Note
Go to the end to download the full example code.
Empirical Mode Decomposition (EMD) and Ensemble Empirical Mode Decomposition (EEMD)#
EMD (Empirical Mode Decomposition) is an adaptive time-space analysis method suitable for processing series that are non-stationary and non-linear. EMD performs operations that partition a series into ‘modes’ (IMFs; Intrinsic Mode Functions) without leaving the time domain. It can be compared to other time-space analysis methods like Fourier Transforms and wavelet decomposition. Like these methods, EMD is not based on physics. However, the modes may provide insight into various signals contained within the data. In particular, the method is useful for analyzing natural signals, which are most often non-linear and non-stationary. Some common examples would include the Southern Oscillation Index (SOI), Niño 3.4 Index, etc.
EEMD (Ensemble EMD) is a noise assisted data analysis method. EEMD consists of “sifting” an ensemble of white noise-added signal. EEMD can separate scales naturally without any a priori subjective criterion selection as in the intermittence test for the original EMD algorithm.
Wu and Huang (2009) state: “White noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As the EMD is a time space analysis method, the white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the signal, which is then treated as the true and more physical meaningful answer.” Further, they state: “EEMD represents a substantial improvement over the original EMD and is a truly noise assisted data analysis (NADA) method.”
See also
https://www.clear.rice.edu/elec301/Projects02/empiricalMode/
Wu, Z., & Huang, N. E. (2009). Ensemble empirical mode decomposition: a noise-assisted data analysis method. Advances in Adaptive Data Analysis, 01(01), 1-41. https://doi.org/10.1142/S1793536909000047
Before proceeding with all the steps, first import some necessary libraries and packages
import xarray as xr
import matplotlib.pyplot as plt
import easyclimate as ecl
Load and inspect Niño 3 SST anomaly data The dataset contains monthly sea surface temperature anomalies in the Niño 3 region, a key indicator for ENSO monitoring and analysis
data = xr.open_dataset("test_input_nino3_wavelet.nc")["nino3"]
data
<xarray.DataArray 'nino3' (time: 504)> Size: 4kB [504 values with dtype=float64] Coordinates: * time (time) datetime64[ns] 4kB 1871-01-31 1871-04-30 ... 1996-10-31
- time: 504
- ...
[504 values with dtype=float64]
- time(time)datetime64[ns]1871-01-31 ... 1996-10-31
array(['1871-01-31T00:00:00.000000000', '1871-04-30T00:00:00.000000000', '1871-07-31T00:00:00.000000000', ..., '1996-04-30T00:00:00.000000000', '1996-07-31T00:00:00.000000000', '1996-10-31T00:00:00.000000000'], shape=(504,), dtype='datetime64[ns]')
- timePandasIndex
PandasIndex(DatetimeIndex(['1871-01-31', '1871-04-30', '1871-07-31', '1871-10-31', '1872-01-31', '1872-04-30', '1872-07-31', '1872-10-31', '1873-01-31', '1873-04-30', ... '1994-07-31', '1994-10-31', '1995-01-31', '1995-04-30', '1995-07-31', '1995-10-31', '1996-01-31', '1996-04-30', '1996-07-31', '1996-10-31'], dtype='datetime64[ns]', name='time', length=504, freq=None))
Perform Empirical Mode Decomposition (EMD) on the time series EMD decomposes the nonlinear, non-stationary signal into intrinsic mode functions (IMFs) representing oscillatory modes embedded in the data at different timescales The time_step=”M” parameter indicates monthly resolution of the input data
imf_result = ecl.filter.filter_emd(data, time_step="M")
imf_result
<xarray.Dataset> Size: 36kB Dimensions: (time: 504) Coordinates: * time (time) datetime64[ns] 4kB 1871-01-31 1871-04-30 ... 1996-10-31 Data variables: input (time) float64 4kB -0.15 -0.3 -0.14 -0.41 ... -0.08 -0.18 -0.06 imf0 (time) float64 4kB 0.0756 -0.09539 0.0829 ... -0.169 -0.03726 imf1 (time) float64 4kB 0.1577 0.1695 0.1588 ... 0.009739 0.1157 0.09504 imf2 (time) float64 4kB 0.4274 0.4314 0.4123 ... -0.5048 -0.5273 -0.514 imf3 (time) float64 4kB -0.4811 -0.4838 -0.4807 ... 0.1399 0.1341 0.1286 imf4 (time) float64 4kB -0.152 -0.1461 -0.1396 ... -0.08033 -0.07906 imf5 (time) float64 4kB -0.1279 -0.1265 -0.125 ... 0.1428 0.1425 0.1421 imf6 (time) float64 4kB -0.04962 -0.04913 -0.04864 ... 0.2044 0.2046
- time: 504
- time(time)datetime64[ns]1871-01-31 ... 1996-10-31
array(['1871-01-31T00:00:00.000000000', '1871-04-30T00:00:00.000000000', '1871-07-31T00:00:00.000000000', ..., '1996-04-30T00:00:00.000000000', '1996-07-31T00:00:00.000000000', '1996-10-31T00:00:00.000000000'], shape=(504,), dtype='datetime64[ns]')
- input(time)float64-0.15 -0.3 -0.14 ... -0.18 -0.06
array([-0.15, -0.3 , -0.14, ..., -0.08, -0.18, -0.06], shape=(504,))
- imf0(time)float640.0756 -0.09539 ... -0.169 -0.03726
array([ 7.55980640e-02, -9.53948308e-02, 8.29018295e-02, -5.78035224e-02, 4.99995012e-02, -7.87136152e-02, 1.35925074e-01, -1.10585523e-01, -2.52977915e-01, -8.54653132e-02, 2.84822191e-01, -4.19953337e-02, -2.23880306e-01, 1.78957729e-01, 6.78380736e-02, -1.01725786e-01, 7.96866801e-02, -7.66437155e-02, 1.28118404e-01, 1.53534105e-01, -2.39367000e-01, -3.67180351e-02, 4.58964727e-01, 1.28455332e-01, -6.20772239e-01, -6.20707332e-01, 1.50123157e-01, 3.87834398e-01, 5.19259633e-01, -4.18398422e-01, -3.65914937e-01, 5.49840146e-02, 1.52515726e-01, -1.47226987e-01, 1.88830576e-01, -2.46099219e-01, -1.10930796e-01, 8.86609870e-02, 2.34163408e-01, 3.67063747e-01, 2.79590245e-01, 3.48834320e-02, -2.11381508e-01, -2.75580965e-01, -1.67906797e-01, 1.96018938e-01, -4.32535720e-02, -4.53467646e-02, -1.09816586e-01, 9.88552735e-02, -9.30140086e-02, 8.33942342e-02, -7.15939894e-02, 8.12877338e-02, -1.42291685e-01, 2.55879665e-01, -3.16898177e-01, -4.97608944e-01, -2.89500986e-01, 5.31108209e-01, -3.07268312e-01, -3.23726228e-01, 1.97196452e-01, -1.41542067e-02, -5.31700465e-02, 6.23027277e-02, -1.01488135e-03, -2.13053695e-01, -3.17134013e-01, -1.76493080e-01, 2.92705963e-02, 6.38968189e-01, 6.58939163e-01, -1.48291273e-01, -6.78379266e-01, -3.53333839e-01, -1.26556899e-02, 5.20569782e-01, 3.21873045e-01, -3.64519240e-01, ... -5.39449383e-02, -7.20407436e-01, -5.49713141e-01, 2.55896269e-01, 3.77088909e-01, -2.66901371e-01, -1.76582452e-01, 1.17359444e-01, -8.15290625e-02, -2.59205501e-02, 6.46853622e-02, 4.99022929e-02, -7.65794666e-02, 8.90693912e-02, -1.09709109e-01, 1.33638869e-01, -1.45993692e-01, 1.44667455e-01, -1.52554161e-01, 9.90289698e-02, 2.66413486e-01, 7.47293127e-02, -4.71097638e-01, 5.73676833e-01, 5.97235129e-01, 5.46122760e-01, -4.69549034e-02, -3.95252525e-01, 2.47201624e-01, -1.24489585e-01, 9.16648814e-02, -1.49125456e-01, 2.34190213e-01, 3.94466807e-02, -3.11144989e-01, 1.94316137e-01, 2.43819091e-01, 1.46478625e-01, -1.68366321e-01, 1.77556106e-01, -7.68745852e-02, -2.89548956e-01, -1.20656956e-01, 4.17123959e-01, 2.65262126e-01, -2.72592884e-01, -4.67246198e-01, -1.90960550e-01, 2.68170877e-01, 3.50597291e-01, -2.97451582e-01, -1.62339982e-01, 8.24841427e-02, 1.65152287e-01, -1.42759432e-01, -1.37096919e-02, 5.82174434e-04, 2.20016014e-01, -3.10753461e-01, 5.23576229e-02, 3.89826820e-01, 5.92541391e-01, -5.96785298e-01, -2.55968076e-01, -2.23262098e-02, 3.68988402e-01, -3.21618181e-01, 7.05881565e-02, 3.74284330e-01, 2.05185734e-01, -5.11223163e-01, 5.42076656e-01, 3.11134972e-01, -1.50759543e-01, -3.97656569e-01, -1.09859861e-01, 2.37310514e-01, 9.47271519e-03, -1.68985875e-01, -3.72638823e-02])
- imf1(time)float640.1577 0.1695 ... 0.1157 0.09504
array([ 1.57710288e-01, 1.69457700e-01, 1.58760688e-01, 5.14880998e-02, -7.34049675e-02, -1.04428512e-01, -1.14984993e-01, -1.24107833e-01, -8.95121938e-02, 1.45741867e-02, 9.65646563e-02, 6.66498255e-02, -3.94248888e-02, -1.34298864e-01, -8.35727511e-02, 1.17228901e-01, 3.47242075e-01, 3.63742470e-01, 5.98972637e-02, -3.58067886e-01, -6.41949820e-01, -6.45789229e-01, -4.04869967e-01, 2.82720504e-03, 4.82621597e-01, 8.99295845e-01, 1.15417013e+00, 1.15293558e+00, 8.24042950e-01, 2.34289459e-01, -4.24124082e-01, -9.58045951e-01, -1.22426779e+00, -1.16667867e+00, -9.93356328e-01, -8.03015099e-01, -5.42561775e-01, -2.32142715e-01, 8.09335863e-02, 3.49358633e-01, 5.25082964e-01, 5.70249354e-01, 4.99098084e-01, 3.39881910e-01, 1.24883147e-01, -8.81856873e-02, -2.07784322e-01, -1.91361858e-01, -7.30818659e-02, 7.86558284e-02, 1.24652534e-01, -2.63597454e-02, -1.75427610e-01, -4.45426443e-02, 2.28154021e-01, 3.59541487e-01, 4.17410732e-01, 4.83836505e-01, 5.67158169e-01, 4.68634806e-01, 3.40972856e-02, -5.03164221e-01, -8.34417212e-01, -8.14079144e-01, -6.03893098e-01, -3.31413682e-01, -2.87627057e-02, 2.88589841e-01, 5.98563448e-01, 8.68994550e-01, 1.03408085e+00, 1.01906786e+00, 7.69074992e-01, 3.30782047e-01, -1.71377665e-01, -6.06617954e-01, -8.76660955e-01, -8.99644105e-01, -6.22852736e-01, -1.24396499e-01, ... 4.46045391e-01, 3.47976753e-01, 1.38804836e-01, -1.36504519e-01, -3.75895260e-01, -5.05877074e-01, -4.63841955e-01, -2.40011967e-01, 1.28484965e-01, 4.90227869e-01, 7.00622813e-01, 6.69449237e-01, 4.86356349e-01, 2.74340947e-01, 6.67871140e-02, -2.46032879e-01, -6.12185909e-01, -8.38120114e-01, -9.38791859e-01, -9.62573099e-01, -8.22507825e-01, -4.36651532e-01, 8.76037047e-02, 5.13047522e-01, 6.39031352e-01, 3.67326151e-01, -5.33264646e-02, -2.87442214e-01, -1.28634612e-01, 9.08660907e-02, 1.93978471e-02, -1.86087802e-02, 4.06508302e-02, -4.04168611e-02, -1.90552002e-01, -2.65153650e-01, -2.18259624e-01, -6.31677326e-02, 1.57267608e-01, 4.01259067e-01, 6.36411656e-01, 7.67100460e-01, 6.93976673e-01, 4.16085901e-01, -1.76953255e-02, -4.60671703e-01, -7.41693237e-01, -7.38152318e-01, -5.18138439e-01, -1.84393658e-01, 1.48031900e-01, 3.36682081e-01, 3.66184437e-01, 2.61139908e-01, 8.14136672e-02, -5.86532699e-02, -1.17369299e-01, -7.84960966e-02, 2.85209728e-02, 7.20751774e-02, 4.17181901e-02, -4.31996757e-02, -1.27602130e-01, -6.48659451e-02, 8.52551845e-02, 2.02033730e-01, 1.70603977e-01, -4.27235266e-02, -2.34204955e-01, -2.06449484e-01, -1.89909948e-02, 1.78197342e-01, 1.95045904e-01, 7.82484522e-02, -6.46318551e-02, -1.41102989e-01, -1.14236944e-01, 9.73897091e-03, 1.15664915e-01, 9.50356010e-02])
- imf2(time)float640.4274 0.4314 ... -0.5273 -0.514
array([ 0.42739013, 0.43139067, 0.41227158, 0.37278878, 0.31693938, 0.2487205 , 0.17212927, 0.0911628 , 0.00873138, -0.07626145, -0.16463632, -0.25624132, -0.34839684, -0.43779134, -0.51968087, -0.58359176, -0.61761746, -0.60985866, -0.55189827, -0.44911249, -0.31032357, -0.14494719, 0.03522729, 0.21781711, 0.39043949, 0.54071166, 0.65625085, 0.72566127, 0.74406944, 0.71774644, 0.65497966, 0.56391697, 0.45270623, 0.32871722, 0.1965265 , 0.06120956, -0.07183886, -0.19722402, -0.30955119, -0.40342563, -0.4734526 , -0.51534188, -0.52943541, -0.5180056 , -0.48340012, -0.42786813, -0.35364174, -0.26379108, -0.16473829, -0.06374352, 0.03193307, 0.11459916, 0.17483367, 0.20522012, 0.20808905, 0.18831216, 0.15117856, 0.10208174, 0.04641517, -0.01027824, -0.06185791, -0.10203384, -0.12451602, -0.12483374, -0.10579346, -0.07202094, -0.02814193, 0.02121781, 0.07143255, 0.11787651, 0.15592396, 0.18149163, 0.19309408, 0.19145529, 0.17754925, 0.15230549, 0.11665354, 0.07152293, 0.01837522, -0.0391999 , -0.09708072, -0.15114551, -0.19760627, -0.23450657, -0.26221095, -0.28158075, -0.29347732, -0.29876199, -0.29824781, -0.29255994, -0.28229657, -0.26806121, -0.25045739, -0.23008863, -0.20755844, -0.18347035, -0.15842787, -0.13303453, -0.10789384, -0.08360932, ... 0.59008997, 0.57792707, 0.50670433, 0.38602753, 0.22988173, 0.05225201, -0.13222817, -0.30594921, -0.44912291, -0.54157847, -0.56844056, -0.5358974 , -0.45495993, -0.33652091, -0.19147309, -0.03135591, 0.12970444, 0.2769351 , 0.39556318, 0.47081582, 0.49160419, 0.46163386, 0.38863597, 0.28083482, 0.14656342, -0.00522116, -0.16255345, -0.31079496, -0.43479495, -0.51940269, -0.55305473, -0.53857593, -0.48253457, -0.39153804, -0.27219367, -0.13110884, 0.02510911, 0.18981175, 0.35490229, 0.50664682, 0.62818966, 0.70221572, 0.71126205, 0.63721349, 0.47078182, 0.23948763, -0.01993112, -0.27086505, -0.4772192 , -0.61035498, -0.67094487, -0.66487674, -0.59330161, -0.47010955, -0.31209899, -0.13351765, 0.04952099, 0.22023588, 0.36184598, 0.45757023, 0.49062758, 0.45042627, 0.3511317 , 0.21309853, 0.05668146, -0.09776486, -0.22988573, -0.32279894, -0.37351208, -0.38254198, -0.350596 , -0.27859222, -0.16904503, -0.03317769, 0.10734551, 0.23277034, 0.32717667, 0.38557319, 0.4057008 , 0.38768591, 0.34119694, 0.27828784, 0.20999381, 0.1432314 , 0.08372364, 0.03577379, -0.00181922, -0.03163255, -0.0569288 , -0.08381496, -0.11949329, -0.17010035, -0.23577497, -0.31018703, -0.38568121, -0.45402185, -0.5047562 , -0.52726821, -0.51397521])
- imf3(time)float64-0.4811 -0.4838 ... 0.1341 0.1286
array([-4.81130913e-01, -4.83773153e-01, -4.80653758e-01, -4.72076549e-01, -4.58469492e-01, -4.40260554e-01, -4.17877701e-01, -3.91748898e-01, -3.62302113e-01, -3.29965312e-01, -2.95166460e-01, -2.58333525e-01, -2.19894471e-01, -1.80277267e-01, -1.39909877e-01, -9.92202680e-02, -5.86368750e-02, -1.85808556e-02, 2.05627648e-02, 5.84181112e-02, 9.46093085e-02, 1.28760482e-01, 1.60495756e-01, 1.89439255e-01, 2.15215106e-01, 2.37447432e-01, 2.55760359e-01, 2.69778012e-01, 2.79124515e-01, 2.83563492e-01, 2.83416565e-01, 2.79142204e-01, 2.71185008e-01, 2.59973792e-01, 2.45934086e-01, 2.29491418e-01, 2.11071319e-01, 1.91099318e-01, 1.70000944e-01, 1.48201728e-01, 1.26127199e-01, 1.04202886e-01, 8.28543190e-02, 6.24765818e-02, 4.33259505e-02, 2.55601623e-02, 9.31993101e-03, -5.25402951e-03, -1.80210054e-02, -2.88402829e-02, -3.75711482e-02, -4.40728873e-02, -4.82047866e-02, -4.98261320e-02, -4.87962099e-02, -4.50895973e-02, -3.91420357e-02, -3.15045573e-02, -2.27281945e-02, -1.33639797e-02, -3.96294511e-03, 4.92387683e-03, 1.27454538e-02, 1.90359300e-02, 2.36752635e-02, 2.66490198e-02, 2.79519300e-02, 2.75949574e-02, 2.55931228e-02, 2.19614474e-02, 1.67149522e-02, 9.86865814e-03, 1.43758635e-03, -8.51878541e-03, -1.97631524e-02, -3.20137529e-02, -4.49888256e-02, -5.84066089e-02, -7.19853412e-02, -8.54432609e-02, ... -2.18513480e-01, -1.87376230e-01, -1.51738295e-01, -1.12011910e-01, -6.91024499e-02, -2.40240240e-02, 2.22092579e-02, 6.85832863e-02, 1.14083951e-01, 1.57697144e-01, 1.98408754e-01, 2.35305367e-01, 2.67915396e-01, 2.96024123e-01, 3.19455877e-01, 3.38034985e-01, 3.51585776e-01, 3.59932576e-01, 3.62899714e-01, 3.60307957e-01, 3.51963832e-01, 3.37665311e-01, 3.17190396e-01, 2.90459979e-01, 2.58047121e-01, 2.20915351e-01, 1.80088844e-01, 1.36591776e-01, 9.14483222e-02, 4.56826580e-02, 3.18958797e-04, -4.36186000e-02, -8.51058430e-02, -1.23241243e-01, -1.57613865e-01, -1.87935424e-01, -2.13917634e-01, -2.35272207e-01, -2.51710858e-01, -2.62945300e-01, -2.68687249e-01, -2.68648416e-01, -2.62710115e-01, -2.51432055e-01, -2.35543541e-01, -2.15773881e-01, -1.92852381e-01, -1.67508349e-01, -1.40471091e-01, -1.12469914e-01, -8.41973598e-02, -5.61554254e-02, -2.86353957e-02, -1.88506915e-03, 2.38477555e-02, 4.83152796e-02, 7.12697045e-02, 9.24632316e-02, 1.11648062e-01, 1.28576398e-01, 1.43062847e-01, 1.55171646e-01, 1.65029440e-01, 1.72762873e-01, 1.78498588e-01, 1.82363231e-01, 1.84487351e-01, 1.85020316e-01, 1.84128175e-01, 1.81980169e-01, 1.78745539e-01, 1.74593525e-01, 1.69693370e-01, 1.64214314e-01, 1.58325598e-01, 1.52196464e-01, 1.45996153e-01, 1.39893906e-01, 1.34058964e-01, 1.28625412e-01])
- imf4(time)float64-0.152 -0.1461 ... -0.07906
array([-0.15202624, -0.14605076, -0.13962781, -0.13278511, -0.12555568, -0.11797253, -0.11006867, -0.10187711, -0.09343086, -0.08476293, -0.07590633, -0.06689408, -0.05775917, -0.04853463, -0.03925347, -0.02994868, -0.02065329, -0.0114003 , -0.00222273, 0.00684641, 0.01577412, 0.02452739, 0.03307319, 0.04137853, 0.0494104 , 0.05713577, 0.06452165, 0.07153502, 0.07814287, 0.08431219, 0.09000997, 0.09520584, 0.09988332, 0.10404171, 0.10768357, 0.11081149, 0.11342805, 0.11553583, 0.11713741, 0.11823536, 0.11883226, 0.1189307 , 0.11853325, 0.11764249, 0.116261 , 0.11439133, 0.11203606, 0.10919775, 0.10587906, 0.10208289, 0.09781227, 0.09307019, 0.08785966, 0.08218367, 0.07604522, 0.06945826, 0.06248043, 0.05518034, 0.04762656, 0.03988768, 0.0320323 , 0.024129 , 0.01624637, 0.008453 , 0.00081238, -0.00663246, -0.01384762, -0.02081544, -0.02752235, -0.03395474, -0.04009901, -0.04594158, -0.05146885, -0.05666722, -0.06152309, -0.06602289, -0.070153 , -0.07389983, -0.0772498 , -0.0801893 , -0.08270474, -0.08478253, -0.0864094 , -0.08757342, -0.08826299, -0.08846651, -0.08817252, -0.08737017, -0.08605498, -0.0842473 , -0.08197372, -0.07926081, -0.07613515, -0.07262332, -0.0687519 , -0.06454746, -0.06003658, -0.05524585, -0.05020183, -0.04493111, ... -0.14101509, -0.14099096, -0.14049253, -0.13951544, -0.13805533, -0.13610781, -0.13366852, -0.13073309, -0.12729715, -0.12335632, -0.11890624, -0.11394253, -0.10846572, -0.1024985 , -0.09607878, -0.08924707, -0.08204388, -0.07450972, -0.0666851 , -0.05861052, -0.05032649, -0.04187352, -0.03329211, -0.02462278, -0.01590602, -0.00718235, 0.00150772, 0.01012369, 0.01862505, 0.0269713 , 0.03512192, 0.04303641, 0.05067426, 0.05799496, 0.06495801, 0.0715229 , 0.07764912, 0.08329617, 0.08842709, 0.09301918, 0.09705829, 0.10055024, 0.10350583, 0.10593589, 0.10785121, 0.10926261, 0.1101809 , 0.1106169 , 0.1105814 , 0.11008523, 0.10913921, 0.10775421, 0.10594113, 0.10371085, 0.10107426, 0.09804226, 0.09462573, 0.09083557, 0.08668257, 0.08217719, 0.07732978, 0.07215428, 0.06667898, 0.06093575, 0.05495648, 0.04877304, 0.04241729, 0.03592112, 0.0293164 , 0.022635 , 0.0159088 , 0.00916967, 0.00244949, -0.00421987, -0.01080653, -0.01727861, -0.02360426, -0.02975158, -0.0356887 , -0.04138376, -0.04680487, -0.05192016, -0.05669776, -0.06110579, -0.06511238, -0.06868955, -0.07182817, -0.07453575, -0.07682302, -0.0787007 , -0.08017952, -0.08127019, -0.08198344, -0.08233044, -0.08232433, -0.08197944, -0.08131031, -0.08033148, -0.07905748])
- imf5(time)float64-0.1279 -0.1265 ... 0.1425 0.1421
array([-1.27923704e-01, -1.26498530e-01, -1.25008157e-01, -1.23454129e-01, -1.21837991e-01, -1.20161287e-01, -1.18425561e-01, -1.16632358e-01, -1.14783221e-01, -1.12879696e-01, -1.10923327e-01, -1.08915657e-01, -1.06858232e-01, -1.04752596e-01, -1.02600293e-01, -1.00402867e-01, -9.81618634e-02, -9.58788256e-02, -9.35552984e-02, -9.11928260e-02, -8.87929529e-02, -8.63572233e-02, -8.38871816e-02, -8.13843722e-02, -7.88503395e-02, -7.62866277e-02, -7.36947813e-02, -7.10763445e-02, -6.84328618e-02, -6.57658775e-02, -6.30769359e-02, -6.03675815e-02, -5.76393585e-02, -5.48938113e-02, -5.21324843e-02, -4.93569218e-02, -4.65686681e-02, -4.37692677e-02, -4.09602649e-02, -3.81432040e-02, -3.53196294e-02, -3.24910855e-02, -2.96591165e-02, -2.68252651e-02, -2.39910670e-02, -2.11580558e-02, -1.83277656e-02, -1.55017301e-02, -1.26815555e-02, -9.86913679e-03, -7.06644146e-03, -4.27543684e-03, -1.49809032e-03, 1.26363074e-03, 4.00775896e-03, 6.73232695e-03, 9.43536735e-03, 1.21149128e-02, 1.47689959e-02, 1.73956492e-02, 1.99929055e-02, 2.25587973e-02, 2.50913573e-02, 2.75886180e-02, 3.00486122e-02, 3.24693724e-02, 3.48489312e-02, 3.71853213e-02, 3.94765753e-02, 4.17207259e-02, 4.39158056e-02, 4.60598471e-02, 4.81508829e-02, 5.01869459e-02, 5.21660684e-02, 5.40862833e-02, 5.59456230e-02, 5.77421203e-02, 5.94738077e-02, 6.11387179e-02, ... -3.30488147e-02, -3.02472648e-02, -2.74028793e-02, -2.45179707e-02, -2.15948518e-02, -1.86358351e-02, -1.56432333e-02, -1.26193589e-02, -9.56652475e-03, -6.48704335e-03, -3.38322736e-03, -2.57389420e-04, 2.88815784e-03, 6.05110178e-03, 9.22912976e-03, 1.24199292e-02, 1.56211873e-02, 1.88305917e-02, 2.20458295e-02, 2.52645882e-02, 2.84845551e-02, 3.17034177e-02, 3.49188632e-02, 3.81285790e-02, 4.13302526e-02, 4.45215712e-02, 4.77002222e-02, 5.08638931e-02, 5.40102710e-02, 5.71370436e-02, 6.02418980e-02, 6.33225084e-02, 6.63764962e-02, 6.94014694e-02, 7.23950360e-02, 7.53548042e-02, 7.82783821e-02, 8.11633776e-02, 8.40073989e-02, 8.68081428e-02, 8.95636614e-02, 9.22720954e-02, 9.49315857e-02, 9.75402730e-02, 1.00096298e-01, 1.02597802e-01, 1.05042925e-01, 1.07429809e-01, 1.09756594e-01, 1.12021420e-01, 1.14222429e-01, 1.16357762e-01, 1.18425559e-01, 1.20423961e-01, 1.22351108e-01, 1.24205143e-01, 1.25984205e-01, 1.27686435e-01, 1.29309974e-01, 1.30852964e-01, 1.32313544e-01, 1.33689855e-01, 1.34980039e-01, 1.36182235e-01, 1.37294586e-01, 1.38315231e-01, 1.39242312e-01, 1.40073969e-01, 1.40808344e-01, 1.41443576e-01, 1.41977806e-01, 1.42409177e-01, 1.42735827e-01, 1.42955899e-01, 1.43067978e-01, 1.43072621e-01, 1.42971566e-01, 1.42766650e-01, 1.42459637e-01, 1.42052093e-01])
- imf6(time)float64-0.04962 -0.04913 ... 0.2044 0.2046
array([-0.04961763, -0.04913109, -0.04864438, -0.04815757, -0.04767075, -0.047184 , -0.04669742, -0.04621108, -0.04572508, -0.04523949, -0.04475441, -0.04426991, -0.04378609, -0.04330303, -0.04282082, -0.04233953, -0.04185927, -0.0413801 , -0.04090213, -0.04042543, -0.03995008, -0.03947619, -0.03900382, -0.03853307, -0.03806402, -0.03759675, -0.03713136, -0.03666793, -0.03620654, -0.03574728, -0.03529024, -0.03483549, -0.03438314, -0.03393325, -0.03348592, -0.03304124, -0.03259928, -0.03216013, -0.03172389, -0.03129063, -0.03086044, -0.03043341, -0.03000962, -0.02958916, -0.02917211, -0.02875856, -0.02834859, -0.02794229, -0.02753975, -0.02714105, -0.02674628, -0.02635552, -0.02596885, -0.02558637, -0.02520816, -0.0248343 , -0.02446488, -0.02409999, -0.02373971, -0.02338412, -0.02303332, -0.02268739, -0.0223464 , -0.02201046, -0.02167964, -0.02135404, -0.02103372, -0.02071879, -0.02040933, -0.02010542, -0.01980715, -0.0195146 , -0.01922786, -0.01894701, -0.01867215, -0.01840335, -0.0181407 , -0.01788429, -0.0176342 , -0.01739052, -0.01715333, -0.01692273, -0.01669879, -0.01648159, -0.01627124, -0.01606781, -0.01587138, -0.01568205, -0.01549989, -0.015325 , -0.01515746, -0.01499735, -0.01484477, -0.01469979, -0.0145625 , -0.01443299, -0.01431135, -0.01419765, -0.01409199, -0.01399444, ... 0.09733781, 0.09894708, 0.10055397, 0.10215821, 0.10375955, 0.10535773, 0.1069525 , 0.10854359, 0.11013075, 0.11171372, 0.11329225, 0.11486608, 0.11643494, 0.11799859, 0.11955677, 0.12110922, 0.12265567, 0.12419589, 0.1257296 , 0.12725655, 0.12877648, 0.13028914, 0.13179427, 0.13329161, 0.1347809 , 0.13626189, 0.13773433, 0.13919794, 0.14065248, 0.14209769, 0.14353331, 0.14495908, 0.14637475, 0.14778006, 0.14917476, 0.15055857, 0.15193126, 0.15329256, 0.15464221, 0.15597996, 0.15730554, 0.15861871, 0.15991921, 0.16120677, 0.16248114, 0.16374206, 0.16498929, 0.16622255, 0.16744159, 0.16864616, 0.16983599, 0.17101084, 0.17217043, 0.17331453, 0.17444286, 0.17555517, 0.17665121, 0.17773072, 0.17879343, 0.1798391 , 0.18086746, 0.18187827, 0.18287125, 0.18384615, 0.18480273, 0.18574071, 0.18665985, 0.18755988, 0.18844055, 0.1893016 , 0.19014277, 0.19096381, 0.19176446, 0.19254446, 0.19330355, 0.19404149, 0.194758 , 0.19545284, 0.19612574, 0.19677645, 0.19740471, 0.19801027, 0.19859287, 0.19915224, 0.19968814, 0.2002003 , 0.20068848, 0.2011524 , 0.20159182, 0.20200647, 0.20239611, 0.20276047, 0.20309929, 0.20341232, 0.2036993 , 0.20396 , 0.20419428, 0.20440205, 0.20458347])
- timePandasIndex
PandasIndex(DatetimeIndex(['1871-01-31', '1871-04-30', '1871-07-31', '1871-10-31', '1872-01-31', '1872-04-30', '1872-07-31', '1872-10-31', '1873-01-31', '1873-04-30', ... '1994-07-31', '1994-10-31', '1995-01-31', '1995-04-30', '1995-07-31', '1995-10-31', '1996-01-31', '1996-04-30', '1996-07-31', '1996-10-31'], dtype='datetime64[ns]', name='time', length=504, freq=None))
Perform Ensemble Empirical Mode Decomposition (EEMD) on the time series EEMD improves upon EMD by adding white noise ensembles to overcome mode mixing The method performs multiple EMD trials (default=100) with different noise realizations and averages the results to obtain more stable IMF components
eimf_result = ecl.filter.filter_eemd(data, time_step="M")
eimf_result
<xarray.Dataset> Size: 40kB Dimensions: (time: 504) Coordinates: * time (time) datetime64[ns] 4kB 1871-01-31 1871-04-30 ... 1996-10-31 Data variables: input (time) float64 4kB -0.15 -0.3 -0.14 -0.41 ... -0.08 -0.18 -0.06 eimf0 (time) float64 4kB 0.0452 -0.09849 0.1185 ... -0.1226 -0.002305 eimf1 (time) float64 4kB 0.1247 0.09537 0.06561 ... -0.006502 -0.006618 eimf2 (time) float64 4kB 0.3061 0.2938 0.256 ... -0.2338 -0.2405 -0.2236 eimf3 (time) float64 4kB -0.3626 -0.3664 -0.3726 ... -0.145 -0.1475 eimf4 (time) float64 4kB -0.1766 -0.1756 -0.1737 ... -0.003443 -0.002258 eimf5 (time) float64 4kB -0.04115 -0.0403 -0.03942 ... 0.1949 0.1932 eimf6 (time) float64 4kB -0.01001 -0.009842 -0.009671 ... 0.1099 0.1098 eimf7 (time) float64 4kB -0.05519 -0.05516 -0.05514 ... 0.04692 0.04706
- time: 504
- time(time)datetime64[ns]1871-01-31 ... 1996-10-31
array(['1871-01-31T00:00:00.000000000', '1871-04-30T00:00:00.000000000', '1871-07-31T00:00:00.000000000', ..., '1996-04-30T00:00:00.000000000', '1996-07-31T00:00:00.000000000', '1996-10-31T00:00:00.000000000'], shape=(504,), dtype='datetime64[ns]')
- input(time)float64-0.15 -0.3 -0.14 ... -0.18 -0.06
array([-0.15, -0.3 , -0.14, ..., -0.08, -0.18, -0.06], shape=(504,))
- eimf0(time)float640.0452 -0.09849 ... -0.002305
array([ 4.51992090e-02, -9.84892176e-02, 1.18494076e-01, -5.07833586e-02, 3.30102721e-02, -1.25410384e-01, 1.79769414e-01, -5.72083336e-02, -2.04933754e-01, -9.67095409e-02, 3.21874535e-01, -4.50482828e-02, -2.24696650e-01, 1.17167055e-01, 8.27513756e-03, -1.53787801e-01, 1.20805353e-01, -6.63568488e-04, 1.15899152e-01, 9.09689499e-02, -3.39634877e-01, -5.93634502e-02, 4.64979686e-01, 2.62552649e-01, -4.21328441e-01, -4.43578828e-01, 1.63591860e-01, 2.73344874e-01, 4.69380441e-01, -3.48065430e-01, -2.07131486e-01, 1.07041867e-01, 1.07262536e-01, -1.80299998e-01, 2.42391887e-01, -1.66818540e-01, -1.02679256e-02, 2.48706397e-03, 1.01153034e-02, 5.33176887e-02, 6.34094261e-02, 1.14510822e-02, -4.07835842e-02, -6.53179868e-02, -4.27330419e-02, 1.66386114e-01, -5.57928288e-02, -3.33120198e-02, -8.98572457e-02, 1.29202380e-01, -1.04344769e-02, 5.37763528e-02, -1.25656949e-01, 2.46126917e-02, -8.72437354e-02, 3.41436132e-01, -1.95043871e-01, -3.88339528e-01, -2.26142612e-01, 5.78863840e-01, -3.06029123e-01, -3.63253491e-01, 7.89959015e-02, -1.75431922e-02, -4.72089036e-02, 9.90807321e-02, 5.62070531e-02, -3.79183631e-02, -1.89636393e-01, -1.27010982e-01, -1.37374940e-01, 3.91339622e-01, 4.34549131e-01, -1.26177165e-01, -5.52026390e-01, -2.45849453e-01, -1.67981804e-02, 3.86933211e-01, 2.02621615e-01, -3.67261499e-01, ... -7.96202193e-03, -6.19036285e-01, -4.20513224e-01, 2.96763706e-01, 3.93531026e-01, -2.78843097e-01, -1.28206244e-01, 1.34259132e-01, -6.58709222e-02, 3.58762883e-03, 7.79415317e-02, 1.54427571e-02, -1.02811223e-01, 8.28286041e-02, -9.24166347e-02, 1.59384856e-01, -1.72222202e-01, 1.30770642e-01, -1.47132625e-01, 3.22889640e-02, 2.09562803e-01, 7.78428024e-02, -5.04480605e-01, 4.46744786e-01, 3.92520803e-01, 2.85282285e-01, -2.32078743e-01, -4.68900212e-01, 3.42055347e-01, 3.56681317e-02, 2.96952024e-02, -2.58099536e-01, 2.45143543e-01, 7.40467103e-02, -2.96389612e-01, 1.39156980e-01, 9.56125476e-02, 3.24625013e-02, -1.88181183e-01, 1.93690597e-01, -2.54889949e-02, -1.76176353e-01, -7.33578787e-02, 3.78864784e-01, 2.29683390e-01, -2.55527744e-01, -4.52304494e-01, -1.77896659e-01, 2.57303749e-01, 3.24488618e-01, -2.33454911e-01, -1.17700521e-01, 6.42720549e-02, 1.48350547e-01, -1.40256867e-01, -2.15787706e-02, 5.58026139e-02, 1.85676984e-01, -3.48667654e-01, -5.87698469e-02, 2.13011234e-01, 5.13462276e-01, -5.21751742e-01, -1.04537143e-01, 3.43300439e-02, 3.95640802e-01, -3.27945595e-01, 3.96129010e-02, 2.84464519e-01, 1.49382221e-01, -5.07293127e-01, 4.84321330e-01, 2.62525377e-01, -1.87567570e-01, -3.98911175e-01, -5.39492525e-02, 2.42569287e-01, 4.29190972e-02, -1.22612015e-01, -2.30527626e-03])
- eimf1(time)float640.1247 0.09537 ... -0.006618
array([ 1.24740593e-01, 9.53651064e-02, 6.56140300e-02, 3.45504155e-02, 5.18639425e-03, -6.87632876e-03, -3.42924721e-02, -7.72419662e-02, -6.73271111e-02, 3.18354054e-02, 1.13047014e-01, 5.92616006e-02, -5.83373049e-02, -1.08439618e-01, -2.78834961e-02, 1.46835235e-01, 3.01129643e-01, 2.79459819e-01, 2.15549585e-02, -3.51340896e-01, -6.52922886e-01, -7.46155848e-01, -6.41663324e-01, -3.95149755e-01, -4.08196471e-02, 3.91441765e-01, 8.03378937e-01, 9.84212759e-01, 7.17307409e-01, 1.32103627e-01, -4.05098623e-01, -6.79433363e-01, -6.99839054e-01, -5.99351858e-01, -5.25100295e-01, -4.92004345e-01, -3.55345884e-01, -7.02923555e-02, 2.49143347e-01, 4.53373176e-01, 4.58750378e-01, 2.91672362e-01, 6.61639163e-02, -9.71642387e-02, -1.54884298e-01, -1.69842002e-01, -1.99059663e-01, -1.88651720e-01, -6.87841999e-02, 8.66824425e-02, 1.30524495e-01, 3.20803002e-02, -7.58937251e-02, -5.08336679e-02, 6.85935386e-02, 1.49426266e-01, 1.80672213e-01, 2.55768757e-01, 3.85266882e-01, 3.46537052e-01, 3.84425736e-04, -4.27329971e-01, -6.88823330e-01, -6.84046546e-01, -5.09565373e-01, -3.00891456e-01, -1.06635993e-01, 9.00454557e-02, 3.37420690e-01, 6.58077401e-01, 9.65318784e-01, 1.05719120e+00, 7.69534917e-01, 1.78631573e-01, -4.37730152e-01, -8.20597246e-01, -9.02522632e-01, -8.02840977e-01, -5.75781523e-01, -2.15190889e-01, ... 4.23903300e-01, 2.90539700e-01, 1.17629512e-01, -7.30422563e-02, -2.66631067e-01, -4.14499566e-01, -4.23703467e-01, -2.58882917e-01, 3.03741473e-02, 3.25191424e-01, 5.02298315e-01, 4.98166732e-01, 3.67902879e-01, 2.08402286e-01, 5.19843165e-02, -1.45755417e-01, -3.91014154e-01, -5.84169581e-01, -6.66862244e-01, -6.48698740e-01, -5.48054516e-01, -3.16280778e-01, 1.09754596e-01, 5.88251382e-01, 7.47756806e-01, 4.31417231e-01, -7.83337554e-02, -4.19906370e-01, -4.14283599e-01, -2.27143405e-01, -9.72983971e-02, -4.34471736e-02, -7.61846976e-02, -2.16690034e-01, -3.31950502e-01, -2.91183636e-01, -1.80771802e-01, -8.01490620e-02, 6.87201644e-02, 2.79052804e-01, 4.55766316e-01, 5.55262022e-01, 5.77063293e-01, 4.33929940e-01, 7.48648285e-02, -3.56819655e-01, -6.34715171e-01, -6.09278408e-01, -3.65212598e-01, -6.30108077e-02, 1.83375593e-01, 3.16782989e-01, 3.33952390e-01, 2.14943592e-01, 8.23876706e-03, -1.50104589e-01, -1.92078783e-01, -1.21860423e-01, 2.24600204e-02, 1.55580090e-01, 1.91202174e-01, 3.43050672e-02, -2.00863577e-01, -2.01030301e-01, 3.30263031e-02, 2.04753385e-01, 1.54622692e-01, -1.77567532e-02, -1.66406945e-01, -1.92498606e-01, -4.83158634e-02, 1.79037942e-01, 2.38964017e-01, 1.00465360e-01, -8.48186278e-02, -1.77926705e-01, -1.50169308e-01, -6.47991715e-02, -6.50214732e-03, -6.61802685e-03])
- eimf2(time)float640.3061 0.2938 ... -0.2405 -0.2236
array([ 3.06087688e-01, 2.93789662e-01, 2.56019989e-01, 1.99566751e-01, 1.33858042e-01, 6.84991273e-02, 1.09854293e-02, -3.52303657e-02, -7.08297413e-02, -1.02355275e-01, -1.39715100e-01, -1.91026954e-01, -2.58606116e-01, -3.41562823e-01, -4.33502572e-01, -5.18762889e-01, -5.75917371e-01, -5.85800872e-01, -5.40483486e-01, -4.43595808e-01, -3.01156249e-01, -1.19685518e-01, 9.02804339e-02, 3.10441385e-01, 5.17078575e-01, 6.84486557e-01, 7.87498993e-01, 8.01445631e-01, 7.14161662e-01, 5.48542867e-01, 3.36622590e-01, 1.10393577e-01, -9.83138924e-02, -2.58344608e-01, -3.44864458e-01, -3.54682118e-01, -3.08419562e-01, -2.33886289e-01, -1.55584411e-01, -9.59404684e-02, -7.19553988e-02, -8.43219564e-02, -1.21841459e-01, -1.71128108e-01, -2.18230709e-01, -2.48632891e-01, -2.51117883e-01, -2.22242936e-01, -1.66303625e-01, -9.30673123e-02, -1.40222435e-02, 6.10772525e-02, 1.26963977e-01, 1.79505651e-01, 2.13841937e-01, 2.24927362e-01, 2.06369800e-01, 1.54681333e-01, 7.42353384e-02, -2.38916626e-02, -1.26419460e-01, -2.16878139e-01, -2.78355444e-01, -2.95915871e-01, -2.65750331e-01, -1.94175231e-01, -9.16528189e-02, 2.83548090e-02, 1.50040163e-01, 2.58636610e-01, 3.42441624e-01, 3.91051966e-01, 3.99124760e-01, 3.74896573e-01, 3.31162214e-01, 2.80685934e-01, 2.33960650e-01, 1.96844708e-01, 1.69995547e-01, 1.45103445e-01, ... 4.10836388e-01, 3.94248590e-01, 3.33087709e-01, 2.41397829e-01, 1.36420933e-01, 3.36092850e-02, -5.63655551e-02, -1.26938073e-01, -1.78662072e-01, -2.16363667e-01, -2.45484433e-01, -2.71298274e-01, -2.96444621e-01, -3.18620686e-01, -3.31291088e-01, -3.23423805e-01, -2.82610195e-01, -2.00615039e-01, -7.37374812e-02, 9.32743914e-02, 2.82867761e-01, 4.67767256e-01, 6.16141777e-01, 6.96797732e-01, 6.79472176e-01, 5.53992391e-01, 3.50413678e-01, 1.05521474e-01, -1.44608451e-01, -3.66495034e-01, -5.30376111e-01, -6.19972992e-01, -6.30611200e-01, -5.68326355e-01, -4.45355484e-01, -2.77930220e-01, -8.41705388e-02, 1.17060327e-01, 3.04993451e-01, 4.58368337e-01, 5.54767879e-01, 5.72715391e-01, 5.02183978e-01, 3.58350856e-01, 1.68907446e-01, -3.55831499e-02, -2.24051980e-01, -3.66705797e-01, -4.49061583e-01, -4.75215245e-01, -4.52027638e-01, -3.86565520e-01, -2.87252893e-01, -1.63659978e-01, -3.01860820e-02, 9.62229768e-02, 2.00603580e-01, 2.74119102e-01, 3.13018015e-01, 3.14732807e-01, 2.78681286e-01, 2.12327642e-01, 1.32632027e-01, 5.82236515e-02, 6.89680591e-04, -3.71810099e-02, -5.43259637e-02, -5.26793427e-02, -3.84839541e-02, -1.81876226e-02, -1.71861133e-03, 1.62396215e-03, -1.57028412e-02, -5.43738274e-02, -1.05691503e-01, -1.59064834e-01, -2.04708687e-01, -2.33750308e-01, -2.40535406e-01, -2.23628541e-01])
- eimf3(time)float64-0.3626 -0.3664 ... -0.145 -0.1475
array([-3.62593908e-01, -3.66351037e-01, -3.72617131e-01, -3.80614087e-01, -3.89246154e-01, -3.97344961e-01, -4.03691310e-01, -4.06965441e-01, -4.05738033e-01, -3.98554778e-01, -3.84123807e-01, -3.61486278e-01, -3.30112697e-01, -2.90113956e-01, -2.42198349e-01, -1.87524641e-01, -1.27514775e-01, -6.36989354e-02, 2.31973278e-03, 6.88654218e-02, 1.34247760e-01, 1.96779197e-01, 2.54763782e-01, 3.06502709e-01, 3.50295399e-01, 3.84438601e-01, 4.07234956e-01, 4.17163740e-01, 4.14033033e-01, 3.99040417e-01, 3.73910267e-01, 3.40495418e-01, 3.00655740e-01, 2.56247914e-01, 2.09118642e-01, 1.60948642e-01, 1.13121310e-01, 6.69541645e-02, 2.37724824e-02, -1.50835554e-02, -4.82821747e-02, -7.47966782e-02, -9.43245030e-02, -1.07091898e-01, -1.13580114e-01, -1.14436164e-01, -1.10427111e-01, -1.02433378e-01, -9.13190049e-02, -7.78679085e-02, -6.27348265e-02, -4.64822107e-02, -2.96487337e-02, -1.27382998e-02, 3.83030101e-03, 1.97063043e-02, 3.45674466e-02, 4.81975870e-02, 6.04780154e-02, 7.12786251e-02, 8.04701038e-02, 8.79314709e-02, 9.35647139e-02, 9.72953065e-02, 9.90181136e-02, 9.85205822e-02, 9.56250545e-02, 9.01920095e-02, 8.20921040e-02, 7.12112433e-02, 5.74843294e-02, 4.09260156e-02, 2.17648038e-02, 4.04766098e-04, -2.27426531e-02, -4.72657168e-02, -7.27243659e-02, -9.86496623e-02, -1.24552526e-01, -1.49908418e-01, ... -1.58334383e-01, -1.29941889e-01, -1.00056119e-01, -6.92203338e-02, -3.80847359e-02, -7.33509652e-03, 2.24608705e-02, 5.11305701e-02, 7.88455300e-02, 1.05832060e-01, 1.32309182e-01, 1.58447198e-01, 1.84258955e-01, 2.09419422e-01, 2.33419863e-01, 2.55748599e-01, 2.75900157e-01, 2.93388328e-01, 3.07661574e-01, 3.18041500e-01, 3.23736516e-01, 3.23938776e-01, 3.17833451e-01, 3.04630938e-01, 2.83811882e-01, 2.55716815e-01, 2.21460752e-01, 1.82251634e-01, 1.39296396e-01, 9.38042824e-02, 4.69996772e-02, 1.32333393e-04, -4.55901513e-02, -8.91983639e-02, -1.30082071e-01, -1.67785676e-01, -2.01857259e-01, -2.31813180e-01, -2.57155785e-01, -2.77387022e-01, -2.92011201e-01, -3.00572965e-01, -3.02869359e-01, -2.99085361e-01, -2.89509062e-01, -2.74441281e-01, -2.54181042e-01, -2.29024190e-01, -1.99285978e-01, -1.65412385e-01, -1.28115261e-01, -8.82999030e-02, -4.69239764e-02, -4.94444853e-03, 3.66830348e-02, 7.69804814e-02, 1.14927502e-01, 1.49499474e-01, 1.79679954e-01, 2.04530063e-01, 2.23311055e-01, 2.35601247e-01, 2.41207411e-01, 2.40068221e-01, 2.32223555e-01, 2.17810487e-01, 1.97184368e-01, 1.71069849e-01, 1.40494994e-01, 1.06663485e-01, 7.08450093e-02, 3.42915757e-02, -1.77616654e-03, -3.61844213e-02, -6.78055237e-02, -9.55396850e-02, -1.18315076e-01, -1.35093178e-01, -1.44985136e-01, -1.47530795e-01])
- eimf4(time)float64-0.1766 -0.1756 ... -0.002258
array([-0.17661072, -0.17558881, -0.17371828, -0.17099827, -0.16744061, -0.16305931, -0.15786967, -0.15188852, -0.14513419, -0.13762605, -0.12938547, -0.12043979, -0.11083239, -0.10063194, -0.08991868, -0.07877439, -0.06728129, -0.05552176, -0.04357822, -0.03153312, -0.01946888, -0.00746793, 0.00438767, 0.01601668, 0.02733962, 0.03827971, 0.04876271, 0.05871922, 0.06809451, 0.07684786, 0.08494651, 0.09236289, 0.09907345, 0.10505769, 0.11029966, 0.11478845, 0.11851704, 0.12147986, 0.12367138, 0.12508615, 0.12571918, 0.12556688, 0.12462852, 0.12291395, 0.12044468, 0.11725111, 0.11336965, 0.108841 , 0.103709 , 0.0980208 , 0.09182444, 0.08517006, 0.07811255, 0.07070842, 0.06301211, 0.05507572, 0.0469504 , 0.03868601, 0.03032766, 0.02191988, 0.01350787, 0.00513607, -0.00315295, -0.01131928, -0.01932718, -0.02714236, -0.03472855, -0.04204989, -0.04907283, -0.05576591, -0.06210079, -0.0680503 , -0.07358805, -0.07868899, -0.08332916, -0.08748582, -0.0911369 , -0.09426113, -0.09683771, -0.09884627, -0.10026726, -0.10108208, -0.10127288, -0.10082264, -0.0997161 , -0.09794008, -0.09548493, -0.09234721, -0.08853288, -0.08406033, -0.07896036, -0.07327383, -0.06704608, -0.06032464, -0.05315758, -0.04559284, -0.03767802, -0.0294606 , -0.02098829, -0.01230915, ... -0.13616399, -0.14212626, -0.14746125, -0.15211087, -0.15601862, -0.15913103, -0.16139832, -0.16277303, -0.16321068, -0.16267067, -0.16112278, -0.15855687, -0.15498639, -0.15043892, -0.14495232, -0.13857319, -0.13135354, -0.12334803, -0.11461298, -0.1052063 , -0.09518756, -0.08461925, -0.07356869, -0.06210862, -0.05031655, -0.03827817, -0.02608609, -0.0138373 , -0.00163078, 0.01043849, 0.02228052, 0.03381186, 0.04496341, 0.05567638, 0.06589842, 0.07558055, 0.08467579, 0.09313849, 0.10092841, 0.10801137, 0.11435947, 0.11995262, 0.12477938, 0.12884109, 0.1321573 , 0.13475667, 0.13667195, 0.13793843, 0.13859189, 0.13866757, 0.1382005 , 0.13722575, 0.13577854, 0.13389247, 0.13159675, 0.12891952, 0.12588884, 0.1225327 , 0.11887926, 0.11495686, 0.11079551, 0.10642639, 0.10187991, 0.0971839 , 0.09236599, 0.08744945, 0.08245411, 0.07740126, 0.07231334, 0.06721296, 0.06212304, 0.0570669 , 0.0520684 , 0.04715177, 0.04234138, 0.03766149, 0.03313625, 0.02878955, 0.02464452, 0.02072326, 0.01704598, 0.01362985, 0.01048623, 0.00761958, 0.00503156, 0.00272377, 0.00069694, -0.00104876, -0.00251372, -0.00369895, -0.00460601, -0.00523712, -0.0055954 , -0.00568468, -0.00550914, -0.00507341, -0.00438272, -0.00344273, -0.00225835])
- eimf5(time)float64-0.04115 -0.0403 ... 0.1949 0.1932
array([-4.11480647e-02, -4.03044811e-02, -3.94242800e-02, -3.85081995e-02, -3.75570437e-02, -3.65716807e-02, -3.55530217e-02, -3.45019611e-02, -3.34193444e-02, -3.23059932e-02, -3.11628296e-02, -2.99911452e-02, -2.87923937e-02, -2.75680241e-02, -2.63194776e-02, -2.50481993e-02, -2.37556349e-02, -2.24432300e-02, -2.11124296e-02, -1.97646789e-02, -1.84014231e-02, -1.70241078e-02, -1.56341763e-02, -1.42330520e-02, -1.28221393e-02, -1.14028413e-02, -9.97656295e-03, -8.54471280e-03, -7.10870038e-03, -5.66994671e-03, -4.22988275e-03, -2.78994155e-03, -1.35149486e-03, 8.42565793e-05, 1.51618204e-03, 2.94314188e-03, 4.36402847e-03, 5.77782035e-03, 7.18349867e-03, 8.58002711e-03, 9.96639477e-03, 1.13416696e-02, 1.27049313e-02, 1.40552793e-02, 1.53918377e-02, 1.67137244e-02, 1.80200022e-02, 1.93097156e-02, 2.05818740e-02, 2.18354482e-02, 2.30694065e-02, 2.42827583e-02, 2.54745451e-02, 2.66437907e-02, 2.77896184e-02, 2.89112300e-02, 3.00079438e-02, 3.10791750e-02, 3.21243455e-02, 3.31428803e-02, 3.41342040e-02, 3.50977293e-02, 3.60328413e-02, 3.69389115e-02, 3.78152119e-02, 3.86608457e-02, 3.94748001e-02, 4.02559756e-02, 4.10032653e-02, 4.17156051e-02, 4.23919476e-02, 4.30312767e-02, 4.36325901e-02, 4.41949048e-02, 4.47172921e-02, 4.51988857e-02, 4.56389140e-02, 4.60367008e-02, 4.63916499e-02, 4.67032452e-02, ... 3.31031402e-02, 3.69102311e-02, 4.07547499e-02, 4.46329991e-02, 4.85412776e-02, 5.24758576e-02, 5.64329533e-02, 6.04087695e-02, 6.43995109e-02, 6.84013810e-02, 7.24105828e-02, 7.64233193e-02, 8.04357935e-02, 8.44442081e-02, 8.84447663e-02, 9.24336709e-02, 9.64071248e-02, 1.00361331e-01, 1.04292492e-01, 1.08196814e-01, 1.12070515e-01, 1.15909905e-01, 1.19711435e-01, 1.23471576e-01, 1.27186799e-01, 1.30853619e-01, 1.34468671e-01, 1.38028659e-01, 1.41530306e-01, 1.44970438e-01, 1.48346024e-01, 1.51654082e-01, 1.54891666e-01, 1.58055841e-01, 1.61143700e-01, 1.64152396e-01, 1.67079126e-01, 1.69921125e-01, 1.72675721e-01, 1.75340294e-01, 1.77912234e-01, 1.80389070e-01, 1.82768595e-01, 1.85048822e-01, 1.87227847e-01, 1.89303890e-01, 1.91275377e-01, 1.93140866e-01, 1.94899067e-01, 1.96548851e-01, 1.98089251e-01, 1.99519389e-01, 2.00838453e-01, 2.02045668e-01, 2.03140281e-01, 2.04121565e-01, 2.04988817e-01, 2.05741352e-01, 2.06378499e-01, 2.06899585e-01, 2.07303969e-01, 2.07591104e-01, 2.07760562e-01, 2.07812094e-01, 2.07745641e-01, 2.07561376e-01, 2.07259777e-01, 2.06841636e-01, 2.06307906e-01, 2.05659670e-01, 2.04898057e-01, 2.04024226e-01, 2.03039420e-01, 2.01945018e-01, 2.00742536e-01, 1.99433602e-01, 1.98019934e-01, 1.96503329e-01, 1.94885615e-01, 1.93168670e-01])
- eimf6(time)float64-0.01001 -0.009842 ... 0.1098
array([-0.0100138 , -0.00984153, -0.00967068, -0.00950128, -0.00933339, -0.00916704, -0.0090023 , -0.00883919, -0.00867778, -0.00851811, -0.00836021, -0.00820415, -0.00804995, -0.00789765, -0.00774731, -0.00759895, -0.00745262, -0.00730835, -0.0071662 , -0.00702619, -0.00688838, -0.00675279, -0.00661947, -0.00648846, -0.0063598 , -0.00623353, -0.00610967, -0.00598828, -0.00586939, -0.00575304, -0.00563927, -0.00552813, -0.00541967, -0.00531392, -0.00521093, -0.00511073, -0.00501338, -0.00491889, -0.00482733, -0.00473872, -0.00465311, -0.00457053, -0.00449103, -0.00441464, -0.0043414 , -0.00427136, -0.00420455, -0.00414101, -0.00408078, -0.0040239 , -0.00397041, -0.00392036, -0.00387377, -0.00383069, -0.00379117, -0.00375523, -0.00372292, -0.00369428, -0.00366935, -0.00364816, -0.00363075, -0.00361716, -0.00360742, -0.00360156, -0.00359961, -0.0036016 , -0.00360756, -0.0036175 , -0.00363145, -0.00364944, -0.00367148, -0.0036976 , -0.00372783, -0.00376218, -0.00380067, -0.00384334, -0.0038902 , -0.00394127, -0.00399657, -0.00405612, -0.00411994, -0.00418804, -0.00426044, -0.00433717, -0.00441823, -0.00450364, -0.00459341, -0.00468754, -0.00478606, -0.00488897, -0.00499628, -0.005108 , -0.00522411, -0.00534464, -0.00546957, -0.00559889, -0.00573261, -0.0058707 , -0.00601316, -0.00615998, ... 0.04534034, 0.04642807, 0.04751357, 0.04859661, 0.04967698, 0.05075446, 0.05182884, 0.05289989, 0.05396741, 0.05503119, 0.056091 , 0.05714663, 0.05819788, 0.05924454, 0.06028638, 0.06132322, 0.06235483, 0.06338101, 0.06440156, 0.06541625, 0.06642489, 0.06742727, 0.06842317, 0.06941239, 0.07039471, 0.07136994, 0.07233786, 0.07329826, 0.07425093, 0.07519567, 0.07613227, 0.07706051, 0.07798019, 0.0788911 , 0.07979304, 0.08068579, 0.08156914, 0.08244289, 0.08330682, 0.08416073, 0.08500441, 0.08583766, 0.08666025, 0.08747199, 0.08827266, 0.08906206, 0.08983997, 0.0906062 , 0.09136052, 0.09210274, 0.09283263, 0.09355 , 0.09425464, 0.09494633, 0.09562487, 0.09629005, 0.09694166, 0.09757949, 0.09820334, 0.09881298, 0.09940823, 0.09998887, 0.10055471, 0.10110554, 0.10164117, 0.10216141, 0.10266607, 0.10315496, 0.10362789, 0.1040847 , 0.1045252 , 0.10494924, 0.10535665, 0.10574731, 0.10612107, 0.10647779, 0.10681733, 0.10713957, 0.10744438, 0.10773164, 0.10800123, 0.10825305, 0.10848703, 0.10870307, 0.10890111, 0.10908112, 0.10924312, 0.10938714, 0.10951327, 0.1096216 , 0.10971222, 0.10978524, 0.10984076, 0.1098789 , 0.10989978, 0.10990351, 0.10989023, 0.10986007, 0.10981317])
- eimf7(time)float64-0.05519 -0.05516 ... 0.04706
array([-5.51877853e-02, -5.51636638e-02, -5.51380186e-02, -5.51108537e-02, -5.50821731e-02, -5.50519811e-02, -5.50202816e-02, -5.49870787e-02, -5.49523766e-02, -5.49161794e-02, -5.48784911e-02, -5.48393158e-02, -5.47986577e-02, -5.47565208e-02, -5.47129092e-02, -5.46678270e-02, -5.46212783e-02, -5.45732671e-02, -5.45237977e-02, -5.44728740e-02, -5.44205003e-02, -5.43666804e-02, -5.43114186e-02, -5.42547190e-02, -5.41965856e-02, -5.41370226e-02, -5.40760339e-02, -5.40136238e-02, -5.39497963e-02, -5.38845555e-02, -5.38179055e-02, -5.37498504e-02, -5.36803942e-02, -5.36095412e-02, -5.35372953e-02, -5.34636606e-02, -5.33886414e-02, -5.33122415e-02, -5.32344652e-02, -5.31553166e-02, -5.30747996e-02, -5.29929185e-02, -5.29096773e-02, -5.28250801e-02, -5.27391309e-02, -5.26518340e-02, -5.25631933e-02, -5.24732131e-02, -5.23818973e-02, -5.22892500e-02, -5.21952754e-02, -5.20999776e-02, -5.20033605e-02, -5.19054284e-02, -5.18061854e-02, -5.17056354e-02, -5.16037827e-02, -5.15006313e-02, -5.13961852e-02, -5.12904487e-02, -5.11834257e-02, -5.10751204e-02, -5.09655368e-02, -5.08546792e-02, -5.07425514e-02, -5.06291577e-02, -5.05145022e-02, -5.03985889e-02, -5.02814219e-02, -5.01630053e-02, -5.00433432e-02, -4.99224398e-02, -4.98002990e-02, -4.96769250e-02, -4.95523219e-02, -4.94264937e-02, -4.92994446e-02, -4.91711787e-02, -4.90417000e-02, -4.89110127e-02, ... 3.26131024e-02, 3.28345929e-02, 3.30552582e-02, 3.32750916e-02, 3.34940869e-02, 3.37122373e-02, 3.39295366e-02, 3.41459782e-02, 3.43615556e-02, 3.45762624e-02, 3.47900921e-02, 3.50030382e-02, 3.52150943e-02, 3.54262538e-02, 3.56365103e-02, 3.58458573e-02, 3.60542883e-02, 3.62617969e-02, 3.64683766e-02, 3.66740209e-02, 3.68787234e-02, 3.70824775e-02, 3.72852768e-02, 3.74871148e-02, 3.76879850e-02, 3.78878810e-02, 3.80867963e-02, 3.82847244e-02, 3.84816588e-02, 3.86775931e-02, 3.88725208e-02, 3.90664354e-02, 3.92593304e-02, 3.94511993e-02, 3.96420358e-02, 3.98318332e-02, 4.00205852e-02, 4.02082852e-02, 4.03949268e-02, 4.05805036e-02, 4.07650089e-02, 4.09484364e-02, 4.11307796e-02, 4.13120319e-02, 4.14921871e-02, 4.16712384e-02, 4.18491795e-02, 4.20260040e-02, 4.22017052e-02, 4.23762768e-02, 4.25497123e-02, 4.27220052e-02, 4.28931490e-02, 4.30631372e-02, 4.32319635e-02, 4.33996212e-02, 4.35661039e-02, 4.37314052e-02, 4.38955186e-02, 4.40584375e-02, 4.42201556e-02, 4.43806663e-02, 4.45399632e-02, 4.46980398e-02, 4.48548896e-02, 4.50105061e-02, 4.51648829e-02, 4.53180135e-02, 4.54698915e-02, 4.56205102e-02, 4.57698633e-02, 4.59179444e-02, 4.60647468e-02, 4.62102642e-02, 4.63544900e-02, 4.64974178e-02, 4.66390412e-02, 4.67793535e-02, 4.69183485e-02, 4.70560195e-02])
- timePandasIndex
PandasIndex(DatetimeIndex(['1871-01-31', '1871-04-30', '1871-07-31', '1871-10-31', '1872-01-31', '1872-04-30', '1872-07-31', '1872-10-31', '1873-01-31', '1873-04-30', ... '1994-07-31', '1994-10-31', '1995-01-31', '1995-04-30', '1995-07-31', '1995-10-31', '1996-01-31', '1996-04-30', '1996-07-31', '1996-10-31'], dtype='datetime64[ns]', name='time', length=504, freq=None))
Visualize the first three IMF components from standard EMD IMFs are ordered from highest frequency (IMF0) to lowest frequency (IMF2) Each IMF must satisfy two conditions:
Number of extrema and zero crossings differs by at most one
Mean of upper and lower envelopes is zero at any point
fig, ax = plt.subplots(4, 1, figsize = (8, 8), sharex=True)
fig.subplots_adjust(hspace=0.2)
axi = ax[0]
imf_result["input"].plot(ax = axi, color = "r")
axi.set_xlabel("")
axi.set_ylabel("Input")
axi.set_title("Input Signal: Niño 3")
axi = ax[1]
imf_result["imf0"].plot(ax = axi)
axi.set_xlabel("")
axi.set_ylabel("IMF 0")
axi = ax[2]
imf_result["imf1"].plot(ax = axi)
axi.set_xlabel("")
axi.set_ylabel("IMF 1")
axi = ax[3]
imf_result["imf2"].plot(ax = axi)
axi.set_xlabel("Time")
axi.set_ylabel("IMF 2")

Text(51.222222222222214, 0.5, 'IMF 2')
Visualize the first three eIMF components from EEMD Ensemble IMFs show improved mode separation compared to standard EMD The noise-assisted approach helps distinguish:
High-frequency noise/oscillations (eIMF0)
Seasonal-to-interannual variability (eIMF1)
Lower frequency trends (eIMF2)
fig, ax = plt.subplots(4, 1, figsize = (8, 8), sharex=True)
fig.subplots_adjust(hspace=0.2)
axi = ax[0]
eimf_result["input"].plot(ax = axi, color = "r")
axi.set_xlabel("")
axi.set_ylabel("Input")
axi.set_title("Input Signal: Niño 3")
axi = ax[1]
eimf_result["eimf0"].plot(ax = axi)
axi.set_xlabel("")
axi.set_ylabel("eIMF 0")
axi = ax[2]
eimf_result["eimf1"].plot(ax = axi)
axi.set_xlabel("")
axi.set_ylabel("eIMF 1")
axi = ax[3]
eimf_result["eimf2"].plot(ax = axi)
axi.set_xlabel("Time")
axi.set_ylabel("eIMF 2")

Text(64.22222222222221, 0.5, 'eIMF 2')
Total running time of the script: (0 minutes 6.002 seconds)