1 A Temporal Type-2 Fuzzy System for Time-dependent Explainable Artiﬁcial

2025-04-30 0 0 758.89KB 16 页 10玖币

侵权投诉

A Temporal Type-2 Fuzzy System for

Time-dependent Explainable Artiﬁcial

Intelligence

Mehrin Kiani, Javier Andreu-Perez∗,Senior Member, IEEE, and Hani Hagras, Fellow, IEEE

Abstract—Explainable Artiﬁcial Intelligence (XAI) is a

paradigm that delivers transparent models and decisions,

which are easy to understand, analyze, and augment by a

non-technical audience. Fuzzy Logic Systems (FLS) based

XAI can provide an explainable framework, while also

modeling uncertainties present in real-world environments,

which renders it suitable for applications where explainabil-

ity is a requirement. However, most real-life processes are

not characterized by high levels of uncertainties alone; they

are inherently time-dependent as well, i.e., the processes

change with time. To account for the temporal component

associated with a process, in this work, we present novel

Temporal Type-2 FLS Based Approach for time-dependent

XAI (TXAI) systems, which can account for the likelihood

of a measurement’s occurrence in the time domain using

(the measurement’s) frequency of occurrence. In Tempo-

ral Type-2 Fuzzy Sets (TT2FSs), a four-dimensional (4D)

time-dependent membership function is developed where

relations are used to construct the inter-relations between

the elements of the universe of discourse and its frequency

of occurrence. The proposed TXAI system with TT2FSs is

exempliﬁed with a step-by-step numerical example and an

empirical study using a real-life intelligent environments

dataset to solve a time-dependent classiﬁcation problem

(predict whether or not a room is occupied depending on the

sensors readings at a particular time of day). The TXAI sys-

tem performance is also compared with other state-of-the-art

classiﬁcation methods with varying levels of explainability.

The TXAI system manifested better classiﬁcation prowess,

with 10-fold test datasets, with a mean recall of 95.40%

than a standard XAI system (based on non-temporal general

type-2 (GT2) fuzzy sets) that had a mean recall of 87.04%.

TXAI also performed signiﬁcantly better than most non-

explainable AI systems between 3.95%, to 19.04% improve-

ment gain in mean recall. Temporal convolution network

(TCN) was marginally better than TXAI (by 1.98% mean

recall improvement) although with a major computational

complexity. In addition, TXAI can also outline the most

likely time-dependent trajectories using the frequency of

occurrence values embedded in the TXAI model; viz. given

a rule at a determined time interval, what will be the next

most likely rule at a subsequent time interval. In this regard,

the proposed TXAI system can have profound implications

for delineating the evolution of real-life time-dependent

processes, such as behavioural or biological processes.

I. Introduction

Over the last few decades, the widespread applica-

tion of artiﬁcial intelligence (AI) systems have enhanced

∗Corresponding author: javier.andreu@essex.ac.uk

M. Kiani, J. Andreu-Perez and H. Hagras are with the School of

Computer and Electronic Engineering, University of Essex, Colchester,

CO4 3SQ, United Kingdom.

many aspects of everyday life from risk management

[1], sky shepherding of sheep [2], medical image seg-

mentation [3], recognition of expertise level [4], mobile

applications [5] to Covid-19 detection based on cough

samples [6]. Although opaque AI systems oﬀer remark-

able prediction accuracy, they are limited by a lack of

explanation behind their predictions. A lack of explana-

tion renders the AI systems untrustworthy, and partic-

ularly inapplicable where users want to understand the

decision process of the AI system. To this end, there is

a growing need for transparent, human-understandable

AI systems called explainable AI (XAI) systems [7].

Several approaches taken towards the development of

XAI systems include: 1) Intrinsic: a method in which

model inference structure is fully transparent such as

short decision trees or sparse linear models, and 2) Post-

hoc: a model-agnostic meta-model is used to decipher

the inference rationale of a black-box model permutation

feature importance can be computed for decision trees.

Within post-hoc methods attempts to unravel a black-

box model into a surrogate intrinsic model have also

been undertaken. A particular category of these are the

anchor-based models.

Although anchor-based approach provides a step to-

wards implementing human-understandable explana-

tions [8], explanatory patterns rest on hard thresholds

and are constrained by Boolean logic. However, real-

life processes are characterised with uncertainty and

therefore hard thresholds based models are not partic-

ular well-suited to model them (real-life processes). In

this regard, another approach to implement XAI systems

is fuzzy logic systems (FLS) [7, 9]. The FLS based XAI

systems are well-suited for explainable modelling of

real-life processes because of FLS capability to handle

uncertainty in the input data, and subsequently improve

the process model and performance. In addition, the use

of conceptual labels (CoLs) that model uncertainty and

axioms of FLS based XAI systems pave way for human-

understandable models for describing complex, real-life

processes.

The FLS based XAI systems handle uncertainty in the

input data using fuzzy sets that convert crisp numbers

(viz. uncertain observations) to CoLs characterised with

membership values [9, 10]. The fuzzy sets are deﬁned

by membership functions (MFs) and represent a given

CoL. The membership value is usually in the range [0,1]

arXiv:2210.12571v1 [cs.AI] 22 Oct 2022

−15 −10 −5 0 5 10 15

0.2

0.4

0.6

0.8

x∈X

Membership Degree µ(x)

−15 −10 −5 0 5 10 15

0.2

0.4

0.6

0.8

x∈X

Membership Degree µ(x)

(a) Type-1 (T1) Fuzzy Set (b) Interval Type-2 (IT2) Fuzzy Set (c) General Type-2 (GT2) Fuzzy Set

Fig. 1: The three types of fuzzy sets: (a) Type-1 (T1) fuzzy sets where each crisp measurement, x∈X, gets assigned a

membership degree, µT1(x)⊆[0,1], but there is no ambiguity in the membership degree, for example as shown by

the red dashed line: µT1(x=1)=0.95. (b) Interval type-2 (IT2) fuzzy sets have lower and upper membership degrees

assigned to each crisp measurement for example µIT 2(x=1)=[0.7,0.95]. (c) General type-2 (GT2) fuzzy sets have

T1 fuzzy sets as membership degree for a crisp measurement i.e. µT2(x=1)={u,µT1(u)∀u∈[0,1],∀µT1∈[0,1]}

where uis called the primary membership degree and µis called the secondary membership degree.

and is a soft measure of the degree of association the

associated fuzzy set has for a given crisp measurement to

belong to the CoL represented by the fuzzy set [10]. For

example, an XAI system modelling the heights of people

in a community using type-1 fuzzy sets may represent

height using CoLs of Tall, Medium, and Short. The MF

associated with each CoL’s MF will assign a crisp number

for the height of a person with a membership grade; for

example, a height of 6ft may get assigned membership

grades of 0.8,0.5,0.1 to represent CoLs of Tall, Medium,

and Short respectively.

In general, fuzzy sets can model uncertainty in the

feature domain at diﬀerent levels: Type 1 (T1), interval

type-2 (IT2), and general type-2 (GT2) fuzzy sets; illus-

trated in Fig. 1. Despite the variability in the extent for

uncertainty modelling amongst the types of fuzzy sets,

all fuzzy sets are modelling uncertainty from a single

time snapshot of the feature domain. More speciﬁcally,

fuzzy sets do not integrate associated temporal informa-

tion in their membership grade calculation. This is a

critical limitation of the fuzzy sets since most real-life

systems are time-variant, i.e., their behaviour changes

with time. To model time-dependent real-life systems

more eﬀectively, in this work, we present the theory

of a new Temporal Type-2 Fuzzy Set (TT2FS) based

approach for time-dependent XAI (TXAI).

The prowess of TXAI system for incorporating time

information for modelling time-variant processes is of

paramount signiﬁcance since the insights provided by

a TXAI system can shed light on both spatial (feature

domain) and temporal behaviour of the time-dependent

process. More speciﬁcally, the TXAI is able to inform

not only about the relation between input features but

can also describe the impact of time on the evolution

of the inter-relation of the features. As an example,

let’s consider a standard XAI system composed of a T1

fuzzy set for modelling thermal sensation ‘Cold’ in the

domain of values of temperature T °C as shown in Fig.

2 (a), and a T1 fuzzy set for the time of occurrence of

concept ‘Cold’ during the months of a year as shown

in Fig. 2 (b). The notion is that the perception of ‘cold’

is mostly associated with the months of winter than in

the months of spring. Hence, using the time information

associated with a fuzzy concept (such as Cold in this

case), a temperature can belong to the concept (Cold)

diﬀerently according to a particular point in time (e.g.,

months of a year).

Crediting a fuzzy membership with its associated time

information is particularly advantageous for the mod-

elling of time-dependent noise-prone processes. More-

over, for dynamic processes, the ability to delineate

its’ (dynamic process) trajectories across time would

inform the evolution of the temporal dynamics of the

process. To this end, our proposed TXAI system has

been designed to integrate temporal information as well

as able to outline the trajectories of a time-dependent

process. To demonstrate the eﬃcacy of TXAI system

for time-dependent process modelling, in this work, an

occupancy dataset is used [11]. Using the values of

temperature, light and carbon dioxide (CO2), and the

time the aforementioned measurements are taken, the

TXAI system is used to make a prediction of whether or

not the room is occupied.

The rest of the paper is organised as follows: in

Section II related works are outlined, Section III presents

the TXAI system deﬁnition and operations, Section IV

outlines the TXAI inference system with a numerical

step-by-step example as well as the evolution of a TXAI

model using temporal trajectories. An empirical study

using TXAI system, as well as state-of-the-art systems

(with varying levels of explainability) for performance

comparison, on the aforementioned occupancy dataset

[11] is presented in Section V, with conclusion and future

research in Section VI.

0 5 10 15 20 25

0.2

0.4

0.6

0.8

Temperature °C

Membership Degree µ

(a) ‘Cold’ membership function in temperature domain.

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

0.2

0.4

0.6

0.8

Months of a year

Membership Degree µ

W inter Spring Summer Autumn

(b) ‘Cold’ membership function in time domain.

Fig. 2: An illustrative type-1 (T1) membership function (MF) for the fuzzy concept of ‘Cold’ in the (a) universe

of temperature in °C and (b) in the universe of time: months of a year. In this case, the membership degree for

experiencing ‘Cold’ at 15 °C is µColdtemp (15°C)=0.2. Likewise considering the prevalence of particular linguistic

variable ’Cold’, viz. the likelihood of observing ‘Cold’ can be diﬀerent in February µColdtime (February)=0.85 than

March µColdtime (March)=0.3. In this regard, the additional information of time can credit the primary membership

in feature-domain through a fuzzy relation.

II. Related Works

Fuzzy sets have enabled explainable models of com-

plex real-life processes which prove too ill-deﬁned for

closed form mathematical analysis. In this regard, al-

though uncertainty in complex processes could be han-

dled by fuzzy sets, the time-variant characteristics of

complex processes have not been integrated into the

modelling by standard XAI systems based on state-of-

the-art fuzzy sets.

There have been few notable attempts in the literature

to model time in the MFs. The work by Garibaldi et

al. [12] on non-stationary fuzzy sets proposed that vari-

ation within a MF can be incorporated by perturbing

the parameters of the MF. Their work aims to develop

non-deterministic fuzzy reason as a way to model the

variability in fuzzy decision making to mimic the vari-

ability in expert opinions. The ability of non-stationary

fuzzy sets to integrate diﬀering experts’ opinions is

a signiﬁcant contribution since it allows for a more

comprehensive model that takes into account all experts’

opinions. However, their work does not incorporate the

variation within a fuzzy concept with respect to time,

which is the aim of the present work, to represent the

time-variant transformation of a same fuzzy linguistic

variable.

Similarly, the work by Kostikova et al. [13] propose

dynamic fuzzy sets by extending the classical fuzzy set

to include a time dimension for representing MF at

diﬀerent time points. They propose four diﬀerent types

of dynamic MFs depending on how many parameters

are changed in the deﬁnition of the dynamic MF. They

simulated their dynamic MFs by using diﬀering expert

assessments on multilevel fuzzy description of a complex

system. However, the dynamic MF is essentially a set of

functions determined at diﬀerent time points with no

bearing on the temporal variation in the fuzzy concept.

In another work by Maeda et al. [14], they propose

dynamic fuzzy reason to deal with the notion of time delay

between premise and consequent. An example of where

a time delay between premise and consequent assumes

critical importance is: ‘If it starts snowing, the traﬃc on

road will increase about 30 minutes later’. They propose

the use of fuzzy relations between a fuzzy concept

and its fuzzy time interval to assign a credit degree to

the concept. The temporal fuzzy reasoning provides a

framework for modelling delay in fuzzy reasoning and

the temporal dynamics of a fuzzy concept. In this work,

we have built on the work of Maeda et al. [14] to credit

the membership grade of a concept based on time.

To the best of the authors’ knowledge, there is no

work in the literature on fuzzy sets that delineates the

incorporation of time-based variation in a fuzzy concept

to compute the membership grade for the crisp values

of the fuzzy concept. In addition, no previous work has

aimed at delineating the trajectories of a time-variant

process with respect to time. To this end, in this work,

we propose TXAI systems that can integrate information

from both the feature domain and time domain. More

details on the proposed TXAI are outlined in Section

III.

III. Time-dependent Explainable Artificial

Intelligence (TXAI) Systems

In this section, we present the TXAI system based

on TT2FS (temporal type-2 fuzzy sets) that incorporate

information from not only the uncertainty in the input

domain of the fuzzy linguistic term, but also from its

time of occurrence. In particular, the information from

the time of occurrence is integrated into the membership

grade of the TT2FS using fuzzy relations such that it (the

membership grade of the TT2FS) varies with respect to

time (time-dependent).

In the next section, we present the most common

fuzzy relations and outline how they can be used for

implementing TT2FS.

TABLE I: Fuzzy relations between the universe of con-

cept X and time domain T.

Name Deﬁnition of the relation

Godel RG(t,x)={1 if µTA(t)≤µA(x)

µA(x)if µTA(t)>µA(x)

Lukasiewicz RL(t,x)=1∧(1−µTA(t)+µA(x))

Gaines-Rescher RGR(t,x)={1 if µTA(t)≤µA(x)

0 if µTA(t)>µA(x)

Mamdani RM(t,x)=µTA(t)∧µA(x)

A. Fuzzy relations between fuzzy linguistic variables and

time related measures

In this work, fuzzy relations are used to interrelate

the information with respect to the degree of truth of a

determined linguistic term or CoL, A, within the domain

X, and time, T, to form TT2FSs such that the likelihood

of occurrence of A in x∈X, i.e. the primary membership

grade µA(x), is credited by a measure that is dependent

on time such as frequency. The application of fuzzy

relation, for constructing TT2FSs, is motivated by the

work on dynamic fuzzy reasoning models in [14]. They

outline fuzzy relations that can be used to model time

dependencies, as noted in Table I.

Before reviewing the diﬀerent relations that can be

applied to construct a TT2FS, the conditions that need

to be fulﬁlled by the associated temporal MF (TMF) are

listed below:

(i) The TMF should be continuous.

(ii) The TMF should be convex.

(iii) The range of the TMF ⊆[0,1].

(iv) The TMF should reﬂect in the value of membership

grade the intrinsic magnitudes of membership grade

in feature domain and in frequency of occurrence

domain, i.e., they should be directly proportional.

For example, if µA(x)is high and the time represen-

tation is also high then the result from the relation

between them should also be high and vice versa.

An illustrative comparison of the TT2FSs formed for the

CoL ‘Cold’ of feature thermal concept using the fuzzy

relations listed in Table I is shown in Fig. 3. The fuzzy re-

lations are applied on hypothetical primary membership

function of ‘Cold’ in feature domain (temperature) and

time domain (months of a year). As can be seen in Fig.

3, the diﬀerent fuzzy relations are encapsulating distinct

inter-dependencies between time and feature domain.

All relations meet the criteria (i) - (iii) listed above

however, only the Mamdani relation meets the criterion

(iv) as well since it gives credit to µCold based on the

variable frequency of occurrence of ‘Cold’ as observed

in diﬀerent months of the year. Hence, in this work, the

Mamdani relation is used to construct the TT2FSs.

B. Conditional relative frequency distribution of a fuzzy

linguistic term

In our TT2FS we employ a measure of conditional

relative frequency between time and the occurrence of a

linguistic term. We denote as Aan instance of a linguistic

term from a set of conceptual labels (also called words of

the universe of discourse), CoLs ∶=[CoL1,CoL2,...,CoLJ]

of a speciﬁc linguistic variable or input.

Deﬁnition III.1 (Discrete conditional relative frequency

with respect to time).The discretized conditional relative

frequency is deﬁned as the likelihood of observing a linguistic

term A based on its membership grade, across time. This is

denoted as gA(tn,µA(x))with time tdiscretised over Ntime

points (tn) such as tn∈[t1,...,tN], and is given by:

gA(tn,µA(x))=∑

x∈X,tn

δnj

max

[t1,...,tN]∑

x∈X,tn

δnj (1)

δnj is a Kronecker delta function [15] (e.g. δab = 0 if a ≠b,

δab = 1 if a=b) that takes the value of 1 when the following

condition applies, ∃argmaxj(µCoLj(xtn)) ∶Colj=A,∀j∈

[1,...,J], and 0 otherwise. Note xtnis a realisation of xat

time tn.

The numerator in (1) ﬁnds the count of occurrences of

a given Afor a determined time point tnacross all data

instances, whereas the denominator is ﬁnding the maxi-

mum value of the count of occurrences of Aacross all N

time points and all data instances. The resultant discrete

conditional relative frequency gA(tn,µA(x)) is interpo-

lated to form a conditional distribution fA(t,µA(x)). For

the sake of notational simplicity, we denote the later

distribution as fAand the discrete conditional relative

frequency as gAfrom here onwards.

Let us assume that the linguistic variable is ther-

mal sensation deﬁned on the input domain (x∈X) of

temperature in °C and the associated CoLs be: [Cold,

Comfortable, Hot]. For a given crisp input of tempera-

ture such as 15°C, the associated primary membership

grade for all three CoLs of Cold, Comfortable, and Hot

be µCold (15 °C)=[0.4], µcomf .(15°C)=[0.3], µhot(15°C)=

[0]respectively. In this illustrative case, the temperature

of 15 °C has a maximum membership grade, amongst

all CoLs, for Cold and hence 15 °C is assigned with the

CoL of Cold. Referring back to (1), for computing the

conditional relative frequency for Cold the numerator

is going to sum all the data instances where the crisp

inputs are assigned with Cold for a given time point tn

such as a particular month of a year. The denominator

ﬁnds the mode of occurrence of Cold across all months.

The result of the division will scale the gCold values to

[0,1].

An illustration for calculating the gCold values using

(1), with a total of 12 time points as the months of a

year is shown in Fig. 4 (b) with continuous values of

fCold , found using interpolation of gCold , plotted in Fig.

4 (c). Please note the associated time intervals, (as listed

in the illustration in Fig. 4 are seasons in a year such

as Winter, Spring, Summer, and Autumn), are for easing

the computational complexity of the four-dimensional

(4D) TT2FSs as will be explained later in section III-D

by taking time interval based slice of the TT2FS.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1ATemporalType-2FuzzySystemforTime-dependentExplainableArticialIntelligenceMehrinKiani,JavierAndreu-Perez,SeniorMember,IEEE,andHaniHagras,Fellow,IEEEAbstractExplainableArticialIntelligence(XAI)isaparadigmthatdeliverstransparentmodelsanddecisions,whichareeasytounderstand,analyze,andaugmentbyanon-...

展开>> 收起<<

1 A Temporal Type-2 Fuzzy System for Time-dependent Explainable Artiﬁcial.pdf

共16页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 A Temporal Type-2 Fuzzy System for Time-dependent Explainable Artiﬁcial

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: