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 0.03819 -0.1196 0.1208 ... -0.1114 0.02148 eimf1 (time) float64 4kB 0.1285 0.1255 0.1013 ... -0.01852 -0.02684 eimf2 (time) float64 4kB 0.2235 0.216 0.1827 ... -0.2813 -0.2824 -0.2611 eimf3 (time) float64 4kB -0.2995 -0.3097 -0.3228 ... -0.09914 -0.1017 eimf4 (time) float64 4kB -0.1685 -0.1689 -0.1684 ... -0.0324 -0.03042 eimf5 (time) float64 4kB -0.02875 -0.0281 -0.02741 ... 0.1929 0.1915 0.19 eimf6 (time) float64 4kB -0.0004404 -0.0003338 ... 0.1291 0.1291 eimf7 (time) float64 4kB -0.06159 -0.06158 -0.06157 ... 0.06632 0.06652
- 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)float640.03819 -0.1196 ... -0.1114 0.02148
array([ 3.81890429e-02, -1.19614945e-01, 1.20772992e-01, -5.32786577e-02, 2.00231526e-02, -1.02583606e-01, 1.68325385e-01, -3.16435566e-02, -2.12674863e-01, -3.24940651e-02, 2.84187392e-01, -1.99496582e-02, -2.11786193e-01, 1.06321116e-01, -1.16775760e-02, -1.27541678e-01, 1.50916190e-01, -1.75692140e-02, 1.17568109e-01, 7.18166549e-02, -3.45768026e-01, -1.25216761e-01, 4.55328836e-01, 2.38121246e-01, -4.04627443e-01, -4.56898049e-01, 1.46634401e-01, 2.46555699e-01, 4.55678907e-01, -3.28245688e-01, -2.34798004e-01, 9.48322609e-02, 8.54021016e-02, -2.01080033e-01, 2.64777494e-01, -1.72348920e-01, -9.47963520e-03, 6.15077180e-03, -2.61133506e-03, 4.22361548e-02, 5.92589557e-02, 6.11761870e-03, -4.16971663e-02, -1.16665421e-01, -5.61214273e-02, 1.98190844e-01, -1.06178484e-01, -4.79499284e-02, -6.44894436e-02, 1.08357737e-01, -3.89108023e-02, 7.63765995e-02, -1.51393580e-01, 5.38871387e-02, -6.15092650e-02, 3.29997583e-01, -2.30643521e-01, -4.00928204e-01, -2.23203835e-01, 5.84135843e-01, -3.11276433e-01, -3.35126070e-01, 1.46928696e-01, 2.19689795e-02, -7.86996381e-02, 6.21985591e-02, 7.24759580e-02, -4.00097714e-02, -1.74381282e-01, -1.47672576e-01, -1.17474682e-01, 4.05668804e-01, 4.27622888e-01, -1.40178366e-01, -5.40082048e-01, -2.02061622e-01, -5.11174306e-02, 3.54243586e-01, 2.02314253e-01, -3.90677820e-01, ... -3.24947744e-02, -6.00753770e-01, -4.32694000e-01, 2.89039694e-01, 3.72902566e-01, -2.34348574e-01, -1.10614998e-01, 1.02695424e-01, -6.10880934e-02, -2.27547359e-02, 8.38243790e-02, 3.24593646e-02, -1.11340860e-01, 7.72560655e-02, -9.01468471e-02, 1.35231799e-01, -1.47388111e-01, 1.27679390e-01, -1.49860189e-01, 3.80326938e-02, 2.51120198e-01, 1.09466114e-01, -4.85112162e-01, 4.29140063e-01, 3.42311177e-01, 3.14149836e-01, -1.58062217e-01, -4.52836693e-01, 3.43192513e-01, 2.61747555e-02, 5.49060923e-02, -2.38014121e-01, 2.04917887e-01, 6.96176045e-02, -2.83322003e-01, 1.31220041e-01, 1.14225088e-01, 4.10104087e-02, -1.78901334e-01, 1.62621877e-01, -8.71036535e-03, -1.82251836e-01, -1.10671697e-01, 3.84384773e-01, 2.31551388e-01, -2.47874698e-01, -4.01457168e-01, -1.96421057e-01, 2.31675922e-01, 3.12624606e-01, -2.55792716e-01, -1.28468034e-01, 6.23935424e-02, 1.77312534e-01, -1.75830896e-01, 3.50613130e-03, 9.82375012e-03, 1.84790271e-01, -3.46137005e-01, -4.89713638e-02, 2.10788074e-01, 4.97732119e-01, -5.44538970e-01, -1.41592687e-01, 7.24286854e-02, 4.03227786e-01, -3.33535837e-01, 1.20709742e-02, 2.69484464e-01, 1.52490061e-01, -5.02085313e-01, 4.84320569e-01, 2.16395188e-01, -1.82130524e-01, -3.40305437e-01, -7.28174225e-02, 2.39651370e-01, 3.68779284e-02, -1.11390258e-01, 2.14755663e-02])
- eimf1(time)float640.1285 0.1255 ... -0.01852 -0.02684
array([ 1.28467672e-01, 1.25458391e-01, 1.01324677e-01, 5.24306113e-02, 7.03755209e-03, -1.99688006e-02, -6.42074353e-02, -1.08040543e-01, -7.88123622e-02, 3.85184426e-02, 1.24609993e-01, 7.31469740e-02, -4.57667608e-02, -9.83842579e-02, -1.79883676e-02, 1.46423734e-01, 2.81781453e-01, 2.53430249e-01, 1.27432690e-02, -3.33515510e-01, -6.13953997e-01, -6.89851656e-01, -5.73833651e-01, -3.28610305e-01, 1.95014246e-02, 4.49703200e-01, 8.55036031e-01, 1.02359706e+00, 7.50431347e-01, 1.60021828e-01, -4.00303603e-01, -7.17218103e-01, -7.75792593e-01, -6.84291237e-01, -6.01632162e-01, -5.52912610e-01, -4.00683010e-01, -1.00907707e-01, 2.41027413e-01, 4.69294047e-01, 4.81226014e-01, 3.08035977e-01, 8.62618956e-02, -5.54640249e-02, -9.56636648e-02, -1.20555755e-01, -1.75954281e-01, -1.66939445e-01, -4.12967147e-02, 8.90967602e-02, 1.00833950e-01, -3.61737694e-03, -1.10469221e-01, -9.04997811e-02, 2.65125486e-02, 1.14151712e-01, 1.41174568e-01, 2.01694414e-01, 3.36195069e-01, 3.20389527e-01, -8.61018049e-03, -4.34443318e-01, -7.00125461e-01, -6.84902504e-01, -4.74847753e-01, -2.21065196e-01, -2.39515727e-02, 1.33530083e-01, 3.53961353e-01, 6.70590898e-01, 9.77191472e-01, 1.07698071e+00, 7.88278736e-01, 1.75052634e-01, -4.66143231e-01, -8.58392593e-01, -9.27766751e-01, -8.03007189e-01, -5.57181076e-01, -1.88983002e-01, ... 4.61268253e-01, 3.12199041e-01, 1.28477432e-01, -7.43295803e-02, -2.89928925e-01, -4.65908349e-01, -4.97651674e-01, -3.28650361e-01, -1.00908756e-02, 3.11161239e-01, 5.00955123e-01, 4.89660750e-01, 3.54131436e-01, 2.11231980e-01, 7.49178544e-02, -1.06415719e-01, -3.32466133e-01, -5.28317798e-01, -6.44419560e-01, -6.57951161e-01, -5.74395473e-01, -3.52940473e-01, 7.66568858e-02, 5.76619897e-01, 7.48984046e-01, 4.14981459e-01, -1.10421749e-01, -4.41886342e-01, -4.00140376e-01, -1.92747398e-01, -5.85466661e-02, 2.77195787e-02, 3.55272985e-02, -9.39274485e-02, -2.38373143e-01, -2.42946806e-01, -1.55283953e-01, -6.48747142e-02, 6.79590661e-02, 2.63745797e-01, 4.47592896e-01, 5.66854504e-01, 5.87616738e-01, 4.26214291e-01, 4.51525457e-02, -4.07356108e-01, -6.86846392e-01, -6.25991835e-01, -3.38772608e-01, -3.30746126e-02, 1.79902443e-01, 3.04017163e-01, 3.53580534e-01, 2.57613339e-01, 4.55771536e-02, -1.02821560e-01, -1.31179339e-01, -8.71991211e-02, 1.03974485e-02, 1.28155497e-01, 1.78820547e-01, 3.43130250e-02, -1.90005158e-01, -1.96967872e-01, 1.39648840e-02, 1.88591717e-01, 1.64842823e-01, 2.40290083e-03, -1.57904145e-01, -2.03118156e-01, -5.69103235e-02, 1.97136867e-01, 2.57536161e-01, 9.23084435e-02, -1.05274964e-01, -1.64675616e-01, -1.08581720e-01, -4.11614064e-02, -1.85167752e-02, -2.68437126e-02])
- eimf2(time)float640.2235 0.216 ... -0.2824 -0.2611
array([ 2.23527604e-01, 2.16022503e-01, 1.82734104e-01, 1.30202830e-01, 6.85547642e-02, 8.27381155e-03, -4.27769713e-02, -8.14604905e-02, -1.09855208e-01, -1.34988605e-01, -1.65832235e-01, -2.09593413e-01, -2.68394349e-01, -3.40824300e-01, -4.20970000e-01, -4.94741438e-01, -5.43671185e-01, -5.50159916e-01, -5.06573439e-01, -4.15486392e-01, -2.82348333e-01, -1.13285477e-01, 8.24422011e-02, 2.90187766e-01, 4.88227345e-01, 6.51170242e-01, 7.54591455e-01, 7.74853520e-01, 6.99445985e-01, 5.49875273e-01, 3.56865539e-01, 1.50846149e-01, -3.91631831e-02, -1.85148369e-01, -2.65790278e-01, -2.81200758e-01, -2.50797201e-01, -1.98519988e-01, -1.44998988e-01, -1.09114211e-01, -1.05880104e-01, -1.34282595e-01, -1.82895769e-01, -2.38397409e-01, -2.86413987e-01, -3.11862689e-01, -3.04972572e-01, -2.65069093e-01, -1.97728985e-01, -1.13095767e-01, -2.21560015e-02, 6.57975433e-02, 1.44801727e-01, 2.10063256e-01, 2.55517833e-01, 2.73749497e-01, 2.58000015e-01, 2.02639927e-01, 1.10646030e-01, -4.34126210e-03, -1.25890861e-01, -2.35036107e-01, -3.12181006e-01, -3.39633585e-01, -3.12805725e-01, -2.39504512e-01, -1.30369667e-01, -6.53234659e-04, 1.31144447e-01, 2.49523755e-01, 3.42427869e-01, 3.98806616e-01, 4.13289295e-01, 3.94378618e-01, 3.54341203e-01, 3.05297265e-01, 2.56803784e-01, 2.14047060e-01, 1.78123051e-01, 1.42146269e-01, ... 3.95085904e-01, 3.92547169e-01, 3.46213738e-01, 2.69667163e-01, 1.78109623e-01, 8.37161229e-02, -3.98675080e-03, -7.84976160e-02, -1.39843096e-01, -1.90809644e-01, -2.34445675e-01, -2.73723378e-01, -3.11291046e-01, -3.47170727e-01, -3.74417749e-01, -3.79076845e-01, -3.47525465e-01, -2.68215331e-01, -1.34986857e-01, 4.71890760e-02, 2.58229384e-01, 4.67050842e-01, 6.36408123e-01, 7.29670123e-01, 7.11234257e-01, 5.68976040e-01, 3.39238095e-01, 6.68085782e-02, -2.05214343e-01, -4.39179906e-01, -6.04496898e-01, -6.90271520e-01, -6.96333633e-01, -6.29731926e-01, -5.00921318e-01, -3.22188045e-01, -1.11500409e-01, 1.09496881e-01, 3.15575777e-01, 4.82366964e-01, 5.87438539e-01, 6.10433509e-01, 5.42178418e-01, 3.97613220e-01, 2.05351216e-01, -2.66699503e-03, -1.94123092e-01, -3.38431436e-01, -4.21491274e-01, -4.46149450e-01, -4.19896284e-01, -3.56081422e-01, -2.67006507e-01, -1.61679794e-01, -4.95424002e-02, 6.09377156e-02, 1.60760932e-01, 2.41980355e-01, 2.96654392e-01, 3.15599897e-01, 2.93785192e-01, 2.35546610e-01, 1.58341608e-01, 8.27408759e-02, 2.26403826e-02, -1.78214977e-02, -3.60549331e-02, -3.44190116e-02, -2.10160663e-02, -5.41410547e-03, 1.63309955e-03, -8.18610669e-03, -4.02832918e-02, -9.22157363e-02, -1.53052704e-01, -2.11259427e-01, -2.56626984e-01, -2.81325909e-01, -2.82426532e-01, -2.61132455e-01])
- eimf3(time)float64-0.2995 -0.3097 ... -0.1017
array([-2.99503201e-01, -3.09652165e-01, -3.22819242e-01, -3.37986485e-01, -3.53921072e-01, -3.69376609e-01, -3.83068732e-01, -3.93654971e-01, -3.99684798e-01, -3.99651717e-01, -3.92190022e-01, -3.76264359e-01, -3.51479697e-01, -3.17870075e-01, -2.75929650e-01, -2.26579736e-01, -1.70936044e-01, -1.10382295e-01, -4.65796291e-02, 1.86799976e-02, 8.36194196e-02, 1.46478548e-01, 2.05492569e-01, 2.58884269e-01, 3.04875345e-01, 3.41686915e-01, 3.67538721e-01, 3.80828513e-01, 3.81315539e-01, 3.70209976e-01, 3.49221097e-01, 3.20160800e-01, 2.84852076e-01, 2.45114578e-01, 2.02739780e-01, 1.59271237e-01, 1.15995321e-01, 7.41838013e-02, 3.51317045e-02, 1.46362932e-04, -2.94740646e-02, -5.28191053e-02, -6.98122053e-02, -8.08335896e-02, -8.65107068e-02, -8.75741219e-02, -8.48018778e-02, -7.89277333e-02, -7.05774115e-02, -6.02852903e-02, -4.84627009e-02, -3.54638929e-02, -2.16440919e-02, -7.34080866e-03, 7.15451750e-03, 2.15636339e-02, 3.55668813e-02, 4.87898664e-02, 6.09208210e-02, 7.17083508e-02, 8.09035203e-02, 8.82666386e-02, 9.35816168e-02, 9.66779447e-02, 9.74385165e-02, 9.57662689e-02, 9.15759231e-02, 8.48014716e-02, 7.53868863e-02, 6.32984790e-02, 4.85331955e-02, 3.11279076e-02, 1.12741734e-02, -1.06554290e-02, -3.42541889e-02, -5.91187108e-02, -8.48439149e-02, -1.11013080e-01, -1.37149339e-01, -1.62724889e-01, ... -1.76737363e-01, -1.43370860e-01, -1.08820124e-01, -7.37505367e-02, -3.89387559e-02, -5.18542495e-03, 2.69466336e-02, 5.73934548e-02, 8.64661586e-02, 1.14572361e-01, 1.42110602e-01, 1.69397157e-01, 1.96457463e-01, 2.22958343e-01, 2.48428948e-01, 2.72364208e-01, 2.94262645e-01, 3.13613890e-01, 3.29766840e-01, 3.41850464e-01, 3.48900708e-01, 3.49939518e-01, 3.43980638e-01, 3.30076242e-01, 3.07572202e-01, 2.76802439e-01, 2.39089219e-01, 1.95883413e-01, 1.48639033e-01, 9.88217411e-02, 4.79397472e-02, -2.47711883e-03, -5.10708111e-02, -9.68074601e-02, -1.39064945e-01, -1.77466722e-01, -2.11692347e-01, -2.41390793e-01, -2.66180882e-01, -2.85677002e-01, -2.99488184e-01, -3.07215693e-01, -3.08685861e-01, -3.04063625e-01, -2.93613484e-01, -2.77610852e-01, -2.56335267e-01, -2.30065493e-01, -1.99118310e-01, -1.64008601e-01, -1.25470693e-01, -8.43327199e-02, -4.14430409e-02, 2.28807573e-03, 4.57926869e-02, 8.78289554e-02, 1.27148313e-01, 1.62650790e-01, 1.93375914e-01, 2.18415250e-01, 2.37000541e-01, 2.48869869e-01, 2.54138708e-01, 2.52981787e-01, 2.45581204e-01, 2.32168845e-01, 2.13161310e-01, 1.89217814e-01, 1.61213190e-01, 1.30196898e-01, 9.73315950e-02, 6.38207263e-02, 3.07945600e-02, -6.81781936e-04, -2.95514152e-02, -5.47599529e-02, -7.53039254e-02, -9.03064776e-02, -9.91407275e-02, -1.01662896e-01])
- eimf4(time)float64-0.1685 -0.1689 ... -0.03042
array([-0.16854414, -0.16890148, -0.16839457, -0.16700844, -0.16473617, -0.16157283, -0.15751793, -0.15258423, -0.1467893 , -0.14015051, -0.13268581, -0.1244196 , -0.11539615, -0.10569276, -0.09539536, -0.08459016, -0.07336364, -0.06180249, -0.04999282, -0.03802065, -0.02597197, -0.01393274, -0.00198873, 0.00977448, 0.02127144, 0.0324173 , 0.04312901, 0.05332736, 0.06294241, 0.07192607, 0.08024251, 0.08786315, 0.09476365, 0.10092419, 0.1063303 , 0.11097062, 0.1148349 , 0.11791341, 0.12019665, 0.12167498, 0.12233873, 0.1221789 , 0.12119317, 0.11939371, 0.11680607, 0.11346244, 0.10940139, 0.10466547, 0.09929924, 0.09334863, 0.08686009, 0.07988265, 0.07246992, 0.06468102, 0.05657992, 0.048234 , 0.03971071, 0.03107721, 0.02239613, 0.01372344, 0.00511227, -0.00338655, -0.01172533, -0.01985817, -0.02774325, -0.03534564, -0.04263627, -0.04958976, -0.05618214, -0.06239143, -0.06819739, -0.07358143, -0.07852595, -0.08301513, -0.08703413, -0.09056856, -0.09360401, -0.09612578, -0.09811963, -0.09957156, -0.10046789, -0.10079544, -0.10054247, -0.09969921, -0.0982572 , -0.09620967, -0.09355374, -0.09029138, -0.08643306, -0.08200201, -0.07703105, -0.07155894, -0.06562654, -0.05927482, -0.05254455, -0.04547675, -0.03811267, -0.03049337, -0.0226597 , -0.01465244, ... -0.13449199, -0.14044416, -0.14580087, -0.15050607, -0.15450601, -0.15774775, -0.16017821, -0.16174418, -0.1623934 , -0.16207656, -0.16075084, -0.15838636, -0.15497686, -0.15054673, -0.14513652, -0.13879639, -0.13158394, -0.12355996, -0.11478677, -0.10532722, -0.09524449, -0.08460236, -0.07346719, -0.06190967, -0.05000456, -0.03783393, -0.02549426, -0.01309078, -0.00073216, 0.01147463, 0.02343047, 0.03504893, 0.046259 , 0.05699626, 0.06720383, 0.07682864, 0.08582101, 0.09413469, 0.10172687, 0.10855969, 0.1146034 , 0.11983673, 0.1242467 , 0.12783279, 0.13060621, 0.13259364, 0.13383058, 0.13435606, 0.13421015, 0.13343323, 0.13206544, 0.13014659, 0.12771613, 0.12481194, 0.12146718, 0.11771397, 0.1135844 , 0.10911046, 0.104324 , 0.09925734, 0.0939446 , 0.08842186, 0.08272406, 0.07688512, 0.07093751, 0.06491331, 0.05884157, 0.05274776, 0.04665668, 0.04059357, 0.03458434, 0.02865511, 0.02283191, 0.01714069, 0.01160731, 0.00625771, 0.00111787, -0.00378625, -0.00842868, -0.01278387, -0.01682754, -0.02053743, -0.02389744, -0.02689914, -0.02953999, -0.03181995, -0.03374031, -0.03530344, -0.036513 , -0.03737299, -0.03788777, -0.03806273, -0.03790445, -0.03742051, -0.03661929, -0.03550978, -0.03410094, -0.03240212, -0.03042189])
- eimf5(time)float64-0.02875 -0.0281 ... 0.1915 0.19
array([-2.87538910e-02, -2.81025146e-02, -2.74148316e-02, -2.66912765e-02, -2.59324581e-02, -2.51391116e-02, -2.43120780e-02, -2.34521890e-02, -2.25601767e-02, -2.16366430e-02, -2.06822278e-02, -1.96979115e-02, -1.86848154e-02, -1.76440525e-02, -1.65767419e-02, -1.54840237e-02, -1.43670446e-02, -1.32269711e-02, -1.20649817e-02, -1.08822582e-02, -9.67998307e-03, -8.45933876e-03, -7.22150856e-03, -5.96767601e-03, -4.69902475e-03, -3.41673850e-03, -2.12200216e-03, -8.16004162e-04, 5.00067872e-04, 1.82502624e-03, 3.15768882e-03, 4.49687470e-03, 5.84141004e-03, 7.19014002e-03, 8.54191596e-03, 9.89558755e-03, 1.12500022e-02, 1.26039921e-02, 1.39563718e-02, 1.53059536e-02, 1.66515319e-02, 1.79918520e-02, 1.93256369e-02, 2.06515895e-02, 2.19684097e-02, 2.32748411e-02, 2.45696349e-02, 2.58515336e-02, 2.71192817e-02, 2.83716369e-02, 2.96073932e-02, 3.08253673e-02, 3.20244013e-02, 3.32033651e-02, 3.43611629e-02, 3.54967224e-02, 3.66089920e-02, 3.76969345e-02, 3.87595290e-02, 3.97957476e-02, 4.08045279e-02, 4.17847775e-02, 4.27353941e-02, 4.36552981e-02, 4.45434888e-02, 4.53990179e-02, 4.62209666e-02, 4.70084542e-02, 4.77606439e-02, 4.84767346e-02, 4.91559133e-02, 4.97973660e-02, 5.04003149e-02, 5.09640063e-02, 5.14876892e-02, 5.19706229e-02, 5.24121133e-02, 5.28114902e-02, 5.31681079e-02, 5.34813959e-02, ... 2.08922198e-02, 2.46873140e-02, 2.85251470e-02, 3.24017015e-02, 3.63129559e-02, 4.02548886e-02, 4.42234780e-02, 4.82147025e-02, 5.22245406e-02, 5.62489707e-02, 6.02839712e-02, 6.43255205e-02, 6.83695970e-02, 7.24121792e-02, 7.64492455e-02, 8.04767742e-02, 8.44907439e-02, 8.84871330e-02, 9.24619215e-02, 9.64111243e-02, 1.00330986e-01, 1.04217974e-01, 1.08068660e-01, 1.11879741e-01, 1.15647951e-01, 1.19370067e-01, 1.23042942e-01, 1.26663488e-01, 1.30228696e-01, 1.33735616e-01, 1.37181338e-01, 1.40563054e-01, 1.43878012e-01, 1.47123496e-01, 1.50296809e-01, 1.53395261e-01, 1.56416174e-01, 1.59356889e-01, 1.62214770e-01, 1.64987271e-01, 1.67671954e-01, 1.70266480e-01, 1.72768637e-01, 1.75176377e-01, 1.77487756e-01, 1.79700895e-01, 1.81814070e-01, 1.83825764e-01, 1.85734597e-01, 1.87539259e-01, 1.89238507e-01, 1.90831177e-01, 1.92316224e-01, 1.93692733e-01, 1.94959825e-01, 1.96116635e-01, 1.97162335e-01, 1.98096226e-01, 1.98917686e-01, 1.99626128e-01, 2.00220990e-01, 2.00701714e-01, 2.01067754e-01, 2.01318735e-01, 2.01454506e-01, 2.01475020e-01, 2.01380273e-01, 2.01170360e-01, 2.00845596e-01, 2.00406487e-01, 1.99853775e-01, 1.99188405e-01, 1.98411571e-01, 1.97524716e-01, 1.96529420e-01, 1.95427388e-01, 1.94220432e-01, 1.92910412e-01, 1.91499261e-01, 1.89989037e-01])
- eimf6(time)float64-0.0004404 -0.0003338 ... 0.1291
array([-4.40356215e-04, -3.33842053e-04, -2.29170810e-04, -1.26382399e-04, -2.55158611e-05, 7.33920690e-05, 1.70300987e-04, 2.65168425e-04, 3.57952572e-04, 4.48611119e-04, 5.37104016e-04, 6.23393297e-04, 7.07441275e-04, 7.89211439e-04, 8.68668990e-04, 9.45779313e-04, 1.02050777e-03, 1.09281958e-03, 1.16268003e-03, 1.23005573e-03, 1.29491374e-03, 1.35722016e-03, 1.41694163e-03, 1.47404509e-03, 1.52849759e-03, 1.58026659e-03, 1.62932052e-03, 1.67562774e-03, 1.71915670e-03, 1.75987587e-03, 1.79775351e-03, 1.83275761e-03, 1.86485547e-03, 1.89401409e-03, 1.92020039e-03, 1.94338126e-03, 1.96352392e-03, 1.98059854e-03, 1.99457780e-03, 2.00543495e-03, 2.01314328e-03, 2.01767628e-03, 2.01900770e-03, 2.01711229e-03, 2.01196603e-03, 2.00354590e-03, 1.99183066e-03, 1.97680021e-03, 1.95843433e-03, 1.93671296e-03, 1.91161621e-03, 1.88312435e-03, 1.85121720e-03, 1.81587444e-03, 1.77707619e-03, 1.73480303e-03, 1.68903566e-03, 1.63975481e-03, 1.58694171e-03, 1.53057836e-03, 1.47064788e-03, 1.40713535e-03, 1.34002851e-03, 1.26931639e-03, 1.19498837e-03, 1.11703420e-03, 1.03544332e-03, 9.50205448e-04, 8.61310339e-04, 7.68747762e-04, 6.72507485e-04, 5.72579275e-04, 4.68953155e-04, 3.61620119e-04, 2.50571391e-04, 1.35797888e-04, 1.72917202e-05, -1.04954369e-04, -2.30947453e-04, -3.60693185e-04, ... 7.84114583e-02, 7.95164239e-02, 8.06152521e-02, 8.17077123e-02, 8.27935741e-02, 8.38726068e-02, 8.49445800e-02, 8.60092631e-02, 8.70664257e-02, 8.81158372e-02, 8.91572670e-02, 9.01904848e-02, 9.12152598e-02, 9.22313617e-02, 9.32385599e-02, 9.42366238e-02, 9.52253230e-02, 9.62044269e-02, 9.71737050e-02, 9.81329268e-02, 9.90818618e-02, 1.00020279e-01, 1.00947949e-01, 1.01864639e-01, 1.02770119e-01, 1.03664158e-01, 1.04546525e-01, 1.05416989e-01, 1.06275319e-01, 1.07121283e-01, 1.07954652e-01, 1.08775193e-01, 1.09582677e-01, 1.10376871e-01, 1.11157545e-01, 1.11924469e-01, 1.12677410e-01, 1.13416138e-01, 1.14140422e-01, 1.14850032e-01, 1.15544737e-01, 1.16224309e-01, 1.16888519e-01, 1.17537148e-01, 1.18169990e-01, 1.18786856e-01, 1.19387559e-01, 1.19971915e-01, 1.20539739e-01, 1.21090850e-01, 1.21625065e-01, 1.22142205e-01, 1.22642091e-01, 1.23124543e-01, 1.23589382e-01, 1.24036429e-01, 1.24465504e-01, 1.24876431e-01, 1.25269032e-01, 1.25643133e-01, 1.25998564e-01, 1.26335157e-01, 1.26652752e-01, 1.26951194e-01, 1.27230332e-01, 1.27490027e-01, 1.27730162e-01, 1.27950645e-01, 1.28151394e-01, 1.28332350e-01, 1.28493471e-01, 1.28634748e-01, 1.28756203e-01, 1.28857882e-01, 1.28939866e-01, 1.29002249e-01, 1.29045139e-01, 1.29068664e-01, 1.29072961e-01, 1.29058171e-01])
- eimf7(time)float64-0.06159 -0.06158 ... 0.06652
array([-6.15902302e-02, -6.15795605e-02, -6.15672290e-02, -6.15532430e-02, -6.15376114e-02, -6.15203433e-02, -6.15014492e-02, -6.14809403e-02, -6.14588284e-02, -6.14351253e-02, -6.14098427e-02, -6.13829926e-02, -6.13545867e-02, -6.13246371e-02, -6.12931549e-02, -6.12601509e-02, -6.12256359e-02, -6.11896203e-02, -6.11521146e-02, -6.11131289e-02, -6.10726728e-02, -6.10307524e-02, -6.09873731e-02, -6.09425400e-02, -6.08962586e-02, -6.08485350e-02, -6.07993782e-02, -6.07487971e-02, -6.06967998e-02, -6.06433938e-02, -6.05885861e-02, -6.05323831e-02, -6.04747888e-02, -6.04158065e-02, -6.03554391e-02, -6.02936894e-02, -6.02305596e-02, -6.01660517e-02, -6.01001679e-02, -6.00329100e-02, -5.99642800e-02, -5.98942799e-02, -5.98229117e-02, -5.97501772e-02, -5.96760785e-02, -5.96006176e-02, -5.95237964e-02, -5.94456162e-02, -5.93660773e-02, -5.92851800e-02, -5.92029244e-02, -5.91193108e-02, -5.90343394e-02, -5.89480103e-02, -5.88603239e-02, -5.87712802e-02, -5.86808794e-02, -5.85891219e-02, -5.84960078e-02, -5.84015373e-02, -5.83057106e-02, -5.82085280e-02, -5.81099895e-02, -5.80100955e-02, -5.79088462e-02, -5.78062416e-02, -5.77022822e-02, -5.75969680e-02, -5.74902993e-02, -5.73822762e-02, -5.72728990e-02, -5.71621679e-02, -5.70500831e-02, -5.69366448e-02, -5.68218532e-02, -5.67057085e-02, -5.65882110e-02, -5.64693607e-02, -5.63491580e-02, -5.62276030e-02, ... 4.55043488e-02, 4.58278784e-02, 4.61502763e-02, 4.64715287e-02, 4.67916215e-02, 4.71105406e-02, 4.74282721e-02, 4.77448018e-02, 4.80601157e-02, 4.83741999e-02, 4.86870402e-02, 4.89986227e-02, 4.93089333e-02, 4.96179579e-02, 4.99256826e-02, 5.02320932e-02, 5.05371759e-02, 5.08409164e-02, 5.11433009e-02, 5.14443152e-02, 5.17439453e-02, 5.20421772e-02, 5.23389969e-02, 5.26343903e-02, 5.29283434e-02, 5.32208421e-02, 5.35118725e-02, 5.38014204e-02, 5.40894719e-02, 5.43760129e-02, 5.46610294e-02, 5.49445073e-02, 5.52264326e-02, 5.55067913e-02, 5.57855693e-02, 5.60627526e-02, 5.63383272e-02, 5.66122790e-02, 5.68845940e-02, 5.71552582e-02, 5.74242575e-02, 5.76915779e-02, 5.79572054e-02, 5.82211261e-02, 5.84833290e-02, 5.87438062e-02, 5.90025503e-02, 5.92595542e-02, 5.95148106e-02, 5.97683123e-02, 6.00200520e-02, 6.02700225e-02, 6.05182165e-02, 6.07646268e-02, 6.10092461e-02, 6.12520673e-02, 6.14930830e-02, 6.17322860e-02, 6.19696690e-02, 6.22052249e-02, 6.24389464e-02, 6.26708262e-02, 6.29008571e-02, 6.31290319e-02, 6.33553432e-02, 6.35797839e-02, 6.38023468e-02, 6.40230244e-02, 6.42418098e-02, 6.44586955e-02, 6.46736743e-02, 6.48867390e-02, 6.50978824e-02, 6.53070972e-02, 6.55143761e-02, 6.57197119e-02, 6.59230975e-02, 6.61245254e-02, 6.63239885e-02, 6.65214796e-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 5.803 seconds)