Ok here is what i need to do:
I want to do some tracking using Kalman filter(possibly adaptive).My measurements(when they are available) are very good with very small error from the real measurements. In some cases though the measurements jump to a value,completely off from the correct position i am looking for, and then after few frames the come back to their correct position.
The problem is that if my filter(not adaptive) has specific values for Measurement Noise Covariance(R) and State Error Covariance(Q) matrices the results are not very accurate,because even for these 1% of cases i have to do a compromise between R and Q.
So i decided to use an adaptive Kalman filter as they do in here: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.367.1747&rep=rep1&type=pdf
They estimate the measurement noise covariance matrix based on the innovation sequences.
Basically, they are using a moving window on previous samples and the calculate the covariance of the error between the previous measurements-prior estimations. For eg 5 past measurements and the 5 prior estimations.When a faulty measurement comes under the window, the covariance increases and thus the R increases also.
But in practice the R increases(but not enough) so in the next step the estimation is still good but just a bit towards the the faulty measurement.In the next step(because now the the previous estimation has moved a bit towards the measurement) the R becomes smaller with result the new estimation to go even closer to the measurements, and so on and so forth.
In the end after a few frames the estimations follow the faulty measurements. Here is a plot to understand better what i mean.
https://www.dropbox.com/s/rkv0tjcm4s54kv3/untitled.tif
Maybe what i am trying to do is completely wrong and can't be done with the adaptive Kalman filter.Maybe someone who has worked extensively with Kalman Filter in the past and he has faced this problem before can help.
Any idea is welcome!
Before the answer, I want to be sure I got the problem you have right.
You have measurements, some of them are good (Low measurement noise) yet others are outliers.
The problem you're having is tuning the measurement noise covariance matrix.
Practically, you tune for the good measurements.
Outliers measurements are rejected by using the Error Covariance.
If the innovation falls outside an ellipse you define using the Error Covariance Matrix the measurement is rejected.
Whenever a measurement is rejected you just apply the prediction step again and wait for another measurement.
Yes the problem is exactly this.
However i manage to solve it without the need to define any ellipse.What i was doing was correct except the fact that was not working if i had a lot of(lets say fifty) consecutive outliers.
This is normal if you think the size of your window.If it is for example only 10 samples and you have 20 outliers obviously it won't work.But for 5 consecutive outliers work perfectly.Generally i haven't used any threshold as you propose("if the innovation falls outside an ellipse") reject the measurements.I keep the measurements but in the same time when i start to have outliers the Error measurement covariance becomes very large.So the estimation is based more in previous estimation than in current measurement.
If i used your method which is indeed more logical(reject the current measurement,if it is an outlier based on a threshold) i have the problem that i have to define this threshold a priori,right?Maybe i am missing something..
Related
I have a question about curve fitting, I have many curves like the one in the picture.
X axis : time
Y axis : temperature
Each sample comes out every 30s.
GOAL : predict the value at the end of the transient
What would you do in this situation?
What I am doing is this :
for every new sample I start a new fitting (and so each fitting is independent from the previous one) and check the value of the fitted curve 2 hours (all curves I have set before 2h) after the start of the measurement. If for a number (let's say 5) of subsequent fitting the value in the future stays more or less the same(+-0.2°C) I so assume that the estimation is the right one.
This approach seems to me far too simple and I think I am not exploiting all information. For example the info of the error I am making punctually (e.g. at minute 4:00 I predict and at 4:30 I see that I am doing an error).
In the picture the red part of the curve is excluded (but the real data in the future passes through it). the estimation is the blue one. You see in this case I don't have a good prediction... In general I have also more flat curves.
Based on the comments above, I tried to formulate an answer as no one else is giving some input.
I think your are using a good basic procedure. Better results may be obtained by using a more appropriate fitting curve, which includes all the dominant dynamics, but avoids overfitting of the data. Based on your figure, the simplest form I could think of is:
s + a(1-e^(-t/tau))
with parameters s (the initial temperature), a (amplitude = steady state value) and tau (dominant time constant). As you mentioned yourself, limiting the allowed range for the parameters may avoid overfitting and increase the quality of your estimation.
Using a random high order function, like you are using now, may give good interpolation results, but are dangerous to use for extrapolation, because strange effects may occur outside the fitting region.
Alternatives
Using the error (eg. correcting for the extrapolated error) may be possible, but is tricky and may not always give good results.
Training a neural network to perform the estimation is probably overkill, but may give better results if applied correctly. Note that you need a lot of training data which should be representative for the data for which you will use the neural network later on.
I am using the Lomb-Scargle code to estimate some frequencies in a short time-series, the time series is shown in the first image. The results of the Lomb-Scargle analysis are shown in the second, and I have zoomed in on a prominent peak at about 2 cycles per day. However this peak is smeared and thus it is proving difficult to resolve the real frequency of this component. Is there any other methods, or improvements to the method I am using, to accurately resolve the important frequency components within this short time-series?
There is some information on the use of methods for short time series here but its not clear whether they need to be regularly sampled. Ideally I am looking for a method that works with irregularly sampled data, from some research it appears that maximum entropy methods are the answer, but I am not sure whether these have been implemented in MATLAB? Although from the this link, it appears that there is an equivalent method, 'The Yule-Walker AR method produces the same results as a maximum entropy estimator. However again its not clear whether the data need to be uniformly sampled?
First for those, who are not familiar with Simulink, there is a imaginable outside-Simulink partial solution:
I need to create a vector satisfying the following conditions:
known initial value a1
known final value a2
it has a pre-defined step size, but the length is not pre-determined
the first derivative over the whole range is limited to v_max resp. -v_max
the second derivative over the whole range is limited to a_max resp. -a_max
the third derivative over the whole range is limited to j_max resp. -j_max
at the first and the final point all derivatives are zero.
Before you ask "what have you tried so far", I just had the idea to solve it outside Simulink and I tried the whole stuff below ;)
But maybe you guys have a good idea, while I keep working on my own solution.
I'd like to generate smooth ramp signals (3rd derivative limited) based on a trigger signal in Simulink.
To get a triggered step I created a triggered subsystem propagating the trigger output. It looks like that:
But I actually don't want a step, I need a very smooth ramp with limited derivatives up to the 3rd order. The math behind is:
displacement: x
speed: v = x'
acceleration: a = v' = x''
jerk: j = a' = v'' = x'''
(If this looks familiar to you, I once had a very similar question. I thought about a bounty on it, but after the necessary edit of the question both answers would have been invalid)
As there are just rate limiters of 1st order, I used two derivates and a double integration to resolve my problem. But there is a mayor drawback, I can not ignore anymore. For the sake of illustration I chose a relatively big step size of 0.1.
The complete minimal example (Fixed Step, stepsize: 0.1, ode4): Download here
It can be seen, that the signal not even reaches the intended step height of 10 and furthermore is not constant at the end.
Over the development process of my whole model, this approach was satisfactory enough for small step sizes. But I reached the point where I really need the smooth ramp as intended. That means I need a finally constant signal at exactly the value, specified by the step height gain.
I already spent days to resolve the problem, and hope to fine some help here now.
Some of my ideas:
dynamically increase the step height over the actual desired value and saturate the final output. If the rate limits,step height and the simulation step size wouldn't be flexible one could probably find a satisfying solution. But as everything has to be flexible, there are too much cases where the acceleration and jerk limit is violated.
I tried to use the Matlab function block and write my own 3rd order rate limiter. Though it seems possible for me for the trigger moment, I have no solution how to smooth the "deceleration" at the end of the ramp. Also I'd need C-compilers, which would make it hard to use my model on other systems without problems. (At least I think so.)
The solver can not be changed siginificantly (either ode3 or ode4) and a fixed step size is mandatory (0.00001 to 0.01).
Currently used, not really useful approach:
For a dynamic amplification of 1.07 I get the following output (all values normalised on their limits):
Though the displacement looks nice, the violation of the acceleration limit is very harmful.
For a dynamic amplification of 1.05 I get the following output (all values normalised on their limits):
The acceleration stays in its boundaries, but the displacement does not reach the intended value. (not really clear in the picture) The jerk is still to big. (I could live with that, but it's not nice)
So it appears to me that a inside-Simulink solutions is far from reality. Any ideas how to create a well-behaving custom function block?
Simulation step size, step height, and the rate limits are known before the simulation starts. (But I have a lot of these triggered smooth ramps in a row, it should feed a event-discrete control). So I could imagine to create the whole smooth ramp outside simulink and save it as a timeseries object and append it on the current signal when the trigger is activated.
The problems you see are because the difference is not conditioned very well.
Taking the difference amplifies the numerical that exists in your simulation.
Also the jerk will always be large if you try to apply an actual step.
I guess for your approach it would be better to work the other way around:
i.e. make a jerk, acceleration and velocity with which your step is achieved.
I think your looking for something like the ref3 block:
http://www.dct.tue.nl/home_of_ref3.htm
Note the disclaimer on the site and that it is a little cumbersome to use.
An easy (yet to be improved) way is to use a rate limiter and then a state space model with a filter. From the filter you get the velocity, which in turn you can apply a rate limiter to. You continue with rate-limiter and filters until you have the desired curve.
Otherwise you can come up with numerical rate-limiters of higher order using ie runge kutta formulas or finite differences. However it was pointed out, that they may suffer from bad conditioning.
What I usually do is to use one rate limiter and a filter of 3rd Order and just tune the time constant (1 tripple pole), such that my needs are met. This works well, esp
Integrator chains of length > 1 are unstable!
There is a huge field of research dealing with trajectory planning. The easiest way might be to use FIR filters (Biagotti et al) or to implement an online trajectory planner (Ezair et al 2014 / Knierim et al 2012).
I'm a neuroscientist, and not a very good one. My colleague has kindly provided me with a noisy voltage measurements of the PY neuron of the Stomatogastric Ganglion of the lobster.
The activity of this neuron is characterised by a slow depolarised plateaux with fast spikes on top (a burst).
Both idealised and noisy versions are presented here for you to peruse at your leisure.
It's my job to extract the spike times from the noisy signal but this is so far beyond my experience level I have no idea where to begin. Fortunately, I am a total ninja at Matlab.
Could someone kindly provide me with the name of the procedure, filter or smoothing function which is best suited for this task. Or even the appropriate forum to ask such an asinine question.
Presumably, it needs to increase the signal to noise ratio? The problem here seems to be determining the difference between noise and a bona fide spike as the margin between the two is quite small.
UPDATE: 02/07/2013
I have tried the following filters in Matlab with mixed results. It's still very hard to say what is noise and what is a spike.
Lowpass Butterworth filter,
median filter,
gaussian,
moving weighted window,
moving average filter,
smooth,
sgolay filter.
This may not be an adequate response for stackoverflow - but one way of increasing a signal to noise ratio in your case is to average parts of the signal.
low pass your signal to remove noise (and spikes), and find the minima of the filtered signal (from your image, one minimum every 600 data points). Keep the indexes of each minimum,
on the noisy signal, for each minimum index, select the consecutive 700 data points. If you have 50 minima, you should have a 50 by 700 matrix,
average your matrix. You should have a 1 by 700 vector.
By averaging parts of the signal (minimum-locked potentials), you will take advantage of two properties: noise is zero-mean (well, it should be), and the signal of interest is repetitive. The first will therefore decrease as you pile up potentials, and the second will increase. With this process however, you will lose the spike times for each slow wave figure, but at least have them for blocks of 50 minima.
This technique is known in neuroscience as event-related potential (http://en.wikipedia.org/wiki/Event-related_potential). It may not fit perfectly your signal, or the result may not give nice spikes, but you may extract the spike times for some periods of interest (given the nature of your signal, I would say that you would need 5 or 10 potentials to see an emerging mean activity).
There are some toolboxes that do part of the job (but I would program it myself given the complexity of the task). These are eeglab or fieldtrip. They have a bunch of filter/decomposition options too, as well as some statistical features.
I am currently working on an indoor navigation system using a Zigbee WSN in star topology.
I currently have signal strength data for 60 positions in an area of 15m by 10 approximately. I want to use ANN to help predict the coordinates for other positions. After going through a number of threads, I realized that normalizing the data would give me better results.
I tried that and re-trained my network a few times. I managed to get the goal parameter in the nntool of MATLAB to the value .000745, but still after I give a training sample as a test input, and then scaling it back, it is giving a value way-off.
A value of .000745 means that my data has been very closely fit, right? If yes, why this anomaly? I am dividing and multiplying by the maximum value to normalize and scale the value back respectively.
Can someone please explain me where I might be going wrong? Am I using the wrong training parameters? (I am using TRAINRP, 4 layers with 15 neurons in each layer and giving a goal of 1e-8, gradient of 1e-6 and 100000 epochs)
Should I consider methods other than ANN for this purpose?
Please help.
For spatial data you can always use Gaussian Process Regression. With a proper kernel you can predict pretty well and GP regression is a pretty simple thing to do (just matrix inversion and matrix vector multiplication) You don't have much data so exact GP regression can be easily done. For a nice source on GP Regression check this.
What did you scale? Inputs or outputs? Did scale input+output for your trainingset and only the output while testing?
What kind of error measure do you use? I assume your "goal parameter" is an error measure. Is it SSE (sum of squared errors) or MSE (mean squared errors)? 0.000745 seems to be very small and usually you should have almost no error on your training data.
Your ANN architecture might be too deep with too few hidden units for an initial test. Try different architectures like 40-20 hidden units, 60 HU, 30-20-10 HU, ...
You should generate a test set to verify your ANN's generalization. Otherwise overfitting might be a problem.