Selection of Temporal Lags for Predicting Riverflow Series from Hydroelectric Plants Using Variable Selection Methods
Abstract
:1. Introduction
- Li et al. [22]—to increase the estimation of growing stem volume of pine using optical images;
- Bonah et al. [23]—to quantitative tracking of foodborne pathogens;
- Xiong et al. [24]—to increase near-infrared spectroscopy quality;
- Speiser et al. [25]—an extensive investigation with 311 datasets to compare several random forest VS methods for classification;
- Rendall et al. [26]—extensive comparison of large scale data driven prediction methods based on VS and machine learning;
- Marcjasz et al. [27]—to electricity price forecasting;
- Santi et al. [28]—to predict mathematics scores of students;
- Karim et al. [29]—to predict post-operative outcomes of cardiac surgery patients;
- Kim and Kang [30]—to faulty wafer detection in semiconductor manufacturing;
- Furmańczyk and Rejchel [31]—to high-dimensional binary classification problems;
- Fouad and Loáiciga [5]—to predict percentile flows using inflow duration curve and regression models;
- Ata Tutkun and Kayhan Atilgan [32]—investigated VS models in Cox regression, a multivariate model;
- Mehmood et al. [33]—compared several VS approaches in partial least-squares regression tasks;
- McGee and Yaffee [34]—provided a study on short multivariate time series and many variations of Least Absolute Shrinkage and Selection Operator (LASSO) for VS;
- Seo [35]—discussed the VS problem together with outlier detection, due to each input affecting the regression task;
- Dong et al. [36]—to wind power generation prediction;
- Sigauke et al. [37]—presented a probabilistic hourly load forecasting framework based on additive quantile regression models;
- Wang et al. [38]—to short-term wind speed forecasting;
- Taormina and Chau [39]—to rainfall-runoff modeling;
- Taormina et al. [40]—to river flow forecasting;
- Cui and Jiang [41]—to chaotic time series prediction;
- Silva et al. [42]—to predict the price of sugarcane derivatives;
- Siqueira et al. [2]—use of VS methods, such as wrappers and filters to predict streamflow series; and
- Kachba et al. [46]—application of wrapper and non-linear filters to estimate the impact of air pollution on human health.
2. Variable Selection
- Relevance: the concept associated with the importance of a given variable may have to the problem, since the information it contains will be the basis of the selection process. The relevance is strong or weak depending on how much its removal degrades the performance of the predictor;
- Redundancy: two or more variables are redundant if their observed values are highly correlated or dependent. The level of this correlation reveals the degree of redundancy; and
- Optimality: a so-called optimal subset of input variables is when there is no other subset that produces better results.
Variable Selection in Streamflow Series Forecasting
3. Filters
3.1. Partial Autocorrelation Function
3.2. Mutual Information
3.3. Partial Mutual Information
3.4. Normalization of Maximum Relevance and Minimum Common Redundancy Mutual Information
4. Wrappers
4.1. Progressive Selection
4.2. Evaluation Functions
5. Bio-Inspired Metaheuristics
5.1. Genetic Algorithm
5.2. Particle Swarm Optimization
6. Case Study
- Training, from January 1st, 1931 to December 31st, 1995 (780 samples);
- Validation, from January 1st, 1996 to December 31st, 2005 (120 samples); and
- Test, from January 1st, 2006 to December 31st, 2015 (120 samples).
6.1. Predictors
6.2. Computational Results
6.3. Discussion
7. Conclusions
- Mean square error (MSE);
- Bayesian information criterion (BIC);
- Bayesian information criterion (AIC).
- The linear filters used were the:
- Partial autocorrelation function (PACF);
- PACF using the Stedinger [52] approach for hydrological series.
- The nonlinear filters addressed were:
- Mutual Information (MI);
- Partial mutual information (PMI); and
- Normalization of maximum relevance and minimum common redundancy mutual information (N-MRMCR-MI).
- Particle swarm optimization (PSO); and
- Genetic algorithm (GA).
- The selected lags were very diverse depending on the method, especially for the monthly case;
- For the annual approaches, some draws could be found;
- The linear models perform better with filters;
- The wrapper is the best choice for the neural network; and
- Regarding the forecasting methods, the monthly ELM achieved the best error values.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
Month | BIC | AIC | W-MSE | PACF | PACF -Sted. | MI | PMI | N-MRMCR -MI | GA | PSO | |
---|---|---|---|---|---|---|---|---|---|---|---|
PAR | J | 1(1) | 1(1) | 5(1,3,4,5,6) | 1(1) | 1(1) | 1(1) | 1(1) | 3(1,5,6) | 5(1,3,4,5,6) | 5(12,3,4,6) |
F | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(2,4,6) | |
M | 1(1) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 5(1,2,3,4,5,5) | 6(1,2,3,4,5,6) | 2(1,5) | |
A | 2(1,2) | 3(1,2,3) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(1,3,5) | |
M | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 3(1,2,3) | 3(1,2,3) | 5(1,2,3,4,5) | 2(1,3) | 4(1,3,4,6) | 6(1,2,3,4,5,6) | 3(1,2,3) | |
J | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,2,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,4) | |
J | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 4(1,2,4,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,4,5,6) | |
A | 1(1) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,3,5,6) | |
S | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,6) | |
O | 2(3,4) | 3(3,4,6) | 6(1,2,3,4,5,6) | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(2) | 6(1,2,3,4,5,6) | 5(1,2,3,4,6) | 4(1,3,4,6) | |
N | 1(1) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,5) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 6(1,2,3,4,5,6) | 5(1,2,3,4,5) | 4(1,2,4,5) | |
D | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,2,6) | 2(1,2) | 4(1,2,3,4) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(4,5) | |
ELM | J | 1(2) | 1(2) | 1(2) | 1(1) | 1(1) | 1(1) | 1(1) | 3(1,5,6) | 2(1,4) | 1(1) |
F | 1(1) | 1(1) | 1(1) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 6(1,2,3,4,5,6) | 2(1,4) | 5(1,3,4,5,6) | |
M | 1(1) | 1(1) | 2(3,6) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 5(1,2,3,4,5,5) | 5(1,2,3,4,5) | 3(1,3,4) | |
A | 1(5) | 1(5) | 1(5) | 2(1,2) | 2(1,2) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,2) | 4(1,2,4,5) | |
M | 1(2) | 1(2) | 1(2) | 3(1,2,3) | 3(1,2,3) | 5(1,2,3,4,5) | 2(1,3) | 4(1,3,4,6) | 2(2,3) | 3(1,2,3) | |
J | 1(1) | 1(1) | 1(1) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,2,5) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | |
J | 1(1) | 1(1) | 1(1) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 4(1,2,4,5) | 6(1,2,3,4,5,6) | 2(1,6) | 1(1) | |
A | 1(1) | 1(1) | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,5) | 1(3) | |
S | 1(1) | 1(1) | 1(1) | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 2(1,3) | |
O | 1(1) | 1(1) | 1(1) | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(2) | 6(1,2,3,4,5,6) | 1(1) | 3(1,2,6) | |
N | 1(5) | 1(5) | 2(2,5) | 2(1,5) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 3(2,3,5) | |
D | 1(2) | 1(2) | 1(2) | 3(1,2,6) | 2(1,2) | 4(1,2,3,4) | 1(1) | 6(1,2,3,4,5,6) | 1(2) | 2(2,6) | |
AR | 2(1,2) | 2(1,2) | 4(1,2,3,5) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,5) | 4(1,2,3,5) | |
ELM | 1(1) | 2(1,4) | 6(1,2,3,4,5,6) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,5) | 5(1,2,4,5,6) |
Month | BIC | AIC | W-MSE | PACF |
PACF -Sted. | MI | PMI |
N-MRMCR -MI | GA | PSO | |
---|---|---|---|---|---|---|---|---|---|---|---|
PAR | J | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,6) | 1(1) | 1(6) | 1(6) | 1(1) | 6(1,2,3,4,5,6) | 3(4,5,6) |
F | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 4(1,2,4,5) | |
M | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 2(1,2) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(3,4) | |
A | 3(1,2,5) | 3(1,2,5) | 6(1,2,3,4,5,6) | 3(1,2,5) | 2(1,2) | 3(1,2,3) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 1(4) | |
M | 4(1,2,3,5) | 4(1,2,3,5) | 6(1,2,3,4,5,6) | 2(1,3) | 1(1) | 5(1,2,3,4,5) | 6(1,2,3,4,5,6) | 3(1,2,3) | 6(1,2,3,4,5,6) | 2(5,6) | |
J | 3(1,3,5) | 4(1,2,3,5) | 5(1,2,3,5,6) | 1(1) | 1(1) | 5(1,2,3,4,5) | 3(1,2,5) | 2(1,2) | 5(1,2,3,5,6) | 2(1,2) | |
J | 3(1,2,6) | 3(1,2,6) | 3(1,2,6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 3(1,2,6) | 1(1) | |
A | 1(1) | 1(1) | 1(1) | 2(1,3) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 1(1) | 1(1) | 2(1,2) | |
S | 1(1) | 1(1) | 2(1,2) | 2(1,3) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 4(1,2,3,4) | 2(1,2) | 2(1,2) | |
O | 1(1) | 1(1) | 3(1,2,3) | 4(1,3,4,6) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 4(1,2,3,4) | 3(1,2,3) | 1(2) | |
N | 2(1,2) | 2(1,2) | 4(1,2,3,4) | 3(1,2,5) | 2(1,2) | 1(1) | 1(1) | 1(1) | 4(1,2,3,4) | 2(4,5) | |
D | 1(1) | 1(1) | 5(1,2,3,4,5) | 3(1,5,6) | 1(1) | 2(1,2) | 2(1,6) | 2(1,2) | 5(1,2,3,4,5) | 3(1,2,3) | |
ELM | J | 1(2) | 1(2) | 3(1,2,6) | 2(1,6) | 1(1) | 1(6) | 1(6) | 1(1) | 4(1,2,3,5) | 3(4,5,6) |
F | 1(1) | 1(1) | 2(1,5) | 1(1) | 1(1) | 1(1) | 1(1) | 1(1) | 2(1,4) | 2(1,3) | |
M | 1(1) | 1(1) | 2(2,3) | 1(1) | 1(1) | 2(1,2) | 1(1) | 1(1) | 1(2) | 5(1,2,4,5,6) | |
A | 1(5) | 1(5) | 1(5) | 3(1,2,5) | 2(1,2) | 3(1,2,3) | 2(1,2) | 2(1,2) | 1(3) | 1(5) | |
M | 1(2) | 1(2) | 1(2) | 2(1,3) | 1(1) | 5(1,2,3,4,5) | 6(1,2,3,4,5,6) | 3(1,2,3) | 2(1,4) | 2(1,5) | |
J | 1(1) | 1(1) | 3(1,5,6) | 1(1) | 1(1) | 5(1,2,3,4,5) | 3(1,2,5) | 2(1,2) | 2(1,5) | 5(1,2,3,4,5) | |
J | 1(1) | 1(1) | 2(1,2) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 2(1,6) | 4(1,3,4,6) | |
A | 1(1) | 1(1) | 3(1,4,5) | 2(1,3) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 1(1) | 2(1,6) | 4(1,2,3,5) | |
S | 1(1) | 1(1) | 3(1,5,6) | 2(1,3) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 4(1,2,3,4) | 5(1,2,4,5,6) | 3(2,5,6) | |
O | 1(1) | 1(1) | 3(1,3,4) | 4(1,3,4,6) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 4(1,2,3,4) | 5(1,2,3,4,5) | 4(1,2,4,6) | |
N | 1(5) | 1(5) | 3(1,4,5) | 3(1,2,5) | 2(1,2) | 1(1) | 1(1) | 1(1) | 2(1,3) | 3(1,2,3) | |
D | 1(2) | 1(2) | 2(2,5) | 3(1,5,6) | 1(1) | 2(1,2) | 2(1,6) | 2(1,2) | 3(1,2,5) | 4(2,4,5,6) | |
AR | 2(1,2) | 2(1,2) | 2(1,2) | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | |
ELM | 1(1) | 2(1,4) | 6(1,2,3,4,5,6) | 4(1,2,3,4) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,4,5,6) |
Month | BIC | AIC | W-MSE | PACF | PACF -Sted. | MI | PMI | N-MRMCR -MI | GA | PSO | |
---|---|---|---|---|---|---|---|---|---|---|---|
PAR | J | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 3(1,2,3) | 1(1) | 4(1,4,5,6) | 6(1,2,3,4,5,6) | 2(3,6) |
F | 3(1,3,5) | 3(1,3,5) | 6(1,2,3,4,5,6) | 3(1,4,5) | 1(1) | 4(1,2,4,5) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,4) | |
M | 1(1) | 1(1) | 5(1,2,3,4,6) | 1(1) | 1(1) | 2(1,2) | 1(1) | 6(1,2,3,4,5,6) | 5(1,2,3,4,6) | 3(1,2,5) | |
A | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 5(1,2,3,4,6) | 1(5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(2,4,6) | |
M | 1(1) | 2(1,3) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 4(1,2,3,4) | 2(1,3) | 5(1,2,4,5,6) | 6(1,2,3,4,5,6) | 3(1,3,4) | |
J | 5(1,2,3,5,6) | 5(1,2,3,5,6) | 6(1,2,3,4,5,6) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | |
J | 5(1,2,3,4,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 6(1,2,3,4,5,6) | 5(1,2,3,5,6) | 2(1,3) | |
A | 1(1) | 1(1) | 5(1,2,3,4,5) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,5,4) | 6(1,2,3,4,5,6) | 5(1,2,3,4,5) | 4(1,2,3,5) | |
S | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,3,4,5) | |
O | 1(1) | 1(1) | 4(1,2,3,4) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,2,6) | 6(1,2,3,4,5,6) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | |
N | 1(1) | 2(1,3) | 2(1,3) | 1(1) | 1(1) | 5(1,2,3,4,5) | 1(1) | 6(1,2,3,4,5,6) | 2(1,3) | 1(1) | |
D | 1(1) | 1(1) | 5(1,2,3,4,5) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 6(1,2,3,4,5,6) | 5(1,2,3,4,5) | 1(6) | |
ELM | J | 1(2) | 1(2) | 1(3) | 1(1) | 1(1) | 3(1,2,3) | 1(1) | 4(1,4,5,6) | 1(3) | 3(1,3,5) |
F | 1(1) | 1(1) | 3(1,2,6) | 3(1,4,5) | 1(1) | 4(1,2,4,5) | 1(1) | 6(1,2,3,4,5,6) | 2(1,6) | 4(1,2,5,6) | |
M | 1(1) | 1(1) | 2(1,3) | 1(1) | 1(1) | 2(1,2) | 1(1) | 6(1,2,3,4,5,6) | 3(1,5,6) | 3(1,4,6) | |
A | 1(5) | 1(5) | 3(1,2,4) | 1(1) | 1(1) | 5(1,2,3,4,6) | 1(5) | 6(1,2,3,4,5,6) | 2(1,6) | 5(1,2,3,5,6) | |
M | 1(2) | 1(2) | 1(1) | 1(1) | 1(1) | 4(1,2,3,4) | 2(1,3) | 5(1,2,4,5,6) | 2(1,3) | 1(1) | |
J | 1(1) | 1(1) | 2(1,4) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,4,5,6) | 3(1,3,6) | |
J | 1(1) | 1(1) | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 2(1,4) | |
A | 1(1) | 1(1) | 1(1) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,5,4) | 6(1,2,3,4,5,6) | 1(1) | 2(1,2) | |
S | 1(1) | 1(1) | 2(1,5) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 3(1,4,5) | |
O | 1(1) | 1(1) | 5(1,2,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,2,6) | 6(1,2,3,4,5,6) | 1(3) | 4(1,4,5,6) | |
N | 1(5) | 1(5) | 2(1,3) | 1(1) | 1(1) | 5(1,2,3,4,5) | 1(1) | 6(1,2,3,4,5,6) | 1(3) | 4(1,4,5,6) | |
D | 1(2) | 1(2) | 4(2,3,5,6) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 6(1,2,3,4,5,6) | 1(2) | 5(1,2,3,4,5) | |
AR | 2(1,3) | 2(1,3) | 2(1,3) | 2(1,3) | 3(1,3,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,3) | 2(1,3) | |
ELM | 1(1) | 2(1,4) | 6(1,2,3,4,5,6) | 1(1) | 3(1,3,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,4) |
Month | BIC | AIC | W-MSE | PACF | PACF -Sted. | MI | PMI | N-MRMCR -MI | GA | PSO | |
---|---|---|---|---|---|---|---|---|---|---|---|
PAR | J | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,5) | 1(1) | 3(1,5,6) | 1(1) | 1(1) | 5(1,3,4,5,6) | 3(3,4,6) |
F | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,6) | 1(1) | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(5,6) | |
M | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 3(3,5,6) | |
A | 3(1,2,6) | 3(1,2,6) | 4(1,2,5,6) | 2(1,2) | 2(1,2) | 3(1,2,4) | 1(1) | 2(1,2) | 4(1,2,5,6) | 2(2,3) | |
M | 4(1,2,3,5) | 4(1,2,3,5) | 5(1,2,3,4,5) | 3(1,2,3) | 3(1,2,3) | 5(1,2,3,4,5) | 2(1,2) | 3(1,2,3) | 5(1,2,3,4,5) | 3(2,5,6) | |
J | 1(1) | 1(1) | 5(1,2,4,5,6) | 1(1) | 1(1) | 5(1,2,3,4,5) | 6(1,2,3,4,5,6) | 2(1,2) | 5(1,2,4,5,6) | 4(2,3,4,5) | |
J | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 1(3) | |
A | 1(1) | 5(1,2,3,4,6) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 1(1) | 6(1,2,3,4,5,6) | 5(2,3,4,5,6) | |
S | 2(1,3) | 2(1,3) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(1) | |
O | 2(1,3) | 3(1,3,6) | 6(1,2,3,4,5,6) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 1(1) | |
N | 1(1) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 4(1,2,5,6) | |
D | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 5(1,2,3,4,5) | 1(1) | 2(1,2) | 6(1,2,3,4,5,6) | 1(4) | |
ELM | J | 1(2) | 1(2) | 2(1,6) | 2(1,5) | 1(1) | 3(1,5,6) | 1(1) | 1(1) | 5(1,2,3,4,6) | 4(1,2,4,6) |
F | 1(1) | 1(1) | 1(1) | 2(1,6) | 1(1) | 1(1) | 1(1) | 1(1) | 4(1,2,3,6) | 4(1,2,4,5) | |
M | 1(1) | 1(1) | 2(1,2) | 2(1,6) | 1(1) | 2(1,2) | 1(1) | 1(1) | 2(1,2) | 6(1,2,3,4,5,6) | |
A | 1(5) | 1(5) | 2(1,2) | 2(1,2) | 2(1,2) | 3(1,2,4) | 1(1) | 2(1,2) | 2(1,2) | 4(1,2,3,5) | |
M | 1(2) | 1(2) | 1(1) | 3(1,2,3) | 3(1,2,3) | 5(1,2,3,4,5) | 2(1,2) | 3(1,2,3) | 4(1,2,4,6) | 4(2,3,5,6) | |
J | 1(1) | 1(1) | 3(1,2,6) | 1(1) | 1(1) | 5(1,2,3,4,5) | 6(1,2,3,4,5,6) | 2(1,2) | 2(2,5) | 3(1,2,4) | |
J | 1(1) | 1(1) | 2(1,2) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | 2(1,2) | 3(1,2,5) | |
A | 1(1) | 1(1) | 3(1,2,5) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 1(1) | 5(1,2,3,4,5) | 2(1,2) | |
S | 1(1) | 1(1) | 2(4,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 4(1,2,3,4) | 2(2,6) | 2(5,6) | |
O | 1(1) | 1(1) | 2(1,6) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,4) | 3(1,3,4) | 3(1,2,5) | |
N | 1(5) | 1(5) | 1(6) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 1(1) | 1(1) | 2(2,5) | 4(2,3,5,6) | |
D | 1(2) | 1(2) | 3(2,3,4) | 1(1) | 1(1) | 5(1,2,3,4,5) | 1(1) | 2(1,2) | 1(2) | 2(2,6) | |
AR | 2(1,2) | 2(1,2) | 2(1,2) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 2(1,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | |
ELM | 1(1) | 2(1,4) | 6(1,2,3,4,5,6) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 2(1,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,4) |
Month | BIC | AIC | W-MSE | PACF | PACF -Sted. | MI | PMI | N-MRMCR -MI | GA | PSO | |
---|---|---|---|---|---|---|---|---|---|---|---|
PAR | J | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 3(1,5,6) | 1(1) | 3(1,2,4) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,4,5,6) |
F | 2(1,3) | 3(1,3,5) | 5(1,3,4,5,6) | 1(1) | 1(1) | 3(1,2,3) | 1(1) | 6(1,2,3,4,5,6) | 5(1,3,4,5,6) | 3(2,5,6) | |
M | 1(1) | 2(1,6) | 5(1,2,3,4,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 6(1,2,3,4,5,6) | 5(1,2,3,4,6) | 2(2,5) | |
A | 1(2) | 1(2) | 6(1,2,3,4,5,6) | 3(1,2,4) | 2(1,2) | 5(1,2,3,4,5) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(1,5,6) | |
M | 1(1) | 2(1,5) | 4(1,3,5,6) | 5(1,2,3,4,6) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 2(1,6) | 6(1,2,3,4,5,6) | 4(1,3,5,6) | 2(2,3) | |
J | 1(1) | 2(1,2) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,5) | 3(1,2,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(1,3,5) | |
J | 3(1,2,4) | 3(1,2,4) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,3,4,6) | 2(1,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(3,4,5,6) | |
A | 1(1) | 3(1,2,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,3,4,5) | 2(1,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(2,3,6) | |
S | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,3) | 2(1,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 3(2,4,5) | |
O | 1(1) | 3(1,2,4) | 5(1,2,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,4) | 2(1,4) | 6(1,2,3,4,5,6) | 5(1,2,4,5,6) | 2(3,4) | |
N | 3(1,2,3) | 4(1,2,3,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,3) | 2(1,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(5,6) | |
D | 1(1) | 2(1,4) | 6(1,2,3,4,5,6) | 5(1,2,3,4,6) | 4(1,2,3,4) | 3(1,2,4) | 1(1) | 6(1,2,3,4,5,6) | 5(1,3,4,5,6) | 1(1) | |
ELM | J | 1(2) | 1(2) | 1(2) | 3(1,5,6) | 1(1) | 3(1,2,4) | 1(1) | 6(1,2,3,4,5,6) | 2(1,4) | 3(1,3,4) |
F | 1(1) | 1(1) | 1(1) | 1(1) | 1(1) | 3(1,2,3) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 5(1,2,3,4,6) | |
M | 1(1) | 1(1) | 2(3,6) | 2(1,2) | 2(1,2) | 6(1,2,3,4,5,6) | 1(1) | 6(1,2,3,4,5,6) | 2(1,2) | 3(1,3,6) | |
A | 1(5) | 1(5) | 1(5) | 3(1,2,4) | 2(1,2) | 5(1,2,3,4,5) | 3(1,2,3) | 6(1,2,3,4,5,6) | 2(1,2) | 3(1,3,4) | |
M | 1(2) | 1(2) | 1(2) | 5(1,2,3,4,6) | 4(1,2,3,4) | 6(1,2,3,4,5,6) | 2(1,6) | 6(1,2,3,4,5,6) | 1(4) | 4(1,2,3,6) | |
J | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,5) | 3(1,2,5) | 6(1,2,3,4,5,6) | 2(2,5) | 1(1) | |
J | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,3,4,6) | 2(1,3) | 6(1,2,3,4,5,6) | 3(2,3,6) | 2(1,5) | |
A | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 5(1,2,3,4,5) | 2(1,5) | 6(1,2,3,4,5,6) | 3(3,4,5) | 3(2,3,5) | |
S | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,3) | 2(1,3) | 6(1,2,3,4,5,6) | 4(1,2,4,5) | 3(2,3,5) | |
O | 1(1) | 1(1) | 1(1) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,4) | 2(1,4) | 6(1,2,3,4,5,6) | 3(1,4,5) | 3(2,4,6) | |
N | 1(5) | 1(5) | 2(2,5) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,3) | 2(1,3) | 6(1,2,3,4,5,6) | 3(1,2,3) | 1(1) | |
D | 1(2) | 1(2) | 1(2) | 5(1,2,3,4,6) | 4(1,2,3,4) | 3(1,2,4) | 1(1) | 6(1,2,3,4,5,6) | 3(1,4,5) | 2(1,3) | |
AR | 2(1,2) | 2(1,2) | 2(1,2) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 2(1,2) | 2(1,2) | |
ELM | 1(1) | 2(1,4) | 6(1,2,3,4,5,6) | 3(1,2,3) | 3(1,2,3) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 6(1,2,3,4,5,6) | 4(1,2,3,6) |
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Subsets | Selected Inputs |
---|---|
1 | v1 |
2 | v2 |
3 | v3 |
4 | v1,v2 |
5 | v1,v3 |
6 | v2,v3 |
7 | v1,v2,v3 |
Complete Series | Test Set | |||
---|---|---|---|---|
Series | Mean (m³/s) | S. Deviation (m³/s) | Mean (m³/s) | S. Deviation (m³/s) |
Furnas | 912.1225 | 613.5036 | 803.6833 | 611.6814 |
Emborcação | 480.6578 | 360.3957 | 447.7333 | 355.7428 |
Sobradinho | 2.6062 × 103 | 1.9412 × 103 | 1.9607 × 103 | 1.5001 × 103 |
Agua Vermelha | 2.0773 × 103 | 1.2957 × 103 | 1.9635 × 103 | 1.2668 × 103 |
Passo Real | 208.6216 | 169.7734 | 228.0083 | 167.1326 |
Variable Selection | Test | Training | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
MSE | MAE | MSEd | MAEd | MSE | MAE | MSEd | MAEd | |||
FURNAS | WR | BIC | 109,014 | 210.77 | 0.4330 | 0.5078 | 96,033 | 196.58 | 0.4490 | 0.4907 |
AIC | 109,014 | 210.77 | 0.4330 | 0.5078 | 96,033 | 196.58 | 0.4490 | 0.4907 | ||
WRAPPER-MSE | 107,962 | 208.81 | 0.4224 | 0.4992 | 97,037 | 195.55 | 0.4459 | 0.4859 | ||
Lf | FACPPe | 107,551 | 209.32 | 0.4259 | 0.5043 | 96,646 | 195.75 | 0.4464 | 0.4871 | |
FACPPe-Sted. | 107,551 | 209.32 | 0.4259 | 0.5043 | 96,646 | 195.75 | 0.4464 | 0.4871 | ||
Nf | MI | 108,083 | 209.65 | 0.4252 | 0.5015 | 96,702 | 195.43 | 0.4452 | 0.4856 | |
PMI | 108,083 | 209.65 | 0.4252 | 0.5015 | 96,702 | 195.43 | 0.4452 | 0.4856 | ||
N-MRMCR-MI | 108,083 | 209.65 | 0.4252 | 0.5015 | 96,702 | 195.43 | 0.4452 | 0.4856 | ||
M | GA | 107,962 | 208.81 | 0.4224 | 0.4992 | 97,037 | 195.55 | 0.4459 | 0.4859 | |
PSO | 107,962 | 208.81 | 0.4224 | 0.4992 | 97,037 | 195.55 | 0.4459 | 0.4859 | ||
EMBORCAÇÃO | WR | BIC | 51,745 | 139.36 | 0.5487 | 0.5716 | 40,456 | 119.79 | 0.4838 | 0.5131 |
AIC | 51,745 | 139.36 | 0.5487 | 0.5716 | 40,456 | 119.79 | 0.4838 | 0.5131 | ||
WRAPPER-MSE | 51,745 | 139.36 | 0.5487 | 0.5716 | 40,456 | 119.79 | 0.4838 | 0.5131 | ||
Lf | FACPPe | 50,408 | 138.01 | 0.5353 | 0.5613 | 39,953 | 119.40 | 0.4790 | 0.5102 | |
FACPPe-Sted. | 50,408 | 138.01 | 0.5353 | 0.5613 | 39,953 | 119.40 | 0.4790 | 0.5102 | ||
Nf | MI | 50,559 | 138.24 | 0.5397 | 0.5602 | 39,880 | 119.20 | 0.4768 | 0.5081 | |
PMI | 50,559 | 138.24 | 0.5397 | 0.5602 | 39,880 | 119.20 | 0.4768 | 0.5081 | ||
N-MRMCR-MI | 50,559 | 138.24 | 0.5397 | 0.5602 | 39,880 | 119.20 | 0.4768 | 0.5081 | ||
M | GA | 51,745 | 139.36 | 0.5487 | 0.5716 | 40,456 | 119.79 | 0.4838 | 0.5131 | |
PSO | 51,745 | 139.36 | 0.5487 | 0.5716 | 40,456 | 119.79 | 0.4838 | 0.5131 | ||
SOBRADINHO | WR | BIC | 836,738 | 568.04 | 0.3071 | 0.4408 | 1,032,181 | 580.83 | 0.3895 | 0.4366 |
AIC | 836,738 | 568.04 | 0.3071 | 0.4408 | 1,032,181 | 580.83 | 0.3895 | 0.4366 | ||
WRAPPER-MSE | 836,738 | 568.04 | 0.3071 | 0.4408 | 1,032,181 | 580.83 | 0.3895 | 0.4366 | ||
Lf | FACPPe | 836,738 | 568.04 | 0.3071 | 0.4408 | 1,032,181 | 580.83 | 0.3895 | 0.4366 | |
FACPPe-Sted. | 863,796 | 577.30 | 0.3196 | 0.4515 | 1,020,492 | 578.13 | 0.3910 | 0.4403 | ||
Nf | MI | 828,142 | 566.45 | 0.3043 | 0.4375 | 994,408 | 573.77 | 0.3837 | 0.4350 | |
PMI | 828,142 | 566.45 | 0.3043 | 0.4375 | 994,408 | 573.77 | 0.3837 | 0.4350 | ||
N-MRMCR-MI | 828,142 | 566.45 | 0.3043 | 0.4375 | 994,408 | 573.77 | 0.3837 | 0.4350 | ||
M | GA | 836,738 | 568.04 | 0.3071 | 0.4408 | 1,032,181 | 580.83 | 0.3895 | 0.4366 | |
PSO | 836,738 | 568.04 | 0.3071 | 0.4408 | 1,032,181 | 580.83 | 0.3895 | 0.4366 | ||
AGUA VERMELHA | WR | BIC | 417,720 | 404.35 | 0.4097 | 0.4826 | 378,866 | 394.30 | 0.4095 | 0.4780 |
AIC | 417,720 | 404.35 | 0.4097 | 0.4826 | 378,866 | 394.30 | 0.4095 | 0.4780 | ||
WRAPPER-MSE | 417,720 | 404.35 | 0.4097 | 0.4826 | 378,866 | 394.30 | 0.4095 | 0.4780 | ||
Lf | FACPPe | 409,613 | 401.00 | 0.4052 | 0.4803 | 379,329 | 392.98 | 0.4078 | 0.4752 | |
FACPPe-Sted. | 409,613 | 401.00 | 0.4052 | 0.4803 | 379,329 | 392.98 | 0.4078 | 0.4752 | ||
Nf | MI | 415,465 | 404.50 | 0.4062 | 0.4828 | 378,197 | 392.12 | 0.4065 | 0.4749 | |
PMI | 413,991 | 403.40 | 0.4119 | 0.4886 | 369,658 | 393.50 | 0.4149 | 0.4819 | ||
N-MRMCR-MI | 413,991 | 403.40 | 0.4119 | 0.4886 | 378,197 | 392.12 | 0.4065 | 0.4749 | ||
M | GA | 415,465 | 404.50 | 0.4062 | 0.4828 | 378,197 | 392.12 | 0.4065 | 0.4749 | |
PSO | 415,465 | 404.50 | 0.4062 | 0.4828 | 378,197 | 392.12 | 0.4065 | 0.4749 | ||
PASSO REAL | WR | BIC | 14,996 | 88.65 | 0.6570 | 0.5969 | 16,637 | 86.70 | 0.6490 | 0.5718 |
AIC | 14,996 | 88.65 | 0.6570 | 0.5969 | 16,637 | 86.70 | 0.6490 | 0.5718 | ||
WRAPPER-MSE | 14,996 | 88.65 | 0.6570 | 0.5969 | 16,637 | 86.70 | 0.6490 | 0.5718 | ||
Lf | FACPPe | 14,523 | 87.74 | 0.6397 | 0.5914 | 16,497 | 86.32 | 0.6415 | 0.5696 | |
FACPPe-Sted. | 14,523 | 87.74 | 0.6397 | 0.5914 | 16,497 | 86.32 | 0.6415 | 0.5696 | ||
Nf | MI | 14,632 | 88.16 | 0.6447 | 0.5956 | 16,478 | 86.08 | 0.6398 | 0.5676 | |
PMI | 14,632 | 88.16 | 0.6447 | 0.5956 | 16,478 | 86.08 | 0.6398 | 0.5676 | ||
N-MRMCR-MI | 14,632 | 88.16 | 0.6447 | 0.5956 | 16,478 | 86.08 | 0.6398 | 0.5676 | ||
M | GA | 14,996 | 88.65 | 0.6570 | 0.5969 | 16,637 | 86.70 | 0.6490 | 0.5718 | |
PSO | 14,996 | 88.65 | 0.6570 | 0.5969 | 16,637 | 86.70 | 0.6490 | 0.5718 |
Variable Selection | Test | Training | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
MSE | MAE | MSEd | MAEd | MSE | MAE | MSEd | MAEd | |||
FURNAS | WR | BIC | 117,347 | 207.37 | 0.4096 | 0.4819 | 94,879 | 190.77 | 0.4124 | 0.4725 |
AIC | 120,629 | 211.50 | 0.4343 | 0.4957 | 93,403 | 189.09 | 0.4041 | 0.4677 | ||
WRAPPER-MSE | 128,415 | 215.49 | 0.4546 | 0.5046 | 87,842 | 183.45 | 0.3846 | 0.4556 | ||
Lf | PACF | 118,744 | 211.52 | 0.4113 | 0.4882 | 102,317 | 198.26 | 0.4499 | 0.4899 | |
PACF-Sted. | 117,144 | 206.70 | 0.4055 | 0.4788 | 94,824 | 190.41 | 0.4112 | 0.4709 | ||
Nf | MI | 120,121 | 211.57 | 0.4302 | 0.4963 | 92,821 | 187.35 | 0.3973 | 0.4620 | |
PMI | 122,682 | 215.72 | 0.4442 | 0.5055 | 96,930 | 193.12 | 0.4502 | 0.4822 | ||
N-MRMCR-MI | 135,260 | 224.64 | 0.4811 | 0.5271 | 95,500 | 187.39 | 0.3999 | 0.4619 | ||
M | GA | 128,680 | 216.09 | 0.4601 | 0.5073 | 87,837 | 183.33 | 0.3845 | 0.4550 | |
PSO | 133,171 | 228.39 | 0.4794 | 0.5279 | 109,291 | 201.15 | 0.4856 | 0.5007 | ||
EMBORCAÇÃO | WR | BIC | 46,356 | 128.98 | 0.5034 | 0.5350 | 35,421 | 112.66 | 0.4377 | 0.4817 |
AIC | 46,356 | 129.00 | 0.5034 | 0.5352 | 35,415 | 112.57 | 0.4370 | 0.4806 | ||
WRAPPER-MSE | 51,529 | 134.64 | 0.5460 | 0.5489 | 33,362 | 109.12 | 0.4188 | 0.4702 | ||
Lf | PACF | 50,251 | 134.96 | 0.5526 | 0.5612 | 42,622 | 123.54 | 0.5596 | 0.5418 | |
PACF-Sted. | 46,195 | 129.23 | 0.5093 | 0.5444 | 35,710 | 113.81 | 0.4535 | 0.4914 | ||
Nf | MI | 47,918 | 130.25 | 0.5094 | 0.5353 | 39,096 | 115.99 | 0.4588 | 0.4851 | |
PMI | 48,392 | 129.72 | 0.5183 | 0.5361 | 40,967 | 118.63 | 0.4913 | 0.5010 | ||
N-MRMCR-MI | 46,088 | 128.22 | 0.4982 | 0.5343 | 35,593 | 113.05 | 0.4419 | 0.4836 | ||
M | GA | 51,529 | 134.64 | 0.5460 | 0.5489 | 33,362 | 109.12 | 0.4188 | 0.4702 | |
PSO | 58,768 | 153.16 | 0.9161 | 0.6795 | 48,079 | 136.61 | 0.8187 | 0.6266 | ||
SOBRADINHO | WR | BIC | 650,374 | 507.00 | 0.2791 | 0.4097 | 850,140 | 530.47 | 0.3560 | 0.4133 |
AIC | 642,631 | 496.61 | 0.2706 | 0.3972 | 847,763 | 528.97 | 0.3530 | 0.4118 | ||
WRAPPER-MSE | 675,439 | 510.13 | 0.3071 | 0.4194 | 829,497 | 523.19 | 0.3450 | 0.4061 | ||
Lf | PACF | 724,099 | 546.38 | 0.3361 | 0.4462 | 1,021,024 | 577.88 | 0.4665 | 0.4523 | |
PACF-Sted. | 666,690 | 513.82 | 0.2896 | 0.4175 | 886,340 | 540.87 | 0.3671 | 0.4207 | ||
Nf | MI | 628,672 | 495.94 | 0.2923 | 0.4124 | 847,892 | 530.37 | 0.3519 | 0.4104 | |
PMI | 665,958 | 511.55 | 0.2948 | 0.4171 | 886,511 | 544.87 | 0.3852 | 0.4340 | ||
N-MRMCR-MI | 682,074 | 513.84 | 0.3034 | 0.4180 | 824,000 | 519.21 | 0.3399 | 0.4010 | ||
M | GA | 675,494 | 510.64 | 0.3074 | 0.4207 | 829,499 | 523.20 | 0.3450 | 0.4061 | |
PSO | 755,372 | 564.36 | 0.3402 | 0.4509 | 1,181,095 | 632.30 | 0.5290 | 0.4889 | ||
AGUA VERMELHA | WR | BIC | 438,074 | 401.41 | 0.4075 | 0.4729 | 357,604 | 380.46 | 0.3774 | 0.4614 |
AIC | 439,586 | 402.88 | 0.4161 | 0.4769 | 356,951 | 379.09 | 0.3742 | 0.4587 | ||
WRAPPER-MSE | 473,968 | 409.23 | 0.4437 | 0.4881 | 335,448 | 366.74 | 0.3564 | 0.4461 | ||
Lf | PACF | 469,836 | 407.75 | 0.4179 | 0.4732 | 404,467 | 393.70 | 0.4056 | 0.4728 | |
PACF-Sted. | 432,799 | 393.57 | 0.3960 | 0.4611 | 359,174 | 382.19 | 0.3817 | 0.4645 | ||
Nf | MI | 459,970 | 408.37 | 0.4300 | 0.4836 | 379,638 | 380.90 | 0.3817 | 0.4566 | |
PMI | 443,387 | 399.30 | 0.4188 | 0.4711 | 360,565 | 384.67 | 0.3838 | 0.4673 | ||
N-MRMCR-MI | 436,274 | 395.68 | 0.4029 | 0.4664 | 357,410 | 380.80 | 0.3781 | 0.4618 | ||
M | GA | 476,365 | 409.64 | 0.4450 | 0.4884 | 335,047 | 366.40 | 0.3561 | 0.4459 | |
PSO | 775,418 | 560.05 | 0.7722 | 0.6972 | 627,743 | 500.78 | 0.7843 | 0.6385 | ||
PASSO REAL | WR | BIC | 16,793 | 93.49 | 0.7945 | 0.6449 | 15,864 | 85.70 | 0.6228 | 0.5684 |
AIC | 15,601 | 88.50 | 0.7664 | 0.6198 | 15,299 | 84.28 | 0.6022 | 0.5590 | ||
WRAPPER-MSE | 15,584 | 91.70 | 0.7454 | 0.6395 | 14,770 | 82.82 | 0.5797 | 0.5474 | ||
Lf | PACF | 15,924 | 90.39 | 0.8031 | 0.6296 | 15,012 | 84.05 | 0.6062 | 0.5605 | |
PACF-Sted. | 15,522 | 89.04 | 0.7557 | 0.6151 | 14,976 | 83.78 | 0.6018 | 0.5576 | ||
Nf | MI | 14,982 | 90.73 | 0.7152 | 0.6300 | 16,107 | 85.61 | 0.6336 | 0.5671 | |
PMI | 15,282 | 87.41 | 0.7423 | 0.6038 | 16,013 | 85.63 | 0.6344 | 0.564 | ||
N-MRMCR-MI | 15,474 | 90.55 | 0.7402 | 0.6306 | 14,638 | 82.25 | 0.5724 | 0.5428 | ||
M | GA | 15,779 | 92.12 | 0.7551 | 0.6424 | 14,769 | 82.99 | 0.5796 | 0.5486 | |
PSO | 17,895 | 101.47 | 0.8774 | 0.7130 | 22,018 | 103.18 | 0.8769 | 0.6816 |
Variable Selection | Monthly Approach | Annual Approach | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
MSE | MAE | MSEd | MAEd | MSE | MAE | MSEd | MAEd | |||
FURNAS | WR | BIC | 124,391 | 212.47 | 0.4320 | 0.4943 | 123,754 | 220.19 | 0.4597 | 0.5307 |
AIC | 123,550 | 210.02 | 0.4264 | 0.4878 | 126,426 | 219.67 | 0.4503 | 0.5195 | ||
WRAPPER-MSE | 119,067 | 206.45 | 0.4060 | 0.4809 | 123,745 | 217.80 | 0.4455 | 0.5171 | ||
Lf | PACF | 126,594 | 222.32 | 0.4623 | 0.5236 | 129,323 | 219.72 | 0.4451 | 0.5138 | |
PACF-Sted. | 121,806 | 212.11 | 0.4433 | 0.4997 | 126,768 | 216.79 | 0.4509 | 0.5110 | ||
Nf | MI | 130,637 | 230.24 | 0.5115 | 0.5656 | 124,105 | 220.33 | 0.4665 | 0.5279 | |
PMI | 126,445 | 217.60 | 0.4769 | 0.5207 | 126,349 | 221.97 | 0.4611 | 0.5287 | ||
N-MRMCR-MI | 140,451 | 238.15 | 0.5424 | 0.5842 | 126,507 | 222.67 | 0.4650 | 0.5330 | ||
M | GA | 137,304 | 230.58 | 0.5063 | 0.5448 | 135,978 | 232.35 | 0.4963 | 0.5594 | |
PSO | 132,599 | 228.87 | 0.4829 | 0.5377 | 131,426 | 226.52 | 0.4823 | 0.5435 | ||
EMBORCAÇÃO | WR | BIC | 38,143 | 114.51 | 0.4495 | 0.4956 | 48,513 | 130.03 | 0.5333 | 0.5586 |
AIC | 41,110 | 116.21 | 0.4657 | 0.4944 | 45,459 | 127.26 | 0.5158 | 0.5501 | ||
WRAPPER-MSE | 37,551 | 118.56 | 0.4335 | 0.5001 | 44,936 | 129.15 | 0.5137 | 0.5557 | ||
Lf | PACF | 44,227 | 124.36 | 0.5122 | 0.5355 | 45,994 | 130.15 | 0.5153 | 0.5563 | |
PACF-Sted. | 48,543 | 130.42 | 0.5395 | 0.5526 | 44,690 | 129.01 | 0.5095 | 0.5556 | ||
Nf | MI | 49,315 | 131.44 | 0.5641 | 0.5662 | 45,169 | 130.89 | 0.5126 | 0.5565 | |
PMI | 52,571 | 132.20 | 0.5707 | 0.5564 | 44,094 | 128.54 | 0.5061 | 0.5511 | ||
N-MRMCR-MI | 47,931 | 129.26 | 0.5398 | 0.5459 | 45,707 | 129.50 | 0.5169 | 0.5558 | ||
M | GA | 54,659 | 142.88 | 0.6225 | 0.6017 | 45,434 | 128.47 | 0.5166 | 0.5515 | |
PSO | 50,211 | 130.82 | 0.5667 | 0.5644 | 45,298 | 130.20 | 0.5170 | 0.5595 | ||
SOBRADINHO | WR | BIC | 642,185 | 519.69 | 0.2954 | 0.4329 | 669,441 | 534.37 | 0.3209 | 0.4632 |
AIC | 590,254 | 492.20 | 0.2979 | 0.4298 | 657,405 | 530.22 | 0.3166 | 0.4532 | ||
WRAPPER-MSE | 587,680 | 495.73 | 0.2945 | 0.4250 | 672,783 | 531.90 | 0.3229 | 0.4567 | ||
Lf | PACF | 696,480 | 530.92 | 0.3376 | 0.4506 | 718,719 | 549.40 | 0.3366 | 0.4676 | |
PACF-Sted. | 747,932 | 550.63 | 0.3561 | 0.4591 | 690,307 | 549.76 | 0.3510 | 0.4872 | ||
Nf | MI | 692,277 | 533.35 | 0.3526 | 0.4587 | 668,656 | 531.83 | 0.3187 | 0.4527 | |
PMI | 746,916 | 546.82 | 0.3647 | 0.4621 | 694,596 | 540.00 | 0.3234 | 0.4549 | ||
N-MRMCR-MI | 828,437 | 582.14 | 0.4819 | 0.3842 | 710,649 | 542.32 | 0.3316 | 0.4583 | ||
M | GA | 728,468 | 563.81 | 0.3400 | 0.4674 | 698,381 | 555.96 | 0.3501 | 0.4869 | |
PSO | 773,147 | 571.99 | 0.3657 | 0.4738 | 712,829 | 558.87 | 0.3655 | 0.4920 | ||
AGUA VERMELHA | WR | BIC | 408,982 | 384.66 | 0.3646 | 0.4528 | 443,959 | 394.45 | 0.4055 | 0.4727 |
AIC | 411,485 | 387.40 | 0.3701 | 0.4583 | 436,790 | 393.10 | 0.3946 | 0.4661 | ||
WRAPPER-MSE | 374,264 | 375.43 | 0.3412 | 0.4429 | 436,981 | 394.69 | 0.3952 | 0.4664 | ||
Lf | PACF | 436,494 | 412.25 | 0.4109 | 0.4864 | 439,692 | 401.17 | 0.4025 | 0.4759 | |
PACF-Sted. | 419,472 | 401.62 | 0.4040 | 0.4813 | 458,381 | 406.68 | 0.4131 | 0.4813 | ||
Nf | MI | 423,961 | 426.85 | 0.4565 | 0.5274 | 453,903 | 420.36 | 0.4288 | 0.5065 | |
PMI | 434,519 | 410.82 | 0.4420 | 0.5026 | 432,417 | 394.64 | 0.3955 | 0.4687 | ||
N-MRMCR-MI | 417,154 | 397.18 | 0.4020 | 0.4787 | 458,725 | 422.00 | 0.4344 | 0.5076 | ||
M | GA | 502,689 | 437.62 | 0.4673 | 0.5201 | 449,919 | 405.97 | 0.4142 | 0.4874 | |
PSO | 478,617 | 440.21 | 0.4526 | 0.5271 | 439,377 | 402.75 | 0.4022 | 0.4806 | ||
PASSO REAL | WR | BIC | 12,768 | 79.70 | 0.6382 | 0.5549 | 15,859 | 89.78 | 0.7367 | 0.6079 |
AIC | 12,964 | 78.81 | 0.6592 | 0.5490 | 15,866 | 89.78 | 0.7366 | 0.6078 | ||
WRAPPER-MSE | 11,828 | 78.42 | 0.6033 | 0.5488 | 15,435 | 87.80 | 0.7277 | 0.5962 | ||
Lf | PACF | 16,288 | 91.41 | 0.7772 | 0.6218 | 15,257 | 86.40 | 0.7278 | 0.5884 | |
PACF-Sted. | 16,435 | 89.65 | 0.7850 | 0.6116 | 15,351 | 86.55 | 0.7320 | 0.5893 | ||
Nf | MI | 15,059 | 86.49 | 0.7196 | 0.5944 | 16,074 | 89.96 | 0.7612 | 0.6170 | |
PMI | 15,400 | 87.35 | 0.7517 | 0.5979 | 16,146 | 91.00 | 0.7584 | 0.6207 | ||
N-MRMCR-MI | 16,635 | 91.48 | 0.7723 | 0.6274 | 16,208 | 90.10 | 0.7683 | 0.6168 | ||
M | GA | 17,035 | 90.31 | 0.8134 | 0.6140 | 16,258 | 91.15 | 0.7626 | 0.6206 | |
PSO | 15,589 | 86.86 | 0.7344 | 0.5964 | 16,270 | 91.04 | 0.7731 | 0.6217 |
Models | ||||||
---|---|---|---|---|---|---|
VS Method | AR Train | AR Test | PAR Train | PAR Test | ELM Annual | ELM Monthly |
BIC | 1(+1) | - | - | - | 1(+3) | - |
AIC | 1(+1) | - | - | - | 2; 1(+8) | - |
WRAPPER-MSE | - | - | 1(+1) | - | 1(+3); 1(+8) | 5 |
PACF | - | 1(+1); 1(+1); 1(+1); 1(+1) | - | - | 1; 1(+8) | - |
PACF-Sted. | - | 1(+1); 1(+1); 1(+1); 1(+1) | - | 2 | 1(+8) | - |
MI | 1(+2); 1(+2); 1(+2) | 1(+2) | - | 2 | 1(+3); 1(+8) | - |
PMI | 1; 1(+2); 1(+2); 1(+2) | 1(+2) | - | - | 1(+3); 1(+8) | - |
N-MRMCR-MI | 1(+2); 1(+2); 1(+2) | 1(+2) | 1 | 1 | 1(+8) | - |
GA | - | - | 3; 1(+1) | - | 1(+8) | - |
PSO | - | - | - | - | 1(+8) | - |
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Siqueira, H.; Macedo, M.; Tadano, Y.d.S.; Alves, T.A.; Stevan, S.L., Jr.; Oliveira, D.S., Jr.; Marinho, M.H.N.; Neto, P.S.G.d.M.; Oliveira, .F.L.d.; Luna, I.; et al. Selection of Temporal Lags for Predicting Riverflow Series from Hydroelectric Plants Using Variable Selection Methods. Energies 2020, 13, 4236. https://doi.org/10.3390/en13164236
Siqueira H, Macedo M, Tadano YdS, Alves TA, Stevan SL Jr., Oliveira DS Jr., Marinho MHN, Neto PSGdM, Oliveira FLd, Luna I, et al. Selection of Temporal Lags for Predicting Riverflow Series from Hydroelectric Plants Using Variable Selection Methods. Energies. 2020; 13(16):4236. https://doi.org/10.3390/en13164236
Chicago/Turabian StyleSiqueira, Hugo, Mariana Macedo, Yara de Souza Tadano, Thiago Antonini Alves, Sergio L. Stevan, Jr., Domingos S. Oliveira, Jr., Manoel H.N. Marinho, Paulo S.G. de Mattos Neto, João F. L. de Oliveira, Ivette Luna, and et al. 2020. "Selection of Temporal Lags for Predicting Riverflow Series from Hydroelectric Plants Using Variable Selection Methods" Energies 13, no. 16: 4236. https://doi.org/10.3390/en13164236
APA StyleSiqueira, H., Macedo, M., Tadano, Y. d. S., Alves, T. A., Stevan, S. L., Jr., Oliveira, D. S., Jr., Marinho, M. H. N., Neto, P. S. G. d. M., Oliveira, . F. L. d., Luna, I., Filho, M. d. A. L., Sarubbo, L. A., & Converti, A. (2020). Selection of Temporal Lags for Predicting Riverflow Series from Hydroelectric Plants Using Variable Selection Methods. Energies, 13(16), 4236. https://doi.org/10.3390/en13164236