Boundary Values of Discrete Monogenic Functions over Bounded Domains in $$ \mathbb{R}^3 $$

2019 ◽  
pp. 149-165
Author(s):  
Paula Cerejeiras ◽  
Uwe Kähler ◽  
Anastasiia Legatiuk ◽  
Dmitrii Legatiuk
Keyword(s):  
Bounded Domains ◽  
Monogenic Functions ◽  
Boundary Values

scholarly journals Sato’s Hyperfunctions and Boundary Values of Monogenic Functions

2014 ◽  
Vol 24 (4) ◽  
pp. 1131-1143
Author(s):  
Irene Sabadini ◽  
Franciscus Sommen ◽  
Daniele C. Struppa
Keyword(s):  
Monogenic Functions ◽  
Boundary Values

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

scholarly journals gbt-HIPS: Explaining the Classifications of Gradient Boosted Tree Ensembles

2021 ◽  
Vol 11 (6) ◽  
pp. 2511
Author(s):  
Julian Hatwell ◽  
Mohamed Medhat Gaber ◽  
R. Muhammad Atif Azad
Keyword(s):  
State Of The Art ◽  
Heuristic Method ◽  
Good Explanation ◽  
Classification Rule ◽  
Data Sets ◽  
Classification Models ◽  
Boundary Values ◽  
Class Label ◽  
Input Space ◽  
Boosted Tree

This research presents Gradient Boosted Tree High Importance Path Snippets (gbt-HIPS), a novel, heuristic method for explaining gradient boosted tree (GBT) classification models by extracting a single classification rule (CR) from the ensemble of decision trees that make up the GBT model. This CR contains the most statistically important boundary values of the input space as antecedent terms. The CR represents a hyper-rectangle of the input space inside which the GBT model is, very reliably, classifying all instances with the same class label as the explanandum instance. In a benchmark test using nine data sets and five competing state-of-the-art methods, gbt-HIPS offered the best trade-off between coverage (0.16–0.75) and precision (0.85–0.98). Unlike competing methods, gbt-HIPS is also demonstrably guarded against under- and over-fitting. A further distinguishing feature of our method is that, unlike much prior work, our explanations also provide counterfactual detail in accordance with widely accepted recommendations for what makes a good explanation.

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