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Journal of Biostatistics and Biometric Applications

ISSN: 2455-765X

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Research Article
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Comparison of Statistical Techniques for Forecasting Malaria Cases in Ghana

Received Date: March 01, 2019 Accepted Date: June 17, 2019 Published Date: June 19, 2019

Copyright: © 2019 Ashraf S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Cuong’s picture fuzzy set (PFS) has more capability to grip the uncertainties in real-life decision-making problems as compare to intuitionistic fuzzy set. The purpose of this paper is to introduce a new Grey approach in which the attribute values takes from the picture fuzzy numbers, attribute weights information is unknown and develop a multi criteria decision making approach to study the interaction between the input argument under the picture fuzzy environment. The main advantage of the proposed technique is that, it can deal with the situations of the positive interaction, negative interaction or non-interaction among the criteria, during decision-making process. Finally, a numerical approach is demonstrated for implementation of proposed technique and show that how proposed technique is reliable and effective is illustrated.

Keywords: Picture Fuzzy Set; Averaging Aggregation Operator; Geometric Aggregation Operator; Normalized Hamming Distance; Grey Approach

Introduction

In the life of human beings’ decision making is a worldwide procedure, which can be designated as final consequence of certain spiritual and intellectual procedures that commanding to variety of the finest alternative. Based on fuzziness circumstances; assigned membership grades to elements of a set in the interval [0,1] by offering the idea of fuzzy set (FS). Zadeh's work in this direction is remarkable as many of set theoretic properties of crisp cases were defined for fuzzy sets. In FSs each element "x"of the domain set contains only one index namely as degree of membership "()"Px which oscillates from 0 to 1. Non-membership degree for the FS is straightforward equivalent to "1()"Px−. FSs got the attentions of researchers and found its applications in decision science, intelligence science, communications, engineering, computer sciences etc [1].

Form last few decades, the intuitionistic fuzzy set; fruitful and broadly utilized by the researchers to grip the ambiguity and imprecision data [2]. To cumulate all the executive of criteria for alternatives, aggregation operators, play vital character throughout the information merging procedure. Xu Z, presented weighted averaging operator while developed geometric aggregation operator for aggregating the different intuitionistic fuzzy numbers [3,4]. In some decision theories, the decision makers deal with the situation of particular attributes where values of their summation of membership degrees exceed 1. In such condition, IFSs has no ability to obtain any satisfactory result. To overcome this situation Yager RR developed the idea of Pythagorean fuzzy set as a generalization of IFS, which satisfies that the value of square summation of its membership degrees is less then or equals to 1. In particular situation where the neutral membership degree calculates independently in real life problems [5]. In such condition, Pythagorean fuzzy set fails to attain any satisfactory result. Based on these circumstances, to overcome this situation, Cuong BC and Kreinovitch V, initiated the idea of picture fuzzy set (PFSs) [6]. They utilized three indexes (membership degree"()"Px, neutral-membership degree "()"Ixand non-membership degree "()"Nx ) in PFS with axiom that is 0≤P(x)+I(x)+N(x)≤1.

PFS is the generalization of FS and IFS. Obviously PFSs are more suitable than IFS to deal with fuzziness and vagueness. Garg H, introduced picture fuzzy weighted averaging operator, Picture fuzzy ordered weighted averaging operator and Picture fuzzy hybrid averaging operator under picture fuzzy environment [7]. In Singh P, introduced a correlation coefficient for the PFS [8]. Wei G, established decision making technique depending on the picture fuzzy weighted cross-entropy and used to differentiate the alternatives [9]. In Ashraf et al., investigated the multiple attribute decision problems based on the picture fuzzy setting, also developed some picture fuzzy geometric operators and discussed their basic properties [10]. In Zeng et al., proposed the application of exponential Jensen picture fuzzy divergence measure in multi-criteria group decision making problems [11]. In Khan et al., proposed the concept of generalized picture fuzzy soft set and discussed their applications in decision making problems. In Ashraf et al., proposed the concept of cubic picture fuzzy set and discussed their applications [13]. In Ashraf et al., proposed the concept of picture fuzzy linguistic fuzzy set and discussed their applications in decision making problems [14].

Sometime, DM utilizes the picture fuzzy information and due to lack of knowledge about domain of problem and time limiting stress, the knowledge about weights is unknown. Existing intuitionistic fuzzy grey method will flop to handle picture fuzzy information where information about weights is unknown [15,16]. Utilizing incomplete weight information of the alternatives as well as the picture fuzzy information to construct weights of the alternative to use grey approach which is a finest and motivational research issue. Therefore, this work upgrades the idea of grey to establish new methodology for resolving decision problems under picture fuzzy setting, in which we assumed incomplete known information about weights of alternative.

The objectives of this paper are: (1) to discuss the picture fuzzy number (PFNs) and related basic operational identities, (2) to suggest score and accuracy functions for comparison, (4) to propose picture aggregation operators and some debate on their properties, (5) to demonstrate a MADM method based on to traditional grey method with incomplete weight information under picture fuzzy setting,

The superfluity of this paper is planned as follows. Section "Preliminaries" gives brief reassess the initial ideas related to IFSs and PFSs and their properties. In next sections "Comparison Rules for PFNs" and "Picture Fuzzy Number Aggregation Operators" define a rule which utilized to rank the alternatives and then present the aggregated operators. In sections "Grey Relation for Decision Making Based on Picture Fuzzy Setting", MADM method is proposed to deal with picture fuzzy setting and in last descriptive examples to express the effectiveness and reliability of the suggested technique, is illustrated.

Preliminaries

The paper gives brief discussion on basic ideas associated to IFS and PFS with their operations and operators. Also discuss more familiarized ideas which utilized in following analysis.

Definition 1: Let the universe set R. Then A is known to be an IFS of R, if

where PA:R→[0,1] and NA:R→[0,1] are said to be positive-membership degree of α in R and negative-membership degree of α in R respectively. Also PA and NA fulfil the following condition,

Definition 2: Let the universe set R. Then A is known to be PFS of R, if

where PA:A→[0,1],IA:A→[0,1] and NA: A→[0,1] are said to be degree of positive membership of α in R, neutral membership degree of α in R and negative-membership degree of α in R respectively [6]. Also PA, IA and NA satisfy the following condition:

and γA(α)=1−(PA(α)+IA(α)+NA(α))said to be refusal membership degree of α in R For PFS ((PA(α),IAα,NA(α)))are said to picture fuzzy number (PFN) and each PFN can be denoted by e = (Pe,IeNe), where ,PeIe and Ne ∈ [0,1], with condition that .

Definition 3: Let be two PFNs, and µ > 0 [6]. Then the operations for PFNs are defined as:

Comparison Rules

Here we define some functions like score function and accuracy function [3]. Which play important role for ranking of PFNs is described as:

Definition 5: Let k ∈ N be the PFNs [10]. Then

where sc(ek) and ac(ek) are said to be score and accuracy functions of the PFNs, the technique for comparison of PFNs can be described in next definition as.

Definition 6: Let be any two PFNs [10]. Then by using the Definition 5, comparison technique can be described as,

Definition 7: Let be any collection of PFNs [7]. Then picture fuzzy number weighted averaging aggregat ing (PFNWAA) operator, PFNWAA: PFNn → PFN, describe as,

in which be the weight vector of

Definition 8: Let be a collection of PFNs [7]. Then the picture fuzzy number order weighted averaging aggregating (PFNOWAA) operator, PFNOWAA : PFNn → PFN describe as,

with dimensions n, where kth biggest weighted value is eη (k) consequently by total order eη(1)≥ eη(2)≥...≥eη(n). τ={τ12,...,τn} are the weight vectors such that τk ≥ 0 (kεN ) and Σk=1n τk = 1.

Definition 9: Let ek=(pek,Iek,Nek) k ∈ N be a collection of PFNs [7]. Then the picture fuzzy number order weighted averaging aggregating (PFNOWAA) operator, PFNOWAA : PFNn → PFN describe as,

With dimensions n, where kth biggest weighted value is are the weight vectors

Step: 1

Weight the attribute value of each alternative by utilizing the operations of PFSs.

Step: 2

Calculate the score function by utilizing the definition 6

Step: 3

Step: 4

According to Picture fuzzy distance, calculate the distance between the alternative Ai and the PFPIS P+ and the distance between the alternative Ai and the PFNIS P-, respectively:

This distance is known to be Normalized Hamming distance d (ej , ek), and construct a Picture fuzzy positive-ideal separation matrix D+ and Picture fuzzy negative-ideal separation matrix D- as follows:

Step: 5

Grey coefficient for each alternative calculated from PIS and NIS by utilizing following below equation. The grey coefficient for each alternative calculated from PIS is provided as:

Where j = 1,2,3,....,m and k = 1,2,3,....,n.
Similarly, the grey coefficient of each alternative calculated from NIS is provided as

Where j = 1,2,3,....,m and k = 1,2,3,....,n the identification coefficient ρ = 0.5

Step: 6

Calculating the grey coefficient degree for each alternative from PIS and NIS by utilizing following below equation, respectively,

The basic principle of the Grey method is that the chosen alternative should have the largest degree of grey relation from the PIS and the smallest degree of grey relation from the NIS. Obviously, for the weights are known, the smaller ξj- and the larger ξj+, the finest alternative Aj. But incomplete information about weights of alternatives is known. So, in this circumstance the ξj- and ξj+ information about weight calculated initially. So, we provide following optimization models for multiple objectives to calculate the information about weight,ξ

Since each alternative is non-inferior, so there exists no preference relation on the all the alternatives. Then, we aggregate the above optimization models with equal weights into the following optimization model of single objective,

To finding solution of OM2, we obtain optimal solution ϑ =(ϑ12 ,...,ϑm) which utilized as weights information’s of provided alternatives. Then, we obtain ξj+ (l = 1, 2,...,m ) and ξj-( j = 1, 2,...,m ) as utilizing above formula, respectively.

Step: 7

Relative degree calculated for each alternative utilizing the following equation from PIS and NIS,

Step: 8

Ranking all the alternatives Aj (j=1,2,3,....,m) and select finest one(s) in accordance with ξj (j=1,2,3,....,m). If any alternative has the highest ξj value, then it is finest alternative according to the criteria.

A Descriptive Example

Suppose a multi-national corporation in Pakistan is scheduling its economic stratagem for the imminent year, according to a group stratagem objective. For this, there are four alternatives attained after their initial scrutiny and are defined as ξ1 to participate in the "Asian markets"; ξ2 to participate in the "Western markets"; ξ3 to participate in the "China markets"; and ξ4 to participate in the" Local markets". This estimation ensues from the four characteristics, namely as g1 "the evolution scrutiny"; g2 "the danger scrutiny"; "the social-political influence scrutiny" and g4 "the environmental influence scrutiny" whose weights information is ω = (0.2,0.3,0.1,0.4)T.

Step: 1

Weight the attribute value of each alternative by utilizing the operations of PFSs as,

Step: 2

Calculate the score function by utilizing the definition 6 as,

Step: 3

Calculate PFPIS by utilizing score values,

and similarly calculate PFNIS by utilizing score values

Step: 4

Utilizing the normalized Hamming distance to calculate Picture fuzzy positive-ideal separation matrix D+ as,

and to calculate Picture fuzzy negative-ideal separation matrix D as,

Step: 5

Calculate the grey coefficient of each alternative from PIS as,

Calculate the grey coefficient of each alternative from NIS as,

Step: 6

Utilize the model (M2) to establish the following

Solve this model, we obtained the weight vectors of the attributes

w = (0.263,0.292,0.095,0.350)

After that, we can obtain the degree of grey coefficient for each alternative from PFPIS and PFNIS respectively, as,

Step: 7

Relative degree of each alternative calculated using the following equation from PIS and NIS,

Step: 8

According to relational degree, we are ranking the alternatives as

Hence, the best alternative is ξ3= "China markets".

Comparison Analysis with Intuitionistic Fuzzy Grey Approach

Intuitionistic fuzzy numbers can describe the ambiguous things from degrees of positive and negative membership. They deliver an active implementation to signifies the indeterminacy of DM. On the one hand, as declared formerly, in IFN that mediating the things from good and bad features of these two varieties of fuzzy numbers can throw away the thoughts of DM exactly. However, dissimilar the PFNs, the IFNs are not serviceable in some conditions. The IFNs must gratify that the sum of the membership and non-membership degree belongs to [0,1]. Thus, in our case analysis, there exists some numbers which cannot handle by IFNs. For example, in the situation that human being needs opinions involving that type of answer: "yes", "abstain", "No" and "Refusal". Picture fuzzy set is useful in that situation. In summary, the PFNs have stronger capability to procedure information in decision theory, as compared to IFNs [18-33].

Conclusion

The grey relational procedure plays a dynamic role throughout the decision-making procedure and hence in this track, the comparative importance of the criteria remains un-reformed in modified problem. Almost all the researchers have operated the IFS by considering the positive and negative membership degrees only. But, it has been detected that in some circumstances, like in situation of voting, human thoughts including more degrees as, yes, neutral, no, refusal, and in which situation, IFS cannot be exactly characterized. For this circumstance, Picture fuzzy set, which is an upgrading of IFS, has utilized in this paper corresponding picture fuzzy aggregated operator are introduced. Several required characteristics of these operators have explored comprehensibly. Finally, a decision problem has established which based on these defined operators, for ranking the dissimilar alternatives by utilizing picture fuzzy environment. The suggested technique has demonstrated with a numerical example for viewing their effectiveness as well as reliability. Thus, the suggested operations give a new easier track to grip the inexact data throughout the decision problem procedure. In future, we utilize different decision making approaches like TOPSIS, TODAM, VIKOR, ELECTRIC-1, ELECTRICT-2 and many more to deal with uncertainty which is in the form of picture fuzzy sets in real life decision making problems.

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Journal of Biostatistics and Biometric Applications

Figure 1: Poly Nomenclature
Figure 2: Triangles Parallelograms and Quandrilaterals
Figure 3: GADJ
Figure 4: Polycon Concepts
Figure 5: Adjacent factors
Figure 6: Polycon GADJ:linear(j, j+1)

Trend Component

Seasonal Component

N (None)

A (Additive)

M (Multiplicative)

N (None)

N, N

N, A

N, M

A (Additive)

A, N

A, A

A, M

Ad (Additive Damped)

Ad, N

Ad, A

Ad, M

M (Multiplicative)

M, N

M, A

M, M

Md(Multiplicative Damped)

Md, N

Md, A

Md, M

Table 1: Variations in the Combination of the Trend and Seasonal Components

Variable

Order of Differencing

P-value

Conclusion

Malaria Cases

I (0) (original data)

0.3315

The data is not stationary

Differenced Malaria Cases

I (1) (differed data)

0.01

The data is stationary

Table 2: ADF Unit root Test on Malaria Cases Data

Model

AICc

(1, 1, 0) (1, 1, 0)

-178.28

(0, 1, 1) (1, 1, 0)

-181.52

(0, 1, 2) (1, 1, 0)

-182.12

(0, 1, 1) (0, 1, 1)

-187.88

(0, 1, 2) (0, 1, 1)*

-188.15

(1, 1, 1) (1, 1, 0)

-180.79

(1, 1, 1) (0, 1, 1)

-187.33

(1, 1, 1) (0, 1, 0)

-162.85

(1, 1, 2) (0, 1, 1)

-186.06

(1, 1, 2) (0, 1, 1)

-180.66

*The chosen model
Table 3: Competing models and their AICc values

Model

AICc

BIC

M, N, M

88.7279

123.7425

M, Ad, M *

-19.0453

21.5477

M, Md, M

-18.6612

21.9317

A, A, N

43.5431

56.3655

A, Ad, A

-18.9906

21.6024

*The chosen model
Table 4: Competing Models with respective AICc and BIC

MODEL

RMSE

MAPE

SARIMA(0, 1, 2) (0, 1, 1)12*

0.07727

0.4569

ETS (M, Ad, M)

0.1231

0.8141

NNAR (1, 1, 2)

0.1818

1.2421

Combination (ARIMA, ETS, NNAR)

0.1207

0.7958

*The chosen model
Table 5: Forecast Accuracy Measures of Competing Forecast Techniques

Variable

Estimates

Standard Error

P – Value

Malaria Cases

MA(1)

-0.2471

0.1042

0.0198

MA(2)

-0.1575

0.0987

0.1140

SMA(1)

-0.6235

 0.1212

0.0000

Table 6: Parameter Estimates of the chosen ARIMA Model

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