence of detections, such as the standard binomial mixture model. The beta‐binomial distribution has been proposed as a method to account for correlated Bernoulli outcomes (Hisakado, Kitsukawa, & Mori 2006).…”

Section: Introductionmentioning

confidence: 99%

Accounting for non‐independent detection when estimating abundance of organisms with a Bayesian approach

et al. 2011

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Summary1. Binomial mixture models use repeated count data to estimate abundance. They are becoming increasingly popular because they provide a simple and cost-effective way to account for imperfect detection. However, these models assume that individuals are detected independently of each other. This assumption may often be violated in the field. For instance, manatees (Trichechus manatus latirostris) may surface in turbid water (i.e. become available for detection during aerial surveys) in a correlated manner (i.e. in groups). However, correlated behaviour, affecting the non-independence of individual detections, may also be relevant in other systems (e.g. correlated patterns of singing in birds and amphibians). 2. We extend binomial mixture models to account for correlated behaviour and therefore to account for non-independent detection of individuals. We simulated correlated behaviour using beta-binomial random variables. Our approach can be used to simultaneously estimate abundance, detection probability and a correlation parameter. 3. Fitting binomial mixture models to data that followed a beta-binomial distribution resulted in an overestimation of abundance even for moderate levels of correlation. In contrast, the beta-binomial mixture model performed considerably better in our simulation scenarios. We also present a goodness-of-fit procedure to evaluate the fit of beta-binomial mixture models. 4. We illustrate our approach by fitting both binomial and beta-binomial mixture models to aerial survey data of manatees in Florida. We found that the binomial mixture model did not fit the data, whereas there was no evidence of lack of fit for the beta-binomial mixture model. This example helps illustrate the importance of using simulations and assessing goodness-of-fit when analysing ecological data with N-mixture models. Indeed, both the simulations and the goodness-of-fit procedure highlighted the limitations of the standard binomial mixture model for aerial manatee surveys. 5. Overestimation of abundance by binomial mixture models owing to non-independent detections is problematic for ecological studies, but also for conservation. For example, in the case of endangered species, it could lead to inappropriate management decisions, such as downlisting. These issues will be increasingly relevant as more ecologists apply flexible N-mixture models to ecological data.

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Correlated binomial models and correlation structures

Cited by 54 publications

References 5 publications

Phase transition of social learning collectives and the echo chamber

Phase transition of social learning collectives and the echo chamber

Correlation function for generalized Pólya urns: Finite-size scaling analysis

Accounting for non‐independent detection when estimating abundance of organisms with a Bayesian approach

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