I have a stream of data measurements with an initial increasing phase that is followed by a plateau. The measurements are noisy without clear bound. I would like to stop ingesting the stream when the plateau is detected:
while (not_const)
{
add_measurement( stream.get() );
not_const = !is_const();
}
Is there a well-known algorithm for dealing with such problem? I know about Kalman-Filters, but not so sure if they are specifically made for this.
The Kalman filter will cover your noise, so long as the variance is calculable. Yes, it can help in this situation. Depending on your application, you may find that the first derivative of a moving average will do as well for you. Kalman merely optimizes some linear parameters to give a "best" prediction of actual (vs observed-through-noise) values.
You still need to handle your interpretation of that prediction series. You need to define what constitutes a "plateau". How close to 0 do you need the computable slope? Does that figure depend on the preceding input? How abrupt is the transition between the increase and the plateau? The latter considerations would suggest looking at the second derivative as well: a quick-change detector of some ilk.
Related
I'm implementing a Kalman Filter which fuses 3d position data (provided from 2 different computer vision algorithms). I am modeling the problem with a 9-dimensional state vector (position, velocity, and acceleration). However, the data from each sensor does not come at the same time. Since I compute the velocity by considering the time step between the reception of the previous data and the current data point, two consecutive data points can be quite different but separated by only a very small time step, thus making it seem like the position has changed rapidly.
I am wondering if anyone has insight or direction on the best way to approach this problem- will the kalman filter itself be tolerant of this behaviour? Or should I place all data received within a time window into a bin, and perform the update/predict cycle less frequently on a batch of data? The resources I've seen for utilizing kalman filter in object tracking have used only one camera (i.e. synchronous data), so I'm having trouble finding information related to my use case.
Any help is very much appreciated! Thank you!
From all what I got to know from your question and our conversation in the comments let me first shortly describe the issue and suggest the solution.
A quick recap
You have a system with two independent sensors, which take measurements with different rates (30Hz and 5Hz) (and maybe have some time jitter). The good news is that each such measurement is completely sufficient to proceed an update step of your kalman filter. Each measurement have a time stamp.
Another important point is, that the measurements (maybe) have poor precision, so that the change in position looks not plausible.
A possible solution
Define a smallest time interval for calling your kalman filter, so that none of the recieved measurements has to wait too long to be processed. It looks for me like a 100Hz rate could be a good first choice. In this case your dt would be 0.01s.
Design your F and Q matrices based on the chosen dt (they both strongly depend on this value).
In each call without measurement execute the prediction step. As soon as a measurement comes, do update. So your call sequence would look like:
call sequence:
init()
predict()
predict()
predict()
predict()
update(sensor1)
predict()
update(sensor2)
update(sensor1)
predict()
predict()
update(sensor1)
predict()
and so on...
To deal with the precision issue you could use a reference signal (the ground truth). Analyze the error in each sensor reading for each signal (x, y, z) compared to the reference. A kalman filter can work well ONLY with readings, whose error is normally distributed with a zero mean. If you see some systematical offset, may be you can get rid of it. From the observed error you can calculate the standard deviation (and the variance), so you can tell your filter how good the measurements are. It will be your R matrix.
If you don't have a reference you can take some measurements while standing still on the same place. So your reference position would be constant and you could have a look at the dispersion of the readings.
Tune elements of your Q matrix and describe the possible dynamic of your state elements. A smaller Q element for position would tell the filter not to change it too fast. So the (possible) poor performance of your sensors will be partially eliminated (think of a low pass filter as intuition).
I hope it can help you. Please correct me if I understood something wrong.
It would be helpful to see a plot of your sensor readings (and if possible of the reference trajectory).
My problem is the following: I need to classify a data stream coming from an sensor. I have managed to get a baseline using the
median of a window and I subtract the values from that baseline (I want to avoid negative peaks, so I only use the absolute value of the difference).
Now I need to distinguish an event (= something triggered the sensor) from the noise near the baseline:
The problem is that I don't know which method to use.
There are several approaches of which I thought of:
sum up the values in a window, if the sum is above a threshold the class should be EVENT ('Integrate and dump')
sum up the differences of the values in a window and get the mean value (which gives something like the first derivative), if the value is positive and above a threshold set class EVENT, set class NO-EVENT otherwise
combination of both
(unfortunately these approaches have the drawback that I need to guess the threshold values and set the window size)
using SVM that learns from manually classified data (but I don't know how to set up this algorithm properly: which features should I look at, like median/mean of a window?, integral?, first derivative?...)
What would you suggest? Are there better/simpler methods to get this task done?
I know there exist a lot of sophisticated algorithms but I'm confused about what could be the best way - please have a litte patience with a newbie who has no machine learning/DSP background :)
Thank you a lot and best regards.
The key to evaluating your heuristic is to develop a model of the behaviour of the system.
For example, what is the model of the physical process you are monitoring? Do you expect your samples, for example, to be correlated in time?
What is the model for the sensor output? Can it be modelled as, for example, a discretized linear function of the voltage? Is there a noise component? Is the magnitude of the noise known or unknown but constant?
Once you've listed your knowledge of the system that you're monitoring, you can then use that to evaluate and decide upon a good classification system. You may then also get an estimate of its accuracy, which is useful for consumers of the output of your classifier.
Edit:
Given the more detailed description, I'd suggest trying some simple models of behaviour that can be tackled using classical techniques before moving to a generic supervised learning heuristic.
For example, suppose:
The baseline, event threshold and noise magnitude are all known a priori.
The underlying process can be modelled as a Markov chain: it has two states (off and on) and the transition times between them are exponentially distributed.
You could then use a hidden Markov Model approach to determine the most likely underlying state at any given time. Even when the noise parameters and thresholds are unknown, you can use the HMM forward-backward training method to train the parameters (e.g. mean, variance of a Gaussian) associated with the output for each state.
If you know even more about the events, you can get by with simpler approaches: for example, if you knew that the event signal always reached a level above the baseline + noise, and that events were always separated in time by an interval larger than the width of the event itself, you could just do a simple threshold test.
Edit:
The classic intro to HMMs is Rabiner's tutorial (a copy can be found here). Relevant also are these errata.
from your description a correctly parameterized moving average might be sufficient
Try to understand the Sensor and its output. Make a model and do a Simulator that provides mock-data that covers expected data with noise and all that stuff
Get lots of real sensor data recorded
visualize the data and verify your assuptions and model
annotate your sensor data i. e. generate ground truth (your simulator shall do that for the mock data)
from what you learned till now propose one or more algorithms
make a test system that can verify your algorithms against ground truth and do regression against previous runs
implement your proposed algorithms and run them against ground truth
try to understand the false positives and false negatives from the recorded data (and try to adapt your simulator to reproduce them)
adapt your algotithm(s)
some other tips
you may implement hysteresis on thresholds to avoid bouncing
you may implement delays to avoid bouncing
beware of delays if implementing debouncers or low pass filters
you may implement multiple algorithms and voting
for testing relative improvements you may do regression tests on large amounts data not annotated. then you check the flipping detections only to find performance increase/decrease
I have the following problem: I have 2 signals over time. They are from the same source so they should be the same. I want to check if they really are.
Complications:
they may be measured with different sample rates
the start / end time do not correlate. The measurement does not start at the same time and end at the same time.
there may be an time offset between the two signals.
My thoughts go along Fourier transformation, convolution and statistical methods for comparison. Can someone post me some links where I can find more information on how to handle this?
You can easily correct for the phase by just shifting them so their centers of mass line up. (Or alternatively, in the Fourier domain just multiplying by the inverse of the phase of the first coefficient.)
Similarly, if you want to line up the images given only partial data, you can just cross correlate and take the maximal value (which is again easy to do in the Fourier domain).
That leaves the only tricky part of this process as dealing with the sampling rates. Now if you know a-priori what the sample rates are, (and if they are related by a rational number), you can just use sinc interpolation/downsampling to rescale them to a common sampling rate:
https://ccrma.stanford.edu/~jos/st/Bandlimited_Interpolation_Time_Limited_Signals.html
If you don't know the sampling rate, you may be a bit screwed. Technically, you can try just brute forcing over all the different rescalings of your signal, but doing this tends to be either slow or else give mediocre results.
As a last suggestion, if you just want to match sounds exactly you can try using the cepstrum and verifying that the peaks of the signal are close enough to within some tolerance. This type of analysis is used a lot in sound and speech recognition, with some refinements to make it operate a bit more locally. It tends to work best with frequency modulated data like speech and music:
http://en.wikipedia.org/wiki/Cepstrum
Fourier transformation does sound like the right way.
There is too much mathematical information for me to just start explaining here so if you really wanna know what's going on with that (cause I don't think you can just use FT without understanding it) you should use this reference from MIT OpenCourseWare: http://ocw.mit.edu/courses/mathematics/18-103-fourier-analysis-theory-and-applications-spring-2004/lecture-notes/
Hope it helped.
If you are working with a linux box and the waveforms that need to be processed have already been recorded, you can try to use the file command to display details about the recording. It gives you the sampling rate when it is invoked on a wav file, though I am not sure what format you are recording in.
If the signals are time-shifted with respect to each other, you may try to convolve one with a delta function with increasing delays and then comparing. On MATLAB, conv and all should be good enough.
These are just 'crude' attempts (almost like hacking at the problem). There may be algorithms that are shift-invariant that may do a better job.
Hope that helps.
For example you measure the data coming from some device, it can be a mass of the object moving on the bridge. Because it is moving the mass will give data which will vibrate in some amplitude depending on the mass of the object. Bigger the mass - bigger the vibrations.
Are there any methods for filtering such kind of noise from that data?
May be using some formulas of vibrations? Have no idea what kind of formulas or algorithms (filters) can be used here. Please suggest anything.
EDIT 2:
Better picture, I just draw it for better understanding:
Not very good picture. From that graph you can see that the frequency is the same every
time, but the amplitude chanbges periodically. Something like that I have when there are no objects on the moving road. (conveyer belt). vibrating near zero value.
When the object moves, I there are the same waves with changing amplitude.
The graph can tell that there may be some force applying to the system and which produces forced occilations. So I am interested in removing such kind of noise. I do not know what force causes such occilations. Soon I hope I will get some data on the non moving road with and without object on it for comparison with moving road case.
What you have in your last plot is basically an amplitude modulated oscillation coming from a function like:
f[x] := 10 * (4 + Sin[x]) * Sin[80 * x]
The constants have been chosen to match your plot (using just a rule of thumb)
The Plot of this function is
That isn't "noise" (although may be some noise is there too), but can be filtered easily.
Let's see your data for the static and moving payloads ....
Edit
Based on your response to several comments, and based in my previous experience with weighting devices:
You are interfacing the physical world, not just getting input from a mouse and keyboard. It is very important for you understand the device, how it works and how it is designed.
You need a calibration procedure. You have to use several master weights to be sure that the device is working properly and linearly in the whole scale, and that the static case is measured much better than your dynamic needs.
You'll not be able to predict if you can measure with several loads in the conveyor until you do some experiments and look very carefully at the resulting plots
You need to be sure that a load placed anywhere in the conveyor shows the same reading. Or at least you should be able to correlate reading and position.
As I said before, you need a lot of info, and it seems that is not available. I always worked as a team with the engineers designing the device.
Don't hesitate to add more info ...
Have you tried filters with lowpass characteristics? There are different approaches for smoothing data (i.e. Savitzky-Golay, Gauss, moving average) but often, a simple N-point median filter is already sufficient.
It really depends on what you're after.
Take a look at this book:
The Scientist and Engineer's Guide to Digital Signal Processing
You can download it for free. In particular, check chapters 14 and 15.
If the frequency changes with mass and you're trying to measure mass, why not measure the frequency of the oscillations and use that as your primary measure?
Otherwise you need a notch filter which is tunable - figure out the frequency of the "noise" and tune the notch filter to that.
Another book to try is Lyons Understanding Digital Signal Processing
In order to smooth the signal, I'd average the previous 2 * n samples where n is the maximum expected wavelength of the vibrations.
This should cause most of the noise to be eliminated.
If you have some idea of the range of frequencies, you could do a simple average as long as the measurement period were sufficiently long to give you the level of accuracy you want to achieve. The more wavelengths worth of data you average against, the smaller the ratio of contributed error from a partial wavelength.
I'd suggest first simulating/modeling this in software like Matlab.
Data you'll need to consider:
The expected range of vibration frequencies
The measurement accuracy you want to achieve
The expected range of mass you'll want to measure
The function of mass to vibration amplitude
You should be able to apply the same principles as noise-cancelling microphones: put two sensors out, then subtract the secondary sensor's (farther away from the good signal source) signal from the primary sensor's (closer to the good signal source) signal.
Obviously, this works best if the "noise" will reach both sensors fairly equally while the "signal" reaches the primary sensor much more strongly.
For things like sound, this is pretty easy to do in the sensor itself, which makes your software a lot easier and more performant. Depending on what you're measuring, this might be easier to do with multiple sets of hardware and doing the cancellation in software.
If you can characterize the frequency spectra of the unwanted vibration noise, you might be able to synthesize a set of (near) minimum phase notch or band reject filter(s) to allow you to acquire your desired signal at your desired S/N ratio with minimized latency or data set size.
Filtering noisy digital signals is straight forward, as previous posters have noted. There are lots of references. You have not however stated what your objectives are clearly, so we cannot point you into a good direction. Are you looking for a single measurement of a single object on a bridge? [Then see other answers].
Are you monitoring traffic on this bridge and weighing each entity as it passes by? Then you need to determine when entities are on the sensor and when they are not. Typically, as long as the sensor's noise floor is significantly lower than the signal you're measuring this can be accomplished by simple thresholding.
Are you trying to measure the vibrations of the bridge caused by other vehicles? In which case you need either a more expensive sensor if you're having problems doing this, or a clearer measuring objective.
I'm interested in image scaling algorithms and have implemented the bilinear and bicubic methods. However, I have heard of the Lanczos and other more sophisticated methods for even higher quality image scaling, and I am very curious how they work.
Could someone here explain the basic idea behind scaling an image using Lanczos (both upscaling and downscaling) and why it results in higher quality?
I do have a background in Fourier analysis and have done some signal processing stuff in the past, but not with relation to image processing, so don't be afraid to use terms like "frequency response" and such in your answer :)
EDIT: I guess what I really want to know is the concept and theory behind using a convolution filter for interpolation.
(Note: I have already read the Wikipedia article on Lanczos resampling but it didn't have nearly enough detail for me)
The selection of a particular filter for image processing is something of a black art, because the main criterion for judging the result is subjective: in computer graphics, the ultimate question is almost always: "does it look good?". There are a lot of good filters out there, and the choice between the best frequently comes down to a judgement call.
That said, I will go ahead with some theory...
Since you are familiar with Fourier analysis for signal processing, you don't really need to know much more to apply it to image processing -- all the filters of immediate interest are "separable", which basically means you can apply them independently in the x and y directions. This reduces the problem of resampling a (2-D) image to the problem of resampling a (1-D) signal. Instead of a function of time (t), your signal is a function of one of the coordinate axes (say, x); everything else is exactly the same.
Ultimately, the reason you need to use a filter at all is to avoid aliasing: if you are reducing the resolution, you need to filter out high-frequency original data that the new, lower resolution doesn't support, or it will be added to unrelated frequencies instead.
So. While you're filtering out unwanted frequencies from the original, you want to preserve as much of the original signal as you can. Also, you don't want to distort the signal you do preserve. Finally, you want to extinguish the unwanted frequencies as completely as possible. This means -- in theory -- that a good filter should be a "box" function in frequency space: with zero response for frequencies above the cutoff, unity response for frequencies below the cutoff, and a step function in between. And, in theory, this response is achievable: as you may know, a straight sinc filter will give you exactly that.
There are two problems with this. First, a straight sinc filter is unbounded, and doesn't drop off very fast; this means that doing a straightforward convolution will be very slow. Rather than direct convolution, it is faster to use an FFT and do the filtering in frequency space...
However, if you actually do use a straight sinc filter, the problem is that it doesn't actually look very good! As the related question says, perceptually there are ringing artifacts, and practically there is no completely satisfactory way to deal with the negative values that result from "undershoot".
Finally, then: one way to deal with the problem is to start out with a sinc filter (for its good theoretical properties), and tweak it until you have something that also solves your other problems. Specifically, this will get you something like the Lanczos filter:
Lanczos filter: L(x) = sinc(pi x) sinc(pi x/a) box(|x|<a)
frequency response: F[L(x)](f) = box(|f|<1/2) * box(|f|<1/2a) * sinc(2 pi f a)
[note that "*" here is convolution, not multiplication]
[also, I am ignoring normalization completely...]
the sinc(pi x) determines the overall shape of the frequency response (for larger a, the frequency response looks more and more like a box function)
the box(|x|<a) gives it finite support, so you can use direct convolution
the sinc(pi x/a) smooths out the edges of the box and (consequently? equivalently?) greatly improves the rejection of undesirable high frequencies
the last two factors ("the window") also tone down the ringing; they make a vast improvement in both the perceptual artifact and the practical incidence of "undershoot" -- though without completely eliminating them
Please note that there is no magic about any of this. There are a wide variety of windows available, which work just about as well. Also, for a=1 and 2, the frequency response does not look much like a step function. However, I hope this answers your question "why sinc", and gives you some idea about frequency responses and so forth.