# Soft Computing Methods with Phase Space Reconstruction for Wind Speed Forecasting—A Performance Comparison

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Materials and Methods

#### 3.1. Data Sets

#### 3.2. Forecasting Techniques

#### 3.2.1. Auto-Regressive Integrated Moving Average

#### 3.2.2. Nearest Neighbors with Differential Evolution Parameter Optimization

#### 3.2.3. Fuzzy Forecasting

#### 3.2.4. Artificial Neural Network (ANN) with Compact Genetic Algorithm Optimization

#### 3.2.5. EvoDAG

## 4. Results

#### 4.1. Auto-Correlation Analysis of the Data Sets

#### 4.2. Lyapunov Dominant Exponent Analysis of the Data Sets

#### 4.3. Experiments

#### Experiment Settings

**ARIMA**—The ARIMA implementation we used was the one included in the R statistical package [66]. The order of the ARIMA model was estimated using the

`auto.arima`function.

**NN**—To determine the NN parameters (m, $\tau $, and $\u03f5$) we used the deterministic approach described by Kantz in [23]. The deterministic approach uses the Mutual Information algorithm to obtain $\tau $ and the False Nearest Neighbors algorithm to find an optimal m; $\u03f5$ is found by testing the number of neighbors found for an arbitrary $\u03f5$ value, which is updated by the rule $\u03f5\leftarrow \u03f5\times 1.2$ when not enough neighbors are found.

**NNDE**—In NNDE the NN parameters are found by a DE optimization, where the evolutionary individuals are vectors of the form [m, $\tau $, $\u03f5$]. Because of the stochastic nature of DE, this optimization process is executed 30 independent times. The set of parameters that yield the least error score is the one used to forecast.

**FF**—This technique compiles a set of fuzzy rules that describe the time series by using delay vectors of dimension m and time delay $\tau $. These parameter values are set to the same as those obtained by the deterministic method used by NN [23,24]. Since the time series contains outliers, FF uses a simple filter which replaces any value greater than $6\sigma $ ($\sigma $ is the standard deviation of the time series) with the missing value indicator.

**ANN-cGA**—This method determines the optimal topology of a MLP using Compact Genetic Algorithms. The optimization process consists in finding the optimal number of inputs (past observations), the number of hidden neurons, and the learning algorithm.

**EvoDAG**—EvoDAG uses its default parameters and m is set to three days behind.

#### 4.4. Performance Analysis

#### 4.5. One Day Ahead Forecasting

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ACF | Auto-Correlation Function |

ANFIS | Adaptive Neuro Fuzzy Inference System |

ANN | Artificial Neural Network |

ANN-cGA | ANN with cGA |

AR | Auto-regressive |

ARMA | AutoRegressive Moving Average |

ARIMA | AutoRegressive Integrated Moving Average |

cGA | Compact Genetic Algorithms |

EvoDAG | Evolving Directed Acyclic Graph |

FF | Fuzzy Forecast |

FLT | Fuzzy Linguistic Terms |

FR | Fuzzy Rules |

GWPPT | Generalized Wind Power Prediction Tool |

MAPE | Mean Average Percentage Error |

MSE | Mean Square Error |

NARX | Nonlinear Auto-regressive Exogenous Artificial Neural Networks |

NN | Nearest Neighbors |

NNDE | Nearest Neoghbors with Differential Evolution |

ODA | One Day Ahead |

OLS | Ordinary Least Squares |

PDE | Partial Differential Equations |

SMAPE | Symmetric Mean Average Percentage Error |

WPPT | Wind Power Prediction Tool |

RMSE | Root Mean Squared Error |

VAR | Vector Auto-regressive |

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**Figure 5.**Artificial Neural Network (ANN) topology which starts with the input layer (m past observations), continues with the hidden layer (h hidden neurons), and ends with an output layer (a single output ${\widehat{y}}_{t+1}$). For general Artificial Neural Network terminology, see [54].

**Figure 8.**Nearest Neoghbors with Differential Evolution (NNDE) Forecast for the Malpais Validation Set.

Time Series | Minimum Lyapunov Exponent | Maximum Lyapunov Exponent | Average Lyapunov Exponent |
---|---|---|---|

20891 | 0.0557 | 0.1974 | 0.1035 |

22641 | 0.0773 | 0.1798 | 0.1290 |

22887 | 0.0444 | 0.1643 | 0.0885 |

23711 | 0.0889 | 0.1388 | 0.1111 |

24908 | 0.1026 | 0.1488 | 0.1263 |

27947 | 0.0197 | 0.1210 | 0.0662 |

28722 | 0.0561 | 0.1767 | 0.1083 |

29231 | 0.0814 | 0.1802 | 0.1140 |

30230 | 0.0789 | 0.1208 | 0.1064 |

37099 | 0.0959 | 0.1480 | 0.1176 |

aristeomercado | 0.1038 | 0.1601 | 0.1298 |

cointzio | 0.1132 | 0.2041 | 0.1520 |

corrales | 0.0638 | 0.1330 | 0.1073 |

elfresno | 0.0644 | 0.1457 | 0.1098 |

lapalma | 0.1037 | 0.1195 | 0.1119 |

lapiedad | 0.0601 | 0.1811 | 0.1174 |

malpais | 0.1112 | 0.1573 | 0.1411 |

markazuza | 0.0971 | 0.2094 | 0.1573 |

melchorocampo | 0.0616 | 0.1864 | 0.1341 |

patzcuaro | 0.0426 | 0.1590 | 0.0965 |

logistic map | 0.3151 | 0.5921 | 0.4832 |

sine | 0.0000 | 0.0000 | 0.0000 |

Time Series | NN Deterministic [m, tau, epsilon] | NNDE MSE [m, τ, ϵ] | NNDE SMAPE [m, τ, ϵ] | ARIMA (p, d, q) | ANN [m, h, TM] | FF [m, τ] |
---|---|---|---|---|---|---|

20891 | [8, 1, 6.318] | [23, 48, 6.121] | [43, 51, 12.713] | (1, 1, 5) | [57, 24, bfgs] | [8, 1, 20] |

22641 | [7, 6, 6.320] | [45, 72, 10.599] | [26, 95, 6.328] | (0, 1, 5) | [59, 39, bfgs] | [7, 6, 20] |

22887 | [8, 13, 4.389] | [36, 16, 7.653] | [9, 1, 3.346] | (3, 1, 2) | [61, 64, bfgs] | [8, 13, 20] |

23711 | [7, 5, 1.021] | [6, 8, 0.000] | [29, 69, 4.468] | (1, 1, 1) | [54, 35, bfgs] | [7, 5, 20] |

24908 | [7, 1, 2.116] | [28, 11, 2.941] | [14, 100, 2.155] | (3, 0, 3) | [62, 59, bfgs] | [7, 1, 20] |

27947 | [6, 1, 2.116] | [18, 46, 14.602] | [14, 50, 13.834] | (2, 1, 4) | [39, 37, bfgs] | [6, 1, 20] |

28722 | [6, 1, 3.048] | [42, 29, 9.348] | [16, 20, 6.194] | (3, 1, 1) | [45, 47, bfgs] | [6, 1, 20] |

29231 | [6, 5, 5.266] | [50, 7, 8.753] | [43, 7, 8.166] | (5, 1, 2) | [41, 35, bfgs] | [6, 5, 20] |

30230 | [6, 1, 3.048] | [17, 4, 9.849] | [39, 29, 8.056] | (1, 1, 5) | [33, 25, bfgs] | [6, 1, 20] |

37099 | [6, 1, 1.021] | [16, 45, 7.132] | [1, 49, 28.025] | (2, 1, 3) | [43, 63, bfgs] | [6, 1, 20] |

aristeomercado | [8, 8, 10.920] | [13, 5, 28.544] | [7, 16, 22.724] | (1, 0, 2) | [16, 4, bfgs] | [8, 8, 20] |

cointzio | [6, 6, 4.389] | [2, 45, 15.820] | [1, 29, 7.759] | (2, 1, 3) | [10, 2, bfgs] | [6, 6, 20] |

corrales | [7, 6, 4.389] | [3, 23, 10.155] | [24, 1, 19.312] | (0, 1, 4) | [5, 60, rprop] | [7, 6, 20] |

elfresno | [6, 9, 0.410] | [5, 75, 16.560] | [5, 36, 18.935] | (3, 1, 4) | [25, 16, gdx] | [6, 9, 20] |

lapalma | [5, 5, 6.320] | [1, 57, 0.018] | [1, 40, 0.000] | (0, 1, 5) | [5, 21, cg] | [5, 5, 20] |

lapiedad | [5, 10, 4.389] | [2, 22, 8.457] | [11, 2, 21.112] | (2, 0, 5) | [32, 6, gdm] | [5, 10, 20] |

malpais | [9, 1, 116.842] | [23, 97, 12.113] | [41, 2, 19.531] | (0, 1, 2) | [64, 5, gdm] | [9, 1, 20] |

markazuza | [5, 1, 2.540] | [26, 1, 10.878] | [24, 1, 10.878] | (3, 1, 4) | [53, 19, bfgs] | [5, 1, 20] |

melchorocampo | [5, 1, 4.389] | [2, 5, 1.269] | [1, 59, 3.140] | (1, 0, 2) | [39, 15, rprop] | [5, 1, 20] |

patzcuaro | [11, 1, 10.920] | [2, 22, 5.827] | [24, 1, 14.179] | (5, 1, 0) | [29, 3, rprop] | [11, 1, 20] |

Station | NNDE | EvoDAG | FF | ANNCGA | NN | ARIMA |
---|---|---|---|---|---|---|

20891 | 33.552 | 42.287 | 47.349 | 177.667 | 40.701 | 52.485 |

22641 | 70.466 | 73.837 | 76.620 | 138.389 | 71.605 | 74.538 |

22887 | 91.541 | 94.351 | 123.836 | 162.067 | 100.261 | 183.042 |

23711 | 86.782 | 106.386 | 120.273 | 144.784 | 111.488 | 91.577 |

24908 | 90.196 | 147.510 | 144.755 | 183.813 | 146.980 | 138.607 |

27947 | 53.114 | 57.073 | 64.736 | 166.419 | 57.268 | 56.960 |

28722 | 74.403 | 83.761 | 88.769 | 167.003 | 84.692 | 82.507 |

29231 | 51.058 | 56.976 | 64.375 | 160.847 | 55.995 | 60.928 |

30230 | 125.952 | 132.347 | 147.326 | 168.304 | 134.553 | 170.713 |

37099 | 40.640 | 41.770 | 49.845 | 116.413 | 42.350 | 42.964 |

aristeomercado | 38.259 | 49.938 | 62.502 | 189.804 | 37.395 | 48.956 |

cointzio | 25.181 | 39.590 | 61.984 | 189.027 | 45.312 | 76.323 |

corrales | 30.205 | 44.388 | 54.242 | 179.650 | 37.860 | 145.898 |

elfresno | 38.378 | 56.528 | 36.649 | 187.003 | 49.125 | 42.472 |

lapalma | 31.136 | 34.783 | 39.919 | 181.602 | 36.503 | 141.309 |

lapiedad | 50.624 | 64.312 | 31.002 | 180.052 | 56.940 | 200.000 |

malpais | 28.856 | 46.742 | 62.484 | 198.142 | 46.046 | 40.755 |

markazuza | 36.916 | 51.388 | 62.080 | 154.063 | 49.003 | 113.731 |

melchorocampo | 27.885 | 35.134 | 40.899 | 185.632 | 29.965 | 117.441 |

patzcuaro | 39.728 | 83.065 | 91.997 | 187.654 | 50.734 | 84.692 |

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**MDPI and ACS Style**

Flores, J.J.; Cedeño González, J.R.; Rodríguez, H.; Graff, M.; Lopez-Farias, R.; Calderon, F. Soft Computing Methods with Phase Space Reconstruction for Wind Speed Forecasting—A Performance Comparison. *Energies* **2019**, *12*, 3545.
https://doi.org/10.3390/en12183545

**AMA Style**

Flores JJ, Cedeño González JR, Rodríguez H, Graff M, Lopez-Farias R, Calderon F. Soft Computing Methods with Phase Space Reconstruction for Wind Speed Forecasting—A Performance Comparison. *Energies*. 2019; 12(18):3545.
https://doi.org/10.3390/en12183545

**Chicago/Turabian Style**

Flores, Juan. J., José R. Cedeño González, Héctor Rodríguez, Mario Graff, Rodrigo Lopez-Farias, and Felix Calderon. 2019. "Soft Computing Methods with Phase Space Reconstruction for Wind Speed Forecasting—A Performance Comparison" *Energies* 12, no. 18: 3545.
https://doi.org/10.3390/en12183545