Related
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.
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...
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.
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?
I am using the FMOD library to apply FFT to an audio stream, providing me with a constantly updating fixed number of frequency bins. Each bin represents an equal frequency range, with a value between 0 and 1 to represent the intensity of this range from the processed audio. FMOD documentation states that these values can be represented in decibels, where val is the value between 0 and 1:
Decibels = 10.0f * (float)log10(val) * 2.0f
I am attempting to make an automated strobe-like beat detecting visualisation. So far, I test at a constant interval to see whether a particular frequency bin's intensity value surpasses a specified boundary - if this is the case, the strobe flashes. Although a pretty crude way of doing this, it works fairly effectively for my requirements.
However, this specified boundary only works effectively when the system/music player's volumes are maximum. When I reduce either volume, the strobe sensitivity is reduced and becomes either very inaccurate or stops flashing completely. I assume that I need to normalise the data in some way so analysis is performed independent of volume, though by scaling the data by 1/value of largest bin the largest value is always maxed out. This surpasses the specified boundary permanently, causing the strobe to flash indefinitely. I can't think how else this can be achieved and have been on a mental block for days - any help or a point in the right direction would be greatly appreciated!
Normalise over a a longer scale. You'll need something like an envelope follower with a long release time.
If you search for 'compressor' source code, or automatic gain control something will definitely turn up.
But broadly in pseudo C++, and working on your incoming audio (the time-domain signal before the FFT):
auto instant_level = std::abs(signal);
peak_level *= 0.99f;
peak_level = peak_level > instant_level ? peak_level : instant_level;
Now peak_level decays slowly over time. And you can use this to calculate a gain factor to normalize your incoming audio. Adjust the 0.99f as required for a sensible decay time and for the correct sample rate.
There's also a Signal Processing stack exchange site where you'll get quicker answers to these kinds of questions (although occasionally with an almost incomprehensible piece of algebra attached :) )
iOS has an issue recording through some USB audio devices. It cannot be reliably reproduced (happens every 1 in ~2000-3000 records in batches and silently disappears), and we currently manually check our audio for any recording issues. It results in small numbers of samples (1-20) being shifted by a small number that sounds like a sort of 'crackle'.
They look like this:
closer:
closer:
another, single sample error elsewhere in the same audio file:
The question is, how can these be algorithmically be detected (assuming direct access to samples) whilst not triggering false positives on high frequency audio with waveforms like this:
Bonus points: after determining as many errors as possible, how can the audio be 'fixed'?
Dirty audio file - pictured
Another dirty audio file
Clean audio with valid high frequency - pictured
More bonus points: what could be causing this issue in the iOS USB audio drivers/hardware (assuming it is there).
I do not think there is an out of the box solution to find the disturbances, but here is one (non standard) way of tackling the problem. Using this, I could find most intervals and I only got a small number of false positives, but the algorithm could certainly use some fine tuning.
My idea is to find the start and end point of the deviating samples. The first step should be to make these points stand out more clearly. This can be done by taking the logarithm of the data and taking the differences between consecutive values.
In MATLAB I load the data (in this example I use dirty-sample-other.wav)
y1 = wavread('dirty-sample-pictured.wav');
y2 = wavread('dirty-sample-other.wav');
y3 = wavread('clean-highfreq.wav');
data = y2;
and use the following code:
logdata = log(1+data);
difflogdata = diff(logdata);
So instead of this plot of the original data:
we get:
where the intervals we are looking for stand out as a positive and negative spike. For example zooming in on the largest positive value in the plot of logarithm differences we get the following two figures. One for the original data:
and one for the difference of logarithms:
This plot could help with finding the areas manually but ideally we want to find them using an algorithm. The way I did this was to take a moving window of size 6, computing the mean value of the window (of all points except the minimum value), and compare this to the maximum value. If the maximum point is the only point that is above the mean value and at least twice as large as the mean it is counted as a positive extreme value.
I then used a threshold of counts, at least half of the windows moving over the value should detect it as an extreme value in order for it to be accepted.
Multiplying all points with (-1) this algorithm is then run again to detect the minimum values.
Marking the positive extremes with "o" and negative extremes with "*" we get the following two plots. One for the differences of logarithms:
and one for the original data:
Zooming in on the left part of the figure showing the logarithmic differences we can see that most extreme values are found:
It seems like most intervals are found and there are only a small number of false positives. For example running the algorithm on 'clean-highfreq.wav' I only find one positive and one negative extreme value.
Single values that are falsely classified as extreme values could perhaps be weeded out by matching start and end-points. And if you want to replace the lost data you could use some kind of interpolation using the surrounding data-points, perhaps even a linear interpolation will be good enough.
Here is the MATLAB-code I used:
function test20()
clc
clear all
y1 = wavread('dirty-sample-pictured.wav');
y2 = wavread('dirty-sample-other.wav');
y3 = wavread('clean-highfreq.wav');
data = y2;
logdata = log(1+data);
difflogdata = diff(logdata);
figure,plot(data),hold on,plot(data,'.')
figure,plot(difflogdata),hold on,plot(difflogdata,'.')
figure,plot(data),hold on,plot(data,'.'),xlim([68000,68200])
figure,plot(difflogdata),hold on,plot(difflogdata,'.'),xlim([68000,68200])
k = 6;
myData = difflogdata;
myPoints = findPoints(myData,k);
myData2 = -difflogdata;
myPoints2 = findPoints(myData2,k);
figure
plotterFunction(difflogdata,myPoints>=k,'or')
hold on
plotterFunction(difflogdata,myPoints2>=k,'*r')
figure
plotterFunction(data,myPoints>=k,'or')
hold on
plotterFunction(data,myPoints2>=k,'*r')
end
function myPoints = findPoints(myData,k)
iterationVector = k+1:length(myData);
myPoints = zeros(size(myData));
for i = iterationVector
subVector = myData(i-k:i);
meanSubVector = mean(subVector(subVector>min(subVector)));
[maxSubVector, maxIndex] = max(subVector);
if (sum(subVector>meanSubVector) == 1 && maxSubVector>2*meanSubVector)
myPoints(i-k-1+maxIndex) = myPoints(i-k-1+maxIndex) +1;
end
end
end
function plotterFunction(allPoints,extremeIndices,markerType)
extremePoints = NaN(size(allPoints));
extremePoints(extremeIndices) = allPoints(extremeIndices);
plot(extremePoints,markerType,'MarkerSize',15),
hold on
plot(allPoints,'.')
plot(allPoints)
end
Edit - comments on recovering the original data
Here is a slightly zoomed out view of figure three above: (the disturbance is between 6.8 and 6.82)
When I examine the values, your theory about the data being mirrored to negative values does not seem to fit the pattern exactly. But in any case, my thought about just removing the differences is certainly not correct. Since the surrounding points do not seem to be altered by the disturbance, I would probably go back to the original idea of not trusting the points within the affected region and instead using some sort of interpolation using the surrounding data. It seems like a simple linear interpolation would be a quite good approximation in most cases.
To answer the question of why it happens -
A USB audio device and host are not clock synchronous - that is to say that the host cannot accurately recover the relationship between the host's local clock and the word-clock of the ADC/DAC on the audio interface. Various techniques do exist for clock-recovery with various degrees of effectiveness. To add to the problem, the bus clock is likely to be unrelated to either of the two audio clocks.
Whilst you might imagine this not to be too much of a concern for audio receive - audio capture callbacks could happen when there is data - audio interfaces are usually bi-directional and the host will be rendering audio at regular interval, which the other end is potentially consuming at a slightly different rate.
In-between are several sets of buffers, which can over- or under-run, which is what looks to be happening here; the interval between it happening certainly seems about right.
You might find that changing USB audio device to one built around a different chip-set (or, simply a different local oscillator) helps.
As an aside both IEEE1394 audio and MPEG transport streams have the same clock recovery requirement. Both of them solve the problem with by embedding a local clock reference packet into the serial bitstream in a very predictable way which allows accurate clock recovery on the other end.
I think the following algorithm can be applied to samples in order to determine a potential false positive:
First, scan for high amount of high frequency, either via FFT'ing the sound block by block (256 values maybe), or by counting the consecutive samples above and below zero. The latter should keep track of maximum consecutive above zero, maximum consecutive below zero, the amount of small transitions around zero and the current volume of the block (0..1 as Audacity displays it). Then, if the maximum consecutive is below 5 (sampling at 44100, and zeroes be consecutive, while outstsanding samples are single, 5 responds to 4410Hz frequency, which is pretty high), or the sum of small transitions' lengths is above a certain value depending on maximum consecutive (I believe the first approximation would be 3*5*block size/distance between two maximums, which roughly equates to period of the loudest FFT frequency. Also it should be measured both above and below threshold, as we can end up with an erroneous peak, which will likely be detected by difference between main tempo measured on below-zero or above-zero maximums, also by std-dev of peaks. If high frequency is dominant, this block is eligible only for zero-value testing, and a special means to repair the data will be needed. If high frequency is significant, that is, there is a dominant low frequency detected, we can search for peaks bigger than 3.0*high frequency volume, as well as abnormal zeroes in this block.
Also, your gaps seem to be either highly extending or plain zero, with high extends to be single errors, and zero errors range from 1-20. So, if there is a zero range with values under 0.02 absolute value, which is directly surrounded by values of 0.15 (a variable to be finetuned) or higher absolute value AND of the same sign, count this point as an error. Single values that stand out can be detected if you calculate 2.0*(current sample)-(previous sample)-(next sample) and if it's above a certain threshold (0.1+high frequency volume, or 3.0*high frequency volume, whichever is bigger), count this as an error and average.
What to do with zero gaps found - we can copy values from 1 period backwards and 1 period forwards (averaging), where "period" is of the most significant frequency of the FFT of the block. If the "period" is smaller than the gap (say we've detected a gap of zeroes in a high-pitched part of the sound), use two or more periods, so the source data will all be valid (in this case, no averaging can be done, as it's possible that the signal 2 periods forward from the gap and 2 periods back will be in counterphase). If there are more than one frequency of about equal amplitude, we can plain sample these with correct phases, cutting the rest of less significant frequencies altogether.
The outstanding sample should IMO just be averaged by 2-4 surrounding samples, as there seems to be only a single sample ever encountered in your sound files.
The discrete wavelet transform (DWT) may be the solution to your problem.
A FFT calculation is not very useful in your case since its an average representation of relative frequency content over the entire duration of the signal, and thus impossible to detect momentary changes. The dicrete short time frequency transform (STFT) tries to tackle this by computing the DFT for short consecutive time-blocks of the signal, the length of which is determine by the length (and shape) of a window, but since the resolution of the DFT is dependent on the data/block-length, there is a trade-off between resolution in freqency OR in time, and finding this magical fixed window-size can be tricky!
What you want is a time-frequency analysis method with good time resolution for high-frequency events, and good frequency resolution for low-frequency events... Enter the discrete wavelet transform!
There are numerous wavelet transforms for different applications and as you might expect, it's computationally heavy. The DWT may not be practical solution to your problem, but it's worth considering. Good luck with your problem. Some friday-evening reading:
http://klapetek.cz/wdwt.html
http://etd.lib.fsu.edu/theses/available/etd-11242003-185039/unrestricted/09_ds_chapter2.pdf
http://en.wikipedia.org/wiki/Wavelet_transform
http://en.wikipedia.org/wiki/Discrete_wavelet_transform
You can try the following super-simple approach (maybe it's enough):
Take each point in your wave-form and subtract its predecessor (look at the changes from one point to the next).
Look at the distribution of these changes and find their standard deviation.
If any given difference is beyond X times this standard deviation (either above or below), flag it as a problem.
Determine the best value for X by playing with it and seeing how well it performs.
Most "problems" should come as a pair of two differences beyond your cutoff, one going up, and one going back down.
To stick with the super-simple approach, you can then fix the data by just interpolating linearly between the last good point before your problem-section and the first good point after. (Make sure you don't just delete the points as this will influence (raise) the pitch of your audio.)
My bike computer can show me various figures such as distance travelled, time elapsed, max speed, average speed, current speed etc. I usually have it set to display the current and average speeds.
You can reset the distance and time (both together) at any point; the max and average speeds are calculated since the last reset. The distance is taken from the wheel sensor (you have to calibrate it initially to tell it the circumference of your wheel) and the time is from its own real-time clock.
Now, quite often while I am cycling along, I will be going at well above the displayed average speed and yet the average speed shown will go down. As a concrete example, this evening I was cycling home and my current speed was holding steady at 19.5 mph; my average was showing 12.6 mph and as I looked at it, it clicked downwards to 12.5.
What I'm trying to work out is what kind of bizarre averaging algorithm it is using that can give this effect. I can't believe it's doing any kind of fancy stuff other than total distance / total time. I guess it must be some sort of rounding / boundary condition but I can't work out what. Any suggestions?
[I asked this around the office at work but nobody had any ideas other than that I should stop worrying about these sorts of details! Hey, I have to think about something when I'm cycling, it's 9 miles each way...]
I'm going to guess that it has a history of a certain number of data points and displays the average over them. As time goes on the older points are pushed off.
If you were going faster at the point far enough back to be the end of the history pushing off a point will lower your average.
It's not a running average, it's supposed to be the average for the whole trip, right? At least that's what I always assumed mine was doing.
I've noticed that effect too. My theory is that both the clock and the distance counter it uses for the average have a fairly low resolution, so sometimes the clock counter ticks up while the distance counter stays steady, and you get the dip. For Example:
dist time spd
8.5 40.1 12.72
8.5 40.2 12.69
If they are using an integer processor and fixed point, truncation would make the drop appear even larger
It's really a motivational technique.
It probably uses something similar to the Remaining time estimation algorithm.
It's timing between rotations of the wheel but it can easily miss a pass of the magnet over the sensor because of a bump in the road or noise.
So you measure a speed half the correct value for that one data point, it then does a running average so that bad point pollutes the speed for the next few revolutions.
The system needs to sample at some (probably constant) rate.
In order to compute a moving average it only stores at most N datapoints.
So in order to update the average it must drop one of its stored points to get a new average, and if the dropped point was faster than your current speed, the moving average would drop.
In many applications, we have some progress bar for a file download, for a compression task, for a search, etc. We all often use progress bars to let users know something is happening. And if we know some details like just how much work has been done and how much is left to do, we can even give a time estimate, often by extrapolating from how much time it's taken to get to the current progress level.
(source: jameslao.com)
But we've also seen programs which this Time Left "ETA" display is just comically bad. It claims a file copy will be done in 20 seconds, then one second later it says it's going to take 4 days, then it flickers again to be 20 minutes. It's not only unhelpful, it's confusing!
The reason the ETA varies so much is that the progress rate itself can vary and the programmer's math can be overly sensitive.
Apple sidesteps this by just avoiding any accurate prediction and just giving vague estimates!
(source: autodesk.com)
That's annoying too, do I have time for a quick break, or is my task going to be done in 2 more seconds? If the prediction is too fuzzy, it's pointless to make any prediction at all.
Easy but wrong methods
As a first pass ETA computation, probably we all just make a function like if p is the fractional percentage that's done already, and t is the time it's taken so far, we output t*(1-p)/p as the estimate of how long it's going to take to finish. This simple ratio works "OK" but it's also terrible especially at the end of computation. If your slow download speed keeps a copy slowly advancing happening overnight, and finally in the morning, something kicks in and the copy starts going at full speed at 100X faster, your ETA at 90% done may say "1 hour", and 10 seconds later you're at 95% and the ETA will say "30 minutes" which is clearly an embarassingly poor guess.. in this case "10 seconds" is a much, much, much better estimate.
When this happens you may think to change the computation to use recent speed, not average speed, to estimate ETA. You take the average download rate or completion rate over the last 10 seconds, and use that rate to project how long completion will be. That performs quite well in the previous overnight-download-which-sped-up-at-the-end example, since it will give very good final completion estimates at the end. But this still has big problems.. it causes your ETA to bounce wildly when your rate varies quickly over a short period of time, and you get the "done in 20 seconds, done in 2 hours, done in 2 seconds, done in 30 minutes" rapid display of programming shame.
The actual question:
What is the best way to compute an estimated time of completion of a task, given the time history of the computation? I am not looking for links to GUI toolkits or Qt libraries. I'm asking about the algorithm to generate the most sane and accurate completion time estimates.
Have you had success with math formulas? Some kind of averaging, maybe by using the mean of the rate over 10 seconds with the rate over 1 minute with the rate over 1 hour? Some kind of artificial filtering like "if my new estimate varies too much from the previous estimate, tone it down, don't let it bounce too much"? Some kind of fancy history analysis where you integrate progress versus time advancement to find standard deviation of rate to give statistical error metrics on completion?
What have you tried, and what works best?
Original Answer
The company that created this site apparently makes a scheduling system that answers this question in the context of employees writing code. The way it works is with Monte Carlo simulation of future based on the past.
Appendix: Explanation of Monte Carlo
This is how this algorithm would work in your situation:
You model your task as a sequence of microtasks, say 1000 of them. Suppose an hour later you completed 100 of them. Now you run the simulation for the remaining 900 steps by randomly selecting 90 completed microtasks, adding their times and multiplying by 10. Here you have an estimate; repeat N times and you have N estimates for the time remaining. Note the average between these estimates will be about 9 hours -- no surprises here. But by presenting the resulting distribution to the user you'll honestly communicate to him the odds, e.g. 'with the probability 90% this will take another 3-15 hours'
This algorithm, by definition, produces complete result if the task in question can be modeled as a bunch of independent, random microtasks. You can gain a better answer only if you know how the task deviates from this model: for example, installers typically have a download/unpacking/installing tasklist and the speed for one cannot predict the other.
Appendix: Simplifying Monte Carlo
I'm not a statistics guru, but I think if you look closer into the simulation in this method, it will always return a normal distribution as a sum of large number of independent random variables. Therefore, you don't need to perform it at all. In fact, you don't even need to store all the completed times, since you'll only need their sum and sum of their squares.
In maybe not very standard notation,
sigma = sqrt ( sum_of_times_squared-sum_of_times^2 )
scaling = 900/100 // that is (totalSteps - elapsedSteps) / elapsedSteps
lowerBound = sum_of_times*scaling - 3*sigma*sqrt(scaling)
upperBound = sum_of_times*scaling + 3*sigma*sqrt(scaling)
With this, you can output the message saying that the thing will end between [lowerBound, upperBound] from now with some fixed probability (should be about 95%, but I probably missed some constant factor).
Here's what I've found works well! For the first 50% of the task, you assume the rate is constant and extrapolate. The time prediction is very stable and doesn't bounce much.
Once you pass 50%, you switch computation strategy. You take the fraction of the job left to do (1-p), then look back in time in a history of your own progress, and find (by binary search and linear interpolation) how long it's taken you to do the last (1-p) percentage and use that as your time estimate completion.
So if you're now 71% done, you have 29% remaining. You look back in your history and find how long ago you were at (71-29=42%) completion. Report that time as your ETA.
This is naturally adaptive. If you have X amount of work to do, it looks only at the time it took to do the X amount of work. At the end when you're at 99% done, it's using only very fresh, very recent data for the estimate.
It's not perfect of course but it smoothly changes and is especially accurate at the very end when it's most useful.
Whilst all the examples are valid, for the specific case of 'time left to download', I thought it would be a good idea to look at existing open source projects to see what they do.
From what I can see, Mozilla Firefox is the best at estimating the time remaining.
Mozilla Firefox
Firefox keeps a track of the last estimate for time remaining, and by using this and the current estimate for time remaining, it performs a smoothing function on the time.
See the ETA code here. This uses a 'speed' which is previously caculated here and is a smoothed average of the last 10 readings.
This is a little complex, so to paraphrase:
Take a smoothed average of the speed based 90% on the previous speed and 10% on the new speed.
With this smoothed average speed work out the estimated time remaining.
Use this estimated time remaining, and the previous estimated time remaining to created a new estimated time remaining (in order to avoid jumping)
Google Chrome
Chrome seems to jump about all over the place, and the code shows this.
One thing I do like with Chrome though is how they format time remaining.
For > 1 hour it says '1 hrs left'
For < 1 hour it says '59 mins left'
For < 1 minute it says '52 secs left'
You can see how it's formatted here
DownThemAll! Manager
It doesn't use anything clever, meaning the ETA jumps about all over the place.
See the code here
pySmartDL (a python downloader)
Takes the average ETA of the last 30 ETA calculations. Sounds like a reasonable way to do it.
See the code here/blob/916f2592db326241a2bf4d8f2e0719c58b71e385/pySmartDL/pySmartDL.py#L651)
Transmission
Gives a pretty good ETA in most cases (except when starting off, as might be expected).
Uses a smoothing factor over the past 5 readings, similar to Firefox but not quite as complex. Fundamentally similar to Gooli's answer.
See the code here
I usually use an Exponential Moving Average to compute the speed of an operation with a smoothing factor of say 0.1 and use that to compute the remaining time. This way all the measured speeds have influence on the current speed, but recent measurements have much more effect than those in the distant past.
In code it would look something like this:
alpha = 0.1 # smoothing factor
...
speed = (speed * (1 - alpha)) + (currentSpeed * alpha)
If your tasks are uniform in size, currentSpeed would simply be the time it took to execute the last task. If the tasks have different sizes and you know that one task is supposed to be i,e, twice as long as another, you can divide the time it took to execute the task by its relative size to get the current speed. Using speed you can compute the remaining time by multiplying it by the total size of the remaining tasks (or just by their number if the tasks are uniform).
Hopefully my explanation is clear enough, it's a bit late in the day.
In certain instances, when you need to perform the same task on a regular basis, it might be a good idea of using past completion times to average against.
For example, I have an application that loads the iTunes library via its COM interface. The size of a given iTunes library generally do not increase dramatically from launch-to-launch in terms of the number of items, so in this example it might be possible to track the last three load times and load rates and then average against that and compute your current ETA.
This would be hugely more accurate than an instantaneous measurement and probably more consistent as well.
However, this method depends upon the size of the task being relatively similar to the previous ones, so this would not work for a decompressing method or something else where any given byte stream is the data to be crunched.
Just my $0.02
First off, it helps to generate a running moving average. This weights more recent events more heavily.
To do this, keep a bunch of samples around (circular buffer or list), each a pair of progress and time. Keep the most recent N seconds of samples. Then generate a weighted average of the samples:
totalProgress += (curSample.progress - prevSample.progress) * scaleFactor
totalTime += (curSample.time - prevSample.time) * scaleFactor
where scaleFactor goes linearly from 0...1 as an inverse function of time in the past (thus weighing more recent samples more heavily). You can play around with this weighting, of course.
At the end, you can get the average rate of change:
averageProgressRate = (totalProgress / totalTime);
You can use this to figure out the ETA by dividing the remaining progress by this number.
However, while this gives you a good trending number, you have one other issue - jitter. If, due to natural variations, your rate of progress moves around a bit (it's noisy) - e.g. maybe you're using this to estimate file downloads - you'll notice that the noise can easily cause your ETA to jump around, especially if it's pretty far in the future (several minutes or more).
To avoid jitter from affecting your ETA too much, you want this average rate of change number to respond slowly to updates. One way to approach this is to keep around a cached value of averageProgressRate, and instead of instantly updating it to the trending number you've just calculated, you simulate it as a heavy physical object with mass, applying a simulated 'force' to slowly move it towards the trending number. With mass, it has a bit of inertia and is less likely to be affected by jitter.
Here's a rough sample:
// desiredAverageProgressRate is computed from the weighted average above
// m_averageProgressRate is a member variable also in progress units/sec
// lastTimeElapsed = the time delta in seconds (since last simulation)
// m_averageSpeed is a member variable in units/sec, used to hold the
// the velocity of m_averageProgressRate
const float frictionCoeff = 0.75f;
const float mass = 4.0f;
const float maxSpeedCoeff = 0.25f;
// lose 25% of our speed per sec, simulating friction
m_averageSeekSpeed *= pow(frictionCoeff, lastTimeElapsed);
float delta = desiredAvgProgressRate - m_averageProgressRate;
// update the velocity
float oldSpeed = m_averageSeekSpeed;
float accel = delta / mass;
m_averageSeekSpeed += accel * lastTimeElapsed; // v += at
// clamp the top speed to 25% of our current value
float sign = (m_averageSeekSpeed > 0.0f ? 1.0f : -1.0f);
float maxVal = m_averageProgressRate * maxSpeedCoeff;
if (fabs(m_averageSeekSpeed) > maxVal)
{
m_averageSeekSpeed = sign * maxVal;
}
// make sure they have the same sign
if ((m_averageSeekSpeed > 0.0f) == (delta > 0.0f))
{
float adjust = (oldSpeed + m_averageSeekSpeed) * 0.5f * lastTimeElapsed;
// don't overshoot.
if (fabs(adjust) > fabs(delta))
{
adjust = delta;
// apply damping
m_averageSeekSpeed *= 0.25f;
}
m_averageProgressRate += adjust;
}
Your question is a good one. If the problem can be broken up into discrete units having an accurate calculation often works best. Unfortunately this may not be the case even if you are installing 50 components each one might be 2% but one of them can be massive. One thing that I have had moderate success with is to clock the cpu and disk and give a decent estimate based on observational data. Knowing that certain check points are really point x allows you some opportunity to correct for environment factors (network, disk activity, CPU load). However this solution is not general in nature due to its reliance on observational data. Using ancillary data such as rpm file size helped me make my progress bars more accurate but they are never bullet proof.
Uniform averaging
The simplest approach would be to predict the remaining time linearly:
t_rem := t_spent ( n - prog ) / prog
where t_rem is the predicted ETA, t_spent is the time elapsed since the commencement of the operation, prog the number of microtasks completed out of their full quantity n. To explain—n may be the number of rows in a table to process or the number of files to copy.
This method having no parameters, one need not worry about the fine-tuning of the exponent of attenuation. The trade-off is poor adaptation to a changing progress rate because all samples have equal contribution to the estimate, whereas it is only meet that recent samples should be have more weight that old ones, which leads us to
Exponential smoothing of rate
in which the standard technique is to estimate progress rate by averaging previous point measurements:
rate := 1 / (n * dt); { rate equals normalized progress per unit time }
if prog = 1 then { if first microtask just completed }
rate_est := rate; { initialize the estimate }
else
begin
weight := Exp( - dt / DECAY_T );
rate_est := rate_est * weight + rate * (1.0 - weight);
t_rem := (1.0 - prog / n) / rate_est;
end;
where dt denotes the duration of the last completed microtask and is equal to the time passed since the previous progress update. Notice that weight is not a constant and must be adjusted according the length of time during which a certain rate was observed, because the longer we observed a certain speed the higher the exponential decay of the previous measurements. The constant DECAY_T denotes the length of time during which the weight of a sample decreases by a factor of e. SPWorley himself suggested a similar modification to gooli's proposal, although he applied it to the wrong term. An exponential average for equidistant measurements is:
Avg_e(n) = Avg_e(n-1) * alpha + m_n * (1 - alpha)
but what if the samples are not equidistant, as is the case with times in a typical progress bar? Take into account that alpha above is but an empirical quotient whose true value is:
alpha = Exp( - lambda * dt ),
where lambda is the parameter of the exponential window and dt the amount of change since the previous sample, which need not be time, but any linear and additive parameter. alpha is constant for equidistant measurements but varies with dt.
Mark that this method relies on a predefined time constant and is not scalable in time. In other words, if the exactly same process be uniformly slowed-down by a constant factor, this rate-based filter will become proportionally more sensitive to signal variations because at every step weight will be decreased. If we, however, desire a smoothing independent of the time scale, we should consider
Exponential smoothing of slowness
which is essentially the smoothing of rate turned upside down with the added simplification of a constant weight of because prog is growing by equidistant increments:
slowness := n * dt; { slowness is the amount of time per unity progress }
if prog = 1 then { if first microtask just completed }
slowness_est := slowness; { initialize the estimate }
else
begin
weight := Exp( - 1 / (n * DECAY_P ) );
slowness_est := slowness_est * weight + slowness * (1.0 - weight);
t_rem := (1.0 - prog / n) * slowness_est;
end;
The dimensionless constant DECAY_P denotes the normalized progress difference between two samples of which the weights are in the ratio of one to e. In other words, this constant determines the width of the smoothing window in progress domain, rather than in time domain. This technique is therefore independent of the time scale and has a constant spatial resolution.
Futher research: adaptive exponential smoothing
You are now equipped to try the various algorithms of adaptive exponential smoothing. Only remember to apply it to slowness rather than to rate.
I always wish these things would tell me a range. If it said, "This task will most likely be done in between 8 min and 30 minutes," then I have some idea of what kind of break to take. If it's bouncing all over the place, I'm tempted to watch it until it settles down, which is a big waste of time.
I have tried and simplified your "easy"/"wrong"/"OK" formula and it works best for me:
t / p - t
In Python:
>>> done=0.3; duration=10; "time left: %i" % (duration / done - duration)
'time left: 23'
That saves one op compared to (dur*(1-done)/done). And, in the edge case you describe, possibly ignoring the dialog for 30 minutes extra hardly matters after waiting all night.
Comparing this simple method to the one used by Transmission, I found it to be up to 72% more accurate.
I don't sweat it, it's a very small part of an application. I tell them what's going on, and let them go do something else.