I am currently writing a program which uses importance sampling. All is going well so far, however I have two minor queries that I would like to know about.
What is the most basic job that we would expect importance sampling to do?
For an arbitrarily large number of samples, what is the difference between a result with importance sampling and one without it?
Related
I have a complicated theoretical Probability Density Function (PDF) that I define in mathematica and that depends on some parameters that I need to estimate from comparison with real data. From a big simulation done on a cluster and not my laptop I have acquired a lot of events (over 10^9).
The way I understand things, given that I know what the PDF is I 'just' need to sum the probability that those events appear for a given set of parameters and maximise this quantity by adjusting the parameters.
However, given the number of events I would rather work with something less computer-time consuming and work for example with something easily generated like an histogram of my data. But then how would my log-likelihood estimator work?
Thanks a lot for your answers!
What I am trying to do is separating the audio sources and extract its pitch from the raw signal.
I modeled this process myself, as represented below:
Each sources oscillate in normal modes, often makes its component peaks' frequency integer multiplication. It's known as Harmonic. And then resonanced, finally combined linearly.
As seen in above, I've got many hints in frequency response pattern of audio signals, but almost no idea how to 'separate' it. I've tried countless of my own models. This is one of them:
FFT the PCM
Get peak frequency bins and amplitudes.
Calculate pitch candidate frequency bins.
For each pitch candidates, using recurrent neural network analyze all the peaks and find appropriate combination of peaks.
Separate analyzed pitch candidates.
Unfortunately, I've got non of them successfully separates the signal until now.
I want any of advices to solve these kind of problem.
Especially in modeling of source separation like my one above.
Because no one has really attempted to answer this, and because you've marked it with the neural-network tag, I'm going to address the suitability of a neural network to this kind of problem. As the question was somewhat non-technical, this answer will also be "high level".
Neural networks require some sort of sample set from which to learn. In order to "teach" a neural net to solve this problem you would essentially need to have a working set of known solutions to work from. Do you have this? If so, read on. If not, a neural is probably not what you are seeking. You stated that you have "many hints" but no real solution. This leads me to believe you probably don't have sample sets. If you can get them, great, otherwise you might be out of luck.
Supposing now that you have a sample set of Raw Signal samples and corresponding Source 1 and Source 2 outputs... Well, now you're going to need a method for deciding on a topology. Assuming you don't know a lot about how neural nets work (and don't want to), and assuming you also don't know the exact degree of complexity of the problem, I would probably recommend the open source NEAT package to get you started. I am not affiliated in any way with this project, but I have used it, and it allows you to (relatively) intelligently evolve neural network topologies to fit the problem.
Now, in terms of how a neural net would solve this specific problem. The first thing that comes to mind is that all audio signals are essentially time-series. That is to say, the information they convey is somehow dependent and related to the data at previous timesteps (e.g. the detection of some waveform cannot be done from a single time-point; it requires information about previous timesteps as well). Again, there's a million ways of solving this problem, but since I'm already recommending NEAT I'd probably suggest you take a look at the C++ NEAT Time Series mod.
If you're going down this route, you'll probably be wanting to use some sort of sliding window to provide information about the recent past at each time step. For a quick and dirty intro to sliding windows, check out this SO question:
Time Series Prediction via Neural Networks
The size of the sliding window can be important, especially if you're not using recurrent neural nets. Recurrent networks allow neural nets to remember previous time steps (at the cost of performance - NEAT is already recurrent so that choice is made for you here). You will probably want the sliding window length (ie. the number of timesteps in the past provided at every time step) to be roughly equal to your conservative guess of the largest number of previous timesteps required to gain enough information to split your waveform.
I'd say this is probably enough information to get you started.
When it comes to deciding how to provide the neural net with the data, you'll first want to normalise the input signals (consider a sigmoid function) and experiment with different transfer functions (sigmoid would probably be a good starting point).
I would imagine you'll want to have 2 output neurons, providing normalised amplitude (which you would denormalise via the inverse of the sigmoid function) as the output representing Source 1 and Source 2 respectively. For the fitness value (the way you judge the ability of each tested network to solve the problem) would be something along the lines of the negative of the RMS error of the output signal against the actual known signal (ie. tested against the samples I was referring to earlier that you will need to procure).
Suffice to say, this will not be a trivial operation, but it could work if you have enough samples to train the network against. What is a good number of samples? Well as a rule of thumb it's roughly a number that is large enough such that a simple polynomial function of order N (where N is the number of neurons in the netural network you require to solve the problem) cannot fit all of the samples accurately. This is basically to ensure you are not simply overfitting the problem, which is a serious issue with neural networks.
I hope this has been helpful! Best of luck.
Additional note: your work to date wouldn't have been in vain if you go down this route. A neural network is likely to benefit from additional "help" in the form of FFTs and other signal modelling "inputs", so you might want to consider taking the signal processing you have already done, organising into an analog, continuous representation and feeding it as an input alongside the input signal.
I have a catalog of 900 applications.
I need to determine how their reliability is distributed as a whole. (i.e. is it normal).
I can measure the reliability of an individual application.
How can I determine the reliability of the group as a whole without measuring each one?
That's a pretty open-ended question! Overall, distribution fitting can be quite challenging and works best with large samples (100's or even 1000's). It's generally better to pick a modeling distribution based on known characteristics of the process you're attempting to model than to try purely empirical fitting.
If you're going to go empirical, for a start you could take a random sample, measure the reliability scores (whatever you're using for that) of your sample, sort them, and plot them vs normal quantiles. If they fall along a relatively straight line the normal distribution is a plausible model, and you can estimate sample mean and variance to parameterize it. You can apply the same idea of plotting vs quantiles from other proposed distributions to see if they are plausible as well.
Watch out for behavior in the tails, in particular. Pretty much by definition the tails occur rarely and may be under-represented in your sample. Like all things statistical, the larger the sample size you can draw on the better your results will be.
I'd also add that my prior belief would be that a normal distribution wouldn't be a great fit. Your reliability scores probably fall on a bounded range, tend to fall more towards one side or the other of that range. If they tend to the high range, I'd predict that they get lopped off at the end of the range and have a long tail to the low side, and vice versa if they tend to the low range.
Forgive me if I may come as ignorant but I would like to ask some questions regarding using Filter Algorithms for Note Onset Detection.
Is 'Detection Function' the same as using Filters on the audio signal? Or generally, what is the difference between Detection Function, Filtering (pre-processing the signal), and Peak-Picking?
I've constantly heard about the Low-Pass (or High-Pass) filter, but I am confused. I read that it works on cancelling out certain frequencies that are below (or above) a certain threshold. However, I am using the Time-Domain for calculating Note Onsets (that is, using the change in signal amplitude/energy). So I am not sure on how I can apply low-pass filtering to the time-domain. Any other good filters for note-onset detection?
What is the difference between, Spectral and Phase energy? (I have an idea that spectral refers to the spectogram or frequencies, but I do not know what Phase is)
I am having difficulties with working with dynamic thresholding. Any suggestions for a good algorithm? For example, I have the following signal:
As shown in the image above, there are note onsets that I have missed. A brief description of my algorithm, I calculate and take note of the energy/amplitude changes that occur in the audio signal. Then I get the maximum 'energy change' and based on the sensitivity, I take a percentage of it and set it as the threshold. So this is where the problem of dealing with varying degrees of amplitude/energy comes in. If I set the sensitivity too low, I come up with 'ghost' onsets and if I set the sensitivity too high, I miss out on some onsets. Any suggestions to improve the algorithm (or suggest a new algorithm) that I am using?
I am sure that it is difficult to have 100% accuracy but I need to have a better algorithm for note onset detection compared with what I have now. I would appreciate all the help I can get. Thank you very much!
One way is to detect sudden increases in the amplitude envelope. One way of calculating the amplitude envelope is to rectify the input signal (i.e. take the absolute value) and then low pass filter it. Check out the code examples in http://www.musicdsp.org for time domain filter examples and envelope followers.
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I have a sample held in a buffer from DirectX. It's a sample of a note played and captured from an instrument. How do I analyse the frequency of the sample (like a guitar tuner does)? I believe FFTs are involved, but I have no pointers to HOWTOs.
The FFT can help you figure out where the frequency is, but it can't tell you exactly what the frequency is. Each point in the FFT is a "bin" of frequencies, so if there's a peak in your FFT, all you know is that the frequency you want is somewhere within that bin, or range of frequencies.
If you want it really accurate, you need a long FFT with a high resolution and lots of bins (= lots of memory and lots of computation). You can also guess the true peak from a low-resolution FFT using quadratic interpolation on the log-scaled spectrum, which works surprisingly well.
If computational cost is most important, you can try to get the signal into a form in which you can count zero crossings, and then the more you count, the more accurate your measurement.
None of these will work if the fundamental is missing, though. :)
I've outlined a few different algorithms here, and the interpolated FFT is usually the most accurate (though this only works when the fundamental is the strongest harmonic - otherwise you need to be smarter about finding it), with zero-crossings a close second (though this only works for waveforms with one crossing per cycle). Neither of these conditions is typical.
Keep in mind that the partials above the fundamental frequency are not perfect harmonics in many instruments, like piano or guitar. Each partial is actually a little bit out of tune, or inharmonic. So the higher-frequency peaks in the FFT will not be exactly on the integer multiples of the fundamental, and the wave shape will change slightly from one cycle to the next, which throws off autocorrelation.
To get a really accurate frequency reading, I'd say to use the autocorrelation to guess the fundamental, then find the true peak using quadratic interpolation. (You can do the autocorrelation in the frequency domain to save CPU cycles.) There are a lot of gotchas, and the right method to use really depends on your application.
There are also other algorithms that are time-based, not frequency based.
Autocorrelation is a relatively simple algorithm for pitch detection.
Reference: http://cnx.org/content/m11714/latest/
I have written c# implementations of autocorrelation and other algorithms that are readable. Check out http://code.google.com/p/yaalp/.
http://code.google.com/p/yaalp/source/browse/#svn/trunk/csaudio/WaveAudio/WaveAudio
Lists the files, and PitchDetection.cs is the one you want.
(The project is GPL; so understand the terms if you use the code).
Guitar tuners don't use FFT's or DFT's. Usually they just count zero crossings. You might not get the fundamental frequency because some waveforms have more zero crossings than others but you can usually get a multiple of the fundamental frequency that way. That's enough to get the note although you might be one or more octaves off.
Low pass filtering before counting zero crossings can usually get rid of the excess zero crossings. Tuning the low pass filter requires some knowlegde of the range of frequency you want to detect though
FFTs (Fast-Fourier Transforms) would indeed be involved. FFTs allow you to approximate any analog signal with a sum of simple sine waves of fixed frequencies and varying amplitudes. What you'll essentially be doing is taking a sample and decomposing it into amplitude->frequency pairs, and then taking the frequency that corresponds to the highest amplitude.
Hopefully another SO reader can fill the gaps I'm leaving between the theory and the code!
A little more specifically:
If you start with the raw PCM in an input array, what you basically have is a graph of wave amplitude vs time.Doing a FFT will transform that to a frequency histogram for frequencies from 0 to 1/2 the input sampling rate. The value of each entry in the result array will be the 'strength' of the corresponding sub-frequency.
So to find the root frequency given an input array of size N sampled at S samples/second:
FFT(N, input, output);
max = max_i = 0;
for(i=0;i<N;i++)
if (output[i]>max) max_i = i;
root = S/2.0 * max_i/N ;
Retrieval of fundamental frequencies in a PCM audio signal is a difficult task, and there would be a lot to talk about it...
Anyway, usually time-based method are not suitable for polyphonic signals, because a complex wave given by the sum of different harmonic components due to multiple fundamental frequencies has a zero-crossing rate which depends only from the lowest frequency component...
Also in the frequency domain the FFT is not the most suitable method, since frequency spacing between notes follow an exponential scale, not linear. This means that a constant frequency resolution, used in the FFT method, may be insufficient to resolve lower frequency notes if the size of the analysis window in the time domain is not large enough.
A more suitable method would be a constant-Q transform, which is DFT applied after a process of low-pass filtering and decimation by 2 (i.e. halving each step the sampling frequency) of the signal, in order to obtain different subbands with different frequency resolution. In this way the calculation of DFT is optimized. The trouble is that also time resolution is variable, and increases for the lower subbands...
Finally, if we are trying to estimate the fundamental frequency of a single note, FFT/DFT methods are ok. Things change for a polyphonic context, in which partials of different sounds overlap and sum/cancel their amplitude depending from their phase difference, and so a single spectral peak could belong to different harmonic contents (belonging to different notes). Correlation in this case don't give good results...
Apply a DFT and then derive the fundamental frequency from the results. Googling around for DFT information will give you the information you need -- I'd link you to some, but they differ greatly in expectations of math knowledge.
Good luck.