Finding the time in which a specific value is reached in time-series data when peaks are found - algorithm

Question

I would like to find the time instant at which a certain value is reached in a time-series data with noise. If there are no peaks in the data, I could do the following in MATLAB.
Code from here
% create example data
d=1:100;
t=d/100;
ts = timeseries(d,t);
% define threshold
thr = 55;
data = ts.data(:);
time = ts.time(:);
ind = find(data>thr,1,'first');
time(ind) %time where data>threshold
But when there is noise, I am not sure what has to be done.
In the time-series data plotted in the above image I want to find the time instant at which the y-axis value 5 is reached. The data actually stabilizes to 5 at t>=100 s. But due to the presence of noise in the data, we see a peak that reaches 5 somewhere around 20 s . I would like to know how to detect e.g 100 seconds as the right time and not 20 s . The code posted above will only give 20 s as the answer. I
saw a post here that explains using a sliding window to find when the data equilibrates. However, I am not sure how to implement the same. Suggestions will be really helpful.
The sample data plotted in the above image can be found here
Suggestions on how to implement in Python or MATLAB code will be really helpful.
EDIT:
I don't want to capture when the peak (/noise/overshoot) occurs. I want to find the time when equilibrium is reached. For example, around 20 s the curve rises and dips below 5. After ~100 s the curve equilibrates to a steady-state value 5 and never dips or peaks.

Answer 1

Precise data analysis is a serious business (and my passion) that involves a lot of understanding of the system you are studying. Here are comments, unfortunately I doubt there is a simple nice answer to your problem at all -- you will have to think about it. Data analysis basically always requires "discussion".
First to your data and problem in general:
When you talk about noise, in data analysis this means a statistical random fluctuation. Most often Gaussian (sometimes also other distributions, e.g. Poission). Gaussian noise is a) random in each bin and b) symmetric in negative and positive direction. Thus, what you observe in the peak at ~20s is not noise. It has a very different, very systematic and extended characteristics compared to random noise. This is an "artifact" that must have a origin, but of which we can only speculate here. In real-world applications, studying and removing such artifacts is the most expensive and time-consuming task.
Looking at your data, the random noise is negligible. This is very precise data. For example, after ~150s and later there are no visible random fluctuations up to fourth decimal number.
After concluding that this is not noise in the common sense it could be a least two things: a) a feature of the system you are studying, thus, something where you could develop a model/formula for and which you could "fit" to the data. b) a characteristics of limited bandwidth somewhere in the measurement chain, thus, here a high-frequency cutoff. See e.g. https://en.wikipedia.org/wiki/Ringing_artifacts . Unfortunately, for both, a and b, there are no catch-all generic solutions. And your problem description (even with code and data) is not sufficient to propose an ideal approach.
After spending now ~one hour on your data and making some plots. I believe (speculate) that the extremely sharp feature at ~10s cannot be a "physical" property of the data. It simply is too extreme/steep. Something fundamentally happened here. A guess of mine could be that some device was just switched on (was off before). Thus, the data before is meaningless, and there is a short period of time afterwards to stabilize the system. There is not really an alternative in this scenario but to entirely discard the data until the system has stabilized at around 40s. This also makes your problem trivial. Just delete the first 40s, then the maximum becomes evident.
So what are technical solutions you could use, please don't be too upset that you have to think about this yourself and assemble the best possible solution for your case. I copied your data in two numpy arrays x and y and ran the following test in python:
Remove unstable time
This is the trivial solution -- I prefer it.
plt.figure()
plt.xlabel('time')
plt.ylabel('signal')
plt.plot(x, y, label="original")
y_cut = y
y_cut[:40] = 0
plt.plot(x, y_cut, label="cut 40s")
plt.legend()
plt.grid()
plt.show()
Note carry on reading below only if you are a bit crazy (about data).
Sliding window
You mentioned "sliding window" which is best suited for random noise (which you don't have) or periodic fluctuations (which you also don't really have). Sliding window just averages over consecutive bins, averaging out random fluctuations. Mathematically this is a convolution.
Technically, you can actually solve your problem like this (try even larger values of Nwindow yourself):
Nwindow=10
y_slide_10 = np.convolve(y, np.ones((Nwindow,))/Nwindow, mode='same')
Nwindow=20
y_slide_20 = np.convolve(y, np.ones((Nwindow,))/Nwindow, mode='same')
Nwindow=30
y_slide_30 = np.convolve(y, np.ones((Nwindow,))/Nwindow, mode='same')
plt.xlabel('time')
plt.ylabel('signal')
plt.plot(x,y, label="original")
plt.plot(x,y_slide_10, label="window=10")
plt.plot(x,y_slide_20, label='window=20')
plt.plot(x,y_slide_30, label='window=30')
plt.legend()
#plt.xscale('log') # useful
plt.grid()
plt.show()
Thus, technically you can succeed to suppress the initial "hump". But don't forget this is a hand-tuned and not general solution...
Another caveat of any sliding window solution: this always distorts your timing. Since you average over an interval in time depending on rising or falling signals your convoluted trace is shifted back/forth in time (slightly, but significantly). In your particular case this is not a problem since the main signal region has basically no time-dependence (very flat).
Frequency domain
This should be the silver bullet, but it also does not work well/easily for your example. The fact that this doesn't work better is the main hint to me that the first 40s of data are better discarded.... (i.e. in a scientific work)
You can use fast Fourier transform to inspect your data in frequency-domain.
import scipy.fft
y_fft = scipy.fft.rfft(y)
# original frequency domain plot
plt.plot(y_fft, label="original")
plt.xlabel('frequency')
plt.ylabel('signal')
plt.yscale('log')
plt.show()
The structure in frequency represent the features of your data. The peak a zero is the stabilized region after ~100s, the humps are associated to (rapid) changes in time. You can now play around and change the frequency spectrum (--> filter) but I think the spectrum is so artificial that this doesn't yield great results here. Try it with other data and you may be very impressed! I tried two things, first cut high-frequency regions out (set to zero), and second, apply a sliding-window filter in frequency domain (sparing the peak at 0, since this cannot be touched. Try and you know why).
# cut high-frequency by setting to zero
y_fft_2 = np.array(y_fft)
y_fft_2[50:70] = 0
# sliding window in frequency
Nwindow = 15
Start = 10
y_fft_slide = np.array(y_fft)
y_fft_slide[Start:] = np.convolve(y_fft[Start:], np.ones((Nwindow,))/Nwindow, mode='same')
# frequency-domain plot
plt.plot(y_fft, label="original")
plt.plot(y_fft_2, label="high-frequency, filter")
plt.plot(y_fft_slide, label="frequency sliding window")
plt.xlabel('frequency')
plt.ylabel('signal')
plt.yscale('log')
plt.legend()
plt.show()
Converting this back into time-domain:
# reverse FFT into time-domain for plotting
y_filtered = scipy.fft.irfft(y_fft_2)
y_filtered_slide = scipy.fft.irfft(y_fft_slide)
# time-domain plot
plt.plot(x[:500], y[:500], label="original")
plt.plot(x[:500], y_filtered[:500], label="high-f filtered")
plt.plot(x[:500], y_filtered_slide[:500], label="frequency sliding window")
# plt.xscale('log') # useful
plt.grid()
plt.legend()
plt.show()
yields
There are apparent oscillations in those solutions which make them essentially useless for your purpose. This leads me to my final exercise to again apply a sliding-window filter on the "frequency sliding window" time-domain
# extra time-domain sliding window
Nwindow=90
y_fft_90 = np.convolve(y_filtered_slide, np.ones((Nwindow,))/Nwindow, mode='same')
# final time-domain plot
plt.plot(x[:500], y[:500], label="original")
plt.plot(x[:500], y_fft_90[:500], label="frequency-sliding window, slide")
# plt.xscale('log') # useful
plt.legend()
plt.show()
I am quite happy with this result, but it still has very small oscillations and thus does not solve your original problem.
Conclusion
How much fun. One hour well wasted. Maybe it is useful to someone. Maybe even to you Natasha. Please be not mad a me...

Answer 2

Let's assume your data is in data variable and time indices are in time. Then
import numpy as np
threshold = 0.025
stable_index = np.where(np.abs(data[-1] - data) > threshold)[0][-1] + 1
print('Stabilizes after', time[stable_index], 'sec')
Stabilizes after 96.6 sec
Here data[-1] - data is a difference between last value of data and all the data values. The assumption here is that the last value of data represents the equilibrium point.
np.where( * > threshold )[0] are all the indices of values of data which are greater than the threshold, that is still not stabilized. We take only the last index. The next one is where time series is considered stabilized, hence the + 1.

Answer 3

If you're dealing with deterministic data which is eventually converging monotonically to some fixed value, the problem is pretty straightforward. Your last observation should be the closest to the limit, so you can define an acceptable tolerance threshold relative to that last data point and scan your data from back to front to find where you exceeded your threshold.
Things get a lot nastier once you add random noise into the picture, particularly if there is serial correlation. This problem is common in simulation modeling(see (*) below), and is known as the issue of initial bias. It was first identified by Conway in 1963, and has been an active area of research since then with no universally accepted definitive answer on how to deal with it. As with the deterministic case, the most widely accepted answers approach the problem starting from the right-hand side of the data set since this is where the data are most likely to be in steady state. Techniques based on this approach use the end of the dataset to establish some sort of statistical yardstick or baseline to measure where the data start looking significantly different as observations get added by moving towards the front of the dataset. This is greatly complicated by the presence of serial correlation.
If a time series is in steady state, in the sense of being covariance stationary then a simple average of the data is an unbiased estimate of its expected value, but the standard error of the estimated mean depends heavily on the serial correlation. The correct standard error squared is no longer s2/n, but instead it is (s2/n)*W where W is a properly weighted sum of the autocorrelation values. A method called MSER was developed in the 1990's, and avoids the issue of trying to correctly estimate W by trying to determine where the standard error is minimized. It treats W as a de-facto constant given a sufficiently large sample size, so if you consider the ratio of two standard error estimates the W's cancel out and the minimum occurs where s2/n is minimized. MSER proceeds as follows:
Starting from the end, calculate s2 for half of the data set to establish a baseline.
Now update the estimate of s2 one observation at a time using an efficient technique such as Welford's online algorithm, calculate s2/n where n is the number of observations tallied so far. Track which value of n yields the smallest s2/n. Lather, rinse, repeat.
Once you've traversed the entire data set from back to front, the n which yielded the smallest s2/n is the number of observations from the end of the data set which are not detectable as being biased by the starting conditions.
Justification - with a sufficiently large baseline (half your data), s2/n should be relatively stable as long as the time series remains in steady state. Since n is monotonically increasing, s2/n should continue decreasing subject to the limitations of its variability as an estimate. However, once you start acquiring observations which are not in steady state the drift in mean and variance will inflate the numerator of s2/n. Hence the minimal value corresponds to the last observation where there was no indication of non-stationarity. More details can be found in this proceedings paper. A Ruby implementation is available on BitBucket.
Your data has such a small amount of variation that MSER concludes that it is still converging to steady state. As such, I'd advise going with the deterministic approach outlined in the first paragraph. If you have noisy data in the future, I'd definitely suggest giving MSER a shot.
(*) - In a nutshell, a simulation model is a computer program and hence has to have its state set to some set of initial values. We generally don't know what the system state will look like in the long run, so we initialize it to an arbitrary but convenient set of values and then let the system "warm up". The problem is that the initial results of the simulation are not typical of the steady state behaviors, so including that data in your analyses will bias them. The solution is to remove the biased portion of the data, but how much should that be?