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 0.05665 -0.06566 0.09766 ... -0.09501 0.01567 eimf1 (time) float64 4kB 0.1212 0.1162 0.08709 ... -0.01978 -0.02795 eimf2 (time) float64 4kB 0.2572 0.254 0.2256 ... -0.2786 -0.2826 -0.2652 eimf3 (time) float64 4kB -0.2941 -0.297 -0.3032 ... -0.1057 -0.1092 eimf4 (time) float64 4kB -0.2069 -0.2058 -0.2036 ... 0.00363 0.005315 eimf5 (time) float64 4kB -0.0555 -0.05467 -0.05378 ... 0.1951 0.1931 eimf6 (time) float64 4kB -0.008067 -0.007914 -0.007763 ... 0.1056 0.1055 eimf7 (time) float64 4kB -0.03737 -0.03736 -0.03735 ... 0.0453 0.04542
- 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)float640.05665 -0.06566 ... 0.01567
array([ 5.66516772e-02, -6.56640800e-02, 9.76576869e-02, -4.56015307e-02, 1.02307261e-02, -8.52345967e-02, 1.74544848e-01, -2.47640864e-02, -2.07525910e-01, -6.67944606e-02, 2.92144850e-01, -1.11658843e-02, -2.46837692e-01, 1.56668111e-01, 1.20927732e-03, -1.57476661e-01, 1.11081420e-01, -2.88098453e-02, 1.44210774e-01, 6.31379989e-02, -3.45010281e-01, -1.54764884e-01, 4.34890162e-01, 2.57965834e-01, -4.03096503e-01, -4.17024791e-01, 2.05839438e-01, 2.67713404e-01, 4.43760164e-01, -3.17851447e-01, -2.53830080e-01, 1.27103240e-01, 8.25742448e-02, -1.85287338e-01, 2.44905534e-01, -1.44663756e-01, -3.74693735e-02, 2.52461665e-03, 1.96475865e-02, 6.47558557e-02, 5.47292179e-02, 1.24282191e-02, -4.31976663e-02, -8.43930425e-02, -7.93740278e-02, 2.02035683e-01, -8.91946213e-02, -6.55686143e-02, -8.81798780e-02, 1.31697505e-01, -2.91943246e-02, 9.52894615e-02, -1.67236690e-01, 6.67323657e-02, -4.60398475e-02, 3.64435445e-01, -2.55435336e-01, -4.49965663e-01, -2.80173602e-01, 5.95602327e-01, -3.52766843e-01, -2.94584173e-01, 1.49729851e-01, -7.66013243e-02, -6.23325230e-02, 6.60404268e-02, 3.43742533e-02, -5.69270152e-02, -1.83895972e-01, -1.55808017e-01, -8.83758910e-02, 4.13931977e-01, 4.50358756e-01, -1.38157252e-01, -5.34894885e-01, -2.05416273e-01, -6.82679300e-02, 3.62369408e-01, 2.39597995e-01, -3.68282744e-01, ... 2.45045694e-03, -5.96143480e-01, -4.52489886e-01, 3.04047581e-01, 3.90188363e-01, -2.43647866e-01, -1.66580530e-01, 1.18331998e-01, -5.60890509e-02, -8.49869838e-03, 9.09845867e-02, 1.79765301e-02, -9.15288338e-02, 9.82733927e-02, -1.08438100e-01, 1.69925895e-01, -1.60499958e-01, 1.49496826e-01, -1.84986087e-01, 3.04809226e-02, 1.80193336e-01, 7.78027522e-02, -4.48934648e-01, 4.38203141e-01, 3.33727987e-01, 3.22109120e-01, -1.43590258e-01, -4.64911317e-01, 3.63115458e-01, -1.00555047e-03, 2.14484492e-02, -2.69902573e-01, 2.15317002e-01, 9.22583089e-02, -2.70613596e-01, 1.15688032e-01, 6.75402782e-02, 6.19714329e-02, -1.85988324e-01, 1.71702008e-01, -1.05344611e-02, -1.52908323e-01, -1.23700809e-01, 3.90387605e-01, 2.47554158e-01, -2.83017551e-01, -4.43683492e-01, -1.55519929e-01, 2.25221463e-01, 3.43921571e-01, -2.43552331e-01, -1.30337169e-01, 9.31118506e-02, 1.49587475e-01, -1.46212067e-01, 2.79445284e-04, -1.59702360e-02, 1.62829452e-01, -3.15419704e-01, -5.37655377e-02, 2.11341662e-01, 5.13451428e-01, -5.43539023e-01, -1.07928776e-01, 7.87437018e-02, 4.33300302e-01, -2.98274872e-01, 9.24863850e-03, 2.60104871e-01, 1.26646748e-01, -5.22037534e-01, 4.83316157e-01, 2.56233872e-01, -1.83110859e-01, -3.79222775e-01, -9.55333408e-02, 2.37998809e-01, 4.00925669e-02, -9.50105422e-02, 1.56713378e-02])
- eimf1(time)float640.1212 0.1162 ... -0.01978 -0.02795
array([ 1.21227044e-01, 1.16218410e-01, 8.70911370e-02, 3.58394829e-02, -8.58867675e-03, -2.50122729e-02, -4.51854590e-02, -7.82486897e-02, -6.05815369e-02, 3.86684079e-02, 1.11344681e-01, 5.38584701e-02, -5.48882180e-02, -9.42765943e-02, -2.15410425e-02, 1.29236622e-01, 2.58941017e-01, 2.29012400e-01, -2.01514385e-02, -3.77840884e-01, -6.57468483e-01, -7.25872912e-01, -6.14049084e-01, -3.94858664e-01, -8.03440122e-02, 3.39954746e-01, 7.74539553e-01, 9.87093239e-01, 7.37719022e-01, 1.59095764e-01, -3.79342896e-01, -6.63274413e-01, -6.88084522e-01, -5.80308699e-01, -5.02041097e-01, -4.73191532e-01, -3.54395673e-01, -8.56660896e-02, 2.32053210e-01, 4.44978220e-01, 4.47131337e-01, 2.60309132e-01, 2.45177560e-02, -1.17719467e-01, -1.48688867e-01, -1.54438740e-01, -1.76529091e-01, -1.51595666e-01, -3.86447085e-02, 8.60108712e-02, 1.02179021e-01, -2.52676593e-03, -9.36157943e-02, -4.89828598e-02, 7.02230414e-02, 1.42161378e-01, 1.88128344e-01, 2.77709187e-01, 3.96965221e-01, 3.32625021e-01, -3.91110567e-02, -4.78483823e-01, -7.35116275e-01, -7.12648035e-01, -5.15562626e-01, -2.84569539e-01, -9.35976182e-02, 8.50325227e-02, 3.35727300e-01, 6.72528435e-01, 9.89591779e-01, 1.08951326e+00, 7.94465827e-01, 1.81274821e-01, -4.52754424e-01, -8.39574762e-01, -9.15917586e-01, -8.15583094e-01, -6.02861782e-01, -2.59120812e-01, ... 4.65014505e-01, 3.30295735e-01, 1.43191949e-01, -6.70036709e-02, -2.77792647e-01, -4.42768525e-01, -4.74354494e-01, -3.29140495e-01, -6.11315597e-02, 2.18270237e-01, 3.98662942e-01, 4.11206269e-01, 3.00025999e-01, 1.65234114e-01, 5.46040106e-02, -7.52764741e-02, -2.53161316e-01, -4.11799625e-01, -4.97644789e-01, -4.95620618e-01, -4.18534211e-01, -2.37281884e-01, 1.29500100e-01, 5.68111329e-01, 6.95167693e-01, 3.45812511e-01, -1.75577523e-01, -4.98065879e-01, -4.40215926e-01, -2.00970601e-01, -4.95788897e-02, 9.53364964e-03, -8.92516468e-03, -1.38671798e-01, -2.67379106e-01, -2.51091132e-01, -1.53010084e-01, -6.83667593e-02, 5.08688215e-02, 2.39347221e-01, 4.05797001e-01, 5.06435917e-01, 5.42359825e-01, 4.18615495e-01, 7.25323159e-02, -3.64405391e-01, -6.44160527e-01, -6.07722693e-01, -3.54282826e-01, -5.88717913e-02, 1.71309107e-01, 3.05170224e-01, 3.40747162e-01, 2.30491973e-01, 1.30913142e-02, -1.39759153e-01, -1.56799908e-01, -8.74137415e-02, 2.87878891e-02, 1.48018472e-01, 1.88966320e-01, 3.83051081e-02, -1.95488215e-01, -2.02810185e-01, 1.52641501e-02, 1.86696090e-01, 1.51501583e-01, -9.98674261e-03, -1.55629227e-01, -1.80253222e-01, -3.98503821e-02, 1.77877138e-01, 2.35886704e-01, 1.01318263e-01, -7.97025383e-02, -1.64729905e-01, -1.34943773e-01, -6.23588263e-02, -1.97841398e-02, -2.79454417e-02])
- eimf2(time)float640.2572 0.254 ... -0.2826 -0.2652
array([ 2.57200944e-01, 2.54001237e-01, 2.25593873e-01, 1.75969352e-01, 1.12676730e-01, 4.45231031e-02, -1.96057731e-02, -7.33835881e-02, -1.15464498e-01, -1.49744635e-01, -1.85097756e-01, -2.30470825e-01, -2.90335168e-01, -3.64698754e-01, -4.47418301e-01, -5.23152065e-01, -5.72022380e-01, -5.75845391e-01, -5.26572841e-01, -4.26972245e-01, -2.82818198e-01, -1.00887624e-01, 1.07435467e-01, 3.24561787e-01, 5.27624703e-01, 6.91281035e-01, 7.90738903e-01, 8.01689896e-01, 7.10878850e-01, 5.41117254e-01, 3.26056893e-01, 9.92864427e-02, -1.06653624e-01, -2.61340214e-01, -3.41214064e-01, -3.41911881e-01, -2.86148798e-01, -2.02842069e-01, -1.17641798e-01, -5.46092403e-02, -3.37734508e-02, -5.53005890e-02, -1.04083875e-01, -1.62889600e-01, -2.15374612e-01, -2.46293471e-01, -2.46304177e-01, -2.14844969e-01, -1.58369112e-01, -8.67221445e-02, -9.52656778e-03, 6.50778202e-02, 1.30603514e-01, 1.81690231e-01, 2.13713680e-01, 2.21086596e-01, 1.98329724e-01, 1.43175046e-01, 6.06928498e-02, -3.77463164e-02, -1.38710142e-01, -2.26586330e-01, -2.85538036e-01, -3.02245728e-01, -2.74634104e-01, -2.10057347e-01, -1.17902507e-01, -9.77570790e-03, 1.00700251e-01, 2.01261731e-01, 2.81543675e-01, 3.31892803e-01, 3.48280176e-01, 3.38133007e-01, 3.11341531e-01, 2.77697530e-01, 2.45597726e-01, 2.20034452e-01, 2.00459070e-01, 1.79272970e-01, ... 3.51491057e-01, 3.53392248e-01, 3.13091985e-01, 2.43209873e-01, 1.59850989e-01, 7.68425863e-02, 5.24294357e-03, -4.76383208e-02, -8.44268151e-02, -1.12136804e-01, -1.38012291e-01, -1.69032314e-01, -2.09815446e-01, -2.60522340e-01, -3.14103103e-01, -3.54065727e-01, -3.57611665e-01, -3.04694081e-01, -1.85399931e-01, -3.60210271e-03, 2.18401973e-01, 4.44460495e-01, 6.33649786e-01, 7.46241993e-01, 7.44236822e-01, 6.15338853e-01, 3.94828482e-01, 1.25398594e-01, -1.51611941e-01, -3.99828221e-01, -5.86866140e-01, -6.93646988e-01, -7.13011174e-01, -6.49974523e-01, -5.18318136e-01, -3.35260002e-01, -1.20633635e-01, 1.05339179e-01, 3.18201550e-01, 4.90755250e-01, 5.96888188e-01, 6.13335612e-01, 5.30277300e-01, 3.66330546e-01, 1.54119299e-01, -7.07844553e-02, -2.72454954e-01, -4.17504801e-01, -4.93001931e-01, -5.06486123e-01, -4.68957066e-01, -3.91865323e-01, -2.85896404e-01, -1.59327403e-01, -2.39269155e-02, 1.04173259e-01, 2.10904852e-01, 2.89633056e-01, 3.36522486e-01, 3.46163027e-01, 3.15712616e-01, 2.51431340e-01, 1.70768093e-01, 9.38436563e-02, 3.45279082e-02, -3.25905107e-03, -1.86656451e-02, -1.55638814e-02, -3.06190973e-03, 8.44654637e-03, 8.87057794e-03, -7.43000056e-03, -4.27779191e-02, -9.41141869e-02, -1.52583746e-01, -2.08320680e-01, -2.52556361e-01, -2.78570765e-01, -2.82567293e-01, -2.65155451e-01])
- eimf3(time)float64-0.2941 -0.297 ... -0.1057 -0.1092
array([-2.94054088e-01, -2.96987818e-01, -3.03244669e-01, -3.12263370e-01, -3.22988798e-01, -3.34124150e-01, -3.44275543e-01, -3.51983219e-01, -3.55695849e-01, -3.53854255e-01, -3.44989169e-01, -3.27867341e-01, -3.01753008e-01, -2.66735437e-01, -2.23566355e-01, -1.73411486e-01, -1.17692626e-01, -5.79295769e-02, 4.25808165e-03, 6.71788543e-02, 1.29134749e-01, 1.88411286e-01, 2.43240981e-01, 2.91843705e-01, 3.32438789e-01, 3.63244194e-01, 3.82478926e-01, 3.88574492e-01, 3.81433413e-01, 3.62424648e-01, 3.33389575e-01, 2.96257972e-01, 2.52955355e-01, 2.05400714e-01, 1.55490293e-01, 1.04954810e-01, 5.52711163e-02, 7.87779437e-03, -3.57668057e-02, -7.42215270e-02, -1.06103413e-01, -1.30411008e-01, -1.47074748e-01, -1.56597255e-01, -1.59671521e-01, -1.57073414e-01, -1.49622707e-01, -1.38149984e-01, -1.23415831e-01, -1.06120374e-01, -8.68765416e-02, -6.62302451e-02, -4.46667560e-02, -2.26764843e-02, -7.65742115e-04, 2.05598919e-02, 4.08040635e-02, 5.95855294e-02, 7.66760922e-02, 9.18905442e-02, 1.05065823e-01, 1.16068762e-01, 1.24785933e-01, 1.31145848e-01, 1.35101300e-01, 1.36516788e-01, 1.35234012e-01, 1.31127611e-01, 1.24091143e-01, 1.14034289e-01, 1.00924030e-01, 8.48347113e-02, 6.59532248e-02, 4.45604905e-02, 2.09557749e-02, -4.57060392e-03, -3.16883710e-02, -5.99722190e-02, -8.88811302e-02, -1.17819765e-01, ... -1.58513596e-01, -1.30689874e-01, -1.02721733e-01, -7.51166273e-02, -4.84126076e-02, -2.31338216e-02, 3.63853949e-04, 2.21894193e-02, 4.29164602e-02, 6.32692617e-02, 8.39641566e-02, 1.05598435e-01, 1.28466040e-01, 1.52365322e-01, 1.76740403e-01, 2.00869455e-01, 2.24028041e-01, 2.45498527e-01, 2.64493763e-01, 2.80083176e-01, 2.91296038e-01, 2.97158781e-01, 2.96694048e-01, 2.88955389e-01, 2.73221232e-01, 2.49677395e-01, 2.19435129e-01, 1.83723139e-01, 1.43784759e-01, 1.00869486e-01, 5.62276038e-02, 1.11022923e-02, -3.33109258e-02, -7.60128565e-02, -1.16286553e-01, -1.53555077e-01, -1.87260030e-01, -2.16843082e-01, -2.41746030e-01, -2.61410810e-01, -2.75279749e-01, -2.82847125e-01, -2.83967253e-01, -2.78959930e-01, -2.68291608e-01, -2.52442218e-01, -2.31890991e-01, -2.07119484e-01, -1.78637957e-01, -1.47032992e-01, -1.12983215e-01, -7.72156874e-02, -4.04532834e-02, -3.45214550e-03, 3.29717730e-02, 6.79923284e-02, 1.00795048e-01, 1.30582079e-01, 1.56556132e-01, 1.77968353e-01, 1.94233348e-01, 2.05078393e-01, 2.10426963e-01, 2.10274676e-01, 2.04652380e-01, 1.93640008e-01, 1.77477565e-01, 1.56722986e-01, 1.32193887e-01, 1.04841947e-01, 7.57078335e-02, 4.58078658e-02, 1.61246467e-02, -1.23847422e-02, -3.87931836e-02, -6.21943386e-02, -8.16943235e-02, -9.64532486e-02, -1.05734480e-01, -1.09245207e-01])
- eimf4(time)float64-0.2069 -0.2058 ... 0.005315
array([-2.06893021e-01, -2.05818774e-01, -2.03554427e-01, -2.00116674e-01, -1.95537368e-01, -1.89849482e-01, -1.83086359e-01, -1.75281759e-01, -1.66470415e-01, -1.56690095e-01, -1.45980029e-01, -1.34384974e-01, -1.21971094e-01, -1.08831293e-01, -9.50695750e-02, -8.07918769e-02, -6.61041096e-02, -5.11117688e-02, -3.59200078e-02, -2.06339024e-02, -5.35852984e-03, 9.80103107e-03, 2.47397107e-02, 3.93526728e-02, 5.35355966e-02, 6.71855293e-02, 8.02022929e-02, 9.24917568e-02, 1.03979674e-01, 1.14618382e-01, 1.24370907e-01, 1.33206714e-01, 1.41102742e-01, 1.48042405e-01, 1.54015717e-01, 1.59015929e-01, 1.63039183e-01, 1.66083030e-01, 1.68145307e-01, 1.69224236e-01, 1.69318119e-01, 1.68426392e-01, 1.66551669e-01, 1.63710191e-01, 1.59935238e-01, 1.55267919e-01, 1.49754303e-01, 1.43443522e-01, 1.36386612e-01, 1.28636079e-01, 1.20248706e-01, 1.11286241e-01, 1.01813793e-01, 9.19023149e-02, 8.16239605e-02, 7.10501333e-02, 6.02532443e-02, 4.93047681e-02, 3.82707670e-02, 2.72113138e-02, 1.61806014e-02, 5.22861029e-03, -5.59935576e-03, -1.62611387e-02, -2.67119865e-02, -3.69039828e-02, -4.67911113e-02, -5.63291590e-02, -6.54781214e-02, -7.42017693e-02, -8.24684055e-02, -9.02505260e-02, -9.75224381e-02, -1.04259566e-01, -1.10438047e-01, -1.16033712e-01, -1.21022789e-01, -1.25381780e-01, -1.29086665e-01, -1.32113439e-01, ... -1.04646923e-01, -9.37442552e-02, -8.23553278e-02, -7.05576865e-02, -5.84310862e-02, -4.60581306e-02, -3.35262481e-02, -2.09296638e-02, -8.36972278e-03, 4.04973813e-03, 1.62261903e-02, 2.80628835e-02, 3.94730022e-02, 5.03845353e-02, 6.07429731e-02, 7.05053785e-02, 7.96351475e-02, 8.81002636e-02, 9.58719249e-02, 1.02925604e-01, 1.09240308e-01, 1.14801008e-01, 1.19600093e-01, 1.23636662e-01, 1.26916005e-01, 1.29452193e-01, 1.31268484e-01, 1.32395630e-01, 1.32868773e-01, 1.32725147e-01, 1.32002212e-01, 1.30737431e-01, 1.28968321e-01, 1.26732366e-01, 1.24066917e-01, 1.21009289e-01, 1.17596841e-01, 1.13866906e-01, 1.09856918e-01, 1.05604381e-01, 1.01146609e-01, 9.65209208e-02, 9.17628840e-02, 8.69061526e-02, 8.19806305e-02, 7.70129709e-02, 7.20273054e-02, 6.70457150e-02, 6.20904648e-02, 5.71826292e-02, 5.23427599e-02, 4.75912916e-02, 4.29485959e-02, 3.84349739e-02, 3.40705678e-02, 2.98752447e-02, 2.58683205e-02, 2.20686028e-02, 1.84943297e-02, 1.51630545e-02, 1.20920343e-02, 9.29707287e-03, 6.78947700e-03, 4.57448005e-03, 2.65336483e-03, 1.02539815e-03, -3.12120873e-04, -1.36366166e-03, -2.13511608e-03, -2.63353484e-03, -2.86681316e-03, -2.84346198e-03, -2.57287576e-03, -2.06510704e-03, -1.33079823e-03, -3.81593214e-04, 7.70173722e-04, 2.11154084e-03, 3.62975721e-03, 5.31510081e-03])
- eimf5(time)float64-0.0555 -0.05467 ... 0.1951 0.1931
array([-5.54979247e-02, -5.46651703e-02, -5.37846267e-02, -5.28569815e-02, -5.18831894e-02, -5.08644538e-02, -4.98020666e-02, -4.86973280e-02, -4.75514873e-02, -4.63656184e-02, -4.51406456e-02, -4.38776414e-02, -4.25779473e-02, -4.12429473e-02, -3.98740038e-02, -3.84724802e-02, -3.70397638e-02, -3.55772540e-02, -3.40863498e-02, -3.25684481e-02, -3.10249456e-02, -2.94572380e-02, -2.78667176e-02, -2.62547774e-02, -2.46228089e-02, -2.29722001e-02, -2.13043414e-02, -1.96206587e-02, -1.79226310e-02, -1.62117441e-02, -1.44895246e-02, -1.27575657e-02, -1.10174660e-02, -9.27082403e-03, -7.51923139e-03, -5.76426241e-03, -4.00749479e-03, -2.25048840e-03, -4.94807590e-04, 1.25793281e-03, 3.00616342e-03, 4.74840622e-03, 6.48320089e-03, 8.20915177e-03, 9.92490807e-03, 1.16291324e-02, 1.33204764e-02, 1.49975480e-02, 1.66589241e-02, 1.83031738e-02, 1.99288772e-02, 2.15346309e-02, 2.31190467e-02, 2.46807606e-02, 2.62184303e-02, 2.77307308e-02, 2.92163523e-02, 3.06739918e-02, 3.21023488e-02, 3.35001257e-02, 3.48660578e-02, 3.61989081e-02, 3.74974517e-02, 3.87604781e-02, 3.99868319e-02, 4.11754731e-02, 4.23254084e-02, 4.34356485e-02, 4.45052057e-02, 4.55331017e-02, 4.65183668e-02, 4.74600483e-02, 4.83572238e-02, 4.92090292e-02, 5.00146415e-02, 5.07733395e-02, 5.14844723e-02, 5.21474711e-02, 5.27618639e-02, 5.33272227e-02, ... 4.18000018e-02, 4.56667884e-02, 4.95598154e-02, 5.34758287e-02, 5.74115737e-02, 6.13637957e-02, 6.53292407e-02, 6.93046585e-02, 7.32868005e-02, 7.72724181e-02, 8.12582629e-02, 8.52410861e-02, 8.92176393e-02, 9.31846738e-02, 9.71389412e-02, 1.01077193e-01, 1.04996179e-01, 1.08892638e-01, 1.12763281e-01, 1.16604806e-01, 1.20413894e-01, 1.24187231e-01, 1.27921498e-01, 1.31613393e-01, 1.35259621e-01, 1.38856918e-01, 1.42402080e-01, 1.45891930e-01, 1.49323285e-01, 1.52692958e-01, 1.55997831e-01, 1.59234869e-01, 1.62401090e-01, 1.65493557e-01, 1.68509375e-01, 1.71445675e-01, 1.74299600e-01, 1.77068304e-01, 1.79748977e-01, 1.82338868e-01, 1.84835358e-01, 1.87235862e-01, 1.89537841e-01, 1.91738840e-01, 1.93836468e-01, 1.95828445e-01, 1.97712624e-01, 1.99486963e-01, 2.01149480e-01, 2.02698219e-01, 2.04131253e-01, 2.05446730e-01, 2.06642928e-01, 2.07718412e-01, 2.08672081e-01, 2.09503080e-01, 2.10210709e-01, 2.10794528e-01, 2.11254425e-01, 2.11590467e-01, 2.11802818e-01, 2.11891687e-01, 2.11857327e-01, 2.11700026e-01, 2.11420247e-01, 2.11018588e-01, 2.10495704e-01, 2.09852354e-01, 2.09089383e-01, 2.08207694e-01, 2.07208266e-01, 2.06092180e-01, 2.04860685e-01, 2.03515281e-01, 2.02057617e-01, 2.00489444e-01, 1.98812613e-01, 1.97029044e-01, 1.95140719e-01, 1.93149704e-01])
- eimf6(time)float64-0.008067 -0.007914 ... 0.1055
array([-0.00806694, -0.00791405, -0.00776303, -0.00761395, -0.00746683, -0.00732174, -0.00717871, -0.00703776, -0.00689895, -0.00676229, -0.00662783, -0.00649559, -0.0063656 , -0.0062379 , -0.00611251, -0.00598947, -0.00586879, -0.00575052, -0.00563467, -0.00552128, -0.00541037, -0.00530197, -0.00519611, -0.00509281, -0.00499211, -0.00489403, -0.0047986 , -0.00470584, -0.0046158 , -0.00452849, -0.00444395, -0.0043622 , -0.00428328, -0.00420721, -0.00413402, -0.00406374, -0.0039964 , -0.00393202, -0.00387064, -0.00381229, -0.00375699, -0.00370477, -0.00365568, -0.00360973, -0.00356696, -0.00352739, -0.00349106, -0.00345799, -0.00342821, -0.00340175, -0.00337862, -0.00335887, -0.00334251, -0.00332957, -0.00332007, -0.00331404, -0.0033115 , -0.00331249, -0.00331701, -0.00332511, -0.00333679, -0.00335208, -0.003371 , -0.00339357, -0.00341981, -0.00344973, -0.00348336, -0.00352071, -0.0035618 , -0.00360664, -0.00365525, -0.00370765, -0.00376384, -0.00382385, -0.00388766, -0.00395531, -0.00402679, -0.0041021 , -0.00418127, -0.0042643 , -0.0043512 , -0.00444197, -0.00453663, -0.00463519, -0.00473764, -0.00484399, -0.00495424, -0.0050684 , -0.00518646, -0.00530842, -0.00543427, -0.00556401, -0.00569761, -0.00583506, -0.00597635, -0.00612147, -0.00627038, -0.00642308, -0.00657954, -0.00673974, ... 0.04397097, 0.04497686, 0.04598136, 0.04698426, 0.04798537, 0.04898448, 0.0499814 , 0.05097591, 0.05196782, 0.05295692, 0.05394302, 0.05492591, 0.05590538, 0.05688124, 0.05785327, 0.05882129, 0.05978507, 0.06074443, 0.06169916, 0.06264905, 0.0635939 , 0.0645335 , 0.06546766, 0.06639617, 0.06731883, 0.06823543, 0.06914577, 0.07004965, 0.07094686, 0.0718372 , 0.07272046, 0.07359645, 0.07446496, 0.07532578, 0.07617872, 0.07702357, 0.07786012, 0.07868817, 0.07950752, 0.08031797, 0.08111931, 0.08191134, 0.08269385, 0.08346665, 0.08422952, 0.08498227, 0.08572469, 0.08645658, 0.08717773, 0.08788794, 0.08858701, 0.08927474, 0.08995092, 0.09061534, 0.09126781, 0.09190812, 0.09253607, 0.09315146, 0.09375409, 0.09434375, 0.09492025, 0.09548338, 0.09603296, 0.09656878, 0.09709065, 0.09759837, 0.09809176, 0.09857061, 0.09903476, 0.099484 , 0.09991816, 0.10033707, 0.10074055, 0.10112842, 0.10150053, 0.10185671, 0.1021968 , 0.1025207 , 0.10282827, 0.10311944, 0.10339412, 0.10365229, 0.10389394, 0.10411907, 0.10432767, 0.10451975, 0.10469535, 0.10485447, 0.10499718, 0.10512351, 0.10523352, 0.10532726, 0.10540479, 0.10546618, 0.10551152, 0.10554089, 0.10555443, 0.10555226, 0.10553448])
- eimf7(time)float64-0.03737 -0.03736 ... 0.04542
array([-3.73747891e-02, -3.73624737e-02, -3.73491606e-02, -3.73348507e-02, -3.73195454e-02, -3.73032457e-02, -3.72859529e-02, -3.72676681e-02, -3.72483924e-02, -3.72281271e-02, -3.72068733e-02, -3.71846322e-02, -3.71614050e-02, -3.71371928e-02, -3.71119968e-02, -3.70858181e-02, -3.70586580e-02, -3.70305176e-02, -3.70013980e-02, -3.69713005e-02, -3.69402262e-02, -3.69081763e-02, -3.68751520e-02, -3.68411543e-02, -3.68061846e-02, -3.67702439e-02, -3.67333334e-02, -3.66954544e-02, -3.66566079e-02, -3.66167951e-02, -3.65760173e-02, -3.65342755e-02, -3.64915710e-02, -3.64479049e-02, -3.64032784e-02, -3.63576926e-02, -3.63111488e-02, -3.62636481e-02, -3.62151916e-02, -3.61657805e-02, -3.61154161e-02, -3.60640994e-02, -3.60118317e-02, -3.59586141e-02, -3.59044477e-02, -3.58493339e-02, -3.57932736e-02, -3.57362681e-02, -3.56783186e-02, -3.56194262e-02, -3.55595921e-02, -3.54988175e-02, -3.54371035e-02, -3.53744513e-02, -3.53108621e-02, -3.52463371e-02, -3.51808774e-02, -3.51144841e-02, -3.50471585e-02, -3.49789017e-02, -3.49097150e-02, -3.48395993e-02, -3.47685561e-02, -3.46965863e-02, -3.46236912e-02, -3.45498719e-02, -3.44751296e-02, -3.43994656e-02, -3.43228808e-02, -3.42453766e-02, -3.41669541e-02, -3.40876144e-02, -3.40073588e-02, -3.39261884e-02, -3.38441043e-02, -3.37611078e-02, -3.36771999e-02, -3.35923820e-02, -3.35066551e-02, -3.34200204e-02, ... 3.26707076e-02, 3.28688222e-02, 3.30661897e-02, 3.32628026e-02, 3.34586532e-02, 3.36537339e-02, 3.38480371e-02, 3.40415553e-02, 3.42342809e-02, 3.44262062e-02, 3.46173237e-02, 3.48076257e-02, 3.49971048e-02, 3.51857532e-02, 3.53735634e-02, 3.55605279e-02, 3.57466389e-02, 3.59318890e-02, 3.61162704e-02, 3.62997757e-02, 3.64823973e-02, 3.66641275e-02, 3.68449587e-02, 3.70248834e-02, 3.72038939e-02, 3.73819827e-02, 3.75591422e-02, 3.77353648e-02, 3.79106429e-02, 3.80849688e-02, 3.82583351e-02, 3.84307341e-02, 3.86021582e-02, 3.87725998e-02, 3.89420513e-02, 3.91105052e-02, 3.92779538e-02, 3.94443896e-02, 3.96098049e-02, 3.97741922e-02, 3.99375439e-02, 4.00998523e-02, 4.02611100e-02, 4.04213092e-02, 4.05804424e-02, 4.07385020e-02, 4.08954804e-02, 4.10513701e-02, 4.12061633e-02, 4.13598526e-02, 4.15124304e-02, 4.16638890e-02, 4.18142208e-02, 4.19634183e-02, 4.21114738e-02, 4.22583798e-02, 4.24041287e-02, 4.25487129e-02, 4.26921248e-02, 4.28343568e-02, 4.29754013e-02, 4.31152506e-02, 4.32538974e-02, 4.33913338e-02, 4.35275523e-02, 4.36625454e-02, 4.37963055e-02, 4.39288249e-02, 4.40600960e-02, 4.41901114e-02, 4.43188633e-02, 4.44463441e-02, 4.45725464e-02, 4.46974624e-02, 4.48210847e-02, 4.49434055e-02, 4.50644174e-02, 4.51841127e-02, 4.53024838e-02, 4.54195231e-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 4.658 seconds)