# A Meta-heuristic Approach to Estimate and Explain Classifier Uncertainty

Andrew Houston  Department of Computer Science, Loughborough University,
Epinal Way, Loughborough, UK  Academic Department of Military Rehabilitation,
Defence Medical Rehabilitation Centre, Stanford Hall, Loughborough, UK
Georgina Cosma  Department of Computer Science, Loughborough University,
Epinal Way, Loughborough, UK

###### Abstract

Trust is a crucial factor affecting the adoption of machine learning (ML)
models. Qualitative studies have revealed that end-users, particularly in the
medical domain, need models that can express their uncertainty in decision-
making allowing users to know when to ignore the model’s recommendations.
However, existing approaches for quantifying decision-making uncertainty are
not model-agnostic, or they rely on complex statistical derivations that are
not easily understood by laypersons or end-users, making them less useful for
explaining the model’s decision-making process. This work proposes a set of
class-independent meta-heuristics that can characterize the complexity of an
instance in terms of factors are mutually relevant to both human and ML
decision-making. The measures are integrated into a meta-learning framework
that estimates the risk of misclassification. The proposed framework
outperformed predicted probabilities in identifying instances at risk of being
misclassified. The proposed measures and framework hold promise for improving
model development for more complex instances, as well as providing a new means
of model abstention and explanation.

###### keywords:

Explainable AI, Uncertainty Quantification, Meta-Learning, Fuzzy Clustering,
Complexity Theory

##  1 Introduction

In recent years, significant advancements have been made in the application of
machine learning (ML) to support clinical decision-making processes. The
medical domain has seen various applications of ML, including the prediction
of surgical outcomes [34, 53, 30] to aid in treatment planning, earlier
diagnoses of cancers [38, 47] to improve patient survival rates, and
prognostic tools for better management of neuro-degenerative conditions [18,
50]. Despite their potential benefits, the adoption of such tools remains a
challenge, with trust being cited as a primary barrier [5]. Complex, ‘black-
box’, algorithms that cannot explain when they may be in correct are often
cited a source of distrust [5]. As the use of AI continues to grow in fields
where incorrect decisions can have serious consequences, there is a growing
need to equip end-users with tools to facilitate an appropriate trust
relationship with ML and AI tools, termed trust calibration.

To address the demands for more interpretable models several approaches have
been proposed, including the use of interpretability and explainability
measures such as LIME [57] and SHAP [45], which offer post-hoc explanations
for model predictions. Another technique is the utilisation of inherently
interpretable models, such as decision trees [55]. However, as highlighted by
Kaur et al. [35], these methods may not take into account the contextual
factors that influence how end-users internalise information. Moreover, in
practice, even explanations of accurate model predictions have been shown to
be insufficient to overcome the internal biases of incorrect end-users [36,
12, 72].

The design of methods to facilitate trust calibration requires consideration
of the needs and perspectives of the end user. In the medical domain,
clinicians have emphasized the importance of models indicating uncertainty or
abstaining when their confidence is low [37, 68]. Global measures of model
performance, while useful for gauging overall reliability, are not enough to
foster trust and sustained use, as they lack insight into individual cases
[68, 56]. Therefore, it is crucial to develop methods that are interpretable
and applicable at the instance level. Amann et al. [3] emphasise that the
effectiveness of explanations depends on the end user’s ability to comprehend
their meaning. Thus, the interpretability of the methods themselves is a
crucial factor in facilitating trust calibration.

This paper presents a novel approach to calibrating trust in machine learning
(ML) models, incorporating the following three key contributions:

  * •

A suite of model-agnostic, interpretable meta-heuristics that aim to
characterise instances in terms of the sources of complexity in decision-
making.

  * •

A meta-learning framework, incorporating a synthetic data generator and
Bayesian-optimized, weighted fuzzy-clustering system to estimate the level of
uncertainty in the decision-making of an ML model, that outperforms predictive
probabilities for characterising misclassification risk.

  * •

Experiments evaluating how the proposed methods could enable ML models to
refrain from making predictions when uncertainty is high and how uncertainty
can be communicated using information derived from the meta-features.

The remainder of paper is structured as follows: Section 2 provides a
descriptive overview of sources of uncertainty in decision making and existing
approaches to characterising the complexity of instances and uncertainty of
decision made by ML models; Section 3 describes the proposed meta-heuristics,
outlines the design of the fuzzy clustering system for estimating uncertainty
and describes the design of a knowledge-base used to improve the performance
of the uncertainty estimation system; Section 4 details the methods for
analysing the relationships between the proposed meta-heuristics and
misclassification events, identifying the optimal method for knowledge-base
construction, and evaluating the performance of the uncertainty estimation
system; Section 5 provides the experimental results; Section 6 explores the
use of the proposed methods for abstention and uncertainty explanation;
Lastly, section 7 provides a discussion of future directions.

##  2 Related Works

This section provides a descriptive overview of the types of uncertainty in
decision making, data-centric sources of complexity in machine learning tasks
and existing methods to characterise such sources of complexity and estimate
the uncertainty of decisions made by ML models.

###  2.1 Types of Uncertainty

Uncertainty can be classed as one of two types, Epistemic or aleatoric.
Epistemic uncertainty can be described as a type of uncertainty originating
from the insufficiency of similar training data [31]. Epistemic uncertainty
can manifest in various forms, such as the absence of underrepresented groups
in facial recognition datasets, resulting in a decline in recognition
performance for these groups [7], or in the occurrence of rare circumstances
within a dataset. Aleatoric uncertainty reflects uncertainty arising from a
degree of randomness which cannot be explained away, such as the roll of a
dice, flip of a coin, noise in a signal or low resolution of an image [64].
In machine learning research, several factors have been identified for
increasing the epistemic and aleatoric uncertainty of classification problems,
which class imbalance, class overlap and outliers, and these are described
below.

#### Class Imbalance

Class imbalance is defined as an unequal distribution of instances between
classes in a dataset and is a common problem in many domains, spanning medical
predictions [30], sentiment analysis [23] and information retrieval [8]. Large
class imbalances can result in models that are highly accurate, but lack
sensitivity [44]. Common approaches for addressing class imbalance include re-
sampling techniques, such as SMOTE [9], that either increase the instances in
the minority class or reduce the majority class, and cost-sensitive learning,
which gives higher weight to errors made on specific classes [21]. However,
class imbalance may have limited impact on the classifier performance,
depending on other factors such as the well-defined class boundaries [71, 63].

#### Class Overlap

Class overlap, defined as the overlap in the feature space between instances
of multiple classes, is a well-recognised challenge in classification tasks
[71, 63]. The complexity of an instance in a region of class overlap is higher
compared to instances in regions dominated by a single class. This overlap can
introduce noise in a classification, increasing the aleatoric uncertainty of
decisions made on such instances. The relationship between class overlap and
class imbalance is often discussed in literature, with some studies suggesting
that the impact of class overlap is greater than class imbalance, while the
influence of class imbalance on the complexity of a classification task
increases in problems with high overlap [71, 63, 65]. In the context of
clinical decision-making, class overlap may present in two forms: a general
overlap of all features, making it difficult to distinguish between the
diagnoses or prognoses of patients, or in patients where different aspects of
their presentation align closely with different class outcomes. This is
depicted in Fig. 1, where instance D represents a patient with features $x1$
and $x2$ falling within the global overlap of the blue and red classes, and
instance B represents a patient where feature $x1$ aligns closely with the
blue class and feature $x2$ aligns with the red class.

#### Outliers

Outliers are instances that lie in a region of low neighborhood density and
can result in higher levels of epistemic uncertainty in predictions due to the
absence of similar instances for comparison. The impact of outliers on the
complexity of a prediction is dependent on various factors, such as the
location of the instance in the feature space and the classifier used. For
instance, in the hypothetical dataset shown in Fig.1, two instances may have
similar levels of outlierness but their location in the feature space may
result in differing levels of complexity. While instance A has high levels of
outlierness, its location in the feature space relative to other instances of
the same class may result in highly accurate predictions. On the other hand,
instance C, though having similar levels of outlierness, may not be predicted
accurately as its location in the feature space aligns closely with the
opposing class. The impact of outliers on decision-making complexity is
therefore specific to the classifier and the dataset [2]. In clinical
decision-making, the availability of past evidence and experience plays a
critical role in supporting predictions. Qualitative work found that outlying,
abstract patient presentations can increase uncertainty in decision-making
[33]. Furthermore, due to the core underpinning of clinical decision-making
being the availability of evidence and past experience [11, 73], in the case
of rare and complex patient presentations, much like the presence of outliers
within the context of ML, uncertainty in the decisions made increases [67].

Figure 1: A hypothetical 2-feature dataset where points A and C are instances
where a prediction could be considered to have a high degree of epistemic
uncertainty, point B reflects an instance where a prediction could be
considered to have both high epistemic and aleatoric uncertainty, and point D
is an instance where a prediction could be considered to have a high degree of
aleatoric uncertainty

###  2.2 Existing Approaches for Characterising Complexity and Uncertainty

#### Meta-heuristic Approaches

In 2011, Smith and Martinez [62] proposed a series of hardness measures to
identify instances that are likely to be misclassified in typical
classification problems and defined thresholds for their use as a means of
data cleaning during model development. A subsequent paper further explored
the application of the proposed measures and their relevance to the complexity
of instance-level decision-making [63] finding class overlap to be most
significant factor in characterizing complex instances. The paper presented
several approaches for integrating the proposed hardness measures into the
learning process, including the augmentation of error functions within multi-
layer perceptrons to reduce the weight assigned to more complex instances and
the removal of complex instances to reduce unwanted noise within the training
set. The advantage of the measures proposed by Smith et al. [63] is that they
allow for the accurate characterization of the complexity of individual
instances, enabling robust model development and improved reasoning as to why
misclassifications occur, as demonstrated in Houston et al. [30].
Additionally, the measures are easy to calculate, and the sources of
complexity they reflect are well-defined, making them easily understood by
end-users. However, post-deployment, most of the proposed measures are limited
in their utility because they rely on class knowledge, which is unavailable in
prospective cases.

In 2022, Barandas et al. [6] proposed a method for characterizing aleatoric
and epistemic uncertainty in traditional classification problems and applied
the methods to aid in abstaining from making a prediction. They applied three
measures to characterize the domains of uncertainty. Entropy was used to
measure aleatoric uncertainty, variation ratios were applied to evaluate
epistemic uncertainty resulting from the model, and a density estimation
technique was applied to measure epistemic uncertainty resulting from a lack
of data. The methods successful in utilizing uncertainty measures to improve
the performance of a series of classifiers using uncertainty-based rejection.
The authors augmented the way two models can be compared by looking at both
actual performance metrics, such as accuracy, and the uncertainty values
associated with the predictions. Additionally, the interpretable nature of the
proposed measurements shows promise in acting as a facilitator of trust to the
end user. However, a primary limiting factor for the application of such
methods is their dependence on the classifier and its parameters. The
difficulty with having model-dependent methods for characterizing instances is
their limitation for use within meta-learning solutions, such as developing
dynamic models and ensembles which, in some cases, relies on meta-information
about an instance [15, 40].

#### Model-Dependent Approaches

The most straightforward way to quantify uncertainty is to use the prediction
probability of the predicted class for the instance. Typically, prediction
probability refers to the conditional probability of the predicted class,
although it may differ depending on the model. For example, for SVM models,
Platt scaling is commonly used to compute the prediction probability [52].
Rechkemmer and Yin [56] investigated the impact of predicted probabilities on
end-users’ decision-making and found that higher predicted probabilities
increased users’ willingness to follow the model’s predictions and improved
their self-reported trust in the model. This simple method for quantifying
uncertainty has the advantage of being applicable to new instances post-
deployment. However, it lacks interpretability as it does not explain why a
model is less confident.

Active learning is an approach designed to reduce the computational cost and
time required to train a learning algorithm by identifying the most useful
instances to train a classifier. A popular method of active learning is
uncertainty sampling, introduced by Lewis 181 and Gale [42], which identifies
instances that a model is most uncertain about, learning the representation of
such challenging cases to create a decision boundary. However, while
uncertainty sampling is effective in identifying instances that a model is
most uncertain about, like prediction probabilities, it does not identify the
reasoning behind the uncertainty. Sharma and Bilgic [61] proposed an evidence-
based framework for explaining uncertainty, targeting two specific sources of
uncertainty which they termed ‘conflicting evidence uncertainty’ and
‘insufficient evidence uncertainty’. ‘Conflicting evidence uncertainty’ refers
to the presence of strong evidence for an instance belonging to more than one
class, whereas ‘insufficient evidence uncertainty’ refers to a lack of
evidence for an instance belonging to any class. Their evaluations on real-
world datasets found that ‘conflicting evidence uncertainty’ appeared to be a
more effective means of active learning, outperforming traditional active
learning and ‘insufficient evidence uncertainty’. The benefit of using such
measures, unlike the hardness measures proposed by Smith et al. [63], is their
lack of reliance on class knowledge. However, like the methods proposed in
Barandas et al. [6], a downside to uncertainty measures derived through active
learning is their lack of independence from a classifier. Additionally, active
learning approaches typically fail to capture concepts such as noise and
outlierness [58].

In 2021, Northcutt et al. [48] proposed the confident learning (CL) algorithm
to identify uncertainty in dataset labels and to detect instances that are
incorrectly labelled. The CL algorithm estimates the joint distribution of
noisy labels and latent labels by utilizing the out-of-sample predicted
probabilities and noisy labels. This is done to characterize the class-
conditional label noise and identify instances with labelling issues. These
instances are then pruned, and the model is trained with re-weighting
instances using the estimated latent priors. Applying CL has the major benefit
of making a model more robust against epistemic error. Recently, Abad and Lee
[1] utilized the CL algorithm to identify instances for which classifications
were uncertain in a study focusing on mortality prediction. The study trained
six ML models on a dataset, and then the CL algorithm was used to identify
instances where class labels were uncertain. An XGBoost model was then
retrained, predicting three classes, including a “challenging” patient class.
Results were mixed, with the model achieving only a 31% precision and 14%
recall for the “challenging” instances. However, after the uncertain patients
were separately labelled, the XGBoost model’s performance for identifying
mortality patients increased from 87% to 96%. The work of Abad and Lee [1]
demonstrates how the application of uncertainty identification can be used to
improve classification performance and notify end-users of uncertain
predictions. Despite the somewhat positive results, the application of the CL
algorithm to identify uncertainty may be challenging to explain to end-users,
similar to the use of conditional probabilities. This limits its applicability
for explainability purposes, leaving the reasoning behind the uncertainty
relatively unknown.

#### Summary

To summarise, current methods for characterizing instance complexity and
quantifying decision-making uncertainty have limitations, including post-
deployment applicability, failure to consider multiple sources of complexity,
and a lack of suitability for end-users to understand why a decision is
uncertain or an instance is complex. Therefore, the proposed methods for
characterising uncertainty take a user-focused approach to characterise
complexity and estimate uncertainty, developing heuristics which reflect the
factors that impact a human-made decision.

###  2.3 Characterising Class Diversity

The hardness measures of Smith et al. [63] rely on class knowledge, often
looking at class disagreement with the new instance, which limits their use
post-deployment. Therefore, a class-independent alternative to disagreement is
required. This study proposes the use of a diversity measure, observing how
diverse the classes of the returned instances are as a measure of complexity.
The calculation of diversity can be thought of in a similar way to calculating
class imbalance.

Several methods have been proposed to quantify class imbalance in binary and
multi-class problems, the most informative measures being empirical
distributions and frequencies of classes within a dataset. However, these
measures can be difficult to examine in highly multi-class problems and are
not single-value representations. One proposed solution, suitable for multi-
class problems is the imbalance ratio, proposed by Prati et al. [54],
providing a single-value summary of class imbalance. The imbalance ratio
measures the number of instances in the most probable class for every instance
in the least probable class. However, despite being effective in binary
problems, the ratio is incapable of describing the class disparity in multi-
class solutions due to its disregard for all classes other than the most and
least probable. In recognising this flaw, Ortigosa-Hern’ndez et al. [49]
proposed the imbalance degree, a measure capable of characterising skewed
class distributions in multi-class problems, using a similarity/distance
function. The limitation of their approach was that the measure was sensitive
to the choice of distance metric, which was prone to change across different
classification problems. Acknowledging this, Zhu et al. [74] proposed a multi-
class imbalanced degree, based on the likelihood-ratio test (LRID). Empirical
evaluations on real and synthetic data sets showed their approach to be
superior to the imbalance ratio and imbalance degree, with the additional
benefit of not having to identify appropriate parameters for each dataset.
Considering this study aims to propose generalisable methods, capable of
identify complex instances across a range of classification tasks, the
likelihood ratio imbalance degree will be applied to reflect the diversity
within two of our proposed meta-heuristics, characterising diversity as:

$Diversity=-2\sum_{c=1}^{C}m_{c}ln\frac{b_{c}}{\hat{p}_{c}}$ (1)

where $C$ is the number of classes in the dataset, $m$ is the number of
instances in each respective class, $ln$ is the natural logarithm, $b$
represents a balanced class distribution and $\hat{p}$ represents the
estimated class distribution. $\hat{p}$ is estimated using:

$\hat{p}=\frac{m_{c}}{M}$ (2)

where $M$ is the total number of instances.

The diversity score is normalised to the worst possible imbalance within a
dataset, for example: if a sample consists of 300 records and 3 classes, then
the worst possible imbalance would be if a single class contained all 300
records. Contextualising this within the problem of estimating diversity, a
normalised LRID of 1, would reflect zero diversity being present within our
sample, as only one class is present.

##  3 Proposed Meta-heuristics and Framework for Uncertainty Estimation

This section presents a series of meta-heuristics used to characterise the
uncertainty of a model’s prediction on a given instance and outlines the
proposed framework for estimating the uncertainty of a prediction made by an
ML model.

###  3.1 Meta-Heuristics for Characterising Instance Complexity

Seven meta-heuristics are proposed each requiring a dataset, $X$, containing
$M$ instances and $N$ features, and an instance for which to characterise the
complexity of, $x$.

####  $k$-Diverse Neighbours

$k$-Diverse Neighbours ($k$DN) reflects the local overlap of an instance
within the task space, relative to its nearest neighbours from the training
set and is calculated according to Eq. 3, as:

$KDN(x)=diversity(KNN(x))$ (3)

where the function $KNN(x)$ returns the class labels of nearest $k$ instances
in $X$ to $x$.

### Disjunct Size

Disjunct size is a class-independent measure proposed by Smith et al. [63] and
calculates the size of the disjunct an instance is classified into by an
unpruned decision tree, formed from a training set. The disjunct size of the
returned instance is then normalised by dividing it by the size of the largest
disjunct within the dataset. Disjunct size is calculated according to Eq. 4,
as:

$DS(x)=\frac{|distjunct(x)|-1}{\max_{d\in D}|disjunct(d)|-1}$ (4)

where $D$ is the set of all disjuncts in $X$, and $|distjunct(x)|-1$ returns
size of the disjunct for instance $x$.

#### Disjunct Class Diversity

Disjunct class diversity (DCD) reflects the diversity of the class of
instances within the disjunct which an instance is classified into, ie those
that are similar based on a subset of their features. Contrary to the methods
of DS, the decision tree applied when calculating DCD is pruned. DCD is
calculated according to Eq. 5, as:

$DCD(x)=diversity(Disjunt(x))$ (5)

where the function $Disjunt(x)$ returns the classes of the instances contained
from dataset, $X$, in the same disjunct as instance, $x$.

#### Outlierness

Outlierness (OL), reflects the degree to which an instance is similar to the
instances contained within the training set. The outlierness of an instance is
calculated using the density-based approach proposed by Tang and He [66],
where the metric captures the ratio of the average neighbourhood density, to
the density of instance $x$, using Eq. 6, as:

$OL(x)=\frac{\sum_{x\in S(x)}p(x)}{|S(x)|p(x)}$ (6)

where $S(x)$ is a set of instances formed of the $k$-nearest neighbours,
$S_{KNN}(x)$, reverse nearest neighbours, $S_{RNN}(x)$, and shared nearest
neighbours, $S_{SNN}(x)$, to instance $x$, termed a neighbourhood. The
function $p(x)$ returns the density of the location of instance $x$, where
density is calculated using Eq. 7, as:

$p(x)=\frac{1}{|S(x)|+1}\sum_{X\in
S(x)\cup\\{x\\}}\frac{1}{h^{N}}K(\frac{X-x}{h})$ (7)

where $K\frac{X-x}{h}$ is a Gaussian kernel function with the kernel width of
$h$ and neighbourhood size of $k$, $N$ is the number of features in $X$ and
$|S(x)|$ is the size of the neighbourhood for instance, $x$.

#### Class-Level Outlierness

Class-Level Outlierness (CL-OL), reflects the disparity in the level
outlierness calculated for an instance, across each class within the dataset.
Given an instance, an outlierness score is calculated for each class in the
dataset. Thereafter, disparity is calculated using a variation of the
diversity Eq. 1 where $m$, originally reflecting the number of instances in a
class, now reflects the percentage of the summed outlierness scores, across
all classes,$M$.

#### Evidence Conflict

Evidence conflict (EC) reflects degree to which an instance’s features fall
within the distribution of a multiple classes. To calculate EC, first, an
$M\times N$ matrix is formed, where $M$ refers to the number of classes within
the dataset and $N$ refers to the number of features. The matrix will
henceforth be referred to as the conflict matrix. To populate the conflict
matrix, the conflicting evidence degree is calculated for each feature and
class-comparison. The conflicting evidence degree (CED) is calculated using
Eq. 8, as:

$CED(x^{n})=\begin{cases}1-1/(f(x_{n},X_{n}^{c})/f(x_{n},X_{n}^{r}))&f(x_{n},X_{n}^{c})>f(x_{n},X_{n}^{r})\\\
-1+1/(f(x_{n},X_{n}^{r})/f(x_{n},X_{n}^{c}))&f(x_{n},X_{n}^{c})>f(x_{n},X_{n}^{r})\end{cases}$
(8)

where the function $f(x_{n},X_{n}^{c})$ returns the probability density of the
class $c$ for feature $n$ at the value of $x_{n}$ and the function
$f(x_{n},X_{n}^{r})$ returns the probability density of a different class in
the dataset, $r$, for feature $n$ at the same value. When calculating EC,
class $c$ remains constant and is the class for which the instance has been
predicted by the classifier. Where a feature is categorical, the function is
calculated as follows is calculated using Eq. 9, as:

$f(x_{n})=\frac{|z\in X_{n}^{c}\wedge X_{n}^{c}=x_{n}|}{|X^{c}|}$ (9)

where z refers to the total instances in the dataset, $X$, of class $c$, and
feature $n$, where the category of the instance is the same as the category of
instance $x$.

Given that difference features carry different levels of importance to the
classifier, with some being more useful than others, the conflict matrix is
weighted according to the Fishers Discriminant Ratio value for the features,
normalised to highest ratio among the features. The total conflict score is
calculated as the sum of the conflict across all features for each class
comparison. The maximum conflict score across all class comparisons is
returned.

###  3.2 Proposed Uncertainty Estimation Framework

The proposed framework for estimating uncertainty is shown in Fig. 2.

Figure 2: A graphical illustration of the framework for forming the knowledge
base from which is used to train uncertainty estimation system.

#### Knowledge Base Formation

To train the fuzzy clustering system, a knowledge base is formed from the
meta-data of real instances from the training set and other classification
tasks. However, given that datasets from public repositories have drawn
criticism for not being representative of the classification problems that may
exist in the real world [46] and that their homogeneity, from a complexity
standpoint, can result in a biasing of algorithm development, the framework
embeds a synthetic data generator, termed Sy:Boid [29].

The process for generation is described fully in Houston and Cosma [29], but
briefly, the synthetic data generator takes inputs relating to the number of
instances, number of features, number of classes, and the desired complexity,
in this paper the F1 and N1 measures of Ho 338 and Basu [27] are used. A
series of random points are generated and labelled, and then, using a modified
boid algorithm points are moved about the feature space and the complexity of
the dataset is measured. The algorithm is embedded within a genetic algorithm
which optimises the rules weightings to generate a dataset that closest meets
the desired complexity.

To generate synthetic datasets to complement the real data within the
uncertainty system, Sy:Boid was tasked with generating datasets for all
combinations of F1 and N1, where both F1 and N1 are values between 0 and 1, N1
is not larger than F1 as this rarely occurs in real-data [29], and both F1 and
N1 increase in increments of 0.2. The dataset generation was repeated for each
dataset in table 1, mimicking the number of instances, dimensionality and the
number of classes. After all datasets were generated, 18020 instances were
randomly selected, equaling the number of real instances.

The meta-heuristics for each instance in the training set, public datasets and
synthetic dataset are calculated, using 5-fold cross-validation, where the
meta-heuristics of the validation set are stored in the knowledge base. A
single prediction is also provided for each instance by the classifier of
interest, trained using a nested 5-fold cross-validation where the inner loop
is used to determine the optimal hyper-parameters and the outer-loop is used
to predict the class of the validation set. The outer loop of the cross-
validation for evaluating the performance of the classifier is seeded to
ensure the folds are identical to those used to calculate the meta-heuristics.
Misclassification events are identified using Eq. 10, as:

$f(x)=\begin{cases}1&y_{pred}\neq y_{true}\\\ 0&y_{pred}=y_{true}\end{cases}$
(10)

where $y_{pred}$ is the predicted class of an instance, and $y_{true}$ is the
actual class of an instance. The misclassification events are then stored in
the knowledge base with their respective meta-heuristics.

#### Uncertainty Estimation

To examine the utility of the proposed measures in prospectively
characterising the complexity of a patient, a weighted fuzzy c-means
clustering approach is applied. Within the clustering system, the
misclassification rate is calculated for each cluster using all instances
contained within each respective cluster. Misclassification rates were
calculated using Eq. 11.

$Misclassification\leavevmode\nobreak\ Rate=1-\frac{TP+TN}{{TP+FP+TN+FN}}$
(11)

where $TP$ refers to true positive prediction, $FN$ refers to a false negative
prediction, $FP$ refers to false positive prediction and $TN$ refers to a true
negative prediction.

When a new patient’s data is input into the system their fuzzy membership is
calculated, returning the membership of that patient to each cluster. To
obtain a single value output, the output is defuzzified using the weighted
average method, by:

$x^{*}=\frac{\sum[\mu_{x}(\bar{x})\times\bar{x}]}{\sum\mu_{x}(\bar{x})}$ (12)

where $\mu_{x}$ is the multiplication of each weighting function of instance
$x$ and $\bar{x}$ is the misclassification rate associated with each
membership value.

To train and tune the clustering system, a 5-fold nested cross-validation
approach is applied. The meta-heuristics proposed in subsection 3.1 were shown
to vary in their strengths of association with misclassification performance
(Fig. 4). Therefore, to improve the performance of the clustering system above
that of one where each heuristic is weighted equally, Bayesian optimisation is
applied within the inner-loop to determine the optimal weightings of the
heuristics and number of clusters to include in the model, in order to
maximise the product of the odds ratio, area under the receiver operating
characteristic curve (AUROC) and area under the precision-recall curve
(AUPRC), where the input is the defuzzified output and the target is the
binary misclassification event. The outer-loop is used to evaluate the quality
of the uncertainty estimates for identifying misclassification. The complete
process for developing and training the predictive model and fuzzy clustering
system is detailed in Algorithm 1.

1 Inputs: $X$ Dataset features, $Y$ Dataset Targets, $KB$ Knowledge Base

2 Divide $X$ and $Y$ into $K$ stratified folds

3 for  _$k_{i}$ in $K$ folds_ do

4 Let $X_{k_{i}}$ and $Y_{k_{i}}$ be the test set features and targets,
respectively

5

6 # Generate classification model, model predictions and meta-heuristics for
the training and testing set

7 for  _$k_{j}$ in $K-1$ folds_ do

8 Let $X_{k_{j}}$ and $Y_{k_{j}}$ be the validation set features and targets,
respectively

9 Train and tune model on remaining $K-2$ folds using Bayesian cross-
validation, selecting parameters which maximise the balanced accuracy score.

10 Retrain optimal model on all $K-2$ folds.

11 Let $y_{pred}$ be the predicted class of the trained model applied to
$X_{k_{j}}$

12 Let $M_{k_{j}}=y_{pred}!=Y_{k_{j}}$

13 Let $C_{k_{j}}$ be the calculate meta-heuristics for $X_{k_{j}}$ with the
remaining $K-2$ folds acting as the training set.

14 end for

15 Retrain optimal model on all $K-1$ folds Let $y_{pred}$ be the predicted
class of the trained model applied to $X_{k_{i}}$

16 Let $M_{k_{i}}=y_{pred}!=Y_{k_{i}}$

17 Let $C_{k_{i}}$ be the calculate meta-heuristics for $X_{k_{i}}$ with the
remaining $K-1$ folds acting as the training set.

18 # Generate clustering system

19 for  _$k_{j}$ in $K-1$ folds_ do

20 Let $C_{k_{j}}$ and $M_{k_{j}}$ be the validation complexity heuristics and
misclassification targets, respectively

21 Train and tune clustering system on remaining $K-2$ folds and $KB$ using
Bayesian cross-validation selecting the parameter weights and number of
clusters which maximise the fitness function on the validation set

22

23 end for

24 Let $C_{k_{i}}$ = $C_{k_{i}}\times$ optimal weights

25 Retrain optimal clustering system on all $K-1$ folds and $KB$

26 Let $U_{k_{i}}$ be the estimated uncertainty of the trained clustering
systems defuzzified output for all instances in $C_{k_{i}}$.

27

28 end for

29# Evaluate Performance

Evaluate performance over all $K$ folds using $U$ and $M$

Algorithm 1 Pseudo-code of the method for developing and training the
predictive model and fuzzy clustering system.

##  4 Experimental Methods

This section describes the experimental methods for determining the optimal
design of the uncertainty estimation system and evaluating its performance on
unseen data.

###  4.1 Dataset Description

Twenty-seven publicly available datasets were used in experiments to evaluate
the statistical relationships between the proposed meta-features and
misclassification events, and to determine the optimal design of the knowledge
base. A description of the datasets and their source is provided in Table 1.
The datasets vary in their number of instances and dimensionality (i.e. the
number of features). The selected datasets include continuous, ordinal and
categorical features, to ensure the proposed meta-heuristics and developed
system for estimating uncertainty are evaluated across a range of datasets.

Dataset Source Size Feature Types Instances Dims. Classes Nominal Ordinal
Continuous Acute Inflammations (Inflammation) [17] 120 6 2 Y N Y Acute
Inflammations (Nephritis) [17] 120 6 2 Y N Y Breast Cancer-C [51] 116 9 2 N N
Y Breast Cancer-W [20] 699 9 2 N N Y Breast Tissue [20] 106 9 6 N N Y CECS
[30] 126 23 2 Y Y Y C-Section [20] 80 5 2 Y N Y Chronic Kidney Disease [20]
351 24 2 Y Y Y Diabetic Retinopathy [4] 1151 18 2 Y N Y Early Stage Diabetes
[32] 520 16 2 Y N Y Echocardiogram [20] 126 8 2 Y N Y Fertility [24] 100 9 2 Y
Y Y HCV [20] 615 12 2 Y N Y Heart Disease-C [20] 303 13 2 Y Y Y Heart Failure
[10] 299 12 2 Y N Y Hepatitis [20] 154 19 2 Y N Y Hypothyroid [20] 3163 19 2 Y
N Y Indian Liver Disease [20] 583 10 2 Y N Y LSVT [69] 127 312 2 Y N Y
Lymphography [20] 142 36 2 Y N Y Maternal Health Risk [20] 1014 6 3 N N Y
Myocardial Infarct (Heart Failure) [25] 1700 111 2 Y Y Y Myocardial Infarct
(Death) [25] 1700 111 2 Y Y Y Myocardial Infarct (Infarction) [25] 1700 111 2
Y Y Y NKI Breast [70] 272 12 2 Y Y Y Parkinsons Speech [59] 756 753 2 Y N Y
Pancreatic Cancer [19] 590 8 3 Y N Y Parkinsons [43] 195 22 2 N N Y Thoracic
Surgery [75] 470 17 2 Y Y Y Vertebral Column (2-Class) [20] 311 6 2 N N Y
Vertebral Column (3-Class) [20] 311 6 3 N N Y

Table 1: Description of the datasets included in this paper.

####  4.1.1 Statistical Evaluation of the Proposed Meta-Heuristics

The intention of each meta-heuristic is to characterise the complexity of an
instance, with the sources of complexity being independent from each other.
Therefore, to assess the independence of each meta-heuristic, a correlation
analysis was performed using Spearman’s Rank Correlation. To assess the
association of each meta-heuristic with misclassifications, independent of
each other, univariate binary logistic regressions were applied for each
model, with misclassifications (binary yes/no) acting as the dependent
variable and each meta-heuristic acting as the independent variable. Prior to
applying the logistic regression, each meta-heuristic underwent z-score
transformation to aid in the of interpretation of the odds-ratios.

###  4.2 Statistical Evaluation of the Knowledge Base Construction

To generate the optimal knowledge base, three things must be known: 1) how
many instances should be included, 2) how diverse from the current instance
should the selected instances be, 3) should the knowledge base be comprised of
real instances, synthetic instances or a combination of both. To identify the
optimal parameters, for each patient, 48 knowledge bases were generated
comprised of $m$ instances randomly selected from the nearest $q$% of
instances to the current instance, where $m$ = 100, 500, 1000 or 2000, and $q$
= 10%, 25%, 50% and 100%, with these instances selected from a meta-database
comprised of all real instances, all synthetic instances and a combination of
real and fake instances. The fuzzy clustering system was trained, according to
Algorithm 1, on each of the knowledge bases each time generating 10
defuzzified outputs for each instance (one for each classifier). For each
model, an odds ratio, AUROC and AUPRC were calculated for the outputs from the
fuzzy clustering system, trained on each of the 48 knowledge bases.

To assess what parameters resulted in the defuzzified outputs being best
suited for identifying misclassification, ANOVAs were applied with the
calculated odds-ratio, AUROC, AUPRC acting as the dependent variables and the
number of instances, diversity of the selected instances and realness of the
knowledge base acting as the independent variables. Additionally, the
interaction effect was calculated between the number of instances and the
diversity of the selected instances. Where significant main effects were
found, post-hoc analyses were performed for all two-variable combinations,
apply Bonferroni’s correction to account for multiple comparisons. For all
statistical tests, the null-hypothesis was rejected when $\alpha<0.05$.

###  4.3 External Validation of the Proposed Methods

To ensure the performance of the proposed methods are evaluated on unseen
data, not used to identify the optimal knowledge base, the dataset described
in Hou et al. [28], comprised of 4559 patients with sepsis-3 from the MIMIC-
III database was replicated. The full process for extracting and pre-
processing the patient data is detailed in Hou et al. [28]. The uncertainty
system was formed and evaluated according to Algorithm 1, with performance
measured in terms of the odds ratio between estimated uncertainty and actual
misclassification events, AUROC and AUPRC.

Given that the ability to identify misclassifications without class knowledge
is limited to instances where there is at least some information that allows
them to be classified correctly. The evaluation of the uncertainty estimations
is repeated twice. First including all patients, then again, removing
instances where their presentation and class target means they should be
misclassified. To determine instances which should be misclassified (ISMs),
the methods of Smith and Martinez [62] are applied. First, each instance is
characterised in terms of DS, Disjunct Class Percentage (DCP) and Class-
Likelihood Difference (CLD), where Disjunct size is calculated using Eq. 4,
and DCP and CLP are calculated using Eqs. 13 and 14, as:

$DCP(x)=\frac{|\\{z:z\in disjunct(x)\wedge t(z)=t(x)\\}|}{|disjunct(x)|}$ (13)
$CLD(x)=CL(x,t(x))-\underset{y\in Y-t(x)}{argmaxCL(x,y)}$ (14)

where the function $disjunct(x)$ returns the disjunct that covers instance
$x$, the function $t(x)$ returns the class label of instance $x$, $z$ refers
to instances contained in the returned disjunct, $Y$ is a set containing each
unique class label in the dataset and the function $CL(x)$ returns the
probability of an instance belonging to a certain class, and is derived using
Eq. (15):

$CL(x)=CL(x,t(x))=\prod_{n}^{N}P(x_{n}|t(x))$ (15)

where $x_{n}$ is the $n$th feature of instance $x$. Thereafter, should an
instance meet the following condition $CLD(x,t(x))<0\leavevmode\nobreak\
$and$\leavevmode\nobreak\ ((DS(x)==0\leavevmode\nobreak\
$and$\leavevmode\nobreak\ DCP(x)<0.5)\leavevmode\nobreak\ $or$DN(x)>0.8)$, the
are considered an ISM.

Due to the increased training time required to calculate the meta-heuristics
and train the fuzzy-clustering system, the performance of the estimated
uncertainty output by the fuzzy clustering system is compared against the
absolute difference of the classification probability for the predicted class
from 0.5. For all models, except SVM, classification probability is calculated
as the conditional probability of the predicted class. For SVM classification
probability is calculated using Platt scaling [52].

##  5 Experimental Results

###  5.1 Statistical Analysis of the Proposed Meta-Heuristics

#### Investigation of Co-Linearity between Meta-Heuristics

Results of the correlation analysis are presented in Fig. 3. Results showed
minimal co-linearity between meta-heuristics, meaning the source of
uncertainty that each meta-heuristic measures are independent of one and
other. The only pair of meta-heuristics which demonstrated co-linearity was DS
and DCD ($\rho=$ -0.60). However, due to both DS and DCD being disjunct-based
measures, it is not unsurprising that a degree of overlap exists within the
domains of uncertainty they capture.

Figure 3: Spearman’s rank correlation matrix showing the relationship between
each meta-heuristic. A negative value, highlighted in blue, represented a
negative correlation and a positive value, highlighted in red, represents a
positive correlation.

#### Investigation of the Independent Associations between Meta-Heuristics and
Misclassification Events

Results of the association-based analyses are presented in Fig. 4. Results
indicate significant associations across all classifiers for almost all
heuristics. A summary of the figure is provided below:

  * •

KDN: An increase in KDN was observed to significantly increase the odds of a
misclassification occurring across all classifiers (p $<$ 0.001), with odds
ratios ranging from 1.780 (95% CI = 1.721 - 1.84) for SVM to 2.993 (95% CI =
2.846 - 1.84) for KNN, meaning that the more diverse the nearby instances are,
in terms of their class labels, the greater chance of a misclassification
occurring.

  * •

DS: An increase in DS was shown to significantly decrease the likelihood of a
misclassification occurring across all classifiers (p $<$ 0.001), with odds
ratios ranging from 0.389 (95% CI = 0.371 - 0.407) for logistic regression, to
0.591 (95% CI = 0.371 - 0.407) for SVM, meaning that the more complex the
decision boundary for a given instance, relative to that of other instances,
the greater the chance of misclassification occurring.

  * •

DCD: An increase in DCD was observed to significantly increase the odds of a
misclassification occurring across all classifiers (p $<$ 0.001), with odds
ratios ranging from 1.808 (95% CI = 1.742 - 1.877) for QDA to 2.366 (95% CI =
2.254 - 2.483) for XGBoost, meaning that the more diverse the nearby instances
are, in terms of their class labels, based on a subset of the available
features, the greater chance of a misclassification occurring.

  * •

OL: An increase in OL affected the likelihood of misclassification occurring
differently for each classifier. For logistic regression, an increase in OL
significantly increased the odds of a misclassification occurring (OR = 1.127,
95% CI = 1.089 - 1.167), that is to say, the more outlying an instance from
the instances in the training set, the greater chance of a misclassification.
Whereas, for Adaboost, QDA and MLP, the opposite is true, with an increase in
OL significantly decreasing the chance of misclassification occurring.
Increases in OL was shown to not significantly affect the likelihood of a
misclassification occurring for SVM, KNN, Random Forest, LDA or XGBoost (p $>$
0.05).

  * •

CL-OL: An increase in CL-OL was found to significantly increase the likelihood
of a misclassification occurring across all classifiers (p $<$ 0.05), with
odds ratios ranging from 1.017 (95% 1.028 - 1.121) for Adaboost to 1.402 (95%
CI 1.347 - 1.458) for SVM, meaning that the more equal an instance’s
outlierness is to all available classes, the greater the chance of a
misclassification occurring.

  * •

HD: An increase in HD was shown to significantly decrease the likelihood of a
misclassification occurring across all classifiers (p $<$ 0.001), with odds
ratios ranging from 0.638 (95% CI = 0.606 - 0.669) for Random Forest, to 0.746
(95% CI = 0.720 - 0.773) for SVM, meaning that the closer to the hyperplane an
instance falls, the greater the chance of a misclassification occurring.

  * •

EC: An increase in EC was observed to significantly increase the odds of a
misclassification occurring across all classifiers (p $<$ 0.001), with odds
ratios ranging from 1.599 (95% CI = 1.541 - 1.660) for QDA to 1.829 (95% CI
1.759 - 1.902) for logistic regression, meaning that the degree to which the
features of an instance point to two potential classes, the greater the chance
of a misclassification occurring.

Figure 4: Odds ratios and respective 95% confidence intervals demonstrating
the independent association of each meta-heuristic a misclassification.
Diamonds indicate the odds ratio, with an odds ratio greater than one
reflecting a positive association with a misclassification.

#### Summary

Results of the statistical analysis demonstrate that each of the proposed
meta-heuristics characterises instances in terms of different aspects of
uncertainty, independent of each other. However, as demonstrated by the
association-based analysis, some meta-heuristics are more strongly associated
with misclassifications than others and therefore, should not be weighted
equally when characterising the overall uncertainty of a classification.

###  5.2 Identification of the Optimal Knowledge Base

#### Effect of Size

Results of the between-group analysis found a significant difference in the
odd ratios obtained from knowledge bases of different sizes (p $<$ 0.001),
with the odds ratios obtained using knowledge bases comprised of only 100
instances (2.29 $\pm$ 0.36) being significantly lower (p = 0.001) than those
obtained by knowledge bases containing 500 (2.70 $\pm$ 0.41), 1000 (2.85 $\pm$
0.47) and 2000 instances (2.93 $\pm$ 0.50). Odds ratios obtained using a
knowledge base comprised of 500 instances were significantly lower than those
obtained by knowledge bases comprised of 1000 and 2000 instances (p = 0.045
and 0.001, respectively). No significant difference was observed in the odds
ratios obtained from knowledge bases comprised of 1000 and 2000 instances (p =
0.502). This observation was mirrored in the results of the analysis of both
the AUROC and AUPRC (p $<$ 0.001 and p = 0.027, respectively), with AUROCs
being significantly lower when only 100 instances were used (0.73 $\pm$ 0.03)
compared to when 500 (0.75 $\pm$ 0.03), 1000 (0.75 $\pm$ 0.03) and 2000 (0.75
$\pm$ 0.03) were used (p = 0.001) and AUPRC being significantly worse when the
knowledge base contained only 100 instances compared to 2000 (0.33 $\pm$ 0.06
vs. 0.35 $\pm$ 0.06, p = 0.033).

#### Effect of Diversity

In terms of the diversity of the knowledge base, the between-group analysis
found significant differences between the four groups for the analysis of odd
ratios (p $<$ 0.001). However, post-hoc analysis reveals the only significant
difference (p = 0.001) in odds ratio was that knowledge bases formed using the
nearest 50% of instances to the current instance (2.43 $\pm$ 0.37), were
significantly worse than those formed by the nearest 10% (2.82 $\pm$ 0.40),
25% (2.77 $\pm$ 0.40) and 100% (2.76 $\pm$ 0.68). Similarly, there was a
significant difference in the AUROCs achieved by different thresholds (p $<$
0.001). The post-hoc analysis revealed that the smaller thresholds of 10% and
25% resulted in significantly higher (p $<$ 0.01) AUROCs (0.75 $\pm$ 0.03)
than the larger thresholds of 50% and 100% (0.74 $\pm$ 0.03 and 0.73 $\pm$
0.04). No significant effect was observed in terms of AUPRC (p = 0.095)

#### Interaction Between Size and Diversity

An interaction effect was observed between the number of instances included in
the knowledge base and the diversity of the knowledge base for both odd-ratios
and AUROCs (p $<$ 0.001). As shown in Fig. 5, when the number of instances is
small, the performance of highly diverse knowledge bases suffers, however as
the number of instances increases, performance improves. In the case of
knowledge bases comprised of less diverse instances, performance plateaus as
more instances are introduced, which is likely the result of a saturated meta-
feature space. No interaction was observed in terms of AUPRC (p = 0.670).

Figure 5: The interaction for (A) odds-ratio and (B) AUROC, between the number
of instances included in the knowledge base and the diversity of the knowledge
base, where each line reflects the different diversity groups.

#### Effect of Realness

Concerning the realness of the knowledge base, between-group differences were
observed in terms of odds ratios (p = 0.003). Post-hoc analyses revealed that
knowledge bases comprised solely of fake instances led to significantly higher
(p = 0.036) odds ratios (2.750 $\pm$ 0.50) than those of solely real instances
(2.611 $\pm$ 0.51). No significant differences were observed between the
knowledge bases comprised of both real and fake instances (2.719 $\pm$ 0.49)
and the other two groups (p $>$ 0.05). The realness of the knowledge base did
not affect the performance in terms of either AUROC or AUPRC (p = 0.210 and
0.242, respectively).

#### Summary

Due to the interaction effect between the threshold and the number of
instances, all experiments will be carried out using a threshold of 100% and
selecting 2000 instances to train the fuzzy clustering system. Additionally,
due to the desire to retain some reality within the knowledge base, and
because of negligible performances differences between the knowledge bases
comprised of fake and a combination of real and fake instances, the knowledge
base will be formed of both real and synthetic instances.

###  5.3 External Validation of the Proposed Uncertainty Estimation System

Table 2 shows the performance of the proposed method, with and without the
inclusion of ISMs, compared against the use of classifier probabilities.
Results show that in terms of odds ratios, the proposed method proved superior
for 8 of the 10 models when ISMs were included and 5 of the 10 models when
classifiers were excluded, demonstrating that the estimated uncertainty is
equally associated with the classifier probabilities for instances which
should not be misclassified. In terms of AUROC, both with and without ISMs
included the proposed method was superior in 9 out of 10 models, with the only
model for which the classifier probabilities proved superior being SVM,
demonstrating that the uncertainty estimates achieved through the proposed
method achieve greater discriminative performance between positive and
negative examples, across the majority of models than classifier uncertainty.
Lastly, in terms of AUPRC, the proposed method proved superior in 8 out of 10
models when ISMs were included and 6 out of 10 models when ISMs were removed,
demonstrating the proposed methods are better at identifying
misclassifications than classifier probability across more models both with
and without the presence of ISMs. Generally, the proposed methods proved more
robust in providing estimations of uncertainty in the presence of ISMs,
however, the performance of both methods was improved by their removal.

| Proposed Method | Classifier Probabilites
---|---|---
Model | Odds Ratio (95% CI) | AUROC |  | AUPRC
---
(Improvement from Random)
Odds Ratio (95% CI) | AUROC |  | AUPRC
---
(Improvement from Random)
Including ISMs
LR | 3.263 (2.993 - 3.558) | 0.79 | 0.52 (0.27) | 2.423 (2.231 - 2.633) | 0.72 | 0.42 (0.18)
SVM | 1.459 (1.372 - 1.552) | 0.61 | 0.45 (0.08) | 2.279 (2.117 - 2.454) | 0.71 | 0.56 (0.19)
KNN | 2.814 (2.588 - 3.06) | 0.76 | 0.5 (0.23) | 2.1 (1.955 - 2.255) | 0.70 | 0.43 (0.16)
RF | 2.743 (2.521 - 2.985) | 0.76 | 0.42 (0.22) | 3.08 (2.775 - 3.419) | 0.75 | 0.4 (0.2)
ADA | 2.819 (2.594 - 3.062) | 0.75 | 0.47 (0.2) | 2.438 (2.241 - 2.651) | 0.72 | 0.44 (0.17)
GNB | 2.24 (2.07 - 2.424) | 0.72 | 0.37 (0.16) | 1.468 (1.372 - 1.571) | 0.59 | 0.31 (0.11)
LDA | 3.139 (2.886 - 3.415) | 0.79 | 0.51 (0.27) | 2.45 (2.258 - 2.659) | 0.73 | 0.43 (0.19)
QDA | 2.643 (2.442 - 2.86) | 0.76 | 0.44 (0.22) | 1.974 (1.833 - 2.125) | 0.69 | 0.37 (0.15)
MLP | 2.568 (2.37 - 2.782) | 0.75 | 0.42 (0.19) | 2.543 (2.338 - 2.766) | 0.74 | 0.41 (0.18)
XGB | 2.409 (2.223 - 2.61) | 0.74 | 0.36 (0.18) | 2.495 (2.285 - 2.726) | 0.74 | 0.37 (0.19)
Excluding ISMs
LR | 4.711 (4.157 - 5.339) | 0.85 | 0.49 (0.33) | 3.991 (3.503 - 4.547) | 0.80 | 0.39 (0.23)
SVM | 1.434 (1.34 - 1.535) | 0.61 | 0.42 (0.08) | 2.784 (2.548 - 3.041) | 0.75 | 0.58 (0.24)
KNN | 3.537 (3.176 - 3.94) | 0.80 | 0.46 (0.25) | 2.705 (2.469 - 2.963) | 0.76 | 0.41 (0.2)
RF | 3.667 (3.273 - 4.108) | 0.83 | 0.4 (0.26) | 4.724 (4.051 - 5.508) | 0.81 | 0.4 (0.26)
ADA | 3.422 (3.08 - 3.802) | 0.79 | 0.44 (0.23) | 3.103 (2.783 - 3.46) | 0.76 | 0.43 (0.21)
GNB | 2.479 (2.21 - 2.781) | 0.76 | 0.23 (0.13) | 2.191 (1.998 - 2.402) | 0.73 | 0.28 (0.18)
LDA | 4.087 (3.65 - 4.577) | 0.84 | 0.46 (0.3) | 3.973 (3.509 - 4.498) | 0.81 | 0.42 (0.26)
QDA | 3.454 (3.094 - 3.857) | 0.82 | 0.38 (0.25) | 3.514 (3.133 - 3.941) | 0.82 | 0.39 (0.26)
MLP | 3.341 (3.009 - 3.711) | 0.81 | 0.39 (0.22) | 3.532 (3.146 - 3.964) | 0.80 | 0.4 (0.23)
XGB | 3.152 (2.833 - 3.507) | 0.81 | 0.34 (0.22) | 3.235 (2.866 - 3.652) | 0.79 | 0.35 (0.22)
Table 2: Performance metrics, with and without the inclusion of ISMs, of the
proposed method for uncertainty estimation, compared against the absolute
difference of the classification probability from 0.5 for the predicted class.
Metrics highlighted in bold reflect the best performance for a given method.

##  6 Applications for Abstention and Explainability

The following section explores the application of the proposed meta-heuristics
and uncertainty estimation system for abstaining from making a decision and
explaining the level of uncertainty in terms of the sources of decision-making
complexity captured by the proposed meta-heuristics.

###  6.1 Abstention

In terms of facilitating trust calibration, the estimated uncertainty can be
used as a method for abstention and built into the deployed model, preventing
decisions from being shown to the end-user when uncertainty surpasses a given
threshold. To demonstrate this concept, an abstention threshold, beginning at
the 5th percentile and then incrementally increasing until the 95th
percentile, was applied to the uncertainty estimates for the sepsis dataset
used in subsection 5.3. At each threshold, the percentage of misclassified
instances was calculated. Fig. 6 demonstrates how by varying the threshold for
abstention the number of misclassifications can be reduced. Although such an
approach can reduce the number of misclassification, the number of instances
for which the model is applicable for also reduces. Therefore, the application
of such a method may not be appropriate for all use cases.

Figure 6: Percentage of misclassified instances for a given abstention
threshold.

###  6.2 Explainability

The second application of the proposed methods is for the improved explanation
of predictive uncertainty. Shapley additive explanations (SHAP) can be applied
to the calculated meta-heuristics and the estimated uncertainty to uncertainty
to demonstrate how each concept of complexity influences the level of
certainty in decision-making. As an example, SHAP values were calculated for a
patient with high predictive uncertainty and a patient with low predictive
uncertainty within the Sepsis-3 dataset used in section 5.3, when an MLP model
is applied.

Fig. 7 shows the force plots for two patients, for patient A the
classification decision was regarded as having a higher degree of certainty
than the mean from the training set, and for patient B, the decision was
regarded as lower certainty. The benefit of the proposed meta-heuristics is
that they are a mathematical derivation of explainable concepts within
humanistic decision-making and therefore can be expressed using natural
language. Therefore, for patient A we know that the decision was easier
primarily because there was less diversity in the class outcomes of similar
patients, as evidenced by KDN and DCD, with the level of outlierness playing a
small role in further reducing uncertainty. The only factor which increased
the uncertainty of the decision for patient A was a higher degree of evidence
supporting both class outcomes, evidenced by the EC score. Regarding patient
(B), the opposite is true, both KDN and DCD are high, which indicates a higher
level of diversity in class outcomes among similar patients, with the level of
outlierness increasing the level of uncertainty. However, there is less
conflicting evidence for patient B which works to lower the level of
uncertainty. The benefit of using interpretable methods to explore AI
uncertainty is that they can be used to further understand why a model may
abstain from making a decision, or prompt further investigation into why
patients may be classified incorrectly.

Figure 7: Force plots demonstrating the impact of the meta-heuristics on the
level of uncertainty of an MLP model for predicting 30-day mortality of
sepsis-3 patients. Red banners indicate the meta-feature value is increasing
the level of uncertainty of a prediction and the blue banner indicates the
meta-heuristic is decreasing the level of uncertainty. The base value
indicates the mean estimated uncertainty for instances in the training set and
the f(x) refers to the uncertainty value estimated for the current patient.

##  7 Discussion

The proposed method has been demonstrated to be effective in estimating the
uncertainty associated with AI decision-making, which can be a crucial factor
in instilling trust in AI applications, particularly in high-stakes domains
such as healthcare.There are several clear avenues for the application of such
methods to improve both the robustness of AI development and the facilitation
of trust-calibration.

Section 6 presents a clear use case for the estimated uncertainty scores to be
used as a means of abstaining from making a prediction. Although the present
study did not fully explore the potential of the proposed methods for model
abstention, prior research has suggested methods for determining the optimal
points for abstention [22] and abstention-specific performance measures [13].
Therefore, future research in this area could consider incorporating the
proposed methods for uncertainty estimation within existing frameworks for
developing classification systems with the option of abstention.

To enhance the robustness of AI development, the proposed meta-heuristics can
also be utilised to identify the competency of classifiers. The use of meta-
information to understand the strengths and weakness of algorithms is a long-
standing practice [63, 29, 60, 41]. The class-independent and highly
explainable nature of the proposed meta-heuristics makes them suitable for
algorithmic development, and their application in this area can offer new
insights into how the algorithm may perform on unseen data.

An emerging area in the field of precision medicine is the personalisation of
model development to the patient instead of the overall task. Studies have
investigated meta-learning techniques for dynamic classifier selection [14]
and ensemble generation [15, 26] in this domain. Within the area of
complexity, such methods could be applied to select models based on their
ability to handle more challenging instances to maximise the likelihood of a
correct classification [16]. In domains where large datasets are scarce, meta-
learning approaches have the potential to overcome data limitations and learn
viable solutions to parameter tuning and model architecture from similar
problems before applying the learned principles to the current problem to
arrive at a more optimal solution.

Qualitative research indicates that different methods are employed when
addressing uncertainty in real-world human decision-making, depending on the
source of the uncertainty, with factors such as past experience and evidence
availability influencing the strategy chosen [39, 73]. The advantage of the
proposed methods is that they are explainable and can identify the causes of
uncertainty within the AI decision-making process, enabling clinicians to
reason with the output. By providing end-users with a more transparent means
of engaging with and comprehending model outputs, it is hypothesized that
trust calibration will be enhanced.

In conclusion, the proposed method of uncertainty estimation was effective in
identifying instances that are more likely to be misclassified. Furthermore,
the proposed meta-heuristics offer additional opportunities to improve the
explainability of AI decision-making. Future research directions include the
application of the proposed meta-heuristics to enhance model development
through meta-learning and studying the impact of natural language explanations
of uncertainty on the trust calibration of end-users.

## Acknowledgements

We wish to acknowledge the financial support of DMRC and Loughborough
University who jointly funded the project.

## Author contributions

A.H. and G.C. contributed to the design of the study. A.H. and G.C. conceived
the experiments, A.H. conducted the experiment(s). A.H. analysed the results.
A.H. and G.C. interpreted the findings. G.C. supervised the project. A.H.
drafted the initial version of the manuscript. A.H. and G.C. reviewed the
manuscript.

## Competing Interests Statement

The authors declare they have no competing interests to disclose.

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## Supplementary File

Model | Hyperparameter
---|---
LR |  | solver: liblinear; saga
---
penalty: l1; l2
regularisation: (0.01, 100)
SVM |  | regularisation: (0.01, 100)
---
gamma: (0.01, 100)
degree: (1,5)
kernel: linear; polynomial; rbf; sigmoid
KNN |  | no. of neighbors: (2,11)
---
algorithm: auto; ball_tree; kd_tree; brute
RF |  | maximum depth: (2,10)
---
no. of estimators: (5,20)
maximum features: (0.25,1)
ADA |  | learning rate: (0.005, 0.9)
---
no. of estimators: (5, 20)
NB | variable smoothing : (0.01,1)
LDA | solver: svd; lsqr; eigen
QDA |  | regularisation parameter: (0.00001, 0.1)
---
tol: (0.0001, 0.1)
MLP |  | activation function: tanh; relu
---
solver: sgd; adam
alpha: (0.0001, 0.05)
learning rate: constant; adaptive
XGB |  | no. of estimators: (5, 20)
---
maximum depth: (2, 15)
learning rate: (0.05, 0.20)
min. child weight: (1, 4)
Table S.1: Hyperparameters tuned using Bayesian Cross-Validation for each ML
algorithm.