MSSA in the Wind Speed Forecasting

MSSA in the Wind Speed Forecasting

Reinaldo Castro SouzaPUC-RIO

Moisés Lima de MenezesPUC-RIO / UFF

José Francisco M. PessanhaUERJ

Summary

Objective

SSA Method

MSSA extension

PAR(P) Models

Proposed Methodology

Case Study

Conclusions

ISF 2013 - SEOUL KOREA - June 23-26, 2013

Objective

Investigate the predictive gain obtained when weuse multi-channel singular spectrum analysis(MSSA) and univariate SSA integrated to thePAR(p) model applied to two-dimensional vectortime series.

The methodology is illustrated with an applicationin the modelling of two wind speed time series intwo anemometric station located in braziliannortheast region.


Singular Spectrum Analysis (SSA)

Trajectory MatrixTime Series






Noise time seriesHence:

ApproximateTime Series

Original Time Series

Multi-channel Singular Spectrum Analysis(MSSA)


Multi-channel Singular Spectrum Analysis(MSSA)


Them

Periodic Autorregressive Models - PAR(p)






Proposed Methodology

Forecasts


Two time series

SSA 1

(SERIES 1)

(SERIES 2)

SSA 2

MSSAFiltering the two series

simultaneously

Filtering the two series separately

PAR (p)Two SSA

approximatetime series

TwoMSSA

approximatetime series

Case Study• Two monthly wind speed series of Brazil northeast: Petrolina and

Pesqueira. 16 years (jan/96 – dec/11) – T=192.


456789

10

jan/96 jul/97 jan/99 jul/00 jan/02 jul/03 jan/05 jul/06 jan/08 jul/09 jan/11Sp

ee

d (

m/

s)

Time (months)

Wind Speed (Petrolina)

456789

1011

jan/96 jul/97 jan/99 jul/00 jan/02 jul/03 jan/05 jul/06 jan/08 jul/09 jan/11Sp

ee

d (

m/

s)

Time (months)

Wind Speed (Pesqueira)

Plots of the 9 First Eigenvectors Associated to Trajectory Matrix from two Monthly Wind Speed Series

(using L optimum equal to 93 in SVD by MSSA)


80

90

100

110

1 9 17 25 33 41 49 57 65 73 81 89 97

1(98.005%)

-12

-2

8

1 9 17 25 33 41 49 57 65 73 81 89 97

2(0.569%)

-12

-2

8

1 9 17 25 33 41 49 57 65 73 81 89 97

3(0.563%)

-5

-3

-1

1

3

1 9 17 25 33 41 49 57 65 73 81 89 97

4(0.065%)

-3

-1

1

3

1 9 17 25 33 41 49 57 65 73 81 89 97

5(0.050%)

-4

-2

0

2

4

6

1 9 17 25 33 41 49 57 65 73 81 89 97

6(0.048%)

-4

1

1 9 17 25 33 41 49 57 65 73 81 89 97

7(0.042%)

-4

-2

0

2

4

1 9 17 25 33 41 49 57 65 73 81 89 97

8(0.042%)

-4

-2

0

2

4

1 9 17 25 33 41 49 57 65 73 81 89 97

9(0.032%)

Scatter Plots of 2 First Paired Eigenvectors Associated to TrajectoryMatrix from two Monthly Wind Speed Series by MSSA


-12

-7

-2

3

8

85 90 95 100 105 110

2 (

0.5

69

%)

1 (98.005%)-15

-12

-9

-6

-3

0

3

6

9

12

15

-15 -12 -9 -6 -3 0 3 6 9 12 15

3 (

0.5

63

%)

2 (0.569%)

Scatter Plots of 2 Paired Harmonic Eigenvectors Associated to Trajectory Matrix from two Monthly Wind Speed Series

The number of vertices of each regularpolygon is equal to the harmonic period ofeach eigenvector of the screen plot.


-15

-12

-9

-6

-3

0

3

6

9

12

15

-15 -12 -9 -6 -3 0 3 6 9 12 15

3 (

0.5

63

%)

2 (0.569%)

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

8 (

0.0

42

%)

7 (0.042%)

Scatter plots of 3 Paired Noise Eigenvectors Associated to Trajectory Matrix from two Monthly Wind Speed Series by MSSA

These eigenvectors were classified statistically(via BDS test) as noise.


-3

-2,5

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

-3 -2 -1 0 1 2

38

(0

.00

8%

)

37 (0.008%)-1,5

-1

-0,5

0

0,5

1

1,5

2

-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2

54

(0

.00

5%

)

53 (0.005%)

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

2,5

-2 -1 0 1 2

66

(0

.00

4%

)

65 (0.004%)

5

6

7

8

1 22 43 64 85 106 127 148 169 190

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

1 22 43 64 85 106 127 148 169 190

-2

-1

0

1

2

1 22 43 64 85 106 127 148 169 190

MSSA - Components from Wind Speed Time Series (Petrolina)

MSSA-Component 1 (Trend).

MSSA - Component 2 (Harmonic).

MSSA - Component 3 (Noise).


W - correlation between the three components of Wind Speed Time Series (Petrolina)


The weighted correlation shows that the components

are unrelated. Therefore, they are well separable

Trend Harmonic Noise

Trend 1 0.001 0.002

Harmonic 0.001 1 0.028

Noise 0.002 0.028 1

Extracted Noise from Time Series from Monthly Wind Speed Time Series (Petrolina)

The null hypothesis of BDS test (independence of the noise time series) is not

rejected at 5% of significance level until the dimension 6.

MSSA - Component 3

Dimension BDS

Statistic

Std. Error Z-Statistic Prob.

2 0.006156 0.004463 1.379423 0.1678

3 0.005718 0.007098 0.805545 0.4205

4 0.004280 0.008457 0.506136 0.6128

5 0.004037 0.008817 0.457835 0.6471

6 0.000435 0.008504 0.051183 0.9592


-1,5

-1

-0,5

0

0,5

1

1,5

1 22 43 64 85 106 127 148 169 190

MSSA - Components from Wind Speed Time Series (Pesqueira)

MSSA-Component 1 (Trend).

MSSA - Component 2 (Harmonic).

MSSA - Component 3 (Noise).


5

6

7

8

9

10

1 22 43 64 85 106 127 148 169 190

-2

-1

0

1

2

1 22 43 64 85 106 127 148 169 190

-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

1 22 43 64 85 106 127 148 169 190

W - correlation between the three components of Wind Speed Time Series (Pesqueira)


The weighted correlation shows that the components

are unrelated. Therefore, they are well separable

Trend Harmonic Noise

Trend 1 0.001 0.003

Harmonic 0.001 1 0.029

Noise 0.003 0.029 1

Extracted Noise from Monthly Wind Speed Time Series (Pesqueira)

The null hypothesis of BDS test (independence of the noise time series) is not rejected at 5% of

significance level until the dimension 6.

MSSA – Component 3

Dimension BDS

Statistic

Std. Error Z-Statistic Prob.

2 0.000525 0.000568 0.925448 0.3547

3 0.001376 0.001234 1.114620 0.2650

4 0.001597 0.002008 0.795097 0.4266

5 0.001581 0.002859 0.553033 0.5802

6 0.005030 0.003764 1.336379 0.1814


-2

-1,5

-1

-0,5

0

0,5

1

1,5

2

1 22 43 64 85 106 127 148 169 190

Filtering time series (Petrolina and Pesqueira) by MSSA and Original time series


4

5

6

7

8

9

10

jan/96 jul/97 jan/99 jul/00 jan/02 jul/03 jan/05 jul/06 jan/08 jul/09 jan/11

Petrolina Petrolina (MSSA)

4

5

6

7

8

9

10

11

12


Pesqueira Pesqueira (MSSA)

Filtering time series (Petrolina and Pesqueira) based on SSA using window

length – L equal to 96.


4

5

6

7

8

9

10


Petrolina Petrolina (SSA)

4

6

8

10

12


Pesqueira Pesqueira (SSA)

Comparison between the filtering by MSSA and SSA (Petrolina)


4

5

6

7

8

9

10


Petrolina Petrolina (SSA)

4

5

6

7

8

9

10


Petrolina Petrolina (MSSA)

Comparison between the filtering by MSSA and SSA (Pesqueira)


4

6

8

10

12


Pesqueira Pesqueira (MSSA)

4

6

8

10

12


Pesqueira Pesqueira (SSA)

Scatter plots with trendline - Petrolina


4

5

6

7

8

9

10

4 5 6 7 8 9 10

Pe

tro

lin

a (

SSA

)

Petrolina

4

5

6

7

8

9

10

4 5 6 7 8 9 10

Pe

tro

lin

a (

MS

SA)

Petrolina

Scatter plots with trendline - Pesqueira


4

5

6

7

8

9

10

11

12

4 6 8 10 12

Pe

squ

eir

a (

SSA

)

Pesqueira

4

5

6

7

8

9

10

11

4 6 8 10 12

Pe

squ

eir

a (

MS

SA)

Pesqueira

Comparison between Mean and Standard Deviation in Original time series, SSA and MSSA of Petrolina


PETROLINA Mean Standard Deviation

Month Original SSA MSSA Original SSA MSSA

January 5.9543 5.8964 5.8769 0.67807 0.53464 0.42529

February 5.9840 5.7253 5.7463 0.78301 0.56048 0.47308

March 5.6773 5.8151 5.8361 0.78543 0.68726 0.51030

April 6.2016 6.2973 6.2043 0.76899 0.65919 0.54709

May 6.8644 6.9539 6.8116 0.68510 0.71811 0.60127

June 7.6413 7.4656 7.4933 0.85914 0.81526 0.67732

July 7.7810 7.8534 7.9906 0.78248 0.79876 0.73271

August 8.2303 8.0778 8.1029 0.83149 0.72173 0.72404

September 7.8183 7.8591 7.7981 0.79368 0.59225 0.64441

October 7.1837 7.2509 7.2285 0.78491 0.62828 0.53599

November 6.5102 6.6287 6.6170 0.56942 0.58426 0.44951

December 6.1359 6.1195 6.1214 0.59486 0.46908 0.41776

Comparison between Mean and Standard Deviation in Original time series, SSA and MSSA of Pesqueira


PESQUEIRA Mean Standard Deviation

Month Original SSA MSSA Original SSA MSSA

January 7.4461 7.5673 7.5564 1.466812 1.27299 1.18675

February 7.4726 7.3121 7.2867 1.505957 1.35201 1.31279

March 7.1856 7.1079 7.1051 1.311712 1.42076 1.36174

April 6.6023 6.8138 6.8231 1.256665 1.27799 1.25004

May 6.5351 6.4296 6.4366 1.268066 1.00404 1.04172

June 6.0355 6.1661 6.1926 0.950773 0.90588 0.87616

July 6.4069 6.3782 6.3835 0.98693 0.87387 0.91598

August 7.1029 7.0080 7.0410 1.210563 1.04558 1.17135

September 7.7801 7.8292 7.8385 1.616289 1.39401 1.42067

October 8.2268 8.3271 8.3246 1.672698 1.45432 1.48573

November 8.2243 8.2442 8.2755 1.543823 1.34718 1.37672

December 7.9871 7.8648 7.8460 1.409316 1.28417 1.24755

Comparison between autorregressive order in modeling PAR(p), PAR(p) – SSA and PAR(p) - MSSA


Petrolina

Pesqueira

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

PAR(p) 1 1 1 1 3 1 1 1 1 1 1 2

PAR(p)- SSA 1 4 1 1 1 1 1 1 1 1 1 3

PAR(p)- MSSA 1 1 1 1 1 1 1 1 1 1 1 1


PAR(p) 6 1 2 1 1 5 1 1 1 1 1 2

PAR(p)- SSA 1 1 2 1 1 2 3 5 1 2 1 2

PAR(p)- MSSA 1 1 1 1 1 1 1 1 1 1 1 1

mouth PAR(p) PAR(p) - SSA PAR(p) - MSSA

January 6.120 3.3072 1.48270

February 8.7405 3.1330 0.87904

March 6.1584 4.1720 0.81247

April 5.9539 2.4138 0.63071

May 4.4594 3.3287 0.51478

June 4.5123 2.1882 0.51592

July 2.9755 1.9037 0.54739

August 3.5905 2.3111 0.61316

September 5.0779 2.5236 0.70418

October 5.635 2.4985 0.73237

November 4.1773 1.9260 0.62408

December 4.3738 2.5884 0.68437

MAPE Statistics (In sample) - Petrolina

We can see that the PAR(p) – MSSA was better in all periods.


MAPE Statistics (In sample) - Petrolina

We can see that the PAR(p) – MSSA was better in all periods.


0123456789

10


PAR(p) PAR(p) - MSSA PAR(p) - SSA

mouth PAR(p) PAR(p) - SSA PAR(p) - MSSA

January 6.1182 4.7278 3.5413

February 5.1540 2.6814 1.1211

March 7.1102 1.9755 1.4781

April 7.4127 3.1508 1.8216

May 5.7543 2.7266 1.7512

June 4.3649 2.4026 1.3590

July 3.8340 1.9483 1.4165

August 4.5714 2.2650 1.7567

September 5.3439 1.7724 1.4193

October 5.9651 2.1032 1.4086

November 7.0626 2.3336 1.2869

December 4.4704 0.8727 1.0379

MAPE Statistics (In sample) - Pesqueira

We can see that the PAR(p) – MSSA was better in almost all periods.


MAPE Statistics (In sample) - Pesqueira

We can see that the PAR(p) – MSSA was better in almost all periods.


0

1

2

3

4

5

6

7

8


PAR(p) PAR(p) - MSSA PAR(p) - SSA

Conclusions

• In this work we compared the performances of threeforecasting methods: PAR(p), PAR(p) - SSA and PAR(p) -MSSA.

• The SSA method with analyzed approaches showedefficient to extraction of noises from the original timeseries such that generating an approximate time series(less noisy regarding to the original time series).

• The MSSA method in association to PAR(p) is better thanSSA and the correlation structure remain when there are atleast two time series.


Bibliography• CAO, L. Y. & SOOFI, A., Nonlinear deterministic forecasting of daily dollar exchange rates,

International Journal of Forecasting, (1999), 15(4): 421 - 430.

• ELSNER, J. B. & TSONIS, A. A. Singular Spectrum Analysis: A New Tool in Time Series Analysis. Plenum Press, New York – London, (2010).

• GOLYANDINA, N., NEKRUTKIN, V., & ZHIGLJAVSKY, A., Analysis of Time Series Structure: SSA and Related Techniques, Chapman & Hall/CRC, New York - London, (2001).

• HASSANI, H. Singular Spectrum Analysis: Methodology and Comparison. Journal of Data Science, 5 (2007), 239 – 257.

• HASSANI, H. & ZHIGLJAVSKY, A. Singular Spectrum Analysis: Methodology andApplication to Economics Data. Journal of Systems Science & Complexity (2009) 22: 372-394.

• HSIEH, D. A., Chaos and nonlinear dynamics: Application to financial markets, Journal of Finance, (1991), 46: 1839 -1877.

• KRANE, S. D., An evaluation of real GDP forecasts: Economic Perspectives, (2003): 1996 -2001. Http://www.chicagofed.org/publications/economicperspectives/2003/1qeppart1.pdf.

• ZHIGLJAVSCKY, A., Hassani, H., Heravi, S. Forecasting European Industrial Productionwith Multivariate Singular Sperctum Analysis (MSSA). 31º. International Symposium onForecasting. Prague. (2011).


http://www.chicagofed.org/publications/economicperspectives/2003/1qeppart1.pdf







Thank you!

Reinaldo C. Souza (PUC-Rio)

[email protected]

Moisés L. Menezes (UFF/PUC-Rio)

[email protected]

José F. M. Pessanha (UERJ)

[email protected]


MSSA in the Wind Speed Forecasting

Documents