Improving polynomial estimation of the Shapley value by stratified random sampling with optimum allocation

Castro, Javier; Gómez, Daniel; Molina, Elisenda; Tejada, Juan

doi:10.1016/j.cor.2017.01.019

Cited by 67 publications

(69 citation statements)

References 15 publications

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“…and Brus, D. J., Kempen, B. & Heuvelink G. B. M. 19, 22 , but deferent from the finding of some other scholars 17, 18 . The equidistant sampling method also has certain shortcomings, i.e., the overall number of units may contain hidden forms or “unqualified samples”, and the investigators may neglect them during sample selection.…”

Section: Discussionmentioning

confidence: 89%

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Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Zhao

Zhu

Cheng

et al. 2017

Sci Rep

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The influence of the equidistant sampling method was explored in a hyperspectral model for the accurate prediction of the water content of apple tree canopy. The relationship between spectral reflectance and water content was explored using the sample partition methods of equidistant sampling and random sampling, and a stepwise regression model of the apple canopy water content was established. The results showed that the random sampling model was Y = 0.4797 − 721787.3883 × Z3 − 766567.1103 × Z5 − 771392.9030 × Z6; the equidistant sampling model was Y = 0.4613 − 480610.4213 × Z2 − 552189.0450 × Z5 − 1006181.8358 × Z6. After verification, the equidistant sampling method was verified to offer a superior prediction ability. The calibration set coefficient of determination of 0.6599 and validation set coefficient of determination of 0.8221 were higher than that of the random sampling model by 9.20% and 10.90%, respectively. The root mean square error (RMSE) of 0.0365 and relative error (RE) of 0.0626 were lower than that of the random sampling model by 17.23% and 17.09%, respectively. Dividing the calibration set and validation set by the equidistant sampling method can improve the prediction accuracy of the hyperspectral model of apple canopy water content.

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Section: Discussionmentioning

confidence: 89%

“…When the water content regression model is constructed, the calibration set and the validation set of the partition model are often sampled randomly 17, 18 . The advantage of this method is that the operation is simple, but it is easy to cause polarization of the system, thus affecting the accuracy of the model and test.…”

Section: Introductionmentioning

confidence: 99%

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Zhao

Zhu

Cheng

et al. 2017

Sci Rep

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“…terms. Computing the Shapley value is #P-complete (Deng and Papadimitriou [7]), although some recent work has explored polynomial algorithms for obtaining approximations to the Shapley value (Castro et al [2,3]).…”

Section: Cooperative Games and The Shapley Valuementioning

confidence: 99%

“…Interestingly, player 3 also receives a slightly increased payoff, from 1 6 to 1 5 , since player 3 contributes zero marginal value to any coalition that already contains player 2, and one such coalition has been removed from consideration. Both players 2 and 3 benefit at the expense of player 1, whose payoff is decreased from 2 ). These take the approach of averaging over feasible permutations, a generalization of (2).…”

Section: 2mentioning

confidence: 99%

Hodge decomposition and the Shapley value of a cooperative game

Stern

Tettenhorst

2019

Games and Economic Behavior

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We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.

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“…However, since the Shapley value is expressible as an average over averages by Equation (11), it is possible to approximate these averages via sampling techniques, and particularly so as these averages are naturally stratified by size. In previously published literature, other techniques have been used to allocate samples in this context, particularly simple sampling (Castro, Gómez and Tejada, 2009), Neyman allocation (Castro et al, 2017;O'Brien, Gamal and Rajagopal, 2015), and allocation to minimize Hoeffding's inequality (Maleki et al, 2013). We assess the benefits of using our bound by comparing its performance to the approaches above in the context of some example cooperative games, as described below.…”

Section: Shapley Value Approximationmentioning

confidence: 99%

Stratied Finite Empirical Bernstein Sampling

M¹,

Chapman²

2019

Preprint

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We derive a concentration inequality for the uncertainty in stratified random sampling. Minimising this inequality leads to an iterated online method for choosing samples from the strata. The inequality is versatile and considers a range of factors including: the data ranges, weights, sizes of the strata, as well as the number of samples taken, the estimated sample variances and whether strata are sampled with or without replacement. We evaluate the improvement this method reliably offers against other methods over sets of synthetic data, and also in approximating the Shapley value of cooperative games. The method is seen to be competitive with the performance of perfect Neyman sampling, even without prior information on strata variances. We supply a multidimensional extension of our inequality and discuss some future applications.MSC 2010 subject classifications: 94A20 91A12 60E15.

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Improving polynomial estimation of the Shapley value by stratified random sampling with optimum allocation

Cited by 67 publications

References 15 publications

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Hodge decomposition and the Shapley value of a cooperative game

Stratied Finite Empirical Bernstein Sampling

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Improving polynomial estimation of the Shapley value by stratified random sampling with optimum allocation

Cited by 67 publications

References 15 publications

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Improving the Accuracy of the Hyperspectral Model for Apple Canopy Water Content Prediction using the Equidistant Sampling Method

Hodge decomposition and the Shapley value of a cooperative game

Stratied Finite Empirical Bernstein&nbsp;Sampling

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Stratied Finite Empirical Bernstein Sampling