Construction of Bivariate Distributions

Balakrishna, N.; Lai, C. D.

doi:10.1007/b101765_6

Cited by 83 publications

(40 citation statements)

References 79 publications

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“…Despite its advantages, there are several aspects in which gazeHMM can be improved: First, a multivariate distribution could be used to account for the correlation between velocity and acceleration signals (for examples, see 2 ). Potential problems of this approach might be choosing the right distribution and convergence issues (due to a large number of parameters).…”

Section: Discussionmentioning

confidence: 99%

Characterising eye movement events with an unsupervised hidden markov model

Lüken

Kucharský²,

Visser³

2022

JEMR

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Eye-tracking allows researchers to infer cognitive processes from eye movements that are classified into distinct events. Parsing the events is typically done by algorithms. Here we aim at developing an unsupervised, generative model that can be fitted to eye-movement data using maximum likelihood estimation. This approach allows hypothesis testing about fitted models, next to being a method for classification. We developed gazeHMM, an algorithm that uses a hidden Markov model as a generative model, has few critical parameters to be set by users, and does not require human coded data as input. The algorithm classifies gaze data into fixations, saccades, and optionally postsaccadic oscillations and smooth pursuits. We evaluated gazeHMM’s performance in a simulation study, showing that it successfully recovered hidden Markov model parameters and hidden states. Parameters were less well recovered when we included a smooth pursuit state and/or added even small noise to simulated data. We applied generative models with different numbers of events to benchmark data. Comparing them indicated that hidden Markov models with more events than expected had most likely generated the data. We also applied the full algorithm to benchmark data and assessed its similarity to human coding and other algorithms. For static stimuli, gazeHMM showed high similarity and outperformed other algorithms in this regard. For dynamic stimuli, gazeHMM tended to rapidly switch between fixations and smooth pursuits but still displayed higher similarity than most other algorithms. Concluding that gazeHMM can be used in practice, we recommend parsing smooth pursuits only for exploratory purposes. Future hidden Markov model algorithms could use covariates to better capture eye movement processes and explicitly model event durations to classify smooth pursuits more accurately.

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Section: Discussionmentioning

confidence: 99%

Characterising eye movement events with an unsupervised hidden markov model

Lüken

Kucharský²,

Visser³

2022

JEMR

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“…In the next section, we study a special case of the bivariate weighted distribution in (1) where the W i 's are Kumaraswamy with parameters a and b i , for i = 0, 1, 2, respectively.…”

Section: Type-i Weighted Bivariate Kumaraswamy Distributionmentioning

confidence: 99%

“…Recently the construction of continuous bivariate distributions have received a considerable amount of interest in the literature. A vast literature on this topic exists (see, the book by Balakrishnan and Lai [1]). Kumaraswamy [2] argued that the beta distribution does not faithfully fit hydrological random variables such as daily rainfall and daily stream flow and introduced an alternative distribution to the beta distribution, which is known as the Kumaraswamy distribution.…”

Section: Introductionmentioning

confidence: 99%

Bivariate and Multivariate Weighted Kumaraswamy Distributions: Theory and Applications

Ghosh¹

2019

JSTA

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Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this paper, we derive the bivariate and multivariate weighted Kumaraswamy distributions via the construction method as discussed in B.C. Arnold, I. Ghosh, A. Alzaatreh, Commun. Stat. Theory Methods. 46 (2017), 8897 8912. Several structural properties of the bivariate weighted distributions including marginals, distributions of the minimum and maximum, reliability parameter, and total positivity of order two are discussed. We provide some multivariate extensions of the proposed bivariate weighted Kumaraswamy model. Two real-life data sets are used to show the applicability of the bivariate weighted Kumaraswamy distributions and is compared with other rival bivariate Kumaraswamy models.

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“…Our research is based on one bivariate gamma distribution model, the SAT model, which was first introduced in [15] and some further statistical issues have been studied in [16] and [17].…”

Section: The Bivariate Gamma Distributionmentioning

confidence: 99%

One Hotelling T<sup>2</sup> chart based on transformed data for simultaneous monitoring the frequency and magnitude of an event

Cheng

Mukherjee

2014

2014 IEEE International Conference on Industrial Engineering and Engineering Management

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Commonly, the event frequency and event magnitude have the dependence feature. However, previous research about event monitoring always assume they are independent variables. This assumption is not reasonable in most real applications. Taken the dependence feature into consideration, we introduce a typical bivariate gamma distribution with certain dependence structure. Based on this distribution, a kind of Hotelling T 2 chart based on the transformed data is constructed for jointly monitoring of the shifts in the frequency and the magnitude of an event. An illustrative example based on a real data is provided to show the implementation of this chart. With Monte Carlo simulation, the performance of this proposed chart is studied.

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Construction of Bivariate Distributions

Cited by 83 publications

References 79 publications

Characterising eye movement events with an unsupervised hidden markov model

Characterising eye movement events with an unsupervised hidden markov model

Bivariate and Multivariate Weighted Kumaraswamy Distributions: Theory and Applications

One Hotelling T<sup>2</sup> chart based on transformed data for simultaneous monitoring the frequency and magnitude of an event

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