Hilbert - Huang - guide

A few examples of how to use these programs are given, with a given dataset “gsta.dat”, which is the annual mean global surface temperature anomaly. In “gsta.dat”, the first column is the time; and the second is the corresponding data value.

1) Load and display data

>> load gsta.dat;

>> plot(gsta(:,1),gsta(:,2)); >> axis([1850 2010 -0.6 0.6]);

>> title('the annual mean global surface temperature anomaly') >> xlabel('year') >> ylabel('Kelvin')

2) Using the program as a traditional EMD tool

The eemd.m can be used as an EMD decomposition tool:

>> year=gsta(:,1); >> inData=gsta(:,2); >> rslt=eemd(inData,0,1); >> plot(year,rslt(:,2)); >> hold on;

>> plot(year,rslt(:,3)-0.3); >> plot(year,rslt(:,4)-0.6); >> plot(year,rslt(:,5)-0.9);

>> plot(year,sum(rslt(:,6:8),2)-1.3,'r-'); >> hold off

>> set(gca,'yTickLabel',[]); >> axis([1850 2010 -1.8 0.3]); >> xlabel('year');

“relt” is a matrix containing the decomposition results, with the first column the original input (“gsta(:,2)”) and the last column the trend. In between are the IMFs of frequencies from high to low.

It should be noted that since in the eemd.m, the total number of IMFs m is specified as log2(N)-1, in some occasions (such as the one in this example), the components may be excessively extracted. In these cases, the sum of the latest columns may already satisfy the definition of a trend.

3) Instantaneous frequency of an IMF

The instantaneous frequency can be obtained through calling ifndq.m function:

>> omega_m3=ifndq(rslt(:,4),1); >> subplot(2,1,1); >> plot(year,rslt(:,4));

>> axis([1850 2010 -0.12 0.12]); >> title('IMF C3'); >> ylabel('Kelvin'); >> grid;

>> subplot(2,1,2);

>> plot(year, omega_m3/2/pi,'r-'); >> grid;

>> xlabel('year');

>> ylabel('cycle/year');

>> title('instantaneous frequency'); >> axis([1850 2010 0 0.12]);

It should be noted that the instantaneous frequency calculation program is not suitable for under sampled oscillations, such as the first IMF (with an averaged period about 3 data points). However, for such under sampled oscillations, the instantaneous frequency is no longer “instantaneous” any way, and any method used to obtain such a quantity will have big errors.

4) Using the program as a EEMD tool

The eemd.m can be used as an EEMD decomposition tool. In this case, the noise assed has an amplitude (standard deviation) of 0.2 of the standard deviation of the linearly detrended annual mean global surface temperature anomaly; and the number of ensemble is 100:

>> rslt=eemd(inData,0.2,100);

>> t(1)=1850; >> t(2)=2010; >> y1(1)=0; >> y1(2)=0; >> y2(1)=-0.3; >> y2(2)=-0.3; >> y3(1)=-0.6; >> y3(2)=-0.6; >> y4(1)=-0.9; >> y4(2)=-0.9; >> y5(1)=-1.2; >> y5(2)=-1.2; >> y6(1)=-1.6; >> y6(2)=-1.6; >> plot(t,y1,'k-'); >> hold on;

>> plot(t,y2,'k-'); >> plot(t,y3,'k-'); >> plot(t,y4,'k-'); >> plot(t,y5,'k-'); >> plot(t,y6,'k-'); >> plot(year,rslt(:,1)); >> plot(year,rslt(:,3)-0.3); >> plot(year,rslt(:,4)-0.6);

>> plot(year,rslt(:,5)-0.9); >> plot(year,rslt(:,6)-1.2);

>> plot(year,sum(rslt(:,7:8),2)-1.6,'r-'); >> set(gca,'yTickLabel',[]);

>> title('EEMD decomposition of GSTA (A_n=0.2; N_e_s_b=100)') >> axis([1850 2010 -2.1 0.2]); >> xlabel('year');

5) Statistical significance test

Since the annual mean global surface temperature anomaly behaves completely different from a white noise series, we use computer generated white noise to illustrate how the significance.m can be used:

>> clear; >> clf;

>> data=randn(512,1); >> rslt=eemd(data,0,1); >> imfs=rslt(:,2:8);

>> [sigline95,logep]=significance(imfs,0.05); >> [sigline99,logep]=significance(imfs,0.01); >> plot(sigline95(:,1),sigline95(:,2)); % 95 percenta line >> hold on

>> plot(sigline99(:,1),sigline99(:,2),'m-'); % 99 percenta line

>> plot(logep(:,1),logep(:,2),'r*'); >> plot(logep(1,1),logep(1,2),'k*'); >> grid;

>> xlabel('LOG2 ( Mean Period )');

>> ylabel('LOG2 ( Mean Normalized Energy )');

>> title('Significance test of IMFs of white noise'); >> axis([0 10 -7 0])

3) Trend and detrending

For example, in the previous decomposition, the sum of the last three columns satisfies the definition of trend well.

>> plot(year, rslt(:,1)); >> hold on;

>> plot(year, sum(rslt(:,7:8),2),'r-'); >> plot(year, sum(rslt(:,6:8),2),'g-'); >> plot(year, sum(rslt(:,5:8),2),'m-'); >> title('Trends of different timescales'); >> ylabel('Kelvin'); >> xlabel('year'); >> grid;

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