A classification rule for ordered exponential populations

The Bayesian classification rule used for the classification of the observations of the (second-order) stationary Gaussian random fields with different means and common factorised covariance matrices is investigated. The influence of the observed data augmentation to the Bayesian risk is examined for three different nonlinear widely applicable spatial correlation models. The explicit expression of the Bayesian risk for the classification of augmented data is derived. Numerical comparison of these models by the variability of Bayesian risk in case of the first-order neighbourhood scheme is performed.

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Quadratic Discriminant Analysis of Spatially Correlated Data

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2001.6.1.15212 ◽

2001 ◽

Vol 6 (2) ◽

pp. 15-28 ◽

Cited By ~ 2

Author(s):

K. Dučinskas ◽

J. Šaltytė

Keyword(s):

Discriminant Function ◽

Error Rate ◽

Gaussian Random Field ◽

Correlated Data ◽

Covariance Matrices ◽

Classification Rule ◽

Spatial Correlations ◽

Linear Quadratic ◽

Spatial Correlation Function ◽

Spatially Correlated

The problem of classification of the realisation of the stationary univariate Gaussian random field into one of two populations with different means and different factorised covariance matrices is considered. In such a case optimal classification rule in the sense of minimum probability of misclassification is associated with non-linear (quadratic) discriminant function. Unknown means and the covariance matrices of the feature vector components are estimated from spatially correlated training samples using the maximum likelihood approach and assuming spatial correlations to be known. Explicit formula of Bayes error rate and the first-order asymptotic expansion of the expected error rate associated with quadratic plug-in discriminant function are presented. A set of numerical calculations for the spherical spatial correlation function is performed and two different spatial sampling designs are compared.

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

Applied Sciences ◽

10.3390/app11062511 ◽

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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