removing baseline signal using fourier transforms - numpy
I have timeseries data for many terms an example of which is below:
term1 = [0.0, 0.0, 0.0, 0.0, 2.2384935833581433e-06, 3.938767914008819e-06, 0.0, 0.0, 1.1961851263949013e-06, 0.0, 2.278384397623645e-06, 1.100158422812885e-06, 0.0, 1.095521835393462e-06, 0.0, 0.0, 1.6933152148605343e-06, 0.0, 8.460737945563612e-07, 8.949410770794851e-07, 0.0, 2.8698467119209605e-06, 0.0, 0.0, 0.0, 3.9163008188985015e-06, 2.2244961516216576e-06, 0.0, 0.0, 1.9407903674692482e-06, 0.0, 0.0, 0.0, 0.0, 9.514657329616274e-07, 1.94463053478312e-06, 0.0, 0.0, 0.0, 2.0373216961518047e-06, 1.8835690620014428e-06, 0.0, 0.0, 0.0, 0.0, 9.707946148081127e-07, 0.0, 0.0, 1.6121985390256838e-06, 1.9547361301697883e-06, 0.0, 2.2876018840689116e-06, 2.208826914114183e-06, 1.9640500282823203e-06, 0.0, 2.6234669115235785e-06, 0.0, 0.0, 0.0, 1.986207773222741e-06, 1.049193537387487e-06, 1.090723073046815e-06, 0.0, 1.0257546476943088e-06, 9.179053033814713e-07, 0.0, 0.0, 0.0, 0.0, 9.335621182897889e-07, 0.0, 0.0, 0.0, 0.0, 2.1267500494469387e-06, 2.215050381320923e-06, 2.163720040591388e-06, 1.937729136470388e-06, 1.6037643556956889e-06, 1.313906783569333e-06, 0.0, 1.0064645216223805e-06, 1.876346865234201e-06, 9.504447606257348e-07, 2.017974095266539e-06, 0.0, 2.120782823355757e-06, 0.0, 0.0, 0.0, 0.0, 9.216394491176685e-07, 0.0, 0.0, 1.0401357169083422e-06, 0.0, 0.0, 0.0, 0.0, 0.0, 2.0089962853658684e-06, 1.8249773702806084e-06, 0.0, 1.2890950295073852e-06, 5.42812725267281e-06, 1.9185480428411778e-06, 2.6955316172381044e-06, 0.0, 0.0, 1.0070239923466176e-06, 0.0, 1.021152145542773e-06, 9.919749228739498e-07, 1.9293082175989564e-06, 9.802489636317832e-07, 1.0483850676418046e-06, 0.0, 0.0, 0.0, 0.0, 0.0, 1.9369409504181854e-06, 0.0, 4.619620451983665e-06, 0.0, 6.0795324434248845e-06, 0.0, 1.5312669396405198e-06, 1.2797051559320733e-06, 1.1002903666277531e-06, 0.0, 1.0054768323055684e-06, 2.060260561153169e-06, 1.0898719291496056e-06, 3.4605907920600203e-06, 3.3500051925080486e-06, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 6.5521496510980315e-06, 0.0, 0.0, 0.0, 3.01862187836765e-06, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.849817053093449e-06, 6.5552277941658475e-06, 1.985771944021089e-06, 1.010233667047188e-06, 9.802307070992228e-07, 5.605931075077432e-06, 3.651067480854715e-06, 0.0, 0.0, 2.9476960807432912e-06, 1.834478659509754e-06, 0.0, 0.0, 0.0, 0.0, 3.3801712394749917e-06, 0.0, 2.2884970981856794e-06, 1.02014792144861e-06, 2.906143199237428e-06, 9.807873564740302e-07, 0.0, 2.106593638087213e-06, 3.0329622335542676e-06, 2.9093758515985565e-06, 0.0, 2.12762335960239e-06, 9.614820669172289e-07, 9.264114341404848e-07, 0.0, 0.0, 9.073611487918033e-07, 0.0, 0.0, 0.0, 6.0360958532021484e-06, 0.0, 4.553288270957079e-06, 2.0712553257152562e-06, 3.292603824030081e-06, 2.690786880261329e-06, 2.301011409565074e-06, 2.029661472762958e-06, 0.0, 9.657114492818003e-07, 9.948942029504583e-07, 1.028682761437152e-06, 2.0694207898151387e-06, 3.845369982272845e-06, 9.048250701691842e-07, 1.7726379156614332e-06, 0.0, 9.238711680133629e-07, 9.231112912203808e-07, 9.422814896339613e-07, 0.0, 1.2123519263665934e-06, 0.0, 0.0, 2.1675188628329036e-06, 0.0, 4.498718989767663e-06, 0.0, 0.0, 2.650273839544471e-06, 1.1954029583832415e-06, 4.180999656112778e-06, 1.9036523473937095e-06, 9.75877289286136e-07, 0.0, 2.093618232902467e-06, 1.032899928523325e-06, 0.0, 4.473312219299659e-06, 8.762705923589204e-07, 0.0, 0.0, 1.792797436299666e-06, 0.0, 0.0, 1.1974513445582422e-06, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.1404264054329915e-06, 3.061324451410658e-06, 9.84829683554526e-07, 2.932895354293759e-06, 2.0897069394988045e-06, 0.0, 2.128187093183736e-06, 0.0, 4.686861415132188e-06, 6.37755683086446e-06, 1.8420463661490824e-06, 2.8347403094402523e-06, 1.9033842171380715e-06, 6.909144746582441e-06, 0.0, 0.0, 1.5479612576256442e-06, 5.621978186724636e-06, 2.087185930697078e-06, 1.3168406359462377e-05, 1.9676130885622652e-05, 1.9988766313331908e-05, 3.1801079228204546e-05, 3.322824899588385e-05, 2.0358501231090545e-05, 1.2383952049337664e-05, 1.8052256532066507e-05, 7.770543617518302e-06, 9.226179797741636e-06, 4.400430362089412e-06, 4.333084180992927e-06, 7.477274426653279e-06, 3.0526428255261993e-06, 4.952368123389242e-06, 1.2578584707962998e-05, 0.0, 2.121750274236223e-06, 0.0, 2.38940918273843e-06, 0.0, 1.5693511988273807e-06, 0.0, 0.0, 4.520448247648237e-06, 4.0303440122522456e-05, 2.8660979509446863e-05, 2.4793768971660722e-05, 3.957070185234852e-05, 2.64488881248099e-05, 6.428381095168035e-05, 5.6557662521419976e-05, 6.855540059858658e-05, 7.079288025889968e-05, 7.135683422742382e-05, 5.5663480860112103e-05, 8.088436527379357e-05, 7.142268494354861e-05, 8.243171356847987e-05, 7.658173644233611e-05, 5.4275733753644613e-05, 2.7329513031804995e-05, 1.8666856995404658e-05, 2.5061514626811264e-05, 9.707359513272993e-06, 2.233654188450612e-05, 2.0577084330035857e-05, 6.037067595033506e-05, 5.358585847760433e-05, 6.353114888415205e-05, 4.913406130358561e-05, 6.253876100291326e-05, 5.783647108547192e-05, 5.29265883017118e-05, 4.295770587763158e-05, 0.00012513639867455526, 0.0001264425725280477, 0.00010075697417828198, 7.700585441944497e-05, 6.390017630639553e-05, 6.862379380485504e-05, 8.118867124374998e-05, 8.928305705187346e-05, 8.923668314113125e-05, 5.0862818355003976e-05, 2.5192448399293734e-05, 1.9491995287268695e-05, 1.1397180337584482e-05, 1.8548131739430545e-05, 2.8274146120787152e-05, 2.9861740143137274e-05, 5.749201435920551e-05, 8.676081065218611e-05, 0.00011692016691003383, 6.18107213073443e-05, 8.31986307882476e-05, 5.661490072734421e-05, 6.637785526376392e-05, 6.189842468509176e-05, 5.077848495281155e-05, 3.7630726455798414e-05, 6.325167842846687e-05, 7.447442335517917e-05, 7.881778491014126e-05, 8.347575938861497e-05, 6.553610066345062e-05, 6.209221186256924e-05, 4.671174184109858e-05, 4.583301504850661e-05, 2.9423292949863758e-05, 1.9969520206001368e-05, 1.3386836054765546e-05, 1.0233804045678584e-05, 2.3371876153986385e-05, 3.701784260013326e-05, 2.6804842191646374e-05, 3.729558727386808e-05, 7.011179438698544e-05, 4.616049584765358e-05, 6.019787395273405e-05, 8.312188292939014e-05, 6.281596430043117e-05, 6.370630077282333e-05, 6.169767733530766e-05, 6.099512039036877e-05, 7.192322709245217e-05, 6.727547574464268e-05, 4.891125624919348e-05, 8.775231227342841e-05, 9.349358010749929e-05, 4.85363097385816e-05, 4.475820776946539e-05, 1.9528637281926147e-05, 1.5243002033035396e-05, 1.4322461630125293e-05, 1.0492122514416176e-05, 1.1956759574674148e-05, 1.5232250274180506e-05, 3.394641638997643e-05, 2.6115894879792267e-05, 4.868559048521277e-05, 5.612535494090208e-05, 3.269545148571978e-05, 4.967751016319062e-05, 4.8382804751191425e-05, 5.1860846075881435e-05, 4.4034258653232213e-05, 5.362193446127224e-05, 6.213052893181175e-05, 8.561827093901839e-05, 5.877682625663455e-05, 0.0, 0.0, 7.105805443046969e-05, 0.0, 0.0, 2.31393994554528e-05, 7.05044594070575e-06, 2.21491300929156e-05, 4.926848615186025e-06, 1.0752514744385843e-05, 1.4745260873155369e-05, 1.976297604068538e-05, 3.094705732168692e-05, 5.068338091939653e-05, 2.655137469742496e-05, 3.0142790705685793e-05, 3.89279249469607e-05, 6.264176821226988e-05, 3.598536226187379e-05, 4.430195278344506e-05, 2.7501831818440764e-05, 1.7243328268903956e-05, 1.2049184772240285e-05, 2.1016880758625327e-05, 3.411070201956675e-05, 3.1893789428697184e-05, 1.8509911029027654e-05, 3.920735117199027e-05, 3.700840501998454e-05, 8.529330234343347e-06, 1.1007881643256571e-05, 4.661265813344272e-06, 7.306007242688513e-06, 2.6772256446090046e-06, 3.0075821145106816e-06, 6.713527085725027e-06, 2.204123915846549e-05, 7.880065404542858e-06, 4.3539870647002475e-05, 6.0898558226633984e-05, 7.956054903697144e-05, 4.80968670903199e-05, 3.476307626484116e-05, 3.233622280581405e-05, 4.097520999795124e-05, 1.6048981491512094e-05, 3.4725910431663494e-05, 2.3840743831207534e-05, 4.194630872483221e-05, 3.472531193096608e-05, 2.9240209403218155e-05, 2.5871727972711297e-05, 1.1918039641386187e-05, 1.2189485552920143e-05, 8.254477280067191e-06, 5.343416003103456e-06, 0.0, 4.795714549478586e-06, 6.705621859254362e-06, 9.484831383410081e-06, 2.503719812292549e-05, 1.9037212038371403e-05, 2.448114104715256e-05, 3.2063674685728836e-05, 2.73499598297465e-05, 2.6255716088190032e-05, 2.930473870366029e-05, 2.490020970307041e-05, 2.4037259675477766e-05, 1.8683888229243836e-05, 9.573344744760269e-06, 2.01589736663327e-05, 2.8955116521484698e-05, 1.934869527601605e-05, 2.1111566182648825e-05, 1.0035410663340645e-05, 4.154485944681635e-06, 8.468739061212046e-06, 8.415088253238056e-06, 1.3883239181832948e-06, 0.0, 2.9995080806747692e-06, 1.6303266848611124e-06, 3.714448088730736e-06, 8.976418947425114e-06, 9.729566693747293e-06, 3.3588780313874236e-05, 1.7154466165266127e-05, 1.9646193877372823e-05, 9.475852684603824e-06, 9.763432041631274e-06, 2.5840349706066022e-05, 1.4272109443725072e-05, 2.262309793162043e-05, 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Each element is the combined signal for an hour and the list is over 24 days. Therefore, should have 24 days * 24 hours = 576 elements.
I am interested in the dynamics of the signal of each term over the course of days. However, confounding to this is the baseline changes of the signal within a day. I also have time series for basic terms that capture this baseline signal during a day such as the following.
baseline = [0.0056738537419516195, 0.005420397434626666, 0.005019676698052322, 0.004214006968007205, 0.004143451622795924, 0.00373395198248036, 0.0037080495988714344, 0.0036409523281401525, 0.003919898659196092, 0.004388163261294729, 0.004595501330006892, 0.005097033972892097, 0.0052221285817481335, 0.005184009325081863, 0.005273633551787361, 0.005053393415305126, 0.004444952439008902, 0.004552838940992971, 0.004808237374463801, 0.004895327691624783, 0.005059086256629757, 0.005598114319387153, 0.005952632334681949, 0.005717805004263755, 0.006126432469142252, 0.005592477569387059, 0.004920585487387107, 0.004318038669070883, 0.003877225571288378, 0.0036583898426795327, 0.0037336953886437474, 0.0037760782770061294, 0.0042338376814351954, 0.004192003050341723, 0.0046450557083186645, 0.004900468947653463, 0.005272546953959605, 0.005265723105151999, 0.0052304537716869855, 0.005121826744125637, 0.005224078793461002, 0.00501027352884918, 0.004871995260153345, 0.004863044486978714, 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i think that a good way for me to extract out the signal I am interested in would be to do a spectral analysis on the timeseries for my terms. The high frequencies should be the daily patterns which I want to get rid of and the lower frequencies should be what I'm interested in. I want to somehow 'divide' my observed signal for a term by the baseline daily signal.
This is my baseline's original signal
and this is my term's original signal
and what i'm trying to do is get something like this in a general way without introducing artefacts. i.e remove the ups and downs that happen every day anyway and capture the general trend.
The naive way I thought of doing this is to first generate ffts for both using numpy(below).
baselines fft
term1s fft
and then create a filter like below
fft2 = fft(term1, n=t)
mgft2=abs(fft2)
plot(mgft2[0:t/2+1])
bp = fft2[:]
for i in range(len(bp)):
if i>=22:
bp[i] = 0
ibp = ifft(bp)
but from what i understand that introduces artefacts, changes the magnitudes and I am not sure how to pick a cutting point. I was hoping for some guidance with respect to implementation in numpy on a better way to divide out my baseline frequencies from my term's frequencies.
thanks
Multiplying (or dividing) in the frequency domain is equivalent to convolving in the time domain. In other words a high-pass FIR filter would remove your low-frequency components directly without going into the frequency domain. If you do go to the frequency domain first be aware that simply removing some frequency components and converting back to time will introduce artifacts. FIR filter design is actually based on choosing filters you can multiply by which also meet your desired frequency specs.
All that said, it sounds like you already know the baselines and you could apply your adjustments directly from your known baselines to your data. The point of filtering would be that it would work without knowing the baselines.
It's one thing to give the code, another to give an explanation. If you just want to look at the spectra, try the following:
import numpy as np
import matplotlib.pyplot as plt
hh = np.hanning(len(term1)) # Use a Hann window to deal with spectral leakage
St = np.fft.rfft(hh*term1)
Sb = np.fft.rfft(hh*baseline)
FSample = 1.0/24 # Sampling frequency is 1/24th of a day
deltaF = 1.0/(len(term1)*FSample) # Frequency resolution is 1/capT = 1/(NumSamples*FSample)
faxis = deltaF*np.arange(len(St))
plt.plot(faxis, np.log10(np.abs(np.array([St, Sb]).T)))
See Wikipedia's explanation of spectral leakage and windowing for details on hh. The x-axis of the plot is calibrated in days. There are two thin peaks in the baseline spectrum at 1 and 2 day periodicity, and a broad peak around 1 day in the term1 spectrum.
It wasn't clear to me from your question if the baseline data already had the real data stripped out. If so, I think this shows that there isn't much structure in one that's distinguishable from the other.
Another thing to keep in mind is that Fourier analysis assumes the signal is stationary. Depending on the physics of what you're measuring this assumption may or may not be true. Certainly, the time-domain plot of term1 doesn't look at all stationary, with a wild change in character starting around day 12:
Having said all that, if you have a way to characterize the baseline data, you might be able to apply noise cancelation algorithms like the Widrow-Hoff LMS algorithm. Wikipedia presents a very theoretical overview, I'm not sure where to find a more practical application oriented explanation.
How did you come up with the term1/baseline separation in your example data?
It looks like both your "baseline signal" and your timeseries data contain information overlapping in spectral content. As such, an FFT doesn't provide any extra help in removing the overlapping portions that end up in the same FFT frequency bins.
As an FFT is a linear operator, subtracting in the frequency domain isn't any different from subtracting directly in the time domain.
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0.56, 0.5866666666666669, 0.6133333333333333, 0.6400000000000001, 0.666666666666667, 0.6933333333333334, 0.7200000000000002, 0.746666666666667, 0.7733333333333334, 0.8000000000000003, 0.8266666666666667, 0.8533333333333335, 0.8800000000000003, 0.9066666666666667, 0.9333333333333336, 0.9600000000000004, 0.9866666666666668, 1.0133333333333336, 1.04, 1.0666666666666669, 1.0933333333333337, 1.12, 1.146666666666667, 1.1733333333333333, 1.2000000000000002, 1.226666666666667, 1.2533333333333334, 1.2800000000000002, 1.306666666666667, 1.3333333333333335, 1.3600000000000003, 1.3866666666666667, 1.4133333333333336, 1.4400000000000004, 1.4666666666666668, 1.4933333333333336, 1.52, 1.5466666666666669, 1.5733333333333337, 1.6, 1.626666666666667, 1.6533333333333338, 1.6800000000000002, 1.706666666666667, 1.7333333333333334, 1.7600000000000002, 1.786666666666667, 1.8133333333333335, 1.8400000000000003, 1.8666666666666667, 1.8933333333333335, 1.9200000000000004, 1.9466666666666668, 1.9733333333333336, 2.0, 2.0266666666666673, 2.0533333333333337, 2.08, 2.1066666666666674, 2.1333333333333337, 2.16, 2.1866666666666665, 2.213333333333334, 2.24, 2.2666666666666666, 2.293333333333334, 2.3200000000000003, 2.3466666666666667, 2.373333333333334, 2.4000000000000004, 2.4266666666666667, 2.453333333333334, 2.4800000000000004, 2.506666666666667, 2.533333333333333, 2.5600000000000005, 2.586666666666667, 2.6133333333333333, 2.6400000000000006, 2.666666666666667, 2.6933333333333334, 2.7200000000000006, 2.746666666666667, 2.7733333333333334, 2.8000000000000007, 2.826666666666667, 2.8533333333333335, 2.88, 2.906666666666667, 2.9333333333333336, 2.96, 2.9866666666666672, 3.0133333333333336, 3.04, 3.0666666666666673, 3.0933333333333337, 3.12, 3.1466666666666674, 3.173333333333334, 3.2, 3.2266666666666666, 3.253333333333334, 3.2800000000000002, 3.3066666666666666, 3.333333333333334, 3.3600000000000003, 3.3866666666666667, 3.413333333333334, 3.4400000000000004, 3.466666666666667, 3.493333333333334, 3.5200000000000005, 3.546666666666667, 3.5733333333333333, 3.6000000000000005, 3.626666666666667, 3.6533333333333333, 3.6800000000000006, 3.706666666666667, 3.7333333333333334, 3.7600000000000007, 3.786666666666667, 3.8133333333333335, 3.8400000000000007, 3.866666666666667, 3.8933333333333335, 3.920000000000001, 3.946666666666667, 3.9733333333333336, 4.0, 4.026666666666667, 4.053333333333334, 4.08, 4.106666666666667, 4.133333333333334, 4.16, 4.186666666666667, 4.213333333333334, 4.24, 4.2666666666666675, 4.293333333333334, 4.32, 4.346666666666667, 4.373333333333334, 4.4, 4.426666666666667, 4.453333333333334, 4.48, 4.506666666666667, 4.533333333333334, 4.5600000000000005, 4.586666666666667, 4.613333333333334, 4.640000000000001, 4.666666666666667, 4.693333333333333, 4.720000000000001, 4.746666666666667, 4.773333333333333, 4.800000000000001, 4.826666666666667, 4.8533333333333335, 4.880000000000001, 4.906666666666667, 4.933333333333334, 4.960000000000001, 4.986666666666667, 5.013333333333334, 5.04, 5.066666666666667, 5.093333333333334, 5.12, 5.146666666666667, 5.173333333333334, 5.2, 5.2266666666666675, 5.253333333333334, 5.28, 5.3066666666666675, 5.333333333333334, 5.36, 5.386666666666667, 5.413333333333334, 5.44, 5.466666666666667, 5.493333333333334, 5.5200000000000005, 5.546666666666667, 5.573333333333334, 5.6000000000000005, 5.626666666666667, 5.653333333333334, 5.680000000000001, 5.706666666666667, 5.733333333333333, 5.760000000000001, 5.786666666666667, 5.8133333333333335, 5.840000000000001, 5.866666666666667, 5.8933333333333335, 5.920000000000001, 5.946666666666667, 5.973333333333334]) p_top = np.array([0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 52.872576794481944, 60.36328831067402, 68.58181676019626, 77.54010551102007, 87.24546077586182, 97.71380116903708, 108.97588971792945, 121.1230000000001, 134.28859358309765, 148.68731887019467, 164.5633871885017, 182.18666227169558, 201.83818349095438, 223.81267643614873, 248.42818042617202, 276.03506399135284, 307.0337209116861, 341.87896590372986, 381.1192679598341, 425.3893723626624, 475.5021283556349, 532.4158997232969, 597.4090000000003, 672.0410742621983, 758.533379251738, 859.7165530846363, 979.6593316087284, 1124.0206760913093, 1301.1409742106393, 1519.6640526904762, 1788.227505227972, 2115.468518070886, 2510.024265626171, 2980.5319397027483, 3535.6285264165717, 4183.950889387946, 4934.135917372832, 5794.82045, 4934.420403031789, 4184.487017193709, 3536.3779199568417, 2981.450849155281, 2511.0636810971228, 2116.574298226707, 1789.340587092605, 1520.720465547997, 1302.0718287570105, 1124.752573196128, 980.1205102605351, 859.8428649405212, 758.281365651945, 671.394251505908, 596.3759999999991, 531.0259774866203, 473.7897372355332, 423.3784182587137, 378.7905732211398, 339.2052390738119, 304.05693724550565, 272.8094869362206, 244.93161029019166, 220.00582429999793, 197.7284586919518, 177.8007936067852, 159.92938926933715, 143.86679302868998, 129.3920578431436, 116.3009999999999, 104.39987683440208, 93.51271797907324, 83.48433620575351, 74.20490406269317, 65.59711336145308, 57.63198789171022, 50.29487274165398, 43.58296879663324, 37.48630089022281, 31.9864346061135, 27.059460686269396, 22.676772829659317, 18.800931214057726, 15.390398961980129, 12.405999999999976, 9.808750398065335, 7.556975821856745, 5.609771290609389, 3.930241859207573, 2.4840486871436225, 1.2405555333284073, 0.17249876364223482, -0.7431963631258247, -1.527178008314559, -2.1976614885762182, -2.7685586963324997, -3.2507065483663036, -3.654521247585481, -3.989819778712002, -4.264, -4.483339654609878, -4.653107428963594, -4.778317791591804, -4.863979954625947, -4.914753159149808, -4.934618850520453, -4.92747858548951, -4.8971925826874845, -4.846723689404206, -4.778134576732593, -4.693446251582014, -4.5947146119860145, -4.483931113002393, -4.363044061334697, -4.234, -4.0985235604644705, -3.9574567255996067, -3.811418823406809, -3.661031677747074, -3.507335845802052, -3.3521869245809506, -3.197532929895521, -3.045306521322895, -2.897111875210079, -2.7542247897630006, -2.6179055748822746, -2.4892499037088855, -2.3679261099625046, -2.2528674758007727, -2.142999999999998, -2.03745860943027, -1.936218329497232, -1.8394781167611673, -1.747441997712195, -1.6600861278953287, -1.5769440991709724, -1.4974984903385995, -1.4212457390197035, -1.3480060418851023, -1.277923339237923, -1.2111555085799302, -1.1478331446399963, -1.0878510870420735, -1.030982650405997, -0.9769999999999986, -0.9257060180247605, -0.8770268056960342, -0.8309191877527421, -0.7873394579322428, -0.7462046341344566, -0.707355543280765, -0.6706241363212588, -0.6358440114180867, -0.6028906779941965, -0.5716816224666781, -0.542135986875562, -0.514173025332713, -0.4877142855378665, -0.4626824379381182, -0.43899999999999995, -0.4165943842589561, -0.3954130579480966, -0.37540839142719296, -0.35653259042310337, -0.3387377083468086, -0.3219756460666744, -0.3061981561026408, -0.2913568256017715, -0.27740309673209135, -0.26428826973514485, -0.25196348850713185, -0.24038028798051297, -0.22949466726113874, -0.21926484416543945, -0.20964892669615007, -0.20060488653428893, -0.19209054169683576, -0.18406358448408644, -0.17648157320150093, -0.16930192351620832, -0.1624822410268282, -0.15599398273189863, -0.1498278946257743, -0.143976027663664, -0.1384303016449746, -0.13318251846105061, -0.1282243553559818, -0.12354735885956175, -0.11914296564285694, -0.115, -0.11110306063818942, -0.10743823387446969, -0.10399057427215586, -0.10074506770538746, -0.09768595719983417, -0.09478686404733155, -0.09203198506265693, -0.08941054762959841, -0.0869117305275437, -0.08452465786411881, -0.08223840650397521, -0.08004200111956421, -0.07792441608596867, -0.07587457925344288, -0.07388137031451006, -0.0719336220194075, -0.07002011928032632, -0.06812960289914699, -0.06625080975713306, -0.06437774354263374, -0.062514670658265, -0.06066699908536895, -0.058840093159610946, -0.05703927761570363, -0.055269833348506214, -0.053537002547862714, -0.05184598638412147, -0.05020194624570422, -0.048609299571983254, -0.0470683062510954, -0.04557769671809728, -0.04413616149903207, -0.04274235499024258, -0.04139489568258407, -0.04009236691353722, -0.03883331623326931, -0.037616257023266514, -0.03643968154837971, -0.03530235422607354, -0.03420331195177105, -0.03314152488588915, -0.032107966203906546, -0.031098116639678777, -0.03011436052645273, -0.029159083111374974, -0.02823466944709844, -0.027343502994046268, -0.026487055881413483, -0.025664282278113962, -0.024873704934188466, -0.024113846492361448, -0.023383229629240988, -0.022680377059935815, -0.022003811622373776, -0.02135205612181401, -0.020723633222147094, -0.020117107909086978, -0.01953141316798061, -0.01896567156149227, -0.018419007242562745, -0.017890544187225972, -0.017379406521364806, -0.016884718251067944, -0.016405603478766607, -0.015941186322196526, -0.015490596420519654, -0.01505321655270402, -0.014628780797528938, -0.014217049100448345, -0.01381778138256858, -0.01343073756640109, -0.013055677585210439, -0.012692361374500867, -0.012340548869974452]) def leps(z, D, alpha, r_eq): return (D*(np.exp(-2*alpha*(z - r_eq)) - 2*np.exp(-alpha*(z - r_eq)))) popt, pcov = curve_fit(leps, z_top, p_top) print(popt) plt.figure(figsize=(20,10)) plt.plot(z_top,p_top,label='data') plt.plot(z_top, leps(z_top, *popt), 'r-',label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt)) plt.ylim(-8,1) EDIT: the fit is very poor, even after providing starting parameters. How do you determine the optimal parameters for a good fit?
CGAL detect_features() creates unprotectable features
Based on this example (Remeshing a Polyhedral Domain with Surfaces in the 3D Mesh Generation Manual), I want to mesh a cube-shaped domain with its midplane protected (see the code below). However, make_mesh_3 throws an assertion violation:
CGAL ERROR: assertion violation!
Expr: minimal_size_ > 0 || sq_d > 0
Nevermind the triviality of the example, it's just to show the problem. Based on this discussion, I think the issue is that detect_features creates polylines that intersect each other (perimeter of the midplane intersects the cube edges, and both are added as features).
Are intersecting polyhedra not allowed in the meshdomain? If so, is there a way to access and manipulate the results of detect_features to process conflicting features? I'm trying to figure out how polylines are stored in the mesh domain, but I'm getting nowhere.
Cube with Midplane
// --- External Includes ---
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Mesh_polyhedron_3.h>
#include <CGAL/Polyhedral_mesh_domain_with_features_3.h>
#include <CGAL/Mesh_triangulation_3.h>
#include <CGAL/Mesh_complex_3_in_triangulation_3.h>
#include <CGAL/Mesh_criteria_3.h>
#include <CGAL/make_mesh_3.h>
// CGAL types
namespace cgal {
using Kernel = CGAL::Exact_predicates_inexact_constructions_kernel;
using ConcurrencyTag = CGAL::Sequential_tag;
using PolyhedronSurface = CGAL::Mesh_polyhedron_3<Kernel>::type;
using MeshDomain = CGAL::Polyhedral_mesh_domain_with_features_3<Kernel>;
using Tr = CGAL::Mesh_triangulation_3<MeshDomain,CGAL::Default,ConcurrencyTag>::type;
using Triangulation = CGAL::Mesh_complex_3_in_triangulation_3<Tr>;
using MeshCriteria = CGAL::Mesh_criteria_3<Tr>;
using Point = cgal::MeshDomain::Point_3;
}
int main()
{
// Define points for the cube and midPlane
std::vector<cgal::Point> points =
{
cgal::Point( 1.0, 0.0, 0.0),
cgal::Point( 1.0, 1.0, 0.0),
cgal::Point( 0.0, 1.0, 0.0),
cgal::Point( 0.0, 0.0, 0.0),
cgal::Point( 1.0, 0.0, 1.0),
cgal::Point( 1.0, 1.0, 1.0),
cgal::Point( 0.0, 1.0, 1.0),
cgal::Point( 0.0, 0.0, 1.0),
cgal::Point( 1.0, 0.0, 0.5),
cgal::Point( 1.0, 1.0, 0.5),
cgal::Point( 0.0, 1.0, 0.5),
cgal::Point( 0.0, 0.0, 0.5)
};
// Create polyhedra
cgal::PolyhedronSurface cube, midPlane;
cube.make_triangle( points[0], points[3], points[1]);
cube.make_triangle( points[1], points[3], points[2]);
cube.make_triangle( points[4], points[5], points[7]);
cube.make_triangle( points[5], points[6], points[7]);
cube.make_triangle( points[0], points[4], points[3]);
cube.make_triangle( points[3], points[4], points[7]);
cube.make_triangle( points[1], points[2], points[5]);
cube.make_triangle( points[2], points[6], points[5]);
cube.make_triangle( points[2], points[3], points[6]);
cube.make_triangle( points[3], points[7], points[6]);
cube.make_triangle( points[0], points[1], points[5]);
cube.make_triangle( points[0], points[5], points[4]);
midPlane.make_triangle( points[8], points[9], points[10]);
midPlane.make_triangle( points[8], points[10], points[11]);
// Triangulation
cgal::MeshDomain meshDomain( midPlane, cube );
meshDomain.detect_features();
cgal::MeshCriteria meshCriteria( CGAL::parameters::facet_angle = 30,
CGAL::parameters::edge_size = 0.2 );
auto triangulation = CGAL::make_mesh_3<cgal::Triangulation>( meshDomain,
meshCriteria );
return 0;
}
From here: This surface must be free of intersection. In your case, the bounding box and midplane intersect. If you want to mesh this geometry, those meshes need to be connected together ahead of meshing. If you look at the example in section 3.3.2 here, you see a similar example with an internal geometry that is enclosed away from the bounding box. detect_features() doesn't intersect non-intersecting inputs: it just looks for features in an existing mesh (and adds them to the domain so meshing will respect them).
For your cube it should be easy to construct a polyhedral complex, that is three surfaces, namely the upper and lower part of the cube, and the mid-square. Have a look at this example.
How can I get a more regular surface quadmesh using gmsh?
I want to be able to generate a quadrilateral surface mesh that is highly regular (each face has, as far as possible, the same area) and aligned with the surface boundary.
The following test .geo file simplifies the type of intended use case:
lc = 0.1;
// vertices.
Point(1) = {0, 0, 0, lc};
Point(2) = {0.5, 0, 0, lc};
Point(3) = {1.0, 0, 0, lc};
Point(4) = {1.0, 0.5, 0.5, lc};
Point(5) = {1.0, 1.0, 1.0, lc};
Point(6) = {1.0, 1.5, 0.5, lc};
Point(7) = {1.0, 2.0, 0.0, lc};
Point(8) = {0.5, 2.0, 0.0, lc};
Point(9) = {0.0, 2.0, 0.0, lc};
Point(10) = {0.0, 1.5, 0.5, lc};
Point(11) = {0.0, 1.0, 1.0, lc};
Point(12) = {0.0, 0.5, 0.5, lc};
// curves.
Spline(1) = {1,2,3};
Spline(2) = {3,4,5,6,7};
Spline(3) = {7,8,9};
Spline(4) = {9,10,11,12,1};
Physical Line("bottom") = {1};
Physical Line("top") = {3};
Curve Loop(1) = {2, 3, 4, 1};
//surface.
Transfinite Curve{1} = 20
Transfinite Surface(1) = {2,3,4,1};
Physical Surface("mysurface") = {1};
When I load this .geo file into gmsh gui (v 4.3.0) and run mesh 1D then 2D (Frontal-Delaunay option) and finally 2D recombination (Blossom option) commands the resulting surface mesh is not that regular:
The console log shows:
Info : Meshing 1D...
Info : Meshing curve 1 (Nurb)
Info : Meshing curve 2 (Nurb)
Info : Meshing curve 3 (Nurb)
Info : Meshing curve 4 (Nurb)
Info : Done meshing 1D (0.008326 s)
Info : 70 vertices 74 elements
Info : Meshing 2D...
Info : Meshing surface 1 (Surface, Frontal)
Info : Done meshing 2D (0.013711 s)
Info : 272 vertices 538 elements
Info : Recombining 2D mesh...
Info : Blossom: 665 internal 62 closed
Info : Blossom recombination completed (0.012128 s): 230 quads, 0 triangles, 0 invalid quads, 0 quads with Q < 0.1, avg Q = 0.799983, min Q = 0.502415
Info : Done recombining 2D mesh (0.012205 s)
I suspect this maybe due to my relative inexperience with geo/gmsh. Advice appreciated.
For this example, I would highly recommend the new (experimental) meshing option in GMSH: 9 – packing for parallelograms.
According to the release notes, this option appeared in GMSH at least in 4.1.1:
4.1.2 (January 21, 2019): fixed full-quad subdivision if Mesh.SecondOrderLinear
is set; fixed packing of parallelograms regression in 4.1.1.
With this code:
Mesh.Algorithm = 9; // Packing for Parallelograms
lc = 0.1;
// vertices.
Point(1) = {0, 0, 0, lc};
Point(2) = {0.5, 0, 0, lc};
Point(3) = {1.0, 0, 0, lc};
Point(4) = {1.0, 0.5, 0.5, lc};
Point(5) = {1.0, 1.0, 1.0, lc};
Point(6) = {1.0, 1.5, 0.5, lc};
Point(7) = {1.0, 2.0, 0.0, lc};
Point(8) = {0.5, 2.0, 0.0, lc};
Point(9) = {0.0, 2.0, 0.0, lc};
Point(10) = {0.0, 1.5, 0.5, lc};
Point(11) = {0.0, 1.0, 1.0, lc};
Point(12) = {0.0, 0.5, 0.5, lc};
// curves.
Spline(1) = {1,2,3};
Spline(2) = {3,4,5,6,7};
Spline(3) = {7,8,9};
Spline(4) = {9,10,11,12,1};
Physical Line("bottom") = {1};
Physical Line("top") = {3};
Curve Loop(1) = {2, 3, 4, 1};
Surface(1) = {1};
Recombine Surface{1};
Physical Surface("mysurface") = {1};
I am able to generate the following mesh:
Notice slight modifications to your GEO file:
I removed Transfinite Surfaces
I explicitly setup the meshing algorithm and recombination inside the GEO
which I had to do to make this model work on my version of GMSH.
Inconsistent results in Tensorflow incremental assignment with scatter_update
I am using Tensorflow's scatter_update to set the values of an array in a simple test. I would expect the following code to set all values in the array to 1.0. pt = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] numOne = tf.constant(1) xi = tf.Variable(0) out = tf.Variable(pt) tf.global_variables_initializer().run() xi_ = xi + numOne out_ = tf.scatter_update(out, [xi], [tf.cast(numOne, tf.float32)]) step = tf.group( xi.assign(xi_), out.assign(out_) ) for i in range(15): step.run() print(out.eval()) Instead I get inconsistent results like: [ 1. 1. 1. 0. 1. 1. 1. 1. 1. 1. 1. 0. 1. 1. 1. 1.] Is there some type of locking mechanism that I am missing?
Ops inside tf.group are executed in random order. If you want specific order, you could use tf.control_dependencies or split into two .run calls, ie step1 = xi.assign(xi_) step2 = out.assign(out_) for i in range(15): print(out.eval()) sess.run(step1) sess.run(step2)
Also, using control_dependencies it would be something like: xi_ = xi + numOne with tf.control_dependencies[xi_]: out_ = tf.scatter_update(out, [xi], [tf.cast(numOne, tf.float32)]) with tf.control_dependencies[out_]: step = tf.group( xi.assign(xi_), out.assign(out_) ) for i in range(15): step.run() print(out.eval())
Is it possible to alias multiple names in a numpy record array?
Suppose I construct a numpy record array like this
num_rows = <whatever>
data = np.zeros(
(num_rows,),
dtype={
'names':['apple', 'banana'],
'formats': ['f8', 'f8']
}
Now I can access data either by name or index.
For example, the following are the same:
data['banana'][0]
and
data[0]['banana']
etc.
Is there a way to alias different names?
For example, can I set things up so that there's another name manzana such that
data['manzana']
is the same thing as
data['apple']
?
['offsets' and 'titles' are 2 mechanisms for giving different names to fields]
There is an offset parameter that can function in this way. Usually it is used to split another field into several pieces (e.g. an int into bytes). But it also works with identical fields. In effect it defines several fields with overlapping data.
In [743]: dt=np.dtype({'names':['apple','manzana','banana','guineo'],
'formats':['f8','f8','f8','f8'],
'offsets':[0,0,8,8]})
In [745]: np.zeros((3,),dtype=dt)
Out[745]:
array([(0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0), (0.0, 0.0, 0.0, 0.0)],
dtype={'names':['apple','manzana','banana','guineo'],
'formats':['<f8','<f8','<f8','<f8'],
'offsets':[0,0,8,8], 'itemsize':16})
In [746]: A=np.zeros((3,),dtype=dt)
In [747]: A['banana']=[1,2,3]
In [748]: A
Out[748]:
array([(0.0, 0.0, 1.0, 1.0),
(0.0, 0.0, 2.0, 2.0),
(0.0, 0.0, 3.0, 3.0)],
dtype={'names':['apple','manzana','banana','guineo'], 'formats':['<f8','<f8','<f8','<f8'], 'offsets':[0,0,8,8], 'itemsize':16})
In [749]: A['guineo']
Out[749]: array([ 1., 2., 3.])
In [750]: A['manzana']=[.1,.2,.3]
In [751]: A['apple']
Out[751]: array([ 0.1, 0.2, 0.3])
In [752]: A
Out[752]:
array([(0.1, 0.1, 1.0, 1.0),
(0.2, 0.2, 2.0, 2.0),
(0.3, 0.3, 3.0, 3.0)],
dtype={'names':['apple','manzana','banana','guineo'], 'formats':['<f8','<f8','<f8','<f8'], 'offsets':[0,0,8,8], 'itemsize':16})
There's another dtype parameter, titles that is better suited to your needs, and easier to understand:
http://docs.scipy.org/doc/numpy/reference/arrays.dtypes.html
In [792]: dt1=np.dtype({'names':['apple','banana'],'formats':['f8','f8'], 'titles':['manzana', 'guineo'], 'offsets':[0,8]})
In [793]: A1=np.zeros((3,),dtype=dt1)
In [794]: A1
Out[794]:
array([(0.0, 0.0), (0.0, 0.0), (0.0, 0.0)],
dtype=[(('manzana', 'apple'), '<f8'), (('guineo', 'banana'), '<f8')])
In [795]: A1['apple']=[1,2,3]
In [796]: A1['guineo']=[.1,.2,.3]
In [797]: A1
Out[797]:
array([(1.0, 0.1), (2.0, 0.2), (3.0, 0.3)],
dtype=[(('manzana', 'apple'), '<f8'), (('guineo', 'banana'), '<f8')])
In [798]: A1['banana']
Out[798]: array([ 0.1, 0.2, 0.3])
I have put the answer of #hpaulj into a simple method and share it here in case someone wants to use it.
def add_alias(arr, original, alias):
"""
Adds an alias to the field with the name original to the array arr.
Only one alias per field is allowed.
"""
if arr.dtype.names is None:
raise TypeError("arr must be a structured array. Use add_name instead.")
descr = arr.dtype.descr
try:
index = arr.dtype.names.index(original)
except ValueError:
raise ValueError("arr does not have a field named '" + str(original)
+ "'")
if type(descr[index][0]) is tuple:
raise ValueError("The field " + str(original) +
" already has an alias.")
descr[index] = ((alias, descr[index][0]), descr[index][1])
arr.dtype = np.dtype(descr)
return arr
def add_name(arr, name):
"""
Adds a name to the data of an unstructured array.
"""
if arr.dtype.names is not None:
raise TypeError("arr must not be a structured array. "
+ "Use add_alias instead.")
arr.dtype = np.dtype([(name, arr.dtype.name)])
return arr