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  1. PO
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    <p>One of the links you provided has the right idea (in fact I am doing pretty much the same thing here)</p> <pre><code>import numpy as np import matplotlib.pyplot as plt from scipy.signal import correlate a,b, N = 0, 10, 1000 #Boundaries, datapoints shift = -3 #Shift, note 3/10 of L = b-a x = np.linspace(a,b,N) x1 = 1*x + shift time = np.arange(1-N,N) #Theoritical definition, time is centered at 0 y1 = sum([np.sin(2*np.pi*i*x/b) for i in range(1,5)]) y2 = sum([np.sin(2*np.pi*i*x1/b) for i in range(1,5)]) #Really only helps with large irregular data, try it # y1 -= y1.mean() # y2 -= y2.mean() # y1 /= y1.std() # y2 /= y2.std() cross_correlation = correlate(y1,y2) shift_calculated = time[cross_correlation.argmax()] *1.0* b/N y3 = sum([np.sin(2*np.pi*i*(x1-shift_calculated)/b) for i in range(1,5)]) print "Preset shift: ", shift, "\nCalculated shift: ", shift_calculated plt.plot(x,y1) plt.plot(x,y2) plt.plot(x,y3) plt.legend(("Regular", "Shifted", "Recovered")) plt.savefig("SO_timeshift.png") plt.show() </code></pre> <p>This has the following output:</p> <pre><code>Preset shift: -3 Calculated shift: -2.99 </code></pre> <p><img src="https://i.stack.imgur.com/CPG0R.png" alt="enter image description here"></p> <p>It might be necessary to check </p> <ol> <li><a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.correlate.html" rel="nofollow noreferrer">Scipy Correlate</a></li> <li><a href="http://en.wikipedia.org/wiki/Cross-correlation#Time_delay_analysis" rel="nofollow noreferrer">Time Delay Analaysis</a></li> </ol> <p>Note that the the argmax() of the correlation shows the position of the alignment, it has to be scaled by the length of <code>b-a = 10-0 = 10</code> and N to get the actual value.</p> <p>Checking the source of correlate <a href="https://github.com/scipy/scipy/blob/v0.11.0/scipy/signal/signaltools.py#L55" rel="nofollow noreferrer">Source</a> it is not entirely obvious what the imported function from sigtools behaves. For large datasets circular correlation (via Fast Fourier Transforms) is much faster than the straight-forward method. I suspect this is what is implemented in sigtools but I cannot tell for sure. A search for the file in my python2.7 folder only returned the compiled C pyd file.</p>
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    1. POEstimating small time shift between two time series
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    1. USarynaq
    1. USarynaq
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    1. COHave you experimented with this as your shift gets really small? For example, what if `shift = (x[1]-x[0])/4.0`. This is a more realistic test when compared with OP's request ("the time shift is smaller than the resolution of the data")
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      1. USmgilson
    2. COIt fails when the shift is smaller than the resolution of the data as the resolution of the time used to find the shift is the same as the datas. Didn't consider that. I wonder what the OPs data looks like when it is downsampled. Else it has to be interpolated.
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