ARIMA Model, Neural Networks and SSA in the Short Term Electric Load Forecast

Transcription

1 ARIMA Model, Neural Networks and SSA in the Short Term Electric Load Forecast Keila CASSIANO Statistics Dept. (UFF) José PESSANHA Institute of Mathematics and Statistics (UERJ) Moisés MENEZES Statistics Dept. (UFF) Rafael SOUZA Accounting Sciences Dept. (UFMG) Luiz Albino JUNIOR Latin-american Institute of Tecnology (UNILA) Reinaldo SOUZA Electrical Engineering Dept. (PUC-Rio)

2 Summary Hierarchical Clustering (HC); Singular Spectrum Analysis Method (with Proposed Approach); Predictive Models: ARIMA and Artificial Neural Networks; Proposed Methodology; Case Study; and Conclusions.

3 Agglomerative Hierarchical Clustering (HC) The method produce partitions by a sucessive series of fusions of the n individuals into clusters. Each iteration pairs of clusters is merged when one moves up the hierarchy. 6 individuals = 6 clusters 5 clusters 4 clusters cluster 2 clusters 3 clusters

4 The SSA methodology Steps: 1) PCA Principal Components Analysis: Y=S+R Choice L based in BDS test aplied to noise series (R). The signal series (S) are considered by the d first principal components observed by graphical analysis. 2) Hierarchical Clustering Analysis: The Hierarchical Clustering aplied to the remaining series (S), generating 3 componets (components 1, 2 and 3) less noisy.

5 How do determine L and R? SSA Using Graphical Analysis The BDS test applied to R Reject null hypothesis SIM! YES! NÃO! NO!

6 Predictive Models Two models are used: ARIMA Model; Artifical Neural Network Model.

7 ARIMA Model According to BOX & JENKINS (1970), the general form of an ARIMA model is given by. ARIMA models can be used to modeling linear structures of temporal dependence in time series.

8 Artificial Neural Networks (ANN) According to HAYKIN (2001), the general form of an feedfoward ANN with a hidden layer is given by T k k k k G X W X b which is called feedfoward ANN with one hidden layer. k N Where: and is a continous function t 1, 0, called sigmoid defined by. t t ANN models can be used to modeling nonlinear structures of temporal dependence in time series.

9 Proposed Methodology Step 1: Generate the approximate time series using SSA approach using PCA. Step 2: Generate the K (K=3) SSA-components from SSA-based approximate time series using HC. Step 3: Model each SSA-component from Step 2 through the ARIMA and ANN models. Indeed, we have an approach of the SSA-ARIMA method and from SSA-ANN method. Step 4: Make the Hybrid Linear Combination (HLC) of SSA-ARIMA and SSA-ANN from Step 3[see the next Slide].

10 Hybrid Linear Combination of SSA-ARIMA and SSA-ANN Methods (SSA-HLC) K K yˆ ˆ ˆ HLC, t yarima, k, t ARIMA, k p1 yann, k, t pann, k p2 c d k 1 k 1 Forecast from HLC of SSA- ARIMA and SSA-ANN methods at t of original time series. Forecast from ARIMA model at t of SSA - component k. Forecast from ANN model at t of SSA-component k. Adaptative weights and constants. The optimum parameters from HLC of SSA-ARIMA and SSA-ANN are such that minimize the mean absolute percentage error statistic (or, simply, MAPE). This optimization problem was resolved by solver from Excel.

11 SSA - PCA HC Comp. 1 Comp. 2 Comp. 3 Original Time Series ARIMA 1 ARIMA 2 ARIMA 3 SSA - ARIMA ANN1 ANN 2 ANN 3 SSA - ANN ARIMA ANN HLC of SSA-ARIMA and SSA-ANN Methods (SSA-HLC)

12 Case Study Time Series: Hourly Electricity Load of Electric Reliability Council of Texas ERCOT North Region (From 01/01/ :00h to 03/25/ :00h) (T=2000) Training Sample: Validation Sample: 200. Test Sample: 200. The aim: Generate forecasts 1 step ahead for all samples. Methods: ARIMA - ANN - SSA-ARIMA SSA-ANN - SSA-HLC

13 Electric Load (KW) Hourly Electric Load Time Series (North Region) Training Sample Validation Sample Test Sample Time (hours)

14 Logarithms of the Eigenvalues Associated to Trajectory Matrix from Electric Load Time Series L=1000 d= 602 Optimum point in the PCA step on SVD expansion

15 Extracted Noise from Electric Load Time Series The null hypothesis of BDS test (independence of the noise time series) is not rejected at 5% of significance level until the dimension 6. Dimension BDS Std. Error z-statistic Prob. Statistic The null hypothesis of Dickey-Fuller test (non stationarity of the noise time series) is rejected at 1%, 5% and 10% of significance level. t-statistic Prob. Augmented Dickey-Fuller Test Statistic Test critical values: 1% level % level % level ,00 30,00 20,00 10,00 0,00-10,00-20,00-30,00-40,00 The first 602 components in the SVD expansion in the PCA step are considered, generating this noise time series. Noise

16 Hierarchical Clustering (HC) using the 602 components remaining , ,00 SSA-Component 1 formed from the sum of 238 SSA-components , ,00 0, , ,00 SSA-Component 2 formed from the sum of 62 SSA-components. 500,00 0,00-500, , ,00 500,00 SSA-Component 3 formed from the sum of 302 SSA-components. 0,00-500, ,00

17 SSA-ARIMA Method ARIMA model to SSA-component 1 (in E-Views): D(D(LOG(SSA-component-1))) = AR(1) MA(1) AR(2) MA(2) AR(3) MA(3) AR(4) MA(4) AR(5) MA(5) AR(6) MA(6) MA(7) AR(10) SAR(12) SMA(24) ARIMA model to SSA-component 2 (in E-Views): SSA-component-2 = AR(1) MA(1) AR(2) MA(2) MA(3) AR(4) MA(4) AR(5) AR(6) MA(6) MA(8) SAR(12) Model ARIMA to SSA-component 3 (in E-Views): SSA-component-3 = AR(1) AR(2) MA(2) MA(3) MA(4) SAR(12) MA(23) SAR(24) SMA(24) MA(26)

18 SSA-ANN Method ANN to SSA-component 1 (in MATLAB): Size of Window: 3; Activation function (hidden layer): tansig; Activation function (output layer): pureling; Number of Hidden Layers: 1; Number of neuron in Hidden Layer: 10. ANN to SSA-component 2 (in MATLAB): Size of Window: 5; Activation function (hidden Layer): tansig; Activation function (output layer): pureling; Number of Hidden Layers: 1; Number of neuron in Hidden Layer: 5. ANN to SSA-component 3 (in MATLAB): Size of Window: 4; Activation function (hidden Layer): tansig; Activation function (output layer): pureling; Number of Hidden Layers: 1; Number of neuron in Hidden Layer: 7.

19 SSA-HLC KSSA LSSA yˆ ˆ ˆ HLC, t yarima, k, t ARIMA, k p1 yann, l, t pann, k p2 c d k 1 l 1 Forecast from HLC model at t to orginal time series. Forecast from ARIMA model at t to SSAcomponent k. Forecast from RNA model at t to SSAcomponent k. Adaptative weights and constants. Optimum adaptative weights and constants, given a SSA decomposition of level 3. p-comp1-ann p-comp2-ann p-comp3-ann 0, , , p-comp1-arima p-comp2-arima p-comp3-arima 0, , , p1 p2 c d 0, , , ,24799E-06

20 Adherence Statistics MAPE (%) ARIMA ANN SSA-ARIMA SSA-ANN SSA-HLC Training Sample Validation Sample Test Sample MSE ARIMA ANN SSA-ARIMA SSA-ANN SSA-HLC Training Sample Validation Sample Test Sample

21 Conclusions In this work we compared the performances of five forecasting methods: ARIMA, ANN, SSA-ARIMA, SSA- ANN and Proposed Methodology SSA-HLC. The SSA method with proposed approach showed efficient to extraction of noises from the original time series such that generating an approximate time series (less noisy regarding to the original time series). The obtained results showed that the Proposed Methodology not only significantly improved the performance over SSA- ARIMA and SSA-ANN but also can straightforwardly be made operational to generate the point forecasts.

22 Bibliography BOX, G. E. P. & JENKINS, G. M. (1970). Time Series Analysis: Forecasting and Control. Holden-Day. HAYKIN, S. (2001). Redes Neurais: Princípios e Prática. Tradução: Paulo Martins Engel. 2º Edição. Porto Alegre: Bookman. HASSANI, H. (2007). Singular Spectrum Analysis: Methodology and Comparison. Journal of Data Science. 5,

23 Keila Cassiano (UFF) Moisés Menezes (UFF) Luiz Albino Júnior (UNILA) José F. M. Pessanha (UERJ) Thank you! Rafael M. Souza (UFMG) Reinaldo C. Souza (PUC-Rio)