Ejemplos de submarinos hundidos y como se procedió a su rescate.

The DEHO method was employed using a computational cluster with 7 CPU’s and the parameters contained in Table 32 using PWFTS as FTS method. The experiment performed 5 executions of DEHO for each dataset and the averaged results are presented in Table 33. A sample of the convergence process of DEHO can be seen in Figure46, for SONDA Wind Speed dataset.

The results showed that, for the studied time series, the convergence was fast, expending about 16 generations on average. In the trade off between the objectives f1 and

f2, the accuracy objective showed to be predominant over the parsimony objective during

the convergence of the method.

The optimized values for the hyperparameters generated parsimonic and accurated forecasting models, whose samples of their performance can be seen in Figure

Figure 45 – Speed up provided by the distributed model by number of CPU’s

Parameter Dataset Value

WL SONDA Wind Speed 600,000 SONDA Solar Radiation 600,000 Malaysia Temperature 10,000 Malaysia Eletric Load 10,000 WI All .5 TS All .9 P S All 20 N G All 30 N Gstop All 10 SR All .5 CR All .5 M R All .2

5.5. Conclusion 127

Dataset Generations k µ α Ω L Metric Value

SONDA Solar Radiation 16.4 ± 7.8 50.8 ± 0.7 2 0.24 ± 0.13 2 [1,2] |M| 613 ± 222 RMSE 93.13 ± 0.62 Time 3221 ± 1505

SONDA Wind Speed 30.0 50 1 0.13

± 0.1 1 [1]

|M| 24 ± 1.45 RMSE 0.34 ± 74e-10−4

Time 3058 ± 891

Malaysia Energy Load 12.5 ± 2.5 50.6 ± 1.2 2 0.22 ± 0.23 2 [1,2] |M| 306.9 ± 137.9 RMSE 2745.5 ± 271.27 Time 3945.09 ± 800.71 Malaysia Temperature 16.6 ± 10.15 52.8 ± 3.18 1 0.24 ± 0.09 1 [1] |M| 73.21 ± 1.08 RMSE 1.08 ± 0.06 Time 3916.58 ± 2042.12

Table 33 – Optimization mean results by dataset

Figure 46 – Sample of DEHO convergence

5.5

Conclusion

Training accurate models for forecasting big time series is a challenging task for traditional and soft-computing methods. Usually the methods are not designed to deal with such high volume of data. When such data volume cannot be grounded on a single machine memory, it demands a distributed architecture of storage and processing. This is particularly problematic when optimizing the hyperparameters of a method, because successive model training and testing processes are required.

This chapter proposed two distributed approaches for FTS model scalability, one for middle sized data and another for big sized data. The first one distributes the data across the nodes of a cluster, where individual models are trained and tested. The second one, the distributed model itself, splits the training of a unique model across several nodes, allowing a big time series model to be trained in pieces and then aggregated into a single model.

These approaches were employed on Distributed Evolutionary Hyperparameter Optimization (DEHO). DEHO method is an adapted genetic algorithm that minimizes two

cost functions, the accuracy function f1 and the parsimony function f2. An exploratory

study was performed in order to measure the feasibility of the proposed distributed models and showed the speed-up provided for big time series. The convergence of DEHO method also was analyzed, showing its effectiveness.

5.5.1

Method limitations

The distributed training method is indicated only for big time series. Using the method for small data might slow down the training time due to the network and model merging overheads. DEHO method took into account only time invariant, rule based, monovariate and high-order methods, not being applicable for time variant, multivariate and first order FTS methods.

Next chapter presents a short review of multivariate methods and proposes a simple approach for extending PWFTS to forecasting multivariate time series.

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Chapter 6

Multivariate Models

Despite the existing approaches in the literature, dealing with multivariate and spatio-temporal time series was always a challenging task for FTS methods, specially because of the complexity growth of the rules as the dimension increases. An important gap in the literature is the absence of multiple input and multiple output (MIMO) methods - the majority of FTS literature consists of basically univariate forecasting methods.

A simple approach is to transform multivariate time series into monovariate time series by using Fuzzy Information Granules (FIG). Each FIG acts as a multivariate fuzzy set, or a composition of individual fuzzy sets from different variables, allowing to replace a vector (the values of one data point) by a scalar (the identification of the FIG with the highest membership of that data point).

This approach is employed on Fuzzy Information Granular Fuzzy Time Series (F IG-FTS), a wrapper method that enables PWFTS to tackle multivariate time series.

It begins by partitioning the Universe of Discourse of each individual variable. Then the crisp values of each variable are fuzzyfied and the corresponding fuzzy sets are combined to create one Fuzzy Information Granule, such that it can be used as a reference of all data points in that same region. This incremental approach creates the FIGs on demand, according to the training data, and its sensibility can be controlled using the method’s hyperparameters.

This chapter presents a short review of multivariate FTS methods and Fuzzy Information Granules in Section 6.1. Section 6.2 presents a conventional method for multivariate FTS (MVFTS), which does not employ FIGs. In Section 6.3 the Fuzzy Information Granule Fuzzy Time Series (F IG-FTS) is proposed to enable PWFTS method be used for multivariate data and allowing the use of its interval and probabilistic forecasting features. In Section 6.4 an exploratory study of the performance of F IG-FTS when compared with previous FTS methods is presented and finally, in Section 6.5, the results are discussed and the conclusions given.