Estabilidad

Alias Parameter Type Description ki

Number of

partitions N

+ The number of fuzzy sets that will be created

in the linguistic variable eVi

µ Membership

function µ : U → [0, 1]

A function that measure the membership of a value y ∈ U to a fuzzy set

α α-cut [0, 1] The minimal membership grade to take ac- count on fuzzyfication process

Table 34 – WMVFTS hyperparameters for each variable Vi ∈ V

clustering methods. In Leite et al. [2011], the authors propose the fuzzy set based granular evolving modeling (FBeM) approach for time series prediction, later extended in Soares et al. [2018] for spatio-temporal data.

There are some notable drawbacks in the previous methods, namely: a) the absence of multivariate forecast (MIMO); b) the use of optimization methods to create the FIGs, which makes the learning process computationally expensive; c) the absence of multivariate FTS methods that could provide both weighted and high order characteristics. To fix these drawbacks this work proposes the F IG-FTS method, a weighted and high-order FTS method that will be discussed in the next sections.

6.2

The Conventional Multivariate Fuzzy Time Series

method

Just as it was done in Chapter 2, this section proposes a consensus model for rule based multivariate FTS that extends the model of Chen [1996] to the multivariate case. The Conventional Multivariate Fuzzy Time Series (MVFTS) method was designed to allow several models to be trained individually with subsets of a greater dataset and later to be merged into a single model, feature that enhances the performance of model creation by enabling its distribution.

For each chosen variable Vi ∈ V on Y , MVFTS also incorporates several features

present in the literature, represented by the hyperparameters in Table 34, giving versatility and flexibility to the model. The method is composed of two procedures: the training procedure and the forecasting procedure.

The MVFTS is a first order point forecaster of type Multiple Input/Single Output (MISO), then for the set of variables V one of them is chosen as the target (or endogenous) variable and the others are referred as the explanatory (or exogenous) variables. From now on, the target variable will be distinguished from the others by an asterisk, as ∗V.

The training procedure, explained in subsection 6.2.1 and illustrated in Figure 47, is a three stage process responsible to create a multivariate weighted FTS model M. The

VARIABLES - 𝓥 FUZZY SETS - Ã_𝓥 CRISP DATA - Y PARTITIONING FUZZYFICATION FUZZYFIED DATA - F FUZZY RULES- 𝓜 RULE INDUCTION α-CUT LEGEND ■ PROCEDURES ■ DATA ■ PARAMETERS ■ HYPERPARAMETERS NUMBER OF PARTITIONS - k PARTITIONING METHOD - Π MEMBERSHIP FUNCTION - μ TARGET VARIABLE - 𝓥*

Figure 47 – MVFTS training procedure

final MVFTS model M consists of a set of variables V, a fuzzy linguistic variable eVi

for each Vi ∈ V and a set of weighted fuzzy rules over the linguistic variables eVi. The

inputs of the training procedure are the crisp time series training data Y and the set of hyperparameters for each Vi ∈ V.

The forecasting procedure, explained in subsection 6.2.2 and illustrated in Fig- ure48, aims to produce a point estimate ˆy(t + 1) for the target variable ∗V, given an input sample Y , using the linguistic variables eVi and the induced fuzzy rules on model M.

6.2.1

Training Procedure

Stage 1 Partitioning:

a) Defining UVi: The Universe of Discourse UVi defines the sample space, i.e., the

known bounds of the variable Vi, such that UVi = [min(Y

Vi_{) − D}

1, max(YVi) +

D2], where D1 = min(YVi)×0.2 and D2 = max(YVi)×0.2 are used to extrapolate

the known bounds as a security margin, ∀Vi ∈ V.

b) UVi Partitioning: Split UVi in ki intervals Uj with midpoints cj, for j = 0..ki,

where all the intervals have the same length;

c) Define the linguistic variable eVi: For each interval Uj ∈ UVi create an overlapping

fuzzy set AVi

j , with the membership function µAVi_j . The midpoint of the fuzzy

set AVi

6.2. The Conventional Multivariate Fuzzy Time Series method 133

∀ j > 0 and j < ki, and l0 = min UVi, lk = max UVi. Each fuzzy set A

Vi

j is a

linguistic term of the linguistic variable eVi;

Stage 2 Fuzzyfication:

Transform the original numeric time series Y into a fuzzy time series F , where each data point f (t) ∈ F is an n × k array with the fuzzyfied values of y(t) ∈ Y with respect to the linguistic terms AVi

j ∈ eVi, where the fuzzy membership is greater than

the predefined α-cut, i.e., f (t) = {AVi

j | µAVi_j (y(t)

Vi_{) ≥ α}

i ∀AVji ∈ eVi};

Stage 3 Rule Induction:

a) Generate the temporal patterns: The fuzzy temporal patterns associate the fuzzyfied values V to a set of possible values of the target variable ∗V, such that V → ∗V, whith the format AV0

j , ..., A Vn

j → A ∗V

j , where the precedent, or

left hand side (LHS), is f (t − 1) = AVi

j , ∀Vi ∈ V, and the consequent, or right

hand side (RHS), is f (t + 1) = A∗V_j , A∗V_j ∈ f∗V.

b) Generate the rule base: Select all temporal patterns with the same precedent and group their consequent sets creating a rule with the format V → wk·

A∗V_k , wj · A∗Vj , ..., where the LHS is f (t − 1) = A Vi

j , ∀Vi ∈ V and the RHS is

f (t + 1) ∈ {A∗V_k , A∗V_j , ...}. Each rule can be understood as the weighted set of possibilities which may happen on time t + 1 (the consequent) when a certain precedent Ai0, ..., AiΩ is identified on previous lag (the precedent).

6.2.2

Forecasting Procedure

Step 1 Fuzzyfication: Compute the membership grade µji for y(t − 1) ∈ Y such that

µji = µ_AVi j

(y(t − 1)), for each AVi

j ∈ eVi, for each Vi ∈ V ;

Step 2 Rule matching: Select the K rules where all fuzzy sets AVi

j on the LHS, for each

Vi ∈ V, have µji > αi; The rule fuzzy membership grade is shown below, using the

minimum function as T-norm.

µq=

\

j∈ eVi; i∈V

µji (6.1)

Step 3 Rule mean points: For each selected rule q, compute the mean point mpq of the

target variable ∗V as below, where cj is the c parameter of the µ function from fuzzy

set A∗V_j :

mpq =

X

j∈∗ eVi

FUZZY SETS - Ã_𝓥 y(t) FUZZYFICATION RULE MATCHING f(t) FUZZY RULES - 𝓜 DE FUZZYFICATION LEGEND ■ PROCEDURES ■ DATA ■ PARAMETERS ■ HYPERPARAMETERS f(t+1) ŷ*(t+1) VARIABLES - 𝓥 α-CUT TARGET VARIABLE - 𝓥*

Figure 48 – MVFTS forecasting procedure

Step 4 Defuzzyfication: Compute the forecast as the weighted sum of the rule mid-points mpq by their membership grades µq for each selected rule j:

ˆ y(t + 1) = P q∈Kµq· mpq P q∈Kµq (6.3)

6.2.3

Interval forecasting for MVFTS

The MVFTS model can be used for interval forecasting following the same approach of [I]F T S method. For this it is needed to change the Steps 3 and 4 of the forecasting procedure presented in Section 6.2.2, as presented below:

Step 3 Rule intervals: For each selected rule q, compute the interval Iq _{of the target variable}

∗V as below, where A∗V

j and A∗Vj are respectively the lower and upper bounds of the

target fuzzy sets A∗V_j :

Iq = [Iqmin, I q max] (6.4) Iqmin = min(A ∗V j ∈ ∗ eVi) (6.5) Iqmax = max(A∗V_j ∈ ∗ eVi) (6.6) (6.7)

6.2. The Conventional Multivariate Fuzzy Time Series method 135

Iq by their membership grades µq for each selected rule j:

I(t + 1) = P j∈∗ eViµqI q P q∈Vµq = P q∈V[µqI q min, µqI q max] P q∈Vµq (6.8)

6.2.4

Weighted Multivariate FTS - WMVFTS

A simple extension of MVFTS to embody weights in its rules can be achieved by changing Stage 3.b of the training procedure presented in Section 6.2.1, where the new step is:

Stage 3.b) Generate the rule base: Select all temporal patterns with the same precedent and group their consequent sets creating a rule with the format V → wk· A∗Vk , wj· A∗Vj , ...,

where the LHS is f (t − 1) = AVi

j , ∀Vi ∈ V and the RHS is f (t + 1) ∈ {A∗Vk , A∗Vj , ...}

and the weights wj, wk, ... are the normalized frequencies of each temporal pattern

such that: wi = #A∗V_j #RHS ∀A ∗V j ∈ RHS (6.9)

where #Ai is the number of occurrences of Ai on temporal patterns with the same

precedent LHS and #RHS is the total number of temporal patterns with the same precedent LHS.

It is also need to change the Step 3 of the forecasting procedure presented in Section 6.2.2, as presented below:

Step 3 Rule mean points: For each selected rule q, compute the mean point mpq of the

target variable ∗V as below, where cj is the c parameter of the µ function from fuzzy

set A∗V_j :

mpq =

X

j∈∗ eVi

wj · cj (6.10)

For the interval forecasting method proposed in Section 6.2.3, a new approach is adopted to create the fuzzy rule intervals, as presented below:

Iq = [Iqmin, I q max] (6.11) Iqmin = X j∈∗ eVi wj· A∗Vj (6.12) Iqmax = X j∈∗ eVi wj· A∗Vj (6.13)

MVFTS and WMVFTS methods take separated partitionings for each variable and its rules contains references for the different variables. In next section a simple approach is

proposed for transforming multivariate time series in monovariate ones, allowing the use of monovariate FTS methods to tackle multivariate time series.