100 periods have been collected from a 3 dimensional periodic signal. The wavelength slightly varies. The noise of the wavelength follows Gaussian distribution with zero mean. A good estimate of the wavelength is known, that is not an issue here. The noise of the amplitude may not be Gaussian and may be contaminated with outliers.
How can I compute a single period that approximates 'best' all of the collected 100 periods?
Time-series, ARMA, ARIMA, Kalman Filter, autoregression and autocorrelation seem to be keywords here.
UPDATE 1: I have no idea how time-series models work. Are they prepared for varying wavelengths? Can they handle non-smooth true signals? If a time-series model is fitted, can I compute a 'best estimate' for a single period? How?
UPDATE 2: A related question is this. Speed is not an issue in my case. Processing is done off-line, after all periods have been collected.
Origin of the problem: I am measuring acceleration during human steps at 200 Hz. After that I am trying to double integrate the data to get the vertical displacement of the center of gravity. Of course the noise introduces a HUGE error when you integrate twice. I would like to exploit periodicity to reduce this noise. Here is a crude graph of the actual data (y: acceleration in g, x: time in second) of 6 steps corresponding to 3 periods (1 left and 1 right step is a period):
My interest is now purely theoretical, as http://jap.physiology.org/content/39/1/174.abstract gives a pretty good recipe what to do.
We have used wavelets for noise suppression with similar signal measured from cows during walking.
I'm don't think the noise is so much of a problem here and the biggest peaks represent actual changes in the acceleration during walking.
I suppose that the angle of the leg and thus accelerometer changes during your experiment and you need to account for that in order to calculate the distance i.e you need to know what is the orientation of the accelerometer in each time step. See e.g this technical note for one to account for angle.
If you need get accurate measures of the position the best solution would be to get an accelerometer with a magnetometer, which also measures orientation. Something like this should work: http://www.sparkfun.com/products/10321.
EDIT: I have looked into this a bit more in the last few days because a similar project is in my to do list as well... We have not used gyros in the past, but we are doing so in the next project.
The inaccuracy in the positioning doesn't come from the white noise, but from the inaccuracy and drift of the gyro. And the error then accumulates very quickly due to the double integration. Intersense has a product called Navshoe, that addresses this problem by zeroing the error after each step (see this paper). And this is a good introduction to inertial navigation.
Periodic signal without noise has the following property:
f(a) = f(a+k), where k is the wavelength.
Next bit of information that is needed is that your signal is composed of separate samples. Every bit of information you've collected are based on samples, which are values of f() function. From 100 samples, you can get the mean value:
1/n * sum(s_i), where i is in range [0..n-1] and n = 100.
This needs to be done for every dimension of your data. If you use 3d data, it will be applied 3 times. Result would be (x,y,z) points. You can find value of s_i from the periodic signal equation simply by doing
s_i(a).x = f(a+k*i).x
s_i(a).y = f(a+k*i).y
s_i(a).z = f(a+k*i).z
If the wavelength is not accurate, this will give you additional source of error or you'll need to adjust it to match the real wavelength of each period. Since
k*i = k+k+...+k
if the wavelength varies, you'll need to use
k_1+k_2+k_3+...+k_i
instead of k*i.
Unfortunately with errors in wavelength, there will be big problems keeping this k_1..k_i chain in sync with the actual data. You'd actually need to know how to regognize the starting position of each period from your actual data. Possibly need to mark them by hand.
Now, all the mean values you calculated would be functions like this:
m(a) :: R->(x,y,z)
Now this is a curve in 3d space. More complex error models will be left as an excersize for the reader.
If you have a copy of Curve Fitting Toolbox, localized regression might be a good choice.
Curve Fitting Toolbox supports both lowess and loess localized regression models for curve and curve fitting.
There is an option for robust localized regression
The following blog post shows how to use cross validation to estimate an optimzal spaning parameter for a localized regression model, as well as techniques to estimate confidence intervals using a bootstrap.
http://blogs.mathworks.com/loren/2011/01/13/data-driven-fitting/
Related
I have been using sklearn to learn on some data. This is a binary classifcation task and I am using a RBF kernel. My data set is quite unbalanced (80:20) and I'm using only 120 samples, with 10ish features (I've been experimenting with a few less). Since I set class_weight="auto" the accuracy I've calculated from a cross validated (10 folds) gridsearch has dropped dramatically. Why??
I will include a couple of validation accuracy heatmaps to demonstrate the difference.
NOTE: top heatmap is before classweight was changed to auto.
Accuracy is not the best metrics to use when dealing with unbalanced dataset. Let's say you have 99 positive examples and 1 negative example, and if you predict all outputs to be positive, still you will get 99% accuracy, whereas you have mis-classified the only negative example. You might have gotten high accuracy in the first case because your predictions will be on the side which has high number of samples.
When you do class weight = auto, it takes the imbalance into consideration and hence, your predictions might have moved towards center, you can cross-check it using plotting the histograms of predictions.
My suggestion is, don't use accuracy as performance metric, use something like F1 Score or AUC.
I'm working on a video stabilizer using Opencv in C++.
At this time of the project I'm correctly able to find the translation between two consecutive frames with 3 different technique (Optical Flow, Phase Correlation, BFMatcher on points of interests).
To obtain a stabilized image I add up all the translation vector (from consecutive frame) to one, which is used in warpAffine function to correct the output image.
I'm having good result on fixed camera but result on camera in translation are really bad : the image disappear from the screen.
I think I have to distinguish jitter movement that I want to remove from panning movement that I want to keep. But I'm open to others solutions.
Actually the whole problem is a bit more complex than you might have thought in the beginning. Let's look a it this way: when you move your camera through the world, things that move close to the camera move faster than the ones in the background - so objects at different depths change their relative distance (look at your finder while moving the head and see how it points to different things). This means the image actually transforms and does not only translate (move in x or y) - so how do you want to accompensate for that? What you you need to do is to infer how much the camera moved (translation along x,y and z) and how much it rotated (with the angles of yaw, pan and tilt). This is a not very trivial task but openCV comes with a very nice package: http://opencv.willowgarage.com/documentation/camera_calibration_and_3d_reconstruction.html
So I recommend you to read as much on Homography(http://en.wikipedia.org/wiki/Homography), camera models and calibration as possible and then think what you actually want to stabilize for and if it is only for the rotation angles, the task is much simpler than if you would also like to stabilize for translational jitters.
If you don't want to go fancy and neglect the third dimension, I suggest that you average the optic flow, high-pass filter it and compensate this movement with a image translation into the oposite direction. This will keep your image more or less in the middle of the frame and only small,fast changes will be counteracted.
I would suggest you the possible approaches (in complexity order):
apply some easy-to-implement IIR low pass filter on the translation vectors before applying the stabilization. This will separate the high frequency (jitter) from the low frequency (panning)
same idea, a bit more complex, use Kalman filtering to track a motion with constant velocity or acceleration. You can use OpenCV's Kalman filter for that.
A bit more tricky, put a threshold on the motion amplitude to decide between two states (moving vs static camera) and filter the translation or not.
Finaly, you can use some elaborate technique from machine Learning to try to identify the user's desired motion (static, panning, etc.) and filter or not the motion vectors used for the stabilization.
Just a threshold is not a low pass filter.
Possible low pass filters (that are easy to implement):
there is the well known averaging, that is already a low-pass filter whose cutoff frequency depends on the number of samples that go into the averaging equation (the more samples the lower the cutoff frequency).
One frequently used filter is the exponential filter (because it forgets the past with an exponential rate decay). It is simply computed as x_filt(k) = a*x_nofilt(k) + (1-a)x_filt(k-1) with 0 <= a <= 1.
Another popular filter (and that can be computed beyond order 1) is the Butterworth filter.
Etc Low pass filters on Wikipedia, IIR filters...
I'm doing some work on tracking moving objects using a ceiling mounted downward facing camera. I've got to the point where I can detect the position of the desired object in each frame.
I'm looking into using a Kalman filter to track the object's position and speed through the scene and I've reached a stumbling block. I've set up my system and have all the required parts of the Kalman filter except the measurement variance.
I want to be able to assign a meaningful variance to each measurement to allow the correction phase to use the new information in a sensible manner. I have several measures assigned to my detected objects which could in theory be useful in determining how accurate the position should be and it seems logical to try and combine them to derive a suitable variance.
Am I approaching this in the right manner and if so, can anyone point me in the right direction to continue?
Any help greatly appreciated.
I think you are right. According to this post:
Sensor fusioning with Kalman filter
determining the variance is 100% experimental. It seems to me you have everything you need to get good estimates of the variance.
sorry for the late reply. I have personally encountered the same problem in my previous project. I found the advice given by Gustaf Hendeby in his Sensor Fusion lecture slides ( Page 10 of the slides) extremely valuable.
To summarize:
(1) The SNR of your measurement noise and your process noise determines your filter behavior. A high process noise/measurement noise ration makes your filter slower (low-pass filter), which will usually allow smoother tracking, vice versa a if you set your measurement noise low, you essentially have a high pass filter, which tends to have more jitter.
(2) There are numerous papers in the literature discuss on how to set these noise model properly. However, usually a lot of "tuning" is needed depends on your application. Usually the measurement noise is what we can measure/characterize based on the hardware specification. Therefore a recommendation is to fix "R" (measurement noise covariance) and tune Q (the process model noise covariance).
gyroscopes which measure rate of rotation of angles when integrated produce angles right?
my question is how do i do this? what im doing so far is just adding all the angles ive detected and that seems to be very wrong
AngleIntegrated = GyroDegPersec * (1/GyroBandWidth);
suggestions are very welcome. thanks
You need to integrate with respect to time. So ideally you should sample the gyroscope at regular (fixed) time intervals, T, and then incorporate that sampling interval, T, into your integral calculation.
Note that T needs to be small enough to satisfy the Nyquist criterion.
You can integrate in discrete domain. Suppose the angular rate is da, it's time integral is a.
k is the discrete step number.
a(k) = a(k-1) + T*0.5*(da(k) + da(k-1))
For example, da(k) is current angular rate reading. da(k-1) is previoud angular rate reading. a(k-1) is the previous step's integration value(rotation angle). T is sampling rate. If the sensor outputs in every 1 millisecond, T becomes 0.001.
You can use this formula when k>0. The initial value, a(0), must be given.
Knowing that of course you can't count on having a correct value in the long run (by integration your error window will always increase over time) what I would do is reading the gyroscope, interpolating the current read and previous few ones to get a smooth curve (e.g. a parabola using current read and previous two) and then computing the integral of that parabola from last read time and current time.
First I am going to broadly state what I'm trying to do and ask for advice. Then I will explain my current approach and ask for answers to my current problems.
Problem
I have an MP3 file of a person speaking. I'd like to split it up into segments roughly corresponding to a sentence or phrase. (I'd do it manually, but we are talking hours of data.)
If you have advice on how to do this programatically or for some existing utilities, I'd love to hear it. (I'm aware of voice activity detection and I've looked into it a bit, but I didn't see any freely available utilities.)
Current Approach
I thought the simplest thing would be to scan the MP3 at certain intervals and identify places where the average volume was below some threshold. Then I would use some existing utility to cut up the mp3 at those locations.
I've been playing around with pymad and I believe that I've successfully extracted the PCM (pulse code modulation) data for each frame of the mp3. Now I am stuck because I can't really seem to wrap my head around how the PCM data translates to relative volume. I'm also aware of other complicating factors like multiple channels, big endian vs little, etc.
Advice on how to map a group of pcm samples to relative volume would be key.
Thanks!
PCM is a time frame base encoding of sound. For each time frame, you get a peak level. (If you want a physical reference for this: The peak level corresponds to the distance the microphone membrane was moved out of it's resting position at that given time.)
Let's forget that PCM can uses unsigned values for 8 bit samples, and focus on
signed values. If the value is > 0, the membrane was on one side of it's resting position, if it is < 0 it was on the other side. The bigger the dislocation from rest (no matter to which side), the louder the sound.
Most voice classification methods start with one very simple step: They compare the peak level to a threshold level. If the peak level is below the threshold, the sound is considered background noise.
Looking at the parameters in Audacity's Silence Finder, the silence level should be that threshold. The next parameter, Minimum silence duration, is obviously the length of a silence period that is required to mark a break (or in your case, the end of a sentence).
If you want to code a similar tool yourself, I recommend the following approach:
Divide your sound sample in discrete sets of a specific duration. I would start with 1/10, 1/20 or 1/100 of a second.
For each of these sets, compute the maximum peak level
Compare this maximum peak to a threshold (the silence level in Audacity). The threshold is something you have to determine yourself, based on the specifics of your sound sample (loudnes, background noise etc). If the max peak is below your threshold, this set is silence.
Now analyse the series of classified sets: Calculate the length of silence in your recording. (length = number of silent sets * length of a set). If it is above your Minimum silence duration, assume that you have the end of a sentence here.
The main point in coding this yourself instead of continuing to use Audacity is that you can improve your classification by using advanced analysis methods. One very simple metric you can apply is called zero crossing rate, it just counts how often the sign switches in your given set of peak levels (i.e. your values cross the 0 line). There are many more, all of them more complex, but it may be worth the effort. Have a look at discrete cosine transformations for example...
Just wanted to update this. I'm having moderate success using Audacity's Silence Finder. However, I'm still interested in this problem. Thanks.
PCM is a way of encoding a sinusoidal wave. It will be encoded as a series of bits, where one of the bits (1, I'd guess) indicates an increase in the function, and 0 indicates a decrease. The function can stay roughly constant by alternating 1 and 0.
To estimate amplitude, plot the sin wave, then normalize it over the x axis. Then, you should be able to estimate the amplitude of the sin wave at different points. Once you've done that, you should be able to pick out the spots where amplitude is lower.
You may also try to use a Fourier transform to estimate where the signals are most distinct.