Filtering signal: how to restrict filter that last point of output must equal the last point of input - python-3.x

Question

Please help my poor knowledge of signal processing.
I want to smoothen some data. Here is my code:
import numpy as np
from scipy.signal import butter, filtfilt
def testButterworth(nyf, x, y):
b, a = butter(4, 1.5/nyf)
fl = filtfilt(b, a, y)
return fl
if __name__ == '__main__':
positions_recorded = np.loadtxt('original_positions.txt', delimiter='\n')
number_of_points = len(positions_recorded)
end = 10
dt = end/float(number_of_points)
nyf = 0.5/dt
x = np.linspace(0, end, number_of_points)
y = positions_recorded
fl = testButterworth(nyf, x, y)
I am pretty satisfied with results except one point:
it is absolutely crucial to me that the start and end point in returned values equal to the start and end point of input. How can I introduce this restriction?
UPD 15-Dec-14 12:04:
my original data looks like this
Applying the filter and zooming into last part of the graph gives following result:
So, at the moment I just care about the last point that must be equal to original point. I try to append copy of data to the end of original list this way:
the result is as expected even worse.
Then I try to append data this way:
And the slice where one period ends and next one begins, looks like that:

Answer 1

To do this, you're always going to cheat somehow, since the true filter applied to the true data doesn't behave the way you require.
One of the best ways to cheat with your data is to assume it's periodic. This has the advantages that: 1) it's consistent with the data you actually have and all your changing is to append data to the region you don't know about (so assuming it's periodic as as reasonable as anything else -- although may violate some unstated or implicit assumptions); 2) the result will be consistent with your filter.
You can usually get by with this by appending copies of your data to the beginning and end of your real data, or just small pieces, depending on your filter.
Since the FFT assumes that the data is periodic anyway, that's often a quick and easy approach, and is fully accurate (whereas concatenating the data is an estimation of an infinitely periodic waveform). Here's an example of the FFT approach for a step filter.
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(0, 128)
y = (np.sin(.22*(x+10))>0).astype(np.float)
# filter
y2 = np.fft.fft(y)
f0 = np.fft.fftfreq(len(x))
y2[(f0<-.25) | (f0>.25)] = 0
y3 = abs(np.fft.ifft(y2))
plt.plot(x, y)
plt.plot(x, y3)
plt.xlim(-10, 140)
plt.ylim(-.1, 1.1)
plt.show()
Note how the end points bend towards each other at either end, even though this is not consistent with the periodicity of the waveform (since the segments at either end are very truncated). This can also be seen by adjusting waveform so that the ends are the same (here I used x+30 instead of x+10, and here the ends don't need to bend to match-up so they stay at level with the end of the data.
Note, also, to have the endpoints actually be exactly equal you would have to extend this plot by one point (at either end), since it periodic with exactly the wavelength of the original waveform. Doing this is not ad hoc though, and the result will be entirely consistent with your analysis, but just representing one extra point of what was assumed to be infinite repeats all along.
Finally, this FFT trick works best with waveforms of length 2n. Other lengths may be zero padded in the FFT. In this case, just doing concatenations to either end as I mentioned at first might be the best way to go.

Answer 2

The question is how to filter data and require that the left endpoint of the filtered result matches the left endpoint of the data, and same for the right endpoint. (That is, in general, the filtered result should be close to most of the data points, but not necessarily exactly match any of them, but what if you need a match at both endpoints?)
To make the filtered result exactly match the endpoints of a curve, one could add a padding of points at either end of the curve and adjust the y-position of this padding so that the endpoints of the valid part of the filter exactly matched the end points of the original data (without the padding).
In general, this can be done by either iterating towards a solution, adjusting the padding y-position until the ends line up, or by calculating a few values and then interpolating to determine the y-positions that would be required for the matched endpoints. I'll do the second approach.
Here's the code I used, where I simulated the data as a sine wave with two flat pieces on either side (note, that these flat pieces are not the padding, but I'm just trying to make data that looks a bit like the OPs).
import numpy as np
from scipy.signal import butter, filtfilt
import matplotlib.pyplot as plt
#### op's code
def testButterworth(nyf, x, y):
#b, a = butter(4, 1.5/nyf)
b, a = butter(4, 1.5/nyf)
fl = filtfilt(b, a, y)
return fl
def do_fit(data):
positions_recorded = data
#positions_recorded = np.loadtxt('original_positions.txt', delimiter='\n')
number_of_points = len(positions_recorded)
end = 10
dt = end/float(number_of_points)
nyf = 0.5/dt
x = np.linspace(0, end, number_of_points)
y = positions_recorded
fx = testButterworth(nyf, x, y)
return fx
### simulate some data (op should have done this too!)
def sim_data():
t = np.linspace(.1*np.pi, (2.-.1)*np.pi, 100)
y = np.sin(t)
c = np.ones(10, dtype=np.float)
z = np.concatenate((c*y[0], y, c*y[-1]))
return z
### code to find the required offset padding
def fit_with_pads(v, data, n=1):
c = np.ones(n, dtype=np.float)
z = np.concatenate((c*v[0], data, c*v[1]))
fx = do_fit(z)
return fx
def get_errors(data, fx):
n = (len(fx)-len(data))//2
return np.array((fx[n]-data[0], fx[-n]-data[-1]))
def vary_padding(data, span=.005, n=100):
errors = np.zeros((4, n)) # Lpad, Rpad, Lerror, Rerror
offsets = np.linspace(-span, span, n)
for i in range(n):
vL, vR = data[0]+offsets[i], data[-1]+offsets[i]
fx = fit_with_pads((vL, vR), data, n=1)
errs = get_errors(data, fx)
errors[:,i] = np.array((vL, vR, errs[0], errs[1]))
return errors
if __name__ == '__main__':
data = sim_data()
fx = do_fit(data)
errors = vary_padding(data)
plt.plot(errors[0], errors[2], 'x-')
plt.plot(errors[1], errors[3], 'o-')
oR = -0.30958
oL = 0.30887
fp = fit_with_pads((oL, oR), data, n=1)[1:-1]
plt.figure()
plt.plot(data, 'b')
plt.plot(fx, 'g')
plt.plot(fp, 'r')
plt.show()
Here, for the padding I only used a single point on either side (n=1). Then I calculate the error for a range of values shifting the padding up and down from the first and last data points.
For the plots:
First I plot the offset vs error (between the fit and the desired data value). To find the offset to use, I just zoomed in on the two lines to find the x-value of the y zero crossing, but to do this more accurately, one could calculate the zero crossing from this data:
Here's the plot of the original "data", the fit (green) and the adjusted fit (red):
and zoomed in the RHS:
The important point here is that the red (adjusted fit) and blue (original data) endpoints match, even though the pure fit doesn't.
Is this a valid approach? Of the various options, this seems the most reasonable since one isn't usually making any claims about the data that isn't being shown, and also for show region has an accurately applied filter. For example, FFTs usually assume the data is zero or periodic beyond the boundaries. Certainly, though, to be precise one should explain what was done.