AND

# Interval-valued aggregation functions based on moderate deviations applied
to Motor-Imagery-Based Brain Computer Interface

Javier Fumanal-Idocin, Zdenko Takáč, Javier Fernández , Jose Antonio Sanz,
Harkaitz Goyena, Ching-Teng Lin, , Yu-Kai Wang, , Humberto Bustince Javier
Fumanal-Idocin, Javier Fernandez, Jose Antonio Sanz, Harkaitz Goyena and
Humberto Bustince are with the Departamento de Estadistica, Informatica y
Matematicas, Universidad Publica de Navarra, Campus de Arrosadia, 31006,
Pamplona, Spain. emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Fernandez and Humberto Bustince are with the
Institute of Smart Cities, Universidad Publica de Navarra, Campus de
Arrosadia, 31006, Pamplona, Spain.Javier Fernandez and Humberto Bustince are
with the Laboratory Navarrabiomed, Hospital Complex of Navarre (CHN),
Universidad Publica de Navarra, IdiSNA, Irunlarrea 3. 31008, Pamplona,
Spain.Z. Takáč is with Institute of Information Engineering, Automation and
Mathematics, Faculty of Chemical and Food Technology, Slovak University of
Technology in Bratislava, Radlinského 9, 812 37 Bratislava 1, Slovak Republic,
(e-mail: zdenko.takac@stuba.sk).Y.-K. Wang and C.-T. Lin are with the Centre
for Artificial Intelligence, Faculty of Engineering and Information
Technology, University of Technology Sydney, Ultimo, NSW 2007, Australia
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we develop moderate deviation functions to measure similarity and
dissimilarity among a set of given interval-valued data to construct interval-
valued aggregation functions, and we apply these functions in two Motor-
Imagery Brain Computer Interface (MI-BCI) systems to classify
electroencephalography signals. To do so, we introduce the notion of interval-
valued moderate deviation function and, in particular, we study those
interval-valued moderate deviation functions which preserve the width of the
input intervals. In order to apply them in a MI-BCI system, we first use fuzzy
implication operators to measure the uncertainty linked to the output of each
classifier in the ensemble of the system, and then we perform the decision
making phase using the new interval-valued aggregation functions. We have
tested the goodness of our proposal in two MI-BCI frameworks, obtaining better
results than those obtained using other numerical aggregation and interval-
valued OWA operators, and obtaining competitive results versus some non
aggregation-based frameworks.

###### Index Terms:

Electroencephalography; Brain-Computer-Interface; Moderate Deviations;
Interval-valued aggregation; Motor Imagery; Admissible orders; Classification;
Signal Processing;

©2021 IEEE. Personal use of this material is permitted. Permission from IEEE
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lists, or reuse of any copyrighted component of this work in other works. DOI:
10.1109/TFUZZ.2021.3092824

##  I Introduction

Brain Computer Interface (BCI) is one of the most popular methods for
controlling devices using variations in the brain dynamics [1, 2, 3]. One
popular BCI method is Motor-Imagery (MI), in which a person imagines a
specific body movement, which produces a reaction in the motor areas of the
brain [4, 5]. BCI systems are composed of some different components, such as
signal detection, feature extraction and command identification, in order to
successfully convert a brain signal into a computer command [6].

Usually, BCI systems use different wave transformations to extract useful
information from the ElectroEncephaloGrahpy (EEG) data [7, 8], such as the
Fast Fourier transform (FFt), to convert the signal in the frequency domain
and the Meyer Wavelet transform. It is also very common to use algorithms such
as Common Spatial Filtering (CSP), to classify the signals or to use its
output as features to feed further classifiers [5, 9]. Some of the most common
classifiers used in BCI systems are Linear Discriminant Analysis (LDA),
Quadratic Discriminant Analysis (QDA), Support Vector Machines (SVM) and
K-Nearest Neighbours (KNN) [10, 11].

In the literature, there are many different approaches to EEG-based BCI
classification systems. In [12], the authors extended the CSP to regression
problems using fuzzy sets, and applied it to measure responsiveness in
psychomotor vigilance tasks. In [13], the authors studied the correlation
between different EEG channels and the target class, in order to select only
meaningful channels to the classification problem, while in [14] the authors
purge the outliers from the signal and then use Dempster-Shafer theory to
discover features with the highest interclass variability. In [15] the authors
studied the effects on visual stimuli, in order to understand how human
perceive other people’s emotions in the cocktail party problem [16]. Also, in
[17], the authors used Bispectrum analysis [18] to select the optimal channels
to perform classification.

One recent approach to BCI research is to focus on the information fusion
processes of the system [19, 20, 21]. Due to the high number of components of
the BCI, it is necessary to combine the output from different elements into a
single numerical value. This process is key to the performance of the system
due to the relevance of these components interactions and correlations. One
possibility to deal with this problem are aggregation functions [22, 23].

Aggregation functions are used to fuse several input values into one single
output value. They have been widely applied in classification systems [24,
25], fuzzy rules-based systems [26, 27] and image processing [28, 29], among
others.

In some cases, there is imprecision in the data to aggregate. For the case of
the EEG signal, the presence of noise and imprecision in the measurements can
significantly affect the performance of the BCI system [7]. One solution to
model that uncertainty is to represent each data as an interval, where its
width represents the uncertainty associated to each observation [30, 31]. The
use of intervals has shown to be a suitable solution to tackle classification
problems [32, 33, 34]. For this reason, large efforts have been devoted to the
development of mechanisms to fuse information in the interval-valued setting
[35, 36, 37, 38].

Taking into account these considerations, the objective of this paper is
double:

  * •

To construct a new MI BCI framework to classify EEG signals where the
uncertainty in each classifier output is modeled using interval-valued data.

  * •

To determine the best aggregation function to be applied to the set of
interval-valued data to obtain the final decision.

The selection of the best aggregation function in an interval-valued setting
is still an open problem. In the case of numerical data, several solutions
have been proposed for this problem. In particular, in this work we consider
the following ones: (i) Penalty-based aggregation functions, which determine
the output from a set of inputs by minimizing a disagreement measure between
the original set of values and the possible outputs [39, 40]. (ii) Deviation-
based aggregation functions that were introduced in [41] based on Darczy’s
deviation functions [42], which aggregate a set of deviation functions to
determine how different is a given value from a set of inputs.

To reach our objective, we first develop the theoretical concept of interval-
valued moderate deviation based aggregation function, studying the special
case where the width of the input intervals is the same for all of them. Then,
using the newly-developed interval-valued aggregation functions, we extend two
aggregation-based MI BCI frameworks, namely: the traditional framework
described in [20] and the Multimodal Fuzzy Fusion (MFF) framework proposed in
[19].

The goodness of our proposal is shown by comparing our results (i) with the
outcome obtained by its numerical counterpart using classical aggregation
functions, (ii) with the new method using as aggregation function the
interval-valued OWA operators proposed in [43] and (iii), with other non
aggregation-based MI BCI frameworks [44, 45, 46].

The structure of this paper is described as follows. In Section II, we explain
some preliminary concepts related to the developed work. In Section III, we
discuss the interval-valued moderate deviations, and, in Section IV, we
discuss the specific case in which the length of all the interval-valued
inputs are the same. Then, in Section V, we explain how to apply these
functions in a MI BCI framework and, in Section VI, we show our experimental
results and comparisons with other aggregation functions. Subsequently, in
Section VII, we show how our system performs compared to non aggregation-based
MI BCI frameworks. Finally, in Section VIII, we summarize the work done and
give the final remarks.

##  II Preliminaries

In this section we introduce the concept of aggregation function, fuzzy
implication function, OWA operator and their interval-valued version.

###  II-A Aggregation functions

Aggregation functions [23] are used to fuse information from $n$ sources into
one single output. A function A: $[0,1]^{n}\rightarrow[0,1]$ is said to be an
n-ary aggregation function if the following conditions hold:

  * •

A is increasing in each argument: $\forall
x_{i},y\in[0,1],i\in\\{1,\ldots,n\\}:x_{i}\leq y\implies
A(x_{1},\ldots,x_{i},\ldots,x_{n})\leq A(x_{1},\ldots,y,\ldots x_{n})$

  * •

$A(0,\dots,0)=0$

  * •

$A(1,\dots,1)=1$

Some examples of classical n-ary aggregation functions are:

  * •

Arithmetic mean: $A(\mathbf{x})=\frac{1}{n}\sum_{i=1}^{n}x_{i}$;

  * •

Median: $A(\mathbf{x})=x_{m}$, where for any permutation
$\sigma:\\{1,\ldots,n\\}$ such that $x_{\sigma(1)}\leq\ldots\leq
x_{\sigma(n)},x_{m}=x_{\sigma(\frac{n+1}{2})}$, if $n$ is odd, and
$x_{m}=\frac{1}{2}(x_{\sigma(\frac{n}{2})}+x_{\sigma(\frac{n}{2}+1)})$,
otherwise.

  * •

Max: $A(\mathbf{x})=\max(x_{1},\dots,x_{n})$;

  * •

Min: $A(\mathbf{x})=\min(x_{1},\dots,x_{n})$;

where $\mathbf{x}=(x_{1},\ldots,x_{n})\in[0,1]^{n}.$

###  II-B Interval-valued fuzzy implication functions

An fuzzy implication function is a function $I:[0,1]^{2}\rightarrow[0,1]$ that
satisfies the following properties, for all $x,z,y\in[0,1]$ [47]:

  * •

$x\leq z$ implies $I(x,y)\geq I(z,y)$;

  * •

$y\leq t$ implies $I(x,y)\leq I(x,t)$;

  * •

$I(0,x)=1$ for all $x\in[0,1]$;

  * •

$I(x,1)=1$ for all $x\in[0,1]$;

  * •

$I(1,0)=0$.

Examples of fuzzy implication functions are:

  * •

Kleene-Dienes: $I(x,y)=\max(1-x,y)$

  * •

Łukasiewicz: $I(x,y)=\min(1,1-x+y)$

  * •

Reichenbach: $I(x,y)=1-x+xy$

where $x,y\in[0,1].$

###  II-C Interval-valued aggregation functions [48]

We consider closed subintervals of the unit interval $[0,1]$:

$L([0,1])=\\{X=[\underline{X},\overline{X}]\mid
0\leq\underline{X}\leq\overline{X}\leq 1\\}.$ (1)

The width of the interval $X\in L([0,1])$, denoted by $w(X)$, is given by
$w(X)=\overline{X}-\underline{X}$. An interval-valued function
$f:(L([0,1]))^{n}\to L([0,1])$ is called $w$-preserving if
$w(X_{1})=\ldots=w(X_{n})$ implies $w(f(X_{1},\ldots,X_{n}))=w(X_{1})$,
$\text{for all }X_{1},\ldots,X_{n}\in L([0,1])$.

An order relation on $L([0,1])$ is a binary relation $\leq$ on $L([0,1])$ such
that, for all $X,Y,Z\in L([0,1])$,

  1. (L1)

$X\leq X$, (reflexivity),

  2. (L2)

$X\leq Y$ and $Y\leq X$ imply $X=Y$, (antisymmetry),

  3. (L3)

$X\leq Y$ and $Y\leq Z$ imply $X\leq Z$, (transitivity).

An order relation on $L([0,1])$ is called total or linear if any two elements
of $L([0,1])$ are comparable, i.e., if for every $X,Y\in L([0,1])$, $X\leq Y$
or $Y\leq X$.

We denote by $\leq_{L}$ any order in $L([0,1])$ (which can be partial or
total) with $0_{L}=[0,0]$ as its minimal element and $1_{L}=[1,1]$ as its
maximal element.

The $K_{a}$ operator is defined, for all $X\in L([0,1])$ and $a\in[0,1]$, by:

$K_{a}(X)=(1-a)\underline{X}+a\overline{X}.$ (2)

For $\alpha,\beta\in[0,1]$ with $\alpha\neq\beta$, the total order
$\leq_{\alpha,\beta}$, induced by $K_{\alpha}$ and $K_{\beta}$, is defined,
for all $X,Y\in L([0,1])$, as: [48]

$X\leq_{\alpha,\beta}Y\hskip 5.69046pt\text{ if }\hskip
5.69046pt\begin{cases}K_{\alpha}(X)<K_{\alpha}(Y)\text{ or }\\\
K_{\alpha}(X)=K_{\alpha}(Y)\text{ and }K_{\beta}(X)\leq
K_{\beta}(Y)\end{cases}$ (3)

A total order on $L([0,1])$ is called an admissible order [48, 38] if it
generalizes the standard product order [49, 50] between intervals, which is a
partial order.

###### Definition 1

Consider $n\geq 2$. An $n$-dimensional interval-valued aggregation function in
$L([0,1])$ with respect to $\leq_{L}$ is a mapping $A:(L([0,1]))^{n}\to
L([0,1])$ which verifies:

  1. (A1)

$A([0,0],\ldots,[0,0])=0_{L}$.

  2. (A2)

$A([1,1],\ldots,[1,1])=1_{L}$.

  3. (A3)

$A$ is a non-decreasing function in each variable with respect to $\leq_{L}$.

###  II-D Interval-valued Ordered Weighted Averaging (OWA) operators

Let $\preceq$ be an admissible order on $L([0,1])$, $[a,b]\in L([0,1])$ and
$\mathbf{w}=(w_{1},\dots,w_{n})\in[0,1]^{n},w_{1}+\dots+w_{n}=1$ a weighting
vector. An interval-valued OWA operator associated with $w$ and $\preceq$ is a
mapping $L([0,1]^{n}\rightarrow L([0,1]))$ defined by [43]:

$OWA([a_{1},b_{1}],\dots,[a_{n},b_{n}])=\sum_{i=1}^{n}w_{i}\cdot[a_{\sigma(i)},b_{\sigma(i)}]$
(4)

where $\sigma$ is a permutation such that, for all $i\in\\{1,\dots,n-1\\}$, it
holds that
$[a_{\sigma(i)},b_{\sigma(i)}]\preceq[a_{\sigma(i+1)},b_{\sigma(i+1)}]$.

The weighting vector can be computed using a quantifier function
$Q:[0,1]\rightarrow[0,1]$, defined here, for all $x\in[0,1]$ and
$a,b\in[0,1]$, by:

$Q_{a,b}(x)=\begin{cases}0,&\mbox{if }x<a\\\ 1,&\mbox{if }x>b\\\
\frac{x-a}{b-a},&\mbox{otherwise}\end{cases}$ (5)

We then define, for $i\in\\{1\dots n\\}$:

$w_{i}=Q(\frac{i}{n})-Q(\frac{i-1}{n})$ (6)

Depending on the value of the parameters $a$ and $b$, different weighting
vectors can be obtained. For this study, we have used the following
configurations:

  * •

OWA1: $a=0.1,b=0.5$

  * •

OWA2: $a=0.5,b=1$

  * •

OWA3: $a=0.3,b=0.8$

###  II-E Moderate deviation functions

Regarding the notion of moderate deviation function, we follow the approach
given in [51].

###### Definition 2

A function $\mathcal{D}:[0,1]^{2}\to\mathbb{R}$ is called a moderate deviation
function, if, for all $x,y\in[0,1]$, it satisfies:

  1. (MD1)

$\mathcal{D}$ is non-decreasing in the second component;

  2. (MD2)

$\mathcal{D}$ is non-increasing in the first component;

  3. (MD3)

$\mathcal{D}(x,y)=0$ if and only if $x=y$.

The set of all moderate deviation functions is denoted by $MD$.

The notion of moderate deviation function is closely related to those of
restricted equivalence function [52] that we recall now.

###### Definition 3

A function $R:[0,1]^{2}\to[0,1]$ is called a restricted equivalence function,
if, for all $x,y,z\in[0,1]$, it satisfies:

  1. (R1)

$R(x,y)=0$ if and only if $\\{x,y\\}=\\{0,1\\}$;

  2. (R2)

$R(x,y)=1$ if and only if $x=y$;

  3. (R3)

$R(x,y)=R(y,x)$;

  4. (R4)

If $x\leq y\leq z$, then $R(x,z)\leq R(x,y)$ and $R(x,z)\leq R(y,z)$.

##  III Interval-valued moderate deviation functions

A moderate deviation function was introduced and corresponding deviation-based
aggregation functions were studied in [51]. We make a similar study for
intervals.

###### Definition 4

Let $\leq_{L}$ be a total order on $L([0,1])$ and

$L(\mathbb{R})=\\{A=[\underline{A},\overline{A}]\mid\underline{A},\overline{A}\in\mathbb{R},\underline{A}\leq\overline{A}\\}$
(7)

A function $D:(L([0,1]))^{2}\to L(\mathbb{R})$ is called an interval-valued
moderate deviation function w.r.t. $\leq_{L}$, if, for $X,Y\in L([0,1])$, it
satisfies:

  1. (MD1)

$D$ is non-decreasing in the second component w.r.t. $\leq_{L}$;

  2. (MD2)

$D$ is non-increasing in the first component w.r.t. $\leq_{L}$;

  3. (MD3)

$D(X,Y)=0_{L}$ if and only if $X=Y$.

The set of all interval-valued moderate deviation functions w.r.t. $\leq_{L}$
is denoted by $IMD$.

Interval-valued aggregation functions based on a given moderate deviation
function can be defined by: $A(X_{1},\ldots,X_{n})=Y$ if and only if the
equation $\sum\limits_{i=1}^{n}D(X_{i},Y)=[0,0]$ is satisfied for
$X_{1},\ldots,X_{n}\in L([0,1])$. However, from Definition 4 it is clear that
the equation may not have a solution, or it may have more than one solution.
Hence, we modify the procedure in a similar way as it was done in [51]. We
adopt the convention $\sup\\{a\in[b,c]\,|\,a\in\emptyset\\}=b$ and
$\inf\\{a\in[b,c]\,|\,a\in\emptyset\\}=c$.

###### Definition 5

Let $n\in\mathbb{N}$, $\leq_{L}$ be a total order on $L([0,1])$,
$A:(L([0,1]))^{2}\to L([0,1])$ be an idempotent interval-valued aggregation
function w.r.t. $\leq_{L}$ and $D:(L([0,1]))^{2}\to L(\mathbb{R})$ be an
interval-valued moderate deviation function w.r.t. $\leq_{L}$. Then the
function $M_{D}:(L([0,1]))^{n}\to L([0,1])$ defined, for
$X_{1},\ldots,X_{n}\in L([0,1])$, by

$M_{D}(X_{1},\ldots,X_{n})=A\left(\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)<_{L}[0,0]\right.\right.\right\\},\\\
\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)>_{L}[0,0]\right\\}\right)$
(8)

is called an interval-valued $D$-mean w.r.t. $\leq_{L}$.

We are going to show that the proposed $D$-mean is a symmetric idempotent
interval-valued aggregation function.

###### Theorem 1

Let $n\in\mathbb{N}$, $\leq_{L}$ be a total order on $L([0,1])$ and
$D:(L([0,1]))^{2}\to L(\mathbb{R})$ be an interval-valued moderate deviation
function w.r.t. $\leq_{L}$. Then the interval-valued $D$-mean
$M_{D}:(L([0,1]))^{n}\to L([0,1])$ given in Definition 5 is an $n$-ary
symmetric idempotent interval-valued aggregation function.

Proof: The symmetry is obvious. Regarding idempotency, let
$X_{1}=\ldots=X_{n}$. Then, since $D(X_{1},Y)<_{L}[0,0]$ if and only if
$Y<_{L}X_{1}$, we have

$\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)<_{L}[0,0]\right.\right\\}=X_{1}$
(9)

and similarly

$\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)>_{L}[0,0]\right.\right\\}=X_{1},$
(10)

hence $M_{D}(X_{1},\ldots,X_{1})=A(X_{1},X_{1})=X_{1}$.

It suffices to prove that $M_{D}$ is an interval-valued aggregation function.
The boundary conditions follow from the idempotency, so it only remains to
show the monotonicity. Let $X_{1},\ldots,X_{n}\in L([0,1])$ and let there
exists $k\in\\{1,\ldots,n\\}$ such that $X_{k}<_{L}[1,1]$. Let $Z>_{L}X_{k}$.
Then, since $D(Z,Y)\leq_{L}D(X_{k},Y)$ for all $Y\in L([0,1])$, we have

$\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)-D(X_{k},Y)+D(Z,Y)<_{L}[0,0]\right.\right\\}\geq_{L}$
$\geq_{L}\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)<_{L}[0,0]\right.\right\\}$
(11)

and

$\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)-D(X_{k},Y)+D(Z,Y)>_{L}[0,0]\right.\right\\}\geq_{L}$
$\geq_{L}\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D(X_{i},Y)>_{L}[0,0]\right.\right\\}.$
(12)

Having in mind the monotonicity of $A$, the monotonicity of $M_{D}$ is proved.
$\Box$

###### Example 1

Let $D:(L([0,1]))^{2}\to L(\mathbb{R})$ be given, for $X,Y\in L([0,1])$, by

$D(X,Y)=\left\\{\begin{array}[]{ll}Z,&\hbox{if }Y>_{L}X;\\\ -Z,&\hbox{if
}Y<_{L}X;\\\\[0.0pt] [0,0],&\hbox{if }Y=X,\end{array}\right.$

for some $Z\in L([0,1])$ such that $Z>_{L}[0,0]$. Then, $D\in IMD$ and the
corresponding interval-valued $D$-mean is median if $A$ is the median if $A$
is the arithmetic mean.

###### Definition 6

Let $n\in\mathbb{N}$, $\leq_{L}$ be a total order on $L([0,1])$,
$A:(L([0,1]))^{2}\to L([0,1])$ be an idempotent interval-valued aggregation
function w.r.t. $\leq_{L}$ and $\mathbb{D}=(D_{1},\ldots,D_{n})\in IMD^{n}$.
Let $\mathbf{w}\in[0,\infty[^{n}$ be a non-zero weighting vector. Then

  * •

The function $M_{\mathbb{D}}:(L([0,1]))^{n}\to L([0,1])$ defined by

$M_{\mathbb{D}}(X_{1},\ldots,X_{n})=$ $=A\left(\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D_{i}(X_{i},Y)<_{L}[0,0]\right.\right\\},\right.\\\
\left.\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}D_{i}(X_{i},Y)>_{L}[0,0]\right\\}\right)\right.$
(13)

is called an interval-valued $\mathbb{D}$-mean w.r.t. $\leq_{L}$.

  * •

The function $M_{\mathbb{D},\mathbf{w}}:(L([0,1]))^{n}\to L([0,1])$ defined by

$M_{\mathbb{D},\mathbf{w}}(X_{1},\ldots,X_{n})=$ $=A\left(\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}w_{i}D_{i}(X_{i},Y)<_{L}[0,0]\right.\right\\}\right.,\\\
\left.\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}w_{i}D_{i}(X_{i},Y)>_{L}[0,0]\right.\right\\}\right),$
(14)

is called an interval-valued weighted $\mathbb{D}$-mean w.r.t. $\leq_{L}$.

  * •

The function $OM_{\mathbb{D},\mathbf{w}}:(L([0,1]))^{n}\to L([0,1])$ defined
by

$OM_{\mathbb{D},\mathbf{w}}(X_{1},\ldots,X_{n})=$ $=A\left(\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}w_{i}D_{i}(X_{(i)},Y)<_{L}[0,0]\right.\right\\},\right.\\\
\left.\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}w_{i}D_{i}(X_{(i)},Y)>_{L}[0,0]\right.\right\\}\right),$
(15)

where $X_{(1)}\geq_{L}\ldots\geq_{L}X_{(n)}$, is called an interval-valued
ordered weighted $\mathbb{D}$-mean w.r.t. $\leq_{L}$.

###### Corollary 1

Under the conditions of Definition 6, the interval-valued $\mathbb{D}$-mean
$M_{\mathbb{D}}$, interval-valued weighted $\mathbb{D}$-mean
$M_{\mathbb{D},\mathbf{w}}$ and interval-valued ordered weighted
$\mathbb{D}$-mean $OM_{\mathbb{D},\mathbf{w}}$ are idempotent interval-valued
aggregation functions. Moreover, $OM_{\mathbb{D},\mathbf{w}}$ is symmetric.

##  IV $w$-Preserving interval-valued moderate deviation function

In this section, a special class of interval-valued moderate deviation
functions is studied, in particular, the functions satisfying the property: if
all the input intervals have the same width, then the output interval has also
the same width. We only consider $K_{\alpha,\beta}$ orders in this section.

###### Definition 7

Let $\alpha,\beta\in[0,1]$ with $\beta\neq\alpha$. A function
$D:(L([0,1]))^{2}\to L(\mathbb{R})$ is called a $w$-preserving interval-valued
moderate deviation function w.r.t. $\leq_{\alpha,\beta}$, if, for $X,Y\in
L([0,1])$, it satisfies:

  1. (MD1)

$D$ is non-decreasing in the second component w.r.t. $\leq_{\alpha,\beta}$;

  2. (MD2)

$D$ is non-increasing in the first component w.r.t. $\leq_{\alpha,\beta}$;

  3. (MD3′)

$K_{\alpha}(D(X,Y))=0$ if and only if $K_{\alpha}(X)=K_{\alpha}(Y)$;

  4. (MD4)

if $w(X)=w(Y)$, then $w(D(X,Y))=w(X)$.

The set of all $w$-preserving interval-valued moderate deviation functions
w.r.t. $\leq_{\alpha,\beta}$ is denoted by $wIMD$.

A weaker form of monotone interval-valued functions will be of use later, so
we define the so-called $w$-monotonicity.

###### Definition 8

An interval-valued function $A:(L([0,1]))^{n}\to L([0,1])$ is said to be
$w$-monotone function w.r.t. order $\leq_{L}$, if it satisfies:

  1. (A3′)

If $X_{1}\leq_{L}Y_{1},\ldots,X_{n}\leq_{L}Y_{n}$ where
$w(X_{1})=\ldots=w(X_{n})=w(Y_{1})=\ldots=w(Y_{n})$, then
$A(X_{1},\ldots,X_{n})\leq_{L}A(Y_{1},\ldots,Y_{n})$.

###### Definition 9

Let $\alpha,\beta\in[0,1]$ with $\beta\neq\alpha$, $n\in\mathbb{N}$ and
$B:[0,1]^{n}\to[0,1]$ be an idempotent symmetric aggregation function. Let
$A:(L([0,1]))^{2}\to L([0,1])$ be an idempotent interval-valued $w$-monotone
function w.r.t. $\leq_{\alpha,\beta}$ and $D:(L([0,1]))^{2}\to L(\mathbb{R})$
be a $w$-preserving interval-valued moderate deviation function w.r.t.
$\leq_{\alpha,\beta}$. Then the function $M_{D}:(L([0,1]))^{n}\to L([0,1])$
defined by

$M_{D}(X_{1},\dots,X_{n})=\\\ =A\left(\sup\biggl{\\{}Y\in
L([0,1])\left|\sum_{i=1}^{n}SK_{\alpha}\left(D(X_{i},Y)\right)<0\right.\biggr{.}\right.\\\
\biggl{.}w(Y)=B(w(X_{1}),\dots,w(X_{n}))\biggr{\\}},\\\ \inf\left\\{Y\in
L([0,1])\left|\sum\limits_{i=1}^{n}K_{\alpha}(D(X_{i},Y)\right.\right.)>0\\\
\biggl{.}\biggl{.}w(Y)=B(w(X_{1}),\dots,w(X_{n}))\biggr{\\}}\biggl{)}$ (16)

is called an interval-valued $wD$-mean w.r.t. $\leq_{\alpha,\beta}$.

The following theorem shows that the proposed $wD$-mean has the similar
properties as $D$-mean defined in Definition 5, i.e. it is a symmetric
idempotent interval-valued function. But unlike $D$-mean, our $wD$-mean is
also $w$-preserving (if $A$ is $w$-preserving); on the other side it does not
satisfy monotonicity, only the $w$-monotonicity.

###### Theorem 2

Let $\alpha,\beta\in[0,1]$ with $\beta\neq\alpha$, $n\in\mathbb{N}$ and
$D:(L([0,1]))^{2}\to L(\mathbb{R})$ be a $w$-preserving interval-valued
moderate deviation function w.r.t. $\leq_{\alpha,\beta}$. Then the interval-
valued $wD$-mean $M_{D}:(L([0,1]))^{n}\to L([0,1])$ given in Definition 9
where $A$ is $w$-preserving, is an $n$-ary $w$-preserving symmetric idempotent
interval-valued $w$-monotone function.

Proof: The symmetry is obvious. Idempotency follows from the equivalence
$K_{\alpha}(D(X,Y))<0$ if and only if $K_{\alpha}(Y)<K_{\alpha}(X)$ (recall
that $w(Y)$ has to be equal to $w(X)$, hence the last inequality is equivalent
to $Y<_{\alpha,\beta}X$) in a similar way as in Theorem 1.

Regarding the $w$-monotonicity of the function $M_{D}$, let
$X_{1},\ldots,X_{n}\in L([0,1])$ with $w(X_{1})=\ldots=w(X_{n})$ and let there
exists $k\in\\{1,\ldots,n\\}$ such that $X_{k}<_{\alpha,\beta}[1,1]$. Let
$Z>_{\alpha,\beta}X_{k}$ where $w(Z)=w(X_{k})$. Then, since
$D(Z,Y)\leq_{\alpha,\beta}D(X_{k},Y)$ for all $Y\in L([0,1])$, we have

$\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}K_{\alpha}(D(X_{i},Y))-K_{\alpha}(D(X_{k},Y))\right.\right.\\\
\biggl{.}+K_{\alpha}(D(Z,Y))<0\,\,\,\&\,\,\,w(Y)=w(X_{1})\biggr{\\}}\geq_{\alpha,\beta}\\\
\geq_{\alpha,\beta}\sup\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}K_{\alpha}(D(X_{i},Y))<0\,\,\,\&\,\,\,w(Y)=w(X_{1})\right.\right\\}$
(17)

and

$\inf\left\\{Y\in
L([0,1])\,\left|\,\sum\limits_{i=1}^{n}K_{\alpha}(D(X_{i},Y))-K_{\alpha}(D(X_{k},Y))+\right.\right.\\\
\left.K_{\alpha}(D(Z,Y))>0\,\,\,\&\,\,\,w(Y)=w(X_{1})\right\\}\geq_{\alpha,\beta}\geq_{\alpha,\beta}\inf\left\\{Y\in
L([0,1])\,\left|\,\right.\right.\\\
\left.\sum\limits_{i=1}^{n}K_{\alpha}(D(X_{i},Y))>0\,\,\,\&\,\,\,w(Y)=w(X_{1})\right\\}$
(18)

Having in mind the $w$-monotonicity of $A$, the $w$-monotonicity of $M_{D}$ is
proved.

Finally, the fact that $M_{D}$ is $w$-preserving directly follows from
Equation (16), idempotency of $B$ and the fact that A is w-preserving. $\Box$

A construction method of $w$-preserving interval-valued moderate deviation
functions is given in the following Theorem.

###### Theorem 3

Let $\alpha,\beta\in[0,1]$ with $\beta\neq\alpha$,
$\mathcal{D}:[0,1]^{2}\to\mathbb{R}$ be a strictly monotone moderate deviation
function and $C:[0,1]^{2}\to[0,1]$ be an idempotent function non-decreasing in
the second component and non-increasing in the first component. Then the
function $D:(L([0,1]))^{2}\to L(\mathbb{R})$ given by:

$\left\\{\begin{array}[]{l}K_{\alpha}(D(X,Y))=\mathcal{D}\left(K_{\alpha}(X),K_{\alpha}(Y)\right),\\\
w(D(X,Y))=C\left(w(X),w(Y)\right)\end{array}\right.$

is a $w$-preserving interval-valued moderate deviation function w.r.t.
$\leq_{\alpha,\beta}$.

Proof: (MD1) Let $Y\leq_{\alpha,\beta}Z$. There are two possibilities:

  1. 1.

$K_{\alpha}(Y)<K_{\alpha}(Z)$, then, since $\mathcal{D}$ is strictly monotone,
we have $K_{\alpha}(D(X,Y))<K_{\alpha}(D(X,Z))$, hence
$D(X,Y)<_{\alpha,\beta}D(X,Z)$, or

  2. 2.

$K_{\alpha}(Y)=K_{\alpha}(Z)$ and $K_{\beta}(Y)\leq K_{\beta}(Z)$, in which
case $K_{\alpha}(D(X,Y))=K_{\alpha}(D(X,Z))$ and

$\left\\{\begin{array}[]{ll}w(D(X,Y))=C\left(w(X),w(Y)\right)\leq\\\
C\left(w(X),w(Z)\right)=w(D(X,Z)),&\textrm{ for }\beta>\alpha,\\\
w(D(X,Y))=C\left(w(X),w(Y)\right)\geq\\\
C\left(w(X),w(Z)\right)=w(D(X,Z)),&\textrm{ for }\beta<\alpha\\\
\end{array}\right.$ (19)

so, in both cases $K_{\beta}(D(X,Y))\leq K_{\beta}(D(X,Z))$ and finally
$D(X,Y)\leq_{\alpha,\beta}D(X,Z)$.

(MD2) can be proved similarly to (MD1). (MD3′) follows from the fact that
$\mathcal{D}$ is a moderate deviation function and (MD4) immediately follows
from the idempotency of $C$. $\Box$

###### Example 2

  1. (i)

Taking $\mathcal{D}:[0,1]^{2}\to L(\mathbb{R})$ defined by
$\mathcal{D}(x,y)=y-x$ and $C:[0,1]^{2}\to[0,1]$ defined by
$C(x,y)=\max\left(0,\min(1,2y-x)\right)$, by Theorem 3 one obtains a
$w$-preserving interval-valued moderate deviation function w.r.t.
$\leq_{\alpha,\beta}$ for any $\alpha,\beta$.

  2. (ii)

A class of $w$-preserving interval-valued moderate deviation functions w.r.t.
$\leq_{\alpha,\beta}$ for any $\alpha,\beta$ can be obtained considering
$\mathcal{D}_{\varepsilon,\delta}:[0,1]^{2}\to L(\mathbb{R})$ defined for
positive constants $\varepsilon,\delta$ by (see Example 3.3 in [51]):

$\mathcal{D}_{\varepsilon,\delta}(x,y)=\left\\{\begin{array}[]{ll}y-x+\varepsilon,&\textrm{
if }y>x,\\\ 0,&\textrm{ if }y=x,\\\ y-x-\delta,&\textrm{ if
}y<x\end{array}\right.$ (20)

and $C:[0,1]^{2}\to[0,1]$ defined by
$C(x,y)=\max\left(0,\min(1,f(y)-f(x)+y)\right)$, where $f:[0,1]\to\mathbb{R}$
is any non-decreasing function.

Note that item (i) is a special case of item (ii) for $\varepsilon=\delta=0$
and $f=id$.

###### Example 3

In [53] (Theorem 6) a construction of a moderate deviation function
$\mathcal{D}:[0,1]^{2}\to[-M_{n},M_{p}]$ was introduced in the following way:

$\mathcal{D}(x,y)=\left\\{\begin{array}[]{ll}M_{p}-M_{p}R_{1}(x,y),&\textrm{
if }x\leq y,\\\\[2.84544pt] M_{n}R_{2}(x,y)-M_{n},&\textrm{ if
}x>y,\end{array}\right.$ (21)

for all $x,y\in[0,1]$, where $M_{n},M_{p}\in]0,\infty[$. For different choices
of restricted dissimilarity functions $R_{1},R_{2}$ we obtain different
moderate deviation functions. In particular, we give five examples, in each of
them the choice of $M_{p},M_{n}$ impact results where the ratio between
$M_{p}$ and $M_{n}$ is important since it expresses the emphasis we put on the
positive ($M_{p}$) or negative ($M_{n}$) deviation:

  1. (i)

If $R_{1}(x,y)=R_{2}(x,y)=1-|y-x|$, then

$\mathcal{D}(x,y)=\left\\{\begin{array}[]{ll}M_{p}(y-x),&\textrm{ if }x\leq
y,\\\\[2.84544pt] M_{n}(y-x),&\textrm{ if }x>y.\end{array}\right.$ (22)

  2. (ii)

If $R_{1}(x,y)=R_{2}(x,y)=1-|y^{2}-x^{2}|$, then

$\mathcal{D}(x,y)=\left\\{\begin{array}[]{ll}M_{p}(y^{2}-x^{2}),&\textrm{ if
}x\leq y,\\\\[2.84544pt] M_{n}(y^{2}-x^{2}),&\textrm{ if
}x>y.\end{array}\right.$ (23)

  3. (iii)

If $R_{1}(x,y)=R_{2}(x,y)=1-(y-x)^{2}$, then

$\mathcal{D}(x,y)=\left\\{\begin{array}[]{ll}M_{p}(y-x)^{2},&\textrm{ if
}x\leq y,\\\\[2.84544pt] -M_{n}(y-x)^{2},&\textrm{ if }x>y.\end{array}\right.$
(24)

  4. (iv)

If $R_{1}(x,y)=1-|y^{2}-x^{2}|$ and $R_{2}(x,y)=1-(y-x)^{2}$, then

$\mathcal{D}(x,y)=\left\\{\begin{array}[]{ll}M_{p}(y^{2}-x^{2}),&\textrm{ if
}x\leq y,\\\\[2.84544pt] -M_{n}(y-x)^{2},&\textrm{ if }x>y.\end{array}\right.$
(25)

  5. (v)

If $R_{1}(x,y)=1-(y-x)^{2}$ and $R_{2}(x,y)=1-|y^{2}-x^{2}|$, then

$\mathcal{D}(x,y)=\left\\{\begin{array}[]{ll}M_{p}(y-x)^{2},&\textrm{ if
}x\leq y,\\\\[2.84544pt] M_{n}(y^{2}-x^{2}),&\textrm{ if
}x>y.\end{array}\right.$ (26)

Based on the approach given in Theorem 3, we can build a $w$-preserving
interval-valued moderate deviation function $D(L([0,1]))^{2}\to L(\mathbb{R})$
in such a way that we combine one of the five restricted dissimilarity
functions $\mathcal{D}$ from items (i)-(v) with a function
$C:[0,1]^{2}\to[0,1]$ defined by

$C(x,y)=\max\left(0,\min(1,f(y)-f(x)+y)\right)$ (27)

for some $f:[0,1]\to\mathbb{R}$ being a non-decreasing function (for example
$Id$,…).

It is worth to point out that $wD$-mean is based on the idea to use moderate
deviation functions in a similar way as penalty functions are used to measure
the similarity or dissimilarity between a given set of data [54, 55]. The main
idea is, given a set of intervals, to determine another interval which
represents all of them and which is the most similar to all of them in the
sense determined by the moderate deviation function. That is, we are looking
for the interval $Y$ which makes the sum $D(X_{1},Y)+\ldots+D(X_{n},Y)$ to be
as close to $[0,0]$ as possible.

In what follows, we use our construction given by Theorem 3 to avoid the
computation of $\sup$ and $\inf$ while obtaining $wD$-mean.

###### Theorem 4

Let $\alpha\in[0,1]$, $n\in\mathbb{N}$, let $M_{p},M_{n}$ be positive real
numbers and $\mathcal{D}:[0,1]^{2}\to[-M_{n},M_{p}]$ be a moderate deviation
function. Let $F:L([0,1])^{n+1}\to\mathbb{R}$ be the function given, for all
$X_{1},\ldots,X_{n},Y\in L([0,1])$ such that
$w(Y)=\min(w(X_{1}),\ldots,w(X_{n}))$, by:

$F(X_{1},\ldots,X_{n},Y)=\mathcal{D}\Big{(}K_{\alpha}(X_{1}),K_{\alpha}(Y)\Big{)}+\ldots+\mathcal{D}\Big{(}K_{\alpha}(X_{n}),K_{\alpha}(Y)\Big{)}.$
(28)

Then

  1. (i)

If $\mathcal{D}$ is continuous, then, for each $n$-tuple
$(X_{1},\ldots,X_{n})\in L([0,1])^{n}$, there exists $Y\in L([0,1])$ such that
$w(Y)=\min(w(X_{1}),\ldots,w(X_{n}))$ and $F(X_{1},\ldots,X_{n},Y)=0$.

  2. (ii)

If $\mathcal{D}$ is strictly increasing in the second component, then, for
each $n$-tuple $(X_{1},\ldots,X_{n})\in L([0,1])^{n}$, there exists at most
one $Y\in L([0,1])$ such that $w(Y)=\min(w(X_{1}),\ldots,w(X_{n}))$ and
$F(X_{1},\ldots,X_{n},Y)=0$.

Proof: (i) Since the continuity of $\mathcal{D}$ implies the continuity of the
function $F(X_{1},\ldots,X_{n},\cdot)$, the proof follows from the observation
that $F(X_{1},\ldots,X_{n},Z_{1})\leq 0$ and $F(X_{1},\ldots,X_{n},Z_{2})\geq
0$ if
$K_{\alpha}(Z_{1})=\min\left(K_{\alpha}(X_{1}),\ldots,K_{\alpha}(X_{n})\right)$
and
$K_{\alpha}(Z_{2})=\max\left(K_{\alpha}(X_{1}),\ldots,K_{\alpha}(X_{n})\right)$.
Note that since $w(Y)$ is fully determined by fixed $n$-tuple
$(X_{1},\ldots,X_{n})$, the continuity of the function
$F(X_{1},\ldots,X_{n},\cdot)$ is considered in the sense of the standard
continuity of a real function with the variable $K_{\alpha}(Y)$.

(ii) Observe that the strict monotonicity of $D$ implies the strict
monotonicity (increasingness) of the function $F(X_{1},\ldots,X_{n},\cdot)$.
Again in the sense of a real function with the variable $K_{\alpha}(Y)$.
$\Box$

###### Remark 1

The previous theorem could be formulated in a more general way: instead of
$w(Y)=\min(w(X_{1}),\ldots,w(X_{n}))$ there can be
$w(Y)=B(w(X_{1}),\ldots,w(X_{n}))$ for any aggregation function $B$, but then
the domain of $Y$ should be extended, in particular, $Y\in L(\mathbb{R})$ in
the case of any $B$ and not $Y\in L([0,1])$ as it is in the case of minimum,
although $K_{\alpha}(Y)\in[0,1]$ in both cases. Since our goal is to show that
$Y$ is an output of $wD$-mean, we used this version of the theorem to keep the
range of $wD$-mean in $L([0,1])$.

The following corollary gives us a method of constructing $wD$-means based on
Theorem 3, Theorem 4 and Example 3.

###### Corollary 2

Under the assumptions of Theorem 4, where $\mathcal{D}:[0,1]^{2}\to\mathbb{R}$
is given by Equation (21) with $R_{1},R_{2}$ being continuous strictly
monotone restricted equivalence functions and $D:(L([0,1]))^{2}\to
L(\mathbb{R})$ is given by Theorem 3, the following statements are equivalent:

  1. (i)

$F(X_{1},\ldots,X_{n},Y)=0$;

  2. (ii)

$M_{D}(X_{1},\ldots,X_{n})=Y$, where the interval-valued $wD$-mean $M_{D}$ is
given by Equation (16) with $B=\min$;

  3. (iii)

$\sum\limits_{i=1}^{k}\bigg{(}M_{p}-M_{p}R_{1}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right),K_{\alpha}\left(Y\right)\Big{)}\bigg{)}\\\
+\sum\limits_{i=k+1}^{n}\bigg{(}M_{n}R_{2}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right),K_{\alpha}\left(Y\right)\Big{)}-M_{n}\bigg{)}=0$
(29)

where $\sigma:\\{1,\ldots,n\\}\to\\{1,\ldots,n\\}$ is a permutation such that
$X_{\sigma(1)}\leq_{\alpha,\beta}\ldots\leq_{\alpha,\beta}X_{\sigma(n)}$ and
$k$ is the greatest number from $\\{1,\ldots,n\\}$ satisfying

$\sum\limits_{i=1}^{n}\mathcal{D}(K_{\alpha}(X_{\sigma(i)}),K_{\alpha}(X_{\sigma(k)}))\leq
0.$ (30)

Moreover,
$K_{\alpha}(Y)\in[K_{\alpha}(X_{\sigma(k)}),K_{\alpha}(X_{\sigma(k+1)})[$
whenever $k<n$ and
$K_{\alpha}(Y)=K_{\alpha}(X_{\sigma(1)})=\ldots=K_{\alpha}(X_{\sigma(n)})$
whenever $k=n$.

Proof: First observe that, due to $(MD1)$ and $(MD2)$, we have: if $k$
satisfies Equation (30), then for all $p\in\\{1,\ldots,k\\}$ and all
$q\in\\{k+1,\ldots,n\\}$ it holds

$\sum\limits_{i=1}^{n}\mathcal{D}(K_{\alpha}(X_{\sigma(i)}),K_{\alpha}(X_{\sigma(p)}))\leq
0\qquad\\\ \textrm{ and
}\qquad\sum\limits_{i=1}^{n}\mathcal{D}(K_{\alpha}(X_{\sigma(i)}),K_{\alpha}(X_{\sigma(q)}))>0.$
(31)

Then the equivalence of (i) and (ii) follows from Theorem 4 and the
equivalence of (i) and (iii) follows from the observation:

$F(X_{1},\ldots,X_{n},Y)=\sum\limits_{i=1}^{k}\bigg{(}M_{p}-M_{p}R_{1}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right),K_{\alpha}\left(Y\right)\Big{)}\bigg{)}+\\\
\sum\limits_{i=k+1}^{n}\bigg{(}M_{n}R_{2}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right),K_{\alpha}\left(Y\right)\Big{)}-M_{n}\bigg{)}.$
(32)

$\Box$

###### Remark 2

Let $\alpha,\beta\in[0,1]$, $\alpha\neq\beta$. We build an IV $wD$-mean
$M_{D}:(L([0,1]))^{n}\to L([0,1])$ w.r.t. $\leq_{\alpha,\beta}$ according to
Definition 9. The construction is based on Theorem 4 and Corollary 2. Let us
denote

$M_{D}(X_{1},\ldots,X_{n})=Y.$

Then we need to obtain $K_{\alpha}(Y)$ and $w(Y)$, from which $Y$ can be
derived:

$Y=\big{[}K_{\alpha}(Y)-\alpha w(Y),K_{\alpha}(Y)+(1-\alpha)w(Y)\big{]}.$ (33)

Firstly we calculate $w(Y)$:

$w(Y)=\min\left(w(X_{1}),\ldots,w(X_{n})\right).$ (34)

Now we calculate $K_{\alpha}(Y)$. Recall that $\sigma$ is a permutation of
$\\{1,\ldots,n\\}$ such that
$X_{\sigma(1)}\leq_{\alpha,\beta}\ldots,\leq_{\alpha,\beta}X_{\sigma(n)}$,
then we proceed as follows:

  1. 1.

We obtain a switch point: it is the greatest number $k\in\\{1,\ldots,n\\}$
satisfying:

$\sum\limits_{i=1}^{n}\mathcal{D}\Big{(}K_{\alpha}(X_{\sigma(i)}),K_{\alpha}(X_{\sigma(k)})\Big{)}\leq
0.$ (35)

  2. 2.

We solve an equation ($K_{\alpha}(Y)$ is the desired solution):

$\sum\limits_{i=1}^{k}\bigg{(}M_{p}-M_{p}R_{1}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right),K_{\alpha}\left(Y\right)\Big{)}\bigg{)}+\sum\limits_{i=k+1}^{n}\bigg{(}M_{n}R_{2}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right),K_{\alpha}\left(Y\right)\Big{)}-M_{n}\bigg{)}=0$
(36)

where
$K_{\alpha}(Y)\in[K_{\alpha}(X_{\sigma(k)}),K_{\alpha}(X_{\sigma(k+1)})[$
whenever $k<n$ and
$K_{\alpha}(Y)=K_{\alpha}(X_{\sigma(1)})=\ldots=K_{\alpha}(X_{\sigma(n)})$
whenever $k=n$.

###### Example 4

For each particular choice of $\mathcal{D}$ (or $R_{1},R_{2}$) in cases
(i)-(v) of Example 3, the Equation (36) has the following form:

  1. (i)

$\sum\limits_{i=1}^{k}\bigg{(}M_{p}\Big{(}K_{\alpha}\left(Y\right)-K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}\bigg{)}+\\\
\sum\limits_{i=k+1}^{n}\bigg{(}M_{n}\Big{(}K_{\alpha}\left(Y\right)-K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}\bigg{)}=0$

and the solution is:

$K_{\alpha}(Y)=\frac{M_{p}\sum\limits_{i=1}^{k}K_{\alpha}\left(X_{\sigma(i)}\right)+M_{n}\sum\limits_{i=k+1}^{n}K_{\alpha}\left(X_{\sigma(i)}\right)}{kM_{p}+(n-k)M_{n}}$

  2. (ii)

$\Big{(}K_{\alpha}\left(Y\right)\Big{)}^{2}\Big{(}kM_{p}+(n-k)M_{n}\Big{)}-\\\
M_{p}\sum\limits_{i=1}^{k}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}-M_{n}\sum\limits_{i=k+1}^{n}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}=0$

and the solution is:

$K_{\alpha}\left(Y\right)=\sqrt{\frac{M_{p}\sum\limits_{i=1}^{k}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}+M_{n}\sum\limits_{i=k+1}^{n}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}}{kM_{p}+(n-k)M_{n}}}$

  3. (iii)

$\Big{(}K_{\alpha}\left(Y\right)\Big{)}^{2}\Big{(}kM_{p}-(n-k)M_{n}\Big{)}+\\\
K_{\alpha}\left(Y\right)\Big{(}2M_{n}\sum\limits_{i=k+1}^{n}K_{\alpha}\left(X_{\sigma(i)}\right)-2M_{p}\sum\limits_{i=1}^{k}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}+\\\
M_{p}\sum\limits_{i=1}^{k}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}-M_{n}\sum\limits_{i=k+1}^{n}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}=0$

  4. (iv)

$\Big{(}K_{\alpha}\left(Y\right)\Big{)}^{2}\Big{(}kM_{p}-(n-k)M_{n}\Big{)}+\\\
K_{\alpha}\left(Y\right)\Big{(}2M_{n}\sum\limits_{i=k+1}^{n}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}+\\\
M_{p}\sum\limits_{i=1}^{k}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}-M_{n}\sum\limits_{i=k+1}^{n}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}=0$

  5. (v)

$\Big{(}K_{\alpha}\left(Y\right)\Big{)}^{2}\Big{(}kM_{p}-(n-k)M_{n}\Big{)}+\\\
K_{\alpha}\left(Y\right)\Big{(}-2M_{p}\sum\limits_{i=1}^{k}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}+\\\
M_{p}\sum\limits_{i=1}^{k}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}-M_{n}\sum\limits_{i=k+1}^{n}\Big{(}K_{\alpha}\left(X_{\sigma(i)}\right)\Big{)}^{2}=0$

###### Remark 3

The expression of IV $wD$-mean $M_{D}:(L([0,1]))^{n}\to L([0,1])$ w.r.t.
$\leq_{\alpha,\beta}$ for applications can be obtained as follows:

  1. 1.

choose $\alpha,\beta\in[0,1]$, $\alpha\neq\beta$, which also determines the
order $\leq_{\alpha,\beta}$

  2. 2.

choose $R_{1},R_{2}$, some of the four expressions given in Example 3 (ii)-(v)

  3. 3.

calculate $w\Big{(}M_{D}(X_{1},\ldots,X_{n})\Big{)}$ using Equation (34)

  4. 4.

calculate switch point $k$ using Equation (35), where $\mathcal{D}$ is given
in Example 3 \- one of the four possibilities in (ii)-(v) according to the
choice of $R_{1},R_{2}$ in step 2)

  5. 5.

calculate $K_{\alpha}\Big{(}M_{D}(X_{1},\ldots,X_{n})\Big{)}$ using Equation
(36) - it means to solve one of the four quadratic equations in Example 4
(ii)-(v) according to the choice of $R_{1},R_{2}$ in step 2); the solution
from interval
$\left[K_{\alpha}\left(X_{\sigma(k)}\right),K_{\alpha}\left(X_{\sigma(k+1)}\right)\right]$
is one we are looking for

  6. 6.

calculate $M_{D}(X_{1},\ldots,X_{n})$ using Equation (33)

Comment: Note that this gives us four different expressions of IV $wD$-mean
$M_{D}$, according to the choice (ii)-(v), and each expression further depends
on the choice of $M_{p},M_{n}$ ($M_{p}$ reflects the strongness of penalty for
positive deviation and $M_{n}$ for negative - so important is their ratio - if
$M_{p}=M_{n}$ it means that the positive deviation has the same impact as the
negative deviation …).

##  V Interval-valued aggregation functions and BCI frameworks

In this section we present the two MI BCI frameworks we have used in our
experimentation: the Traditional Framework in Section V-A and the Multimodal
Fuzzy Fusion framework in Section V-B. Then, we explain how we apply the
Interval-valued moderate deviations in both cases, in Section V-C.

###  V-A Traditional BCI framework

Figure 1: Components of the Traditional and Multimodal Fusion BCI Frameworks.

The traditional BCI system was proposed in [20]. Its structure includes four
parts (Fig. 1):

  1. 1.

The first step is acquiring the EEG data from a EEG device and performing
band-pass filtering and artefact removal on the collected EEG signals.

  2. 2.

The second step is EEG feature transformation and feature extraction. Usually,
the FFt is used to rapidly transform the EEG signals into different frequency
components [56]. The FFt analysis transforms the time-series EEG signals in
each channel into the specified frequency range, which in our case is from 1
to 30 Hz, covering the delta ($\delta$) 1-3 Hz, theta ($\theta$) 4-7 Hz, alpha
($\alpha$) 8-13 Hz, beta ($\beta$) 14-30 Hz and All 1-30Hz bands using a
50-point moving window segment overlapping 25 data points. Although some
redundancy is included in the system, the All band is considered in order to
study possible interactions among non-adjacent frequencies and to gather
additional features for the subsequent CSP and classifiers.

  3. 3.

Subsequently, the CSP is used for feature extraction to extract the maximum
spatial separability from the different EEG signals corresponding to the
control commands. The CSP is a well-known supervised mathematical procedure
commonly used in EEG signal processing. The CSP is used to transform
multivariate EEG signals into well-separated subcomponents with maximum
spatial variation using the labels for each example [9], [57], [58].

  4. 4.

Last, pattern classification is performed on the CSP features signals using an
ensemble of classifiers to differentiate the commands. Each base classifier is
trained using a different wave band (for instance, if the base classifier is
the LDA, the ensemble would be composed of: $\delta-LDA$, $\theta-LDA$,
$\alpha-LDA$, $\beta-LDA$, and $All-LDA$) and the final decision is taken
combining all of them. The most common way of obtaining the final decision is
to compute the arithmetic mean of the outputs of all the base classifiers
(each one provides a probability for each class), and take the class with
highest aggregated value. Some of the most common classifiers used for this
task are the LDA, QDA and KNN [59]. This part would correspond to the orange
box in Fig. 1, labeled as “Traditional”.

For this work, we have used LDA classifiers in the traditional framework, as
they are very popular in the BCI literature [5, 60, 61, 62, 63].

###  V-B Multimodal Fuzzy Fusion BCI framework

The Multi-modal Fuzzy Framework (MFF) is proposed in [19]. It follows a
similar structure to the one in the traditional BCI framework (Fig. 1): it
starts with the EEG measurements, it computes the FFt transformation to the
frequency domain and it uses the CSP transform to obtain a set of features to
train the classifiers.

However, in the MFF it is necessary to train not one, but three classifiers
for each wave band: a LDA, a QDA and a KNN. We name the classifiers according
to their type of classifier and the wave band used to train it. For instance,
for the $\delta$ band we would have $\delta-LDA$, $\delta-QDA$ and $\delta-
KNN$.

Then, the decision making phase is performed in two phases (This part would
correspond to the green box in Fig. 1, labeled as “Multimodal Fusion
Framework”):

  1. 1.

Frequency phase: since we got a LDA, QDA and KNN for each wave band, the first
step is to fuse the outputs of these classifiers in each wave band. For
example, in the case of the LDA classifiers, we have a $\delta-LDA$, $\theta-
LDA$, $\alpha-LDA$, $\beta-LDA$ and $All-LDA$ that will be fused them using an
aggregation function to obtain a vector, $FF-LDA$. That is, the same process
explained for the traditional framework is applied but without making the
final decision. We do the same with the QDA and KNN classifiers. The result of
this phase is a list of collective vectors (one for each type of classifier).

  2. 2.

Classifier phase: in this phase, the input is the list of collective vectors
given by each different kind of classifier ($FF-LDA$, $FF-LDA$, $FF-KNN$)
computed in the frequency phase. We fuse the three vectors according to the
classes, and the result is a vector containing the score for each class for
the given sample. As in the traditional framework, the decision is made in
favour to the class associated with the largest value.

We must point out that the same aggregation is used for both the frequency
phase and the classifier phases.

The aggregation functions tested in the MFF are the Choquet integral, the CF
integral using the Hamacher T-norm, the $CF_{min,min}$ generalizations, the
Sugeno integral and the Hamacher Sugeno integral [19, 26, 64].

###  V-C Interval-valued Moderate Deviations applied in the BCI frameworks

We have tested the interval-valued moderate deviation functions in the two
different MI BCI frameworks previously introduced. The idea in both cases is
to replace the existing aggregations for the classifier outputs (the
arithmetic mean in the traditional and the fuzzy integrals in the MFF) for our
new developed ones.

First, we construct the intervals from the probability for each class obtained
from each classifier. We use the length of the intervals to measure the
inaccuracies or uncertainties related to these classifiers’ outputs. To do so,
we have used the mapping in [65]:

$F(x,y)=[\underline{I(x,y)},\overline{I(x,y)+y}]$ (37)

where I is an fuzzy implication function, $x$ is the probability for each
class obtained from the classifier and we set $y$ as $0.3$. We crop the values
so that they are contained in the $[0,1]$ interval. We have tried three
different fuzzy implication functions to construct the intervals: the
Łukasiewicz fuzzy implication function, the Reichenbach fuzzy implication
function and the Kleene-Dienes fuzzy implication function.

Then, we aggregate the interval-valued logits from the classifiers using an
interval-valued moderate deviation based function. We have constructed the
deviation function using the Eq. (21). We have named $MD_{1}$ to the interval-
valued moderate deviation using $R_{1}(x,y)=R_{2}(x,y)=1-\left|y-x\right|$ and
the $MD_{2}$ setting $R_{1}=1-(y-x)^{2}$ and $R_{2}(x,y)=1-|y^{2}-x^{2}|$.

In the traditional BCI framework, we follow this interval construction
algorithm and the $MD_{1}$ and $MD_{2}$ respectively to substitute the
“Average” block in Figure 1. In the case of the MFF we use them to substitute
all the “Integral” blocks in Fig. 1.

To construct the interval-valued moderate deviation functions we need to set
the parameters $M_{p}$ and $M_{n}$. We have optimized them by taking 200
samples of different pairs in the $[1,100]$ range and testing the accuracy
against the training set. We have opted for this method as it seemed to
surpass other optimization algorithms in terms of computational time,
obtaining similar accuracy results.

##  VI Experimental Results in Motor-Imagery based BCI

In this section we discuss the behaviour of our new approaches in the BCI
competition IV dataset 2a (IV-2a) and the BCI competition IV dataset 2b
(IV-2b), which are detailed in [66]. The IV-2a dataset consists of four
classes of MI tasks: tongue, foot, left-hand and right-hand performed by 9
volunteers. For each task, 22 EEG channels were collected. There is a total of
288 trials for each participants, equally distributed among the 4 classes. The
IV-2b dataset consists of three different EEG channels for each subject, who
performed two different motor imagery tasks: moving the right hand or moving
the left hand.

For our experimental setup, we have used 4 out of the 22 channels available in
the IV-2a dataset, (C3, C4, CP3, CP4), as they reported good results in [19].
For the IV-2b dataset, we have used the three channels available. As features,
we have used the FFt to obtain the $\delta$, $\theta$, $\alpha$, $\beta$,
SensoriMotor Rhythm (SMR) ($13-15$Hz) and All, and we have used a CSP filter
with 25 components. From each subject, we have generated twenty partitions
(50% train and 50% test). So, this produces a total of 180 datasets. The
accuracy for each framework is computed as the average for all of them.

For the IV-2a, we have studied the full four classes task, and we have also
studied the binary classification task Left hand vs Right hand, as it is
common practice in the literature [19, 67, 68]. Results for each individual
subject are available at https://github.com/Fuminides/interval_md_bci_results.

###  VI-A Left/Right hand task

In Table I we have shown the results obtained using the $MD_{1}$ aggregation
in the Left/Right hand task. We have used the three considered fuzzy
implication functions and the two possible MI BCI frameworks. We found that
the Łukasiewicz fuzzy implication function is the one that works best for both
frameworks in the IV-2a dataset. For the IV-b dataset, we found the
Reichenbach operator is the best one in the traditional framework and the
three operators gave very similar results for the MFF.

In Table II we have displayed the results using the $MD_{2}$ aggregation
function. We found that in this case that in the IV-a dataset the best fuzzy
implication function is the Reichenbach fuzzy implication function in the
traditional framework and the Łukasiewicz one for the MFF. Results are higher
or equal in all cases using this function compared to the MD1. The best result
is the 0.8251 obtained using the MD2 traditional framework with the
Reichenbach fuzzy implication function. In the IV-2b dataset, we found very
similar performance for any of the implication operators and we also noted
that the MFF performed better in all cases than the traditional framework.

Table I: Accuracies for the binary task using the $MD_{1}$ as aggregation function. | Trad. | MFF
---|---|---
Kleene-Dienes | $0.7443\pm 0.1840$ | $0.7513\pm 0.1906$
Lukasiewicz | $0.7548\pm 0.1962$ | $0.7566\pm 0.1871$
Reichenbach | $0.7443\pm 0.1840$ | $0.7513\pm 0.1906$
Table II: Accuracies for the binary task using the $MD_{2}$ as aggregation function. | Trad. | MFF
---|---|---
Kleene-Dienes | $0.7443\pm 0.1840$ | $0.7513\pm 0.1906$
Lukasiewicz | $0.8146\pm 0.0541$ | $0.8191\pm 0.0395$
Reichenbach | $0.8251\pm 0.0538$ | $0.8184\pm 0.0415$

###  VI-B Four classes task

In Table III we have shown the results obtained using the $MD_{1}$ aggregation
in the left hand, right hand, tongue and foot task for the three considered
fuzzy implication functions and the two frameworks. In this case we obtained
the best result using the traditional framework and the Łukasiewicz fuzzy
implication function.

In Table IV we have shown the results obtained using the $MD_{2}$ aggregation
for all the fuzzy implication functions and the two frameworks. The MFF
framework achieved its best result using the Łukasiewicz fuzzy implication
function. The MD2 always performed better the the MD1 in this case too.

The best results for this task, $0.6943$, were achieved using the MFF, the
Łukasiewicz and the MD2.

Table III: Accuracies for the four classes task using the $MD_{1}$ as aggregation function. | Trad. | MFF
---|---|---
Kleene-Dienes | $0.6256\pm 0.0352$ | $0.6217\pm 0.1904$
Lukasiewicz | $0.6358\pm 0.0366$ | $0.6073\pm 0.2112$
Reichenbach | $0.6256\pm 0.0352$ | $0.6217\pm 0.1904$
Table IV: Accuracies for the four classes task using the $MD_{2}$ as aggregation function. | Trad. | MFF
---|---|---
Kleene-Dienes | $0.6256\pm 0.0352$ | $0.6217\pm 0.1904$
Lukasiewicz | $0.6406\pm 0.0396$ | $0.6943\pm 0.0338$
Reichenbach | $0.6471\pm 0.0359$ | $0.6905\pm 0.0328$

###  VI-C Comparison to the non-interval-valued case

We have compared the best interval-valued MD aggregation, the MD2, with the
standard arithmetic average aggregation in the traditional framework and the
Choquet integral in the MFF framework, as it was the best aggregation for the
MFF in [19].

In Table V there are the results for the Traditional and MFF frameworks in the
Left/Right hand task without using interval-valued aggregations, and the best
result found in Sections VI-A and VI-B. In this case, we obtained a moderate
increase in accuracy compared to the numerical results for the case of IV-2a
dataset, but not in the case of the IV-b, in which the Choquet integral
performed best. We also performed an analogous study for the four classes
task, reported in Table VI. We found in this case that all the configurations
performed almost equally, although the MD2 performed slightly better than the
Choquet MFF.

In Tab. VII, we have computed the statistical differences for the left/right
hand task among the numerical aggregations and the MD2. We have found
significant differences favoring the MD2 aggregation for the case of the IV-2a
dataset and the Choquet MFF for the IV-2b dataset using Kruskal test and the
Anderson post-hoc to compute the respective $P$-values. We have performed the
analogous study for the four classes task in Tab. VIII. We have found in that
case that the difference favouring the MD2 is not significant compared to the
Choquet integral in the MFF, but it is in the case of the arithmetic mean in
the traditional framework.

Table V: Performance for the Left/Right classes task using different aggregation functions. Dataset | Classifier | Accuracy
---|---|---
| Trad. Average | $0.8119\pm 0.0384$
IV-2a | MFF Choquet | $0.8049\pm 0.0417$
|  MD2 (Trad.) | $0.8251\pm 0.0538$
| Trad. Average | $0.7213\pm 0.0824$
IV-2b | MFF Choquet | $0.7440\pm 0.0058$
|  MD2 (Trad.) | $0.7366\pm 0.0708$
Table VI: Performance for the four classes task using different aggregation functions. Dataset | Classifier | Accuracy
---|---|---
| Average (Trad) | $0.6494\pm 0.0369$
IV-2a | Choquet (MFF) | $0.6919\pm 0.0314$
|  MD2 (MFF) | $0.6943\pm 0.0338$
Table VII: $P$-values for the Left/Right hand task, comparing numerical aggregations to the MD2. $*$ marks statistical differences ($P$-value lesser than 0.05). Dataset |  | Trad. Average | MFF Choquet
---|---|---|---
IV-2a | MFF Choquet | $.07$ |
|  Md2 | $*$ | $*$
|  | Trad. Average | MFF Choquet
IV-2b | MFF Choquet | $*$ |
|  Md2 | $*$ | $*$
Table VIII: $P$-values for the four classes task, comparing numerical aggregations to the MD2. $*$ marks statistical differences ($P$-value lesser than 0.05). Dataset |  | Trad. Average | MFF Choquet
---|---|---|---
IV-2a | MFF Choquet | $*$ |
|  Md2 | $*$ | $.25$

###  VI-D Comparison against interval-valued OWA operators

To compare the results obtained using the interval-valued moderate deviations
with other interval-valued aggregations, we have compared them against the
results obtained using interval-valued OWA operators. We have used three OWA
operators: OWA1, OWA2 and OWA3, as described in Section II-D. We have computed
the Kruskal statistical test and the Anderson post-hoc to look for significant
differences among each method, using a level of significance of 0.05.

In Tab. IX we show the comparison among the interval-valued OWAs and the MD2
for the left/right hand task. We show for each aggregation the result for the
best configuration (each configuration is composed of one the three different
fuzzy implication functions and one of the two different frameworks) for each
aggregation. We found that the $MD_{2}$ is the best aggregation for this task,
followed by $OWA_{2}$, $OWA_{1}$, $OWA_{3}$. The Kruskal test found
statistical differences among the different aggregations, and the results for
the Anderson post-hoc are in Tab. X, which shows that the MD2 performs
statistically better than the rest of the aggregations.

In Tab. XI we display the comparison among the interval-valued OWAs and the
MD2 for the four classes classification task. The results do not change much
when it comes to decide which aggregation is the best, and the MD2, is again
the best aggregation. The Kruskal test found statistical differences among the
different aggregations, and the results for the post-hoc are in Tab. XII,
which shows that the MD2 performs statistically better than the rest of the
aggregations, just as in the Left/Right hand task.

Table IX: Best accuracy for each different aggregation functions in the Left/Right hand task. Dataset | Classifier | Accuracy
---|---|---
IV-2a |  MD2 | $0.8251\pm 0.0415$
|  OWA1 | $0.8044\pm 0.0476$
|  OWA2 | $0.8086\pm 0.0465$
|  OWA3 | $0.7993\pm 0.0483$
IV-2b |  MD2 | $0.7366\pm 0.0708$
|  OWA1 | $0.7300\pm 0.0800$
|  OWA2 | $0.7312\pm 0.0285$
|  OWA3 | $0.7296\pm 0.0786$
Table X: $P$-values for the Left/Right hand task comparison for the interval-
valued aggregations. $*$ marks statistical differences ($P$-value lesser than
0.05).

Dataset | Trad. |  MD2 |  OWA1 |  OWA2 |  OWA2
---|---|---|---|---|---
IV-2a |  MD2 |  | $*$ | $*$ | $*$
|  OWA1 |  |  | $.25$ | $.25$
|  OWA2 |  |  |  | $.25$

MFF |  MD2 |  OWA1 |  OWA2 |  OWA2
---|---|---|---|---
MD2 |  | $*$ | $*$ | $*$
OWA1 |  |  | $.25$ | $.25$
OWA2 |  |  |  | $.13$

Table XI: Best accuracy for each different aggregation functions in the four classes hand task. Dataset | Classifier | Accuracy
---|---|---
IV-2a |  MD2 | $0.6943\pm 0.0338$
|  OWA1 | $0.6686\pm 0.0353$
|  OWA2 | $0.6731\pm 0.0344$
|  OWA3 | $0.6494\pm 0.0374$
Table XII: $P$-values for four classes task comparison for the interval-valued
aggregations. $*$ marks statistical differences ($P$-value lesser than 0.05).

Dataset | Trad. |  MD2 |  OWA1 |  OWA2 |  OWA3
---|---|---|---|---|---
IV-2a |  MD2 |  | $*$ | $*$ | $*$
|  OWA1 |  |  | $.25$ | $.25$
|  OWA2 |  |  |  | $.13$

MFF |  MD2 |  OWA1 |  OWA2 |  OWA3
---|---|---|---|---
MD2 |  | $*$ | $*$ | $*$
OWA1 |  |  | $.25$ | $*$
OWA2 |  |  |  | $*$

##  VII Comparison with other BCI frameworks

In this Section we have tested the results obtained using the interval-valued
MD2 with other non aggregation-based MI BCI frameworks. We have compared our
proposal with the results in [44] where the authors use multiscale temporal
features applied to CSP and Riemannian features; with the works in [46], where
the authors use two Convolutional Neural Networks (CNNs) to solve different
BCI tasks, one with two blocks of Convolution and Image reduction, and the
other with four; and with [45] where the authors propose a new Deep Learning
architecture with different types of convolutions.

We have computed these comparisons using the same evaluation process as in
Section VI, with 180 train/test partitions for both datasets. The results are
displayed in Table XIII. For the case of the IV-2a dataset, we have found our
results to be inferior to the CSP and Riemannian Multiscale features, and
superior to those obtained using CNNs, probably because the CNNs require a
much larger number of observations to satisfactory train the model. For the
IV-2b dataset, the performance of the compared methods is more similar than in
the other dataset. In this problem, we found our approach to be the best,
closely followed by the CSP multiscale feature and the EEG net.

Table XIII: Comparison with non aggregation-based frameworks.

Framework IV-2a IV-2b Riemannian Multi Cov. [44] $0.7328\pm 0.1325$ $0.6763\pm
0.1102$ CSP Multi Cov. [44] $0.7350\pm 0.1507$ $0.7258\pm 0.1181$ Shallow CNN
[46] $0.4862\pm 0.1225$ $0.6996\pm 0.1387$ Deep CNN [46] $0.3956\pm 0.0717$
$0.7045\pm 0.1259$ EEG Net [45] $0.5747\pm 0.1063$ $0.7229\pm 0.1281$
Łukasiewicz $MD_{2}$ MFF $0.6943\pm 0.0338$ $0.7310\pm 0.0470$ Reinchenbach
$MD_{2}$ MFF $0.6905\pm 0.0328$ $0.7366\pm 0.0708$

##  VIII Conclusions

In this work we have presented the interval-valued moderate deviations as a
means to aggregate interval-valued data. We have extended the notion of
moderate deviation function to the interval-valued setting, we have analyzed
different properties and we have proposed different construction methods. We
have, in particular, studied the case where the width of all the input
interval-valued data is the same, and those interval-valued moderate deviation
functions which preserve it. We have applied the interval-valued moderate
deviation functions in the decision making phase of two MI BCI frameworks,
using fuzzy implication functions to measure the effects of noise in the EEG
measurements in each classifier output. We have studied two different tasks:
to discriminate between left hand and right hand classes and among left hand,
right hand, tongue and foot classes. We found that the results using interval-
valued moderate deviation functions outperform the rest of decision making
schemes using other numerical and interval-valued aggregations, except for the
case of the Choquet integral in the IV-2b dataset using the MFF.

Regarding non aggregation-based BCI frameworks, we found our proposal to beat
CNN approaches, but we found our results not as good as the MI BCI framework
that used CSP Multiscale Covariance. Since this method focuses on feature
extraction, while ours is devoted to improve the decision making phase, we
think that combining both approaches can be studied in order to further
improve the current results.

In our future works we intend to develop moderate deviation based-aggregation
functions for $n$-component vectors, and to further explore the combination of
aggregation-based MI BCI frameworks with other MI BCI paradigms.

## Acknowledgment

Javier Fumanal Idocin’s, Jose Antonio Sanz’s, Javier Fernandez’s, Harkaitz
Goyena’s and Humberto Bustince’s research has been supported by the project
PID2019-108392GB I00 (AEI/10.13039/501100011033).

Z. Takáč acknowledges the support of the grant VEGA 1/0545/20.

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|  Javier Fumanal Idocin holds a B.Sc in Computer Science at the University of
Zaragoza, Spain (2017) and a M.Sc in Data Science and Computer Engineering at
the University of Granada, Spain (2018). He is now a PhD Student of the Public
University of Navarre, Spain in the department of Statistics, Informatics and
Mathematics. His research interests include machine intelligence, fuzzy logic,
social networks and Brain-Computer Interfaces.
---|---
|  Zdenko Takáč received the Graduate degree in teaching mathematics and
physics from the Faculty of Mathematics and Physics, Bratislava, Slovakia, in
1998, and the Ph.D. degree in teaching mathematics with the thesis analysis of
mathematical proof from Pavol Jozef afrik University, Koice, Slovakia, in
2007.,Since 1999, he has been a member of the Department of Mathematics,
Faculty of Education, Catholic University in Ruomberok, Ruomberok, Slovakia,
and since 2010, he has been a member of the Department of Mathematics,
Institute of Information Engineering, Automation and Mathematics, Slovak
University of Technology in Bratislava, Bratislava, Slovakia. His research
interests include uncertainty modeling, fuzzy sets and fuzzy logic,
aggregation operators, interval-valued, and type-2 fuzzy sets.
---|---
|  Javier Fernandez received the M.Sc. and Ph.Degrees in Mathematics from the
University of Zaragoza, Saragossa, and the University of The Basque Country,
Spain, in 1999 and 2003, respectively. He is currently an Associate Lecturer
with the Department of Statistics, Computer Science and Mathematics Public
University of Navarre, Pamplona, Spain. He is the author or coauthor of
approximately 45 original articles and is involved with teaching artificial
intelligence and computational mathematics for students of the computer
sciences and data science. His research interests include fuzzy techniques for
image processing, fuzzy sets theory, interval-valued fuzzy sets theory,
aggregation functions, fuzzy measures, deep learning, stability, evolution
equation, and unique continuation.
---|---
|  José Antonio Sanz José Antonio Sanz received the M.Sc. and Ph.D. degrees in
computer sciences in 2008 and 2011 respectively, both from the Public
University of Navarra, Spain. He is currently an assistant professor at the
Department of Statistics, Computer Science and Mathematics, Public University
of Navarra. He is the author of 36 published original articles in
international journals and more than 60 contributions to conferences. He is a
member of the European Society for Fuzzy Logic and Technology (EUSFLAT) and
the Spanish Association of Artificial Intelligence (AEPIA). He received the
best paper award in the FLINS 2012 international conference and the Pepe Millá
award in 2014. His research interests include fuzzy techniques for
classification problems, interval-valued fuzzy sets, genetic fuzzy systems and
medical applications using soft computing techniques.
---|---
|  Harkaitz Goyena Harkaitz Goyena received the Graduate degree in Computer
Science, specializing in Computer Science and Intelligent Systems from the
Public University of Navarra, Pamplona, Spain, in 2020. He is currently
studying a master’s degree in mathematical investigation and modeling,
statistics and computing.
---|---
|  Chin-Teng Lin received the B.S. degree from National Chiao-Tung University
(NCTU), Taiwan in 1986, and the Master and Ph.D. degree in electrical
engineering from Purdue University, USA in 1989 and 1992, respectively. He is
currently the Distinguished Professor of Faculty of Engineering and
Information Technology, and Co-Director of Center for Artificial Intelligence,
University of Technology Sydney, Australia. He is also invited as Honorary
Chair Professor of Electrical and Computer Engineering, NCTU, International
Faculty of University of California at San-Diego (UCSD), and Honorary
Professorship of University of Nottingham. Dr. Lin was elevated to be an IEEE
Fellow for his contributions to biologically inspired information systems in
2005, and was elevated International Fuzzy Systems Association (IFSA) Fellow
in 2012. Dr. Lin received the IEEE Fuzzy Systems Pioneer Awards in 2017. He
served as the Editor-in-chief of IEEE Transactions on Fuzzy Systems from 2011
to 2016. He also served on the Board of Governors at IEEE Circuits and Systems
(CAS) Society in 2005-2008, IEEE Systems, Man, Cybernetics (SMC) Society in
2003-2005, IEEE Computational Intelligence Society in 2008-2010, and Chair of
IEEE Taipei Section in 2009-2010. Dr. Lin was the Distinguished Lecturer of
IEEE CAS Society from 2003 to 2005 and CIS Society from 2015-2017. He serves
as the Chair of IEEE CIS Distinguished Lecturer Program Committee in 2018
2019. He served as the Deputy Editor-in-Chief of IEEE Transactions on Circuits
and Systems-II in 2006-2008. Dr. Lin was the Program Chair of IEEE
International Conference on Systems, Man, and Cybernetics in 2005 and General
Chair of 2011 IEEE International Conference on Fuzzy Systems. Dr. Lin is the
coauthor of Neural Fuzzy Systems (Prentice-Hall), and the author of Neural
Fuzzy Control Systems with Structure and Parameter Learning (World
Scientific). He has published more than 330 journal papers (Total Citation:
22,418, H-index: 67, i10-index: 284) in the areas of neural networks, fuzzy
systems, brain computer interface, multimedia information processing, and
cognitive neuro-engineering, including about 120 IEEE journal papers.
---|---
|  Yu-Kai Wang (M’13) received the B.S. degree in mathematics education from
National Taichung University of Education, Taichung, Taiwan, in 2006, the M.S.
degree in biomedical engineering from National Chiao Tung University (NCTU),
Hsinchu Taiwan, in 2009, and the Ph.D. degree in computer science from NCTU,
Hsinchu Taiwan, in 2015. He is currently a Lecturer of Faculty of Engineering
and Information Technology, and Co-Director of Motion Platform and Mixed
Reality Lab, University of Technology Sydney, Australia. His current research
interests include computational neuroscience, human performance modeling,
biomedical signal processing, and the brain–computer interface.
---|---
|  Humberto Bustince received the Graduate degree in physics from the
University of Salamanca in 1983 and Ph.D. in mathematics from the Public
University of Navarra, Pamplona, Spain, in 1994. He is a Full Professor of
Computer Science and Artificial Intelligence in the Public University of
Navarra, Pamplona, Spain where he is the main researcher of the Artificial
Intelligence and Approximate Reasoning group, whose main research lines are
both theoretical (aggregation functions, information and comparison measures,
fuzzy sets, and extensions) and applied (image processing, classification,
machine learning, data mining, and big data). He has led 11 I+D public-funded
research projects, at a national and at a regional level. He is currently the
main researcher of a project in the Spanish Science Program and of a
scientific network about fuzzy logic and soft computing. He has been in charge
of research projects collaborating with private companies. He has taken part
in two international research projects. He has authored more than 210 works,
according to Web of Science, in conferences and international journals, with
around 110 of them in journals of the first quartile of JCR. Moreover, five of
these works are also among the highly cited papers of the last ten years,
according to Science Essential Indicators of Web of Science. Dr. Bustince is
the Editor-in-Chief of the online magazine Mathware & Soft Computing of the
European Society for Fuzzy Logic and technologies and of the Axioms journal.
He is an Associated Editor of the IEEE Transactions on Fuzzy Systems Journal
and a member of the editorial board of the Journals Fuzzy Sets and Systems,
Information Fusion, International Journal of Computational Intelligence
Systems and Journal of Intelligent & Fuzzy Systems. He is the coauthor of a
monography about averaging functions and coeditor of several books. He has
organized some renowned international conferences such as EUROFUSE 2009 and
AGOP. Honorary Professor at the University of Nottingham, National Spanish
Computer Science Award in 2019 and EUSFLAT Excellence Research Award in 2019.
---|---