# A New Perspective for Multivariate Time Series Decision Making through a Nested Computational Approach Using Type-2 Fuzzy Integration

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Basic Concepts

#### 3.1. Type-2 Fuzzy Systems

#### 3.2. Multi-Criteria Decision Making

## 4. Problem Description

## 5. Proposed Method

_{i}(t) are the outputs of the rules, i = 1, …27, ${\mathsf{\mu}}_{\mathrm{i}}$ are the membership function values at the outputs of the rules, i = 1, …27 and y(t) is the total output.

## 6. Experimental Results

## 7. Discussion of Results

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Country Name | Country Code | Country Name | Country Code |
---|---|---|---|

Australia | AUS | Japan | JPN |

Austria | AUT | Republic of Korea, Rep. | KOR |

Belgium | BEL | Latvia | LVA |

Canada | CAN | Lithuania | LTU |

Chile | CHL | Luxembourg | LUX |

Colombia | COL | Mexico | MEX |

Costa Rica | CRI | Netherlands | NLD |

Czech Republic | CZE | New Zealand | NZL |

Denmark | DNK | Norway | NOR |

Estonia | EST | Poland | POL |

Finland | FIN | Portugal | PRT |

France | FRA | Slovak Republic | SVK |

Germany | DEU | Slovenia | SVN |

Greece | GRC | Spain | ESP |

Hungary | HUN | Sweden | SWE |

Iceland | ISL | Switzerland | CHE |

Ireland | IRL | Turkey | TUR |

Israel | ISR | United Kingdom | GBR |

Italy | ITA | United States | USA |

Code Attribute | Attribute Name |
---|---|

att1 | Country |

att2 | IDCountry |

att3 | Criterion |

att4 | IDCriterion |

att5 | IDYear |

att6 | ValueCriterion |

Code Attribute | Attribute Name |
---|---|

att1 | Country |

att2 | IDCountry |

att3 | Criterion |

att4 | IDCriterion |

att5 | IDYear |

att6 | ValueCriterion |

Code Attribute | Attribute Name |
---|---|

att1 | Country |

att2 | IDCountry |

att3 | Criterion |

att4 | IDCriterion |

att5 | IDYear |

att6 | ValueCriterion |

Code Attribute | Attribute Name |
---|---|

att1 | Country |

att2 | IDCountry |

att3 | IDYear |

att4 | ValueCriterion |

Variables Type | Variables Name | Membership Functions |
---|---|---|

Input | Population | LW, MM, HH |

GNI | LW, MM, HH | |

Output | Criterion 1 | LW, MM, HH |

Variables Type | Variables Name | Membership Functions |
---|---|---|

Inputs | GNI1 | L, LM, UM, H |

GNI2 | L, LM, UM, H | |

GNI3 | L, LM, UM, H | |

GNI4 | L, LM, UM, H | |

Output | Criterion 2 | L, LM, UM, H |

Variables Type | Variables Name | Membership Functions |
---|---|---|

Input | Inflation | LW, MM, HH |

OECD cr | LW, MM, HH | |

Output | Criterion 3 | LW, MM, HH |

Variables Type | Variables | Membership Functions |
---|---|---|

Input | Criterion 1 | LW, MM, HH |

Criterion 2 | LW, MM, HH | |

Criterion 3 | LW, MM, HH | |

Output | Criteria | LW, MM, HH |

Variables | Membership Functions | Parameter a | Parameter b | Parameter c |
---|---|---|---|---|

Input 1 Population | LW | 0.000 | 0.100 | 0.300 |

MM | 0.200 | 0.600 | 1.200 | |

HH | 0.900 | 2.500 | 3.000 | |

Input 2 GNI | LW | 0.000 | 0.100 | 0.300 |

MM | 0.200 | 0.600 | 1.200 | |

HH | 0.900 | 2.500 | 3.000 | |

Output 1 Criterion 1 | LW | 0.000 | 0.600 | 1.200 |

MM | 0.900 | 1.600 | 2.200 | |

HH | 2.000 | 2.500 | 3.000 |

Country Code | Criterion 1 | Country Code | Criterion 1 |
---|---|---|---|

AUS | MM | JPN | MM |

AUT | LW | KOR | LW |

BEL | LW | LVA | LW |

CAN | LW | LTU | LW |

CHL | MM | LUX | MM |

COL | MM | MEX | MM |

CRI | MM | NLD | MM |

CZE | MM | NZL | MM |

DNK | LW | NOR | LW |

EST | MM | POL | MM |

FIN | LW | PRT | LW |

FRA | LW | SVK | LW |

DEU | MM | SVN | MM |

GRC | MM | ESP | MM |

HUN | MM | SWE | MM |

ISL | LW | CHE | LW |

IRL | MM | TUR | MM |

ISR | MM | GBR | MM |

ITA | LW | USA | LW |

Variables | Membership Functions | Parameter a | Parameter b | Parameter c |
---|---|---|---|---|

Input1 GNI1 | L | 0 | 0.002 | 0.010 |

LM | 0.008 | 0.144 | 0.244 | |

UM | 0.234 | 0.274 | 0.915 | |

HM | 0.910 | 1.2 | 1.5 | |

Input 2 GNI2 | L | 0 | 0.002 | 0.011 |

LM | 0.009 | 0.144 | 0.245 | |

UM | 0.235 | 0.274 | 0.916 | |

HM | 0.911 | 1.2 | 1.5 | |

Input 3 GNI3 | L | 0 | 0.002 | 0.012 |

LM | 0.010 | 0.144 | 0.246 | |

UM | 0.236 | 0.274 | 0.917 | |

HM | 0.912 | 1.2 | 1.5 | |

Input 4 GNI4 | L | 0 | 0.002 | 0.013 |

LM | 0.011 | 0.144 | 0.247 | |

UM | 0.237 | 0.274 | 0.918 | |

HM | 0.914 | 1.2 | 1.5 | |

Output 1 Criterion 2 | L | 0 | 0.002 | 0.010 |

LM | 0.008 | 0.144 | 0.250 | |

UM | 0.230 | 0.280 | 0.890 | |

HM | 0.790 | 1.2 | 1.5 |

Country Code | Criterion 2 | Country Code | Criterion 2 |
---|---|---|---|

AUS | H | JPN | H |

AUT | H | KOR | H |

BEL | H | LVA | H |

CAN | H | LTU | UM |

CHL | UM | LUX | H |

COL | UM | MEX | UM |

CRI | UM | NLD | H |

CZE | H | NZL | H |

DNK | H | NOR | H |

EST | H | POL | UM |

FIN | H | PRT | H |

FRA | H | SVK | UM |

DEU | H | SVN | H |

GRC | UM | ESP | H |

HUN | UM | SWE | H |

ISL | H | CHE | H |

IRL | H | TUR | UM |

ISR | H | GBR | H |

ITA | H | USA | H |

Variables | Membership Functions | Parameter a | Parameter b | Parameter c |
---|---|---|---|---|

Input 1 Inflation | LW | −3.000 | 0.100 | 0.300 |

MM | 0.200 | 0.600 | 1.200 | |

HH | 0.900 | 2.500 | 3.000 | |

Input 2 OECD cr | LW | −3.000 | 0.100 | 0.300 |

MM | 0.200 | 0.600 | 1.200 | |

HH | 0.900 | 2.500 | 3.000 | |

Output 1 Criterion 3 | LW | −3.000 | 0.600 | 1.200 |

MM | 0.900 | 1.600 | 2.200 | |

HH | 2.000 | 2.500 | 3.000 |

Country Code | Criterion 3 | Country Code | Criterion 3 |
---|---|---|---|

AUS | LW | JPN | LW |

AUT | LW | KOR | LW |

BEL | LW | LVA | LW |

CAN | LW | LTU | LW |

CHL | LW | LUX | LW |

COL | LW | MEX | LW |

CRI | LW | NLD | LW |

CZE | LW | NZL | LW |

DNK | LW | NOR | LW |

EST | LW | POL | LW |

FIN | LW | PRT | LW |

FRA | LW | SVK | LW |

DEU | LW | SVN | LW |

GRC | LW | ESP | LW |

HUN | LW | SWE | LW |

ISL | LW | CHE | LW |

IRL | LW | TUR | LW |

ISR | LW | GBR | LW |

ITA | LW | USA | LW |

Variables | Membership Function | a | b | c | Lower Scale | Lower Lag | |
---|---|---|---|---|---|---|---|

Input 1 Criterion 1 | LW | 0.1223 | 0.6223 | 1.2980 | 1.000 | 0.2000 | 0.2000 |

MM | 0.894 | 1.544 | 2.166 | 1.000 | 0.2000 | 0.2000 | |

HH | 2.000 | 2.500 | 3.000 | 1.000 | 0.2000 | 0.2000 | |

Input 2 Criterion 2 | LW | 0.1223 | 0.6223 | 1.2980 | 1.000 | 0.2000 | 0.2000 |

MM | 0.894 | 1.544 | 2.166 | 1.000 | 0.2000 | 0.2000 | |

HH | 2.000 | 2.500 | 3.000 | 1.000 | 0.2000 | 0.2000 | |

Input 3 Criterion 3 | LW | 0.1223 | 0.6223 | 1.2980 | 1.000 | 0.2000 | 0.2000 |

MM | 0.894 | 1.544 | 2.166 | 1.000 | 0.2000 | 0.2000 | |

HH | 2.000 | 2.500 | 3.000 | 1.000 | 0.2000 | 0.2000 | |

Output 1 Criteria | LW | 0.000 | 0.500 | 1.200 | 1.000 | 0.2000 | 0.2000 |

MM | 1.000 | 1.500 | 2.200 | 1.000 | 0.2000 | 0.2000 | |

HH | 2.000 | 2.500 | 3.000 | 1.000 | 0.2000 | 0.2000 |

Country Code | Criteria | Country Code | Criteria |
---|---|---|---|

AUS | MM | JPN | MM |

AUT | MM | KOR | MM |

BEL | MM | LVA | MM |

CAN | MM | LTU | MM |

CHL | MM | LUX | MM |

COL | MM | MEX | LW |

CRI | MM | NLD | MM |

CZE | MM | NZL | MM |

DNK | MM | NOR | MM |

EST | MM | POL | MM |

FIN | MM | PRT | MM |

FRA | MM | SVK | MM |

DEU | MM | SVN | MM |

GRC | MM | ESP | MM |

HUN | MM | SWE | MM |

ISL | MM | CHE | MM |

IRL | MM | TUR | LW |

ISR | MM | GBR | MM |

ITA | MM | USA | MM |

Country Code | T1 FIS1 Criterion 1 | T1 FIS2 Criterion 2 | T1 FIS3 Criterion 3 | T2 FIS1 Criteria |
---|---|---|---|---|

AUS | MM | H | LW | MM |

AUT | LW | H | LW | MM |

BEL | LW | H | LW | MM |

CAN | LW | H | LW | MM |

CHL | MM | UM | LW | MM |

COL | MM | UM | LW | MM |

CRI | MM | UM | LW | MM |

CZE | MM | H | LW | MM |

DNK | LW | H | LW | MM |

EST | MM | H | LW | MM |

FIN | LW | H | LW | MM |

FRA | LW | H | LW | MM |

DEU | MM | H | LW | MM |

GRC | MM | UM | LW | MM |

HUN | MM | UM | LW | MM |

ISL | LW | H | LW | MM |

IRL | MM | H | LW | MM |

ISR | MM | H | LW | MM |

ITA | LW | H | LW | MM |

JPN | MM | H | LW | MM |

KOR | LW | H | LW | MM |

LVA | LW | H | LW | MM |

LTU | LW | UM | LW | MM |

LUX | MM | H | LW | MM |

MEX | MM | UM | LW | LW |

NLD | MM | H | LW | MM |

NZL | MM | H | LW | MM |

NOR | LW | H | LW | MM |

POL | MM | UM | LW | MM |

PRT | LW | H | LW | MM |

SVK | LW | UM | LW | MM |

SVN | MM | H | LW | MM |

ESP | MM | H | LW | MM |

SWE | MM | H | LW | MM |

CHE | LW | H | LW | MM |

TUR | MM | UM | LW | LW |

GBR | MM | H | LW | MM |

USA | LW | H | LW | MM |

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**MDPI and ACS Style**

Ramirez, M.; Melin, P.
A New Perspective for Multivariate Time Series Decision Making through a Nested Computational Approach Using Type-2 Fuzzy Integration. *Axioms* **2023**, *12*, 385.
https://doi.org/10.3390/axioms12040385

**AMA Style**

Ramirez M, Melin P.
A New Perspective for Multivariate Time Series Decision Making through a Nested Computational Approach Using Type-2 Fuzzy Integration. *Axioms*. 2023; 12(4):385.
https://doi.org/10.3390/axioms12040385

**Chicago/Turabian Style**

Ramirez, Martha, and Patricia Melin.
2023. "A New Perspective for Multivariate Time Series Decision Making through a Nested Computational Approach Using Type-2 Fuzzy Integration" *Axioms* 12, no. 4: 385.
https://doi.org/10.3390/axioms12040385