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 0.03235 -0.1011 0.1258 ... -0.05669 0.0309 eimf1 (time) float64 4kB 0.1107 0.1081 0.08481 ... -0.03636 -0.03344 eimf2 (time) float64 4kB 0.2768 0.2727 0.2422 ... -0.2468 -0.2527 -0.2341 eimf3 (time) float64 4kB -0.3352 -0.3388 -0.3453 ... -0.1639 -0.1653 eimf4 (time) float64 4kB -0.1858 -0.1845 -0.1824 ... 0.0177 0.01863 eimf5 (time) float64 4kB -0.04084 -0.0402 -0.03952 ... 0.1861 0.1844 eimf6 (time) float64 4kB -0.002706 -0.002583 -0.002463 ... 0.1183 0.1183 eimf7 (time) float64 4kB -0.05927 -0.05924 -0.05922 ... 0.05453 0.0547
- 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)float640.03235 -0.1011 ... -0.05669 0.0309
array([ 3.23476019e-02, -1.01096204e-01, 1.25759519e-01, -5.54525102e-02, 2.38729812e-02, -1.01514637e-01, 1.53253580e-01, -5.30808306e-02, -1.80816057e-01, -7.19462852e-02, 2.82142918e-01, -5.06206229e-02, -2.26216208e-01, 1.40940599e-01, 2.83376940e-02, -1.30704140e-01, 1.16020374e-01, -3.15896123e-02, 1.32601560e-01, 4.27324302e-02, -3.44019260e-01, -1.00662113e-01, 4.41084228e-01, 2.52332568e-01, -3.91522884e-01, -4.01402044e-01, 1.87122920e-01, 2.98712806e-01, 4.49129121e-01, -3.33180871e-01, -2.64282845e-01, 5.19918389e-02, 7.19327308e-02, -1.65529825e-01, 2.30167165e-01, -1.76967982e-01, -6.58904792e-02, 2.38665695e-02, 1.02796690e-03, 9.13614933e-02, 6.98333702e-02, 1.38998713e-02, -5.57971713e-02, -7.89833119e-02, -3.49669678e-02, 1.87164253e-01, -9.57014395e-02, -7.33282422e-02, -9.87474369e-02, 1.46287750e-01, 6.69515833e-04, 5.99542800e-02, -1.68703932e-01, 6.06814282e-02, -7.14679719e-02, 3.63095415e-01, -2.15576096e-01, -4.65561866e-01, -2.06723144e-01, 5.89108031e-01, -2.60115146e-01, -3.50456360e-01, 1.47392007e-01, -6.22998007e-02, -5.31642296e-02, 5.37911570e-02, 1.59273848e-02, -1.01202673e-01, -1.67699348e-01, -1.52244226e-01, -8.68739479e-02, 3.73367464e-01, 4.67508164e-01, -2.21479935e-01, -5.69561308e-01, -2.91159556e-01, 2.99627841e-02, 4.51362621e-01, 2.51893407e-01, -3.87226974e-01, ... 2.06985410e-02, -5.92279053e-01, -4.13245554e-01, 2.82665494e-01, 4.02377820e-01, -2.17654637e-01, -1.69351981e-01, 1.58758267e-01, -2.99502336e-02, -4.54884831e-02, 4.34677487e-02, 3.11924485e-02, -1.12853057e-01, 5.07354727e-02, -6.17009334e-02, 1.60361704e-01, -1.81069855e-01, 1.51696975e-01, -1.68569650e-01, 3.50150604e-02, 2.32371167e-01, 9.13418847e-02, -4.74117652e-01, 3.89474882e-01, 3.43224368e-01, 3.18905838e-01, -1.98040138e-01, -4.42891302e-01, 3.35460552e-01, 3.78447286e-02, 4.22850814e-02, -2.27714065e-01, 2.19576712e-01, 9.42149743e-02, -2.82595499e-01, 1.27095311e-01, 8.79773704e-02, 2.71110223e-02, -1.88686672e-01, 1.86991713e-01, -1.20367720e-02, -1.82178936e-01, -5.68777726e-02, 3.80102995e-01, 2.37415419e-01, -2.72308351e-01, -4.42453665e-01, -1.68080728e-01, 1.92455590e-01, 3.71367402e-01, -2.58504899e-01, -1.22930816e-01, 7.80311436e-02, 1.43254847e-01, -1.30364710e-01, 1.23734537e-02, -2.23159876e-02, 2.00018365e-01, -2.98990864e-01, -7.56227694e-02, 1.91004670e-01, 4.97869643e-01, -5.12619007e-01, -1.31581782e-01, 7.51635475e-02, 3.88120351e-01, -2.92610382e-01, 2.65232182e-02, 2.64512979e-01, 1.41762522e-01, -4.90679558e-01, 4.95181974e-01, 2.08230065e-01, -2.50619422e-01, -3.93193983e-01, -1.29838049e-01, 2.53967290e-01, 1.52286129e-02, -5.66917387e-02, 3.08967822e-02])
- eimf1(time)float640.1107 0.1081 ... -0.03636 -0.03344
array([ 0.11070107, 0.10812684, 0.0848079 , 0.04776514, 0.01571188, -0.0089314 , -0.04989498, -0.09778251, -0.08522673, 0.026795 , 0.12316136, 0.07230606, -0.05866511, -0.13186668, -0.07680693, 0.08763337, 0.25900907, 0.27033931, 0.04232795, -0.31328006, -0.60283821, -0.68265045, -0.57129048, -0.33859918, -0.01179002, 0.40225863, 0.81304712, 0.99943083, 0.7533211 , 0.1902609 , -0.36739448, -0.69873846, -0.76678443, -0.67744864, -0.5806471 , -0.50560336, -0.34302538, -0.05443809, 0.26276979, 0.46890317, 0.48324101, 0.32794753, 0.10518951, -0.06063224, -0.13037562, -0.15974024, -0.1737705 , -0.13591631, -0.02559047, 0.09325291, 0.11810487, 0.03307463, -0.0639648 , -0.05157824, 0.05811225, 0.13452025, 0.1573463 , 0.22079592, 0.34725549, 0.31649922, -0.02190657, -0.45482606, -0.72640298, -0.71071468, -0.49877822, -0.23590325, -0.00803299, 0.18245445, 0.4113816 , 0.72382395, 1.02305283, 1.12202099, 0.84674579, 0.24726113, -0.39638942, -0.83647821, -0.99207702, -0.91689693, -0.64793566, -0.22546825, 0.24687406, 0.60647973, 0.70720251, 0.56154727, 0.289776 , 0.01314677, -0.2116898 , -0.31989988, -0.25711737, -0.12134383, -0.05799418, -0.06261055, -0.04213054, 0.0103869 , 0.01457529, -0.03978484, -0.06470107, -0.02547231, 0.00659335, -0.00227677, ... 0.66189982, 1.04847098, 1.04037984, 0.53571567, -0.17018648, -0.69827045, -0.75512879, -0.38887405, 0.11514408, 0.44729579, 0.48954008, 0.32393514, 0.04700349, -0.29193795, -0.5104574 , -0.42736773, -0.07656309, 0.30266073, 0.48439369, 0.44378728, 0.29865116, 0.12882199, -0.05320688, -0.25393565, -0.42089685, -0.45239641, -0.30061842, -0.02387338, 0.25879622, 0.43824567, 0.44800044, 0.32723198, 0.19013823, 0.0565942 , -0.13837617, -0.36782403, -0.52305468, -0.59800557, -0.60264204, -0.51131933, -0.28265135, 0.1330044 , 0.61312982, 0.77900392, 0.43965264, -0.1019242 , -0.45908916, -0.45923407, -0.28555375, -0.16311774, -0.09741339, -0.10552203, -0.2189469 , -0.32197763, -0.28309256, -0.16413019, -0.04487422, 0.10313287, 0.29397844, 0.46256736, 0.57698209, 0.61170744, 0.45517181, 0.06884784, -0.3834465 , -0.65855637, -0.6243804 , -0.39373879, -0.12163585, 0.11235673, 0.27221186, 0.33284703, 0.22934562, 0.01329726, -0.12869009, -0.13559573, -0.07431506, 0.02752797, 0.14908428, 0.19450014, 0.03801461, -0.19745049, -0.20374032, 0.01804298, 0.20061667, 0.17608223, -0.0035311 , -0.18211101, -0.22584397, -0.06772879, 0.1906999 , 0.26550336, 0.11991421, -0.07171462, -0.15583415, -0.13270976, -0.07144743, -0.03636424, -0.03343854])
- eimf2(time)float640.2768 0.2727 ... -0.2527 -0.2341
array([ 2.76795084e-01, 2.72693929e-01, 2.42221825e-01, 1.90851519e-01, 1.27674048e-01, 6.33988132e-02, 7.03773761e-03, -3.66507333e-02, -6.86533545e-02, -9.60048179e-02, -1.29034662e-01, -1.76739944e-01, -2.41862883e-01, -3.22412880e-01, -4.12063546e-01, -4.96304214e-01, -5.55255073e-01, -5.70149118e-01, -5.31613638e-01, -4.41983679e-01, -3.07116905e-01, -1.33779738e-01, 6.72847029e-02, 2.78962568e-01, 4.79991536e-01, 6.45901813e-01, 7.52738920e-01, 7.77601223e-01, 7.08189209e-01, 5.64228445e-01, 3.74036447e-01, 1.65915673e-01, -3.14522322e-02, -1.89509443e-01, -2.86037236e-01, -3.17607763e-01, -2.99755359e-01, -2.54717807e-01, -2.02712094e-01, -1.62596517e-01, -1.48900827e-01, -1.61926431e-01, -1.92843703e-01, -2.31521995e-01, -2.67198101e-01, -2.88285474e-01, -2.85288387e-01, -2.54753983e-01, -2.00331382e-01, -1.29713905e-01, -5.13849145e-02, 2.76246027e-02, 1.02204836e-01, 1.67527211e-01, 2.16589649e-01, 2.39826400e-01, 2.28485927e-01, 1.79478921e-01, 9.90140716e-02, 2.77679656e-04, -1.03020193e-01, -1.95500359e-01, -2.61197832e-01, -2.85051435e-01, -2.62888734e-01, -2.01264311e-01, -1.10244133e-01, -2.22794057e-03, 1.09258344e-01, 2.12400007e-01, 2.97383987e-01, 3.54954852e-01, 3.80258625e-01, 3.78425187e-01, 3.57854775e-01, 3.26854270e-01, 2.91721256e-01, 2.55643882e-01, 2.18639440e-01, 1.75776894e-01, ... 3.98883474e-01, 3.82221004e-01, 3.20223501e-01, 2.29209773e-01, 1.27783954e-01, 3.08995594e-02, -5.09442053e-02, -1.11678578e-01, -1.53694120e-01, -1.83670953e-01, -2.08309315e-01, -2.33709077e-01, -2.64283407e-01, -3.01182225e-01, -3.35967331e-01, -3.52289890e-01, -3.34481692e-01, -2.69217206e-01, -1.50047489e-01, 1.85906733e-02, 2.18868152e-01, 4.21705691e-01, 5.92173891e-01, 6.95025855e-01, 6.95632622e-01, 5.80566576e-01, 3.82124294e-01, 1.40322009e-01, -1.06325915e-01, -3.23966526e-01, -4.84062402e-01, -5.73609307e-01, -5.89618884e-01, -5.38226417e-01, -4.30036125e-01, -2.78359418e-01, -9.92934481e-02, 8.89433775e-02, 2.65331319e-01, 4.08276099e-01, 4.96681586e-01, 5.10697784e-01, 4.41730615e-01, 3.04768224e-01, 1.26243327e-01, -6.30238989e-02, -2.31534599e-01, -3.51527121e-01, -4.14057012e-01, -4.25640180e-01, -3.96262920e-01, -3.37048067e-01, -2.56170655e-01, -1.58771681e-01, -5.30592115e-02, 5.08794376e-02, 1.44625382e-01, 2.21580146e-01, 2.73646826e-01, 2.92442738e-01, 2.73466969e-01, 2.21095241e-01, 1.51468597e-01, 8.32267288e-02, 2.98200845e-02, -4.14261175e-03, -1.65059369e-02, -1.08839538e-02, 4.92858077e-03, 2.29042373e-02, 3.36153353e-02, 2.91962723e-02, 2.68991940e-03, -4.55329486e-02, -1.05553152e-01, -1.65951182e-01, -2.16112619e-01, -2.46826072e-01, -2.52679153e-01, -2.34057137e-01])
- eimf3(time)float64-0.3352 -0.3388 ... -0.1639 -0.1653
array([-0.33518771, -0.33883532, -0.34531471, -0.35390577, -0.3635483 , -0.37306501, -0.38120392, -0.38660706, -0.38779073, -0.38320338, -0.37133593, -0.35108194, -0.32202543, -0.28442866, -0.23909001, -0.18711873, -0.12983237, -0.06872548, -0.00546377, 0.05826388, 0.12082567, 0.18061435, 0.23603563, 0.28549906, 0.32741363, 0.36018756, 0.38223017, 0.39212128, 0.38978915, 0.37644919, 0.35377631, 0.32354738, 0.28754738, 0.24755466, 0.20533313, 0.16245449, 0.12013719, 0.07952857, 0.04177284, 0.0080137 , -0.02062321, -0.04329016, -0.05983493, -0.07056593, -0.07599123, -0.07670492, -0.07339015, -0.066739 , -0.05744722, -0.04613597, -0.03330232, -0.01941636, -0.00496335, 0.00958473, 0.02379654, 0.03729053, 0.0497412 , 0.06085458, 0.07037894, 0.0781468 , 0.08402374, 0.08788664, 0.08963364, 0.08919511, 0.08651746, 0.08152569, 0.07422145, 0.06465176, 0.05288511, 0.03900511, 0.02310383, 0.00528715, -0.01425251, -0.03522846, -0.0573307 , -0.08024679, -0.10366648, -0.12724944, -0.15058796, -0.17324034, -0.19477361, -0.21476157, -0.23278182, -0.24846565, -0.26149652, -0.27157304, -0.27839889, -0.28168009, -0.28112558, -0.27647517, -0.26765382, -0.25485519, -0.23839425, -0.21863679, -0.19596638, -0.17076923, -0.14343262, -0.11434478, -0.08389574, -0.05248216, ... 0.00436595, -0.02589717, -0.05878709, -0.09347812, -0.12898391, -0.16429373, -0.19833232, -0.22999713, -0.2580922 , -0.28131222, -0.29836179, -0.30817609, -0.31048524, -0.30566696, -0.29433708, -0.27731205, -0.25551899, -0.2298897 , -0.20135564, -0.17084708, -0.13913454, -0.10667258, -0.07381324, -0.0410081 , -0.00881627, 0.02229193, 0.05225496, 0.08136847, 0.10999807, 0.13849337, 0.1671455 , 0.19600522, 0.22464141, 0.25227994, 0.27809874, 0.30129428, 0.32110228, 0.33679649, 0.34757724, 0.35260366, 0.35102755, 0.34200042, 0.32473411, 0.29874121, 0.26461472, 0.22392055, 0.1783183 , 0.12947011, 0.07904948, 0.02872046, -0.01991639, -0.06537959, -0.10656641, -0.14325298, -0.17562293, -0.20373751, -0.22754658, -0.24698103, -0.26197099, -0.27245459, -0.27840036, -0.27981871, -0.27664598, -0.26887791, -0.25653164, -0.23962288, -0.21817857, -0.19236314, -0.16262985, -0.12957256, -0.09387743, -0.05626824, -0.01752721, 0.02137611, 0.05929439, 0.09506255, 0.12758062, 0.15592146, 0.17925587, 0.19680687, 0.20802274, 0.21265139, 0.21061961, 0.20200885, 0.18707073, 0.16619401, 0.14003167, 0.10958054, 0.07605242, 0.04075792, 0.00500281, -0.02998989, -0.0630751 , -0.09316018, -0.11922972, -0.14030048, -0.1554383 , -0.16387633, -0.16531135])
- eimf4(time)float64-0.1858 -0.1845 ... 0.0177 0.01863
array([-1.85752183e-01, -1.84466974e-01, -1.82369735e-01, -1.79456541e-01, -1.75729713e-01, -1.71199967e-01, -1.65883039e-01, -1.59795046e-01, -1.52955254e-01, -1.45387677e-01, -1.37117952e-01, -1.28176357e-01, -1.18609179e-01, -1.08481880e-01, -9.78687565e-02, -8.68460099e-02, -7.54903770e-02, -6.38787592e-02, -5.20880290e-02, -4.01946262e-02, -2.82748177e-02, -1.64048118e-02, -4.66083010e-03, 6.88104756e-03, 1.81453540e-02, 2.90573634e-02, 3.95441005e-02, 4.95372789e-02, 5.89786652e-02, 6.78250298e-02, 7.60399529e-02, 8.35912575e-02, 9.04498249e-02, 9.65933183e-02, 1.02007588e-01, 1.06680785e-01, 1.10603709e-01, 1.13768882e-01, 1.16169466e-01, 1.17798739e-01, 1.18650070e-01, 1.18717812e-01, 1.18001318e-01, 1.16512689e-01, 1.14278576e-01, 1.11331184e-01, 1.07704218e-01, 1.03432790e-01, 9.85531290e-02, 9.31037587e-02, 8.71267426e-02, 8.06689548e-02, 7.37826651e-02, 6.65238742e-02, 5.89501307e-02, 5.11190591e-02, 4.30894817e-02, 3.49209104e-02, 2.66726517e-02, 1.84024385e-02, 1.01657558e-02, 2.01379025e-03, -6.00565500e-03, -1.38462774e-02, -2.14636353e-02, -2.88171219e-02, -3.58707218e-02, -4.25919876e-02, -4.89516838e-02, -5.49249533e-02, -6.04912670e-02, -6.56331856e-02, -7.03343969e-02, -7.45783973e-02, -7.83487443e-02, -8.16297094e-02, -8.44059729e-02, -8.66623785e-02, -8.83845282e-02, -8.95593969e-02, ... -1.07544604e-01, -9.75807714e-02, -8.71221738e-02, -7.62369685e-02, -6.49970776e-02, -5.34789807e-02, -4.17627641e-02, -2.99349543e-02, -1.80850699e-02, -6.30351552e-03, 5.32184110e-03, 1.67092804e-02, 2.77853475e-02, 3.84879797e-02, 4.87637033e-02, 5.85648457e-02, 6.78460908e-02, 7.65643937e-02, 8.46800881e-02, 9.21572806e-02, 9.89653743e-02, 1.05080520e-01, 1.10485890e-01, 1.15174907e-01, 1.19150229e-01, 1.22426569e-01, 1.25028644e-01, 1.26987315e-01, 1.28337123e-01, 1.29115142e-01, 1.29360488e-01, 1.29112796e-01, 1.28412127e-01, 1.27297920e-01, 1.25805392e-01, 1.23967102e-01, 1.21815519e-01, 1.19381827e-01, 1.16696635e-01, 1.13789790e-01, 1.10690109e-01, 1.07428075e-01, 1.04034311e-01, 1.00534743e-01, 9.69526016e-02, 9.33050609e-02, 8.96055146e-02, 8.58672598e-02, 8.21028220e-02, 7.83214144e-02, 7.45289763e-02, 7.07325426e-02, 6.69412529e-02, 6.31686629e-02, 5.94314312e-02, 5.57480160e-02, 5.21374342e-02, 4.86186606e-02, 4.52104100e-02, 4.19309513e-02, 3.87989436e-02, 3.58331361e-02, 3.30487378e-02, 3.04571826e-02, 2.80652135e-02, 2.58782115e-02, 2.39018826e-02, 2.21410623e-02, 2.06002016e-02, 1.92835200e-02, 1.81943655e-02, 1.73348938e-02, 1.67058466e-02, 1.63066353e-02, 1.61358591e-02, 1.61917925e-02, 1.64724336e-02, 1.69753108e-02, 1.76972911e-02, 1.86337434e-02])
- eimf5(time)float64-0.04084 -0.0402 ... 0.1861 0.1844
array([-0.04083752, -0.0401978 , -0.03952315, -0.03881376, -0.0380701 , -0.03729283, -0.03648259, -0.03564005, -0.03476586, -0.03386061, -0.03292499, -0.03195988, -0.03096622, -0.02994491, -0.02889689, -0.02782308, -0.02672438, -0.02560173, -0.02445605, -0.02328827, -0.02209933, -0.02089014, -0.01966167, -0.01841483, -0.01715057, -0.01586983, -0.01457354, -0.01326263, -0.01193806, -0.01060078, -0.00925175, -0.00789192, -0.00652229, -0.00514387, -0.00375773, -0.00236498, -0.0009667 , 0.000436 , 0.00184196, 0.00325007, 0.00465918, 0.0060681 , 0.00747561, 0.00888045, 0.01028137, 0.0116771 , 0.01306638, 0.01444793, 0.01582047, 0.0171827 , 0.01853339, 0.01987128, 0.0211951 , 0.02250362, 0.02379564, 0.02506997, 0.02632543, 0.02756088, 0.02877517, 0.02996717, 0.03113574, 0.03227975, 0.03339804, 0.03448946, 0.03555286, 0.03658706, 0.03759093, 0.03856329, 0.03950301, 0.04040897, 0.04128005, 0.04211515, 0.04291323, 0.04367328, 0.04439433, 0.04507553, 0.04571607, 0.04631525, 0.0468724 , 0.04738689, 0.04785817, 0.0482859 , 0.04866997, 0.04901048, 0.0493076 , 0.04956155, 0.04977263, 0.04994119, 0.05006765, 0.05015251, 0.05019626, 0.05019945, 0.05016261, 0.05008629, 0.04997105, 0.04981752, 0.0496263 , 0.04939802, 0.04913331, 0.0488328 , ... -0.0263914 , -0.02361468, -0.02077699, -0.01787837, -0.01491893, -0.01189889, -0.00881876, -0.00567922, -0.00248119, 0.00077396, 0.00408434, 0.00744761, 0.01086114, 0.01432216, 0.01782781, 0.02137513, 0.02496112, 0.02858274, 0.03223693, 0.03592056, 0.03963052, 0.04336366, 0.04711684, 0.05088691, 0.05467073, 0.05846516, 0.06226706, 0.06607329, 0.06988071, 0.07368618, 0.07748657, 0.08127874, 0.08505954, 0.08882585, 0.09257452, 0.09630245, 0.1000065 , 0.10368356, 0.10733054, 0.11094445, 0.11452237, 0.1180614 , 0.12155863, 0.1250112 , 0.12841626, 0.13177095, 0.13507256, 0.13831847, 0.14150612, 0.14463294, 0.14769639, 0.15069402, 0.15362342, 0.15648223, 0.15926812, 0.16197872, 0.16461172, 0.16716476, 0.16963552, 0.17202169, 0.17432101, 0.17653134, 0.1786506 , 0.18067679, 0.18260798, 0.18444232, 0.18617803, 0.18781336, 0.18934666, 0.19077638, 0.192101 , 0.19331903, 0.19442908, 0.19543004, 0.19632111, 0.19710172, 0.19777134, 0.19832948, 0.19877566, 0.19910947, 0.19933059, 0.19943887, 0.19943428, 0.19931681, 0.19908662, 0.19874395, 0.1982892 , 0.19772287, 0.19704556, 0.19625813, 0.19536165, 0.19435725, 0.19324611, 0.19202943, 0.19070843, 0.18928453, 0.18775934, 0.18613469, 0.18441255])
- eimf6(time)float64-0.002706 -0.002583 ... 0.1183
array([-2.70560571e-03, -2.58331250e-03, -2.46300465e-03, -2.34472386e-03, -2.22851427e-03, -2.11442162e-03, -2.00249215e-03, -1.89277212e-03, -1.78530757e-03, -1.68014468e-03, -1.57732939e-03, -1.47690617e-03, -1.37891748e-03, -1.28340419e-03, -1.19040714e-03, -1.09996730e-03, -1.01212501e-03, -9.26920315e-04, -8.44392888e-04, -7.64581385e-04, -6.87523706e-04, -6.13257651e-04, -5.41821018e-04, -4.73251605e-04, -4.07587213e-04, -3.44865638e-04, -2.85124680e-04, -2.28401760e-04, -1.74733179e-04, -1.24154579e-04, -7.67006935e-05, -3.24067391e-05, 8.69156356e-06, 4.65584777e-05, 8.11582754e-05, 1.12455692e-04, 1.40416305e-04, 1.65006006e-04, 1.86191381e-04, 2.03942516e-04, 2.18231592e-04, 2.29031322e-04, 2.36315247e-04, 2.40057218e-04, 2.40231085e-04, 2.36810757e-04, 2.29771374e-04, 2.19089251e-04, 2.04740751e-04, 1.86702311e-04, 1.64951405e-04, 1.39466505e-04, 1.10226243e-04, 7.72097425e-05, 4.03969511e-05, -2.31762602e-07, -4.46961805e-05, -9.30158457e-05, -1.45209350e-04, -2.01294692e-04, -2.61289467e-04, -3.25211191e-04, -3.93077717e-04, -4.64907616e-04, -5.40720046e-04, -6.20534547e-04, -7.04370249e-04, -7.92246112e-04, -8.84180559e-04, -9.80190807e-04, -1.08029321e-03, -1.18450378e-03, -1.29283829e-03, -1.40531192e-03, -1.52193891e-03, -1.64273270e-03, -1.76770633e-03, -1.89687246e-03, -2.03024344e-03, -2.16783154e-03, ... 7.00773555e-02, 7.11192975e-02, 7.21553223e-02, 7.31852252e-02, 7.42088015e-02, 7.52258465e-02, 7.62361556e-02, 7.72395240e-02, 7.82357471e-02, 7.92246202e-02, 8.02059386e-02, 8.11794977e-02, 8.21450927e-02, 8.31025191e-02, 8.40515720e-02, 8.49920469e-02, 8.59237390e-02, 8.68464437e-02, 8.77599563e-02, 8.86640721e-02, 8.95585865e-02, 9.04432947e-02, 9.13179921e-02, 9.21824740e-02, 9.30365357e-02, 9.38799725e-02, 9.47125799e-02, 9.55341530e-02, 9.63444872e-02, 9.71433778e-02, 9.79306201e-02, 9.87060092e-02, 9.94693401e-02, 1.00220408e-01, 1.00959010e-01, 1.01684945e-01, 1.02398012e-01, 1.03098008e-01, 1.03784732e-01, 1.04457983e-01, 1.05117559e-01, 1.05763258e-01, 1.06394880e-01, 1.07012224e-01, 1.07615089e-01, 1.08203279e-01, 1.08776596e-01, 1.09334843e-01, 1.09877825e-01, 1.10405350e-01, 1.10917224e-01, 1.11413257e-01, 1.11893260e-01, 1.12357048e-01, 1.12804441e-01, 1.13235260e-01, 1.13649337e-01, 1.14046503e-01, 1.14426610e-01, 1.14789512e-01, 1.15135068e-01, 1.15463142e-01, 1.15773611e-01, 1.16066360e-01, 1.16341288e-01, 1.16598309e-01, 1.16837348e-01, 1.17058342e-01, 1.17261236e-01, 1.17445986e-01, 1.17612564e-01, 1.17760963e-01, 1.17891190e-01, 1.18003279e-01, 1.18097282e-01, 1.18173264e-01, 1.18231302e-01, 1.18271484e-01, 1.18293905e-01, 1.18298668e-01])
- eimf7(time)float64-0.05927 -0.05924 ... 0.0547
array([-5.92705178e-02, -5.92447533e-02, -5.92174556e-02, -5.91886277e-02, -5.91582725e-02, -5.91263931e-02, -5.90929923e-02, -5.90580731e-02, -5.90216384e-02, -5.89836913e-02, -5.89442347e-02, -5.89032715e-02, -5.88608046e-02, -5.88168371e-02, -5.87713719e-02, -5.87244119e-02, -5.86759601e-02, -5.86260195e-02, -5.85745929e-02, -5.85216834e-02, -5.84672940e-02, -5.84114274e-02, -5.83540868e-02, -5.82952751e-02, -5.82349952e-02, -5.81732500e-02, -5.81100426e-02, -5.80453759e-02, -5.79792528e-02, -5.79116763e-02, -5.78426494e-02, -5.77721749e-02, -5.77002559e-02, -5.76268953e-02, -5.75520961e-02, -5.74758611e-02, -5.73981935e-02, -5.73190960e-02, -5.72385718e-02, -5.71566237e-02, -5.70732546e-02, -5.69884676e-02, -5.69022656e-02, -5.68146515e-02, -5.67256283e-02, -5.66351990e-02, -5.65433664e-02, -5.64501337e-02, -5.63555036e-02, -5.62594792e-02, -5.61620634e-02, -5.60632592e-02, -5.59630696e-02, -5.58614974e-02, -5.57585456e-02, -5.56542172e-02, -5.55485153e-02, -5.54414429e-02, -5.53330033e-02, -5.52231996e-02, -5.51120349e-02, -5.49995125e-02, -5.48856354e-02, -5.47704068e-02, -5.46538300e-02, -5.45359080e-02, -5.44166440e-02, -5.42960412e-02, -5.41741027e-02, -5.40508317e-02, -5.39262313e-02, -5.38003048e-02, -5.36730552e-02, -5.35444858e-02, -5.34145996e-02, -5.32833999e-02, -5.31508898e-02, -5.30170724e-02, -5.28819510e-02, -5.27455286e-02, ... 3.76308687e-02, 3.78883086e-02, 3.81448822e-02, 3.84005819e-02, 3.86554003e-02, 3.89093296e-02, 3.91623624e-02, 3.94144911e-02, 3.96657080e-02, 3.99160058e-02, 4.01653767e-02, 4.04138132e-02, 4.06613077e-02, 4.09078527e-02, 4.11534406e-02, 4.13980638e-02, 4.16417148e-02, 4.18843859e-02, 4.21260697e-02, 4.23667586e-02, 4.26064449e-02, 4.28451211e-02, 4.30827797e-02, 4.33194130e-02, 4.35550136e-02, 4.37895738e-02, 4.40230861e-02, 4.42555428e-02, 4.44869365e-02, 4.47172596e-02, 4.49465044e-02, 4.51746635e-02, 4.54017292e-02, 4.56276940e-02, 4.58525503e-02, 4.60762906e-02, 4.62989073e-02, 4.65203927e-02, 4.67407394e-02, 4.69599398e-02, 4.71779863e-02, 4.73948713e-02, 4.76105873e-02, 4.78251266e-02, 4.80384818e-02, 4.82506453e-02, 4.84616094e-02, 4.86713667e-02, 4.88799095e-02, 4.90872303e-02, 4.92933215e-02, 4.94981755e-02, 4.97017849e-02, 4.99041419e-02, 5.01052391e-02, 5.03050688e-02, 5.05036235e-02, 5.07008957e-02, 5.08968777e-02, 5.10915620e-02, 5.12849411e-02, 5.14770073e-02, 5.16677531e-02, 5.18571708e-02, 5.20452531e-02, 5.22319922e-02, 5.24173806e-02, 5.26014108e-02, 5.27840752e-02, 5.29653661e-02, 5.31452761e-02, 5.33237975e-02, 5.35009229e-02, 5.36766445e-02, 5.38509549e-02, 5.40238466e-02, 5.41953118e-02, 5.43653431e-02, 5.45339328e-02, 5.47010735e-02])
- timePandasIndex
PandasIndex(DatetimeIndex(['1871-01-31', '1871-04-30', '1871-07-31', '1871-10-31', '1872-01-31', '1872-04-30', '1872-07-31', '1872-10-31', '1873-01-31', '1873-04-30', ... '1994-07-31', '1994-10-31', '1995-01-31', '1995-04-30', '1995-07-31', '1995-10-31', '1996-01-31', '1996-04-30', '1996-07-31', '1996-10-31'], dtype='datetime64[ns]', name='time', length=504, freq=None))
Visualize the first three IMF components from standard EMD IMFs are ordered from highest frequency (IMF0) to lowest frequency (IMF2) Each IMF must satisfy two conditions:
Number of extrema and zero crossings differs by at most one
Mean of upper and lower envelopes is zero at any point
fig, ax = plt.subplots(4, 1, figsize = (8, 8), sharex=True)
fig.subplots_adjust(hspace=0.2)
axi = ax[0]
imf_result["input"].plot(ax = axi, color = "r")
axi.set_xlabel("")
axi.set_ylabel("Input")
axi.set_title("Input Signal: Niño 3")
axi = ax[1]
imf_result["imf0"].plot(ax = axi)
axi.set_xlabel("")
axi.set_ylabel("IMF 0")
axi = ax[2]
imf_result["imf1"].plot(ax = axi)
axi.set_xlabel("")
axi.set_ylabel("IMF 1")
axi = ax[3]
imf_result["imf2"].plot(ax = axi)
axi.set_xlabel("Time")
axi.set_ylabel("IMF 2")

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

Text(64.22222222222221, 0.5, 'eIMF 2')
Total running time of the script: (0 minutes 6.859 seconds)