Have done fft (see earlier posting if you are interested!) and got a result, which helps me. Would like to analyse the noisiness / spikiness of an array (actually a vb.nre collection of single). Um, how to explain ...
When signal is good, fft power results is 512 data points (frequency buckets) with low values in all but maybe 2 or 3 array entries, and a decent range (i.e. the peak is high, relative to the noise value in the nearly empty buckets. So when graphed, we have a nice big spike in the values in those few buckets.
When signal is poor/noisy, data values spread (max to min) is low, and there's proportionally higher noise in many more buckets.
What's a good, computationally non-intensive was of analysing the noisiness of this data set? Would some kind of statistical method, standard deviations or something help ?
The key is defining what is noise and what is signal, for which modelling assumptions must be made. Often an assumption is made of white noise (constant power per frequency band) or noise of some other power spectrum, and that model is fitted to the data. The signal to noise ratio can then be used to measure the amount of noise.
Fitting a noise model depends on the nature of your data: if you know that the real signal will have no power in the high frequency components, you can look there for an indication of the noise level, and use the model to predict what the noise will be at the lower frequency components where there is both signal and noise. Alternatively, if your signal is constant in time, taking multiple FFTs at different points in time and comparing them to get a standard deviation for each frequency band can give the level of noise present.
I hope I'm not patronising you to mention the issues inherent with windowing functions when performing FFTs: these can have the effect of introducing spurious "noise" into the frequency spectrum which is in fact an artifact of the periodic nature of the FFT. There's a tradeoff between getting sharp peaks and 'sideband' noise - more here www.ee.iitm.ac.in/~nitin/_media/ee462/fftwindows.pdf
Calculate a standard deviation and then you decide the threshold that will indicate noise. In practice this is usually easy and allows you to easily tweak the "noise level" as needed.
There is a nice single pass stddev algorithm in Knuth. Here is link that describes an implementation.
Standard Deviation
calculate the signal to noise ratio
http://en.wikipedia.org/wiki/Signal-to-noise_ratio
you could also check the stdev for each point and if it's under some level you choose then the signal is good else it's not.
wouldn't the spike be
treated as a noise glitch in SNR, an
outlier to be discarded, as it were?
If it's clear from the time-domain data that there are such spikes, then they will certainly create a lot of noise in the frequency spectrum. Chosing to ignore them is a good idea, but unfortunately the FFT can't accept data with 'holes' in it where the spikes have been removed. There are two techniques to get around this. The 'dirty trick' method is to set the outlier sample to be the average of the two samples on either site, and compute the FFT with a full set of data.
The harder but more-correct method is to use a Lomb Normalised Periodogram (see the book 'Numerical Recipes' by W.H.Press et al.), which does a similar job to the FFT but can cope with missing data properly.
Related
I have a signal 3M samples long. I want to subsample it. I know that to avoid aliasing I need to filter off frequencies higher than Nyquist frequency. I know that I can do that by convolution with certain filters (e.g. Butterworth), but I know that way some of the high frequencies is preserved.
I wonder whether I can just zero the unwanted frequencies in the frequency domain and use inverse FFT to go back to time domain. Is such approach numerically correct? I know that certain libraries speed-up convolution with use of FFT.
It's not really correct. Zeroing out the top frequencies in the FFT will only zero out frequencies with wavelengths that divide the FFT length. If you were to frequency-shift your signal by half a bin and do another FFT, you'd find that there is some leakage and the upper frequencies are not all zero.
The result will be pretty close, but taking an FFT of the whole signal is a very expensive way to get just pretty close.
You should just use a normal filter. As long as you leave a reasonable amount of room between the filter cut-off frequency and the Nyquist frequency, it's easy to ensure that any aliasing error will be much smaller than quantization error and other noise.
Here's the situation I am worrying about.
Let me say I have a model trained with min-max scaled data. I want to test my model, so I also scaled the test dataset with my old scaler which was used in the training stage. However, my new test data's turned out to be the newer minimum, so the scaler returned negative value.
As far as I know, minimum and maximum aren't that stable value, especially in the volatile dataset such as cryptocurrency data. In this case, should I update my scaler? Or should I retrain my model?
I happen to disagree with #Sharan_Sundar. The point of scaling is to bring all of your features onto a single scale, not to rigorously ensure that they lie in the interval [0,1]. This can be very important, especially when considering regularization techniques the penalize large coefficients (whether they be linear regression coefficients or neural network weights). The combination of feature scaling and regularization help to ensure your model generalizes to unobserved data.
Scaling based on your "test" data is not a great idea because in practice, as you pointed out, you can easily observe new data points that don't lie within the bounds of your original observations. Your model needs to be robust to this.
In general, I would recommend considering different scaling routines. scikitlearn's MinMaxScaler is one, as is StandardScaler (subtract mean and divide by standard deviation). In the case where your target variable, cryptocurrency price can vary over multiple orders of magnitude, it might be worth using the logarithm function for scaling some of your variables. This is where data science becomes an art -- there's not necessarily a 'right' answer here.
(EDIT) - Also see: Do you apply min max scaling separately on training and test data?
Ideally you should scale first and then only split into test and train. But its not preferable to use minmax scaler with data which can have dynamically varying min and max values with significant variance in realtime scenario.
I am working on a n-body gravitational simulator that takes input and produces output in metric MKS units. This involves dealing with some very large numbers (like solar masses expressed in kilograms, semimajor axes of planetary orbits expressed in meters, and timescales of years expressed in seconds), which get multiplied by some very small numbers (notably, the gravitational constant, which is 6.67384e-11 in MKS units), and also the occasional very small number getting added to or subtracted from a very large number (mainly when summing up pairwise accelerations), which gets me concerned about the effects of rounding errors.
I've already taken the step of replacing all masses m by Gm (premultiplying by the gravitational constant), which significantly reduces the total number of multiplies, and makes the mass numbers much smaller, and that seems to have had a positive effect on both efficiency and accuracy, as judged by how well the simulator conserves energy.
I am wondering, however: is potentially it worth trying to do some internal re-scaling into different units to further minimize floating point errors? And if so, what kind of range (for double-precision floats) should I be trying to get my numbers centered on for maximum accuracy?
In general if you want precise results in physical based rendering you don't want to use floats or doubles since they have massive rounding problems and thus introduce errors in your simulation.
If you need or want to stick with floats/double you probably should rescale around zero. The reason is that often floating point representations have a higher "density" of values around this point and tend to have fewer on the min/max sides. Image example from google
I would suggest that you change all values to integer based number variables. This erases rounding errors (over/underflow can still happen!) and speeds up the calculation process by an order of magnitude because normal CPUs work faster with integer operations. In case of GPU its basically the same but thats another story all by its own...
But before you take such an effort to further improve your accuracy i would strongly advise an arbitrary precision number library. This may come with an performance loss but should be way easier and yield better results than a rescaling of your values.
Most of the numerical mathematicians come across this problem.
At first let me remind you that you can not deal with numbers (or phsycal values) smaller than the machine epsilon for each calculation. Unfortunately the epsilon depends around which number you are analyzing. You can try eps(a) for any value of a in MATLAB, as far as I remember eps(1.0)~=2.3e-16 and eps(0)~1e-298.
That's why in numerical methods you avoid calculations using very different scaled numbers. Because one is just an ignored (smaller than its epsilon) by the other value and rounding errors are inevitable.
But what else people do? If they encounter such physical problems, before coding, mathematicians analyse the problem theoritically, they make simplifications to use similarly scaled numbers.
I'm reading data from the microphone and want to perform some analysis on it. I'm attempting to generate a spectrum analyser something like this:
What I have at the moment is this:
My understanding is that I need to perform a Fourier analysis - a Fast Fourier Transform ? - to extract the component frequencies and their amplitudes.
Can someone confirm my understanding is correct and exactly what type of Fourier transform I need to apply?
At the moment, I'm getting frames containing 4k samples from the mic (using NAudio). The buffer I've got is 16bits/sample (Signed Short). For reference, the above plot shows approx half a frame
I'm coding in VB so any .Net libraries/examples (preferably on NuGet) would be of most use. I believe implementations vary considerably so the less I have to massage my data, the better.
The top plot is that of a spectrograph, where each vertical time line is colored based on the magnitudes of the result from an FFT (likely windowed) of a slice in time (possibly overlapped) of the input waveform. The number of vertical points to plot (the frequency resolution) is related to the length of the FFT. Almost any FFT will do. If you use the most common complex-to-complex FFT, just set the imaginary portion of each complex input sample to zero, copy a slice in time of samples of your input waveform to the "real" part, FFT, and take the magnitude or log magnitude of each complex result bin, then map these values to colors per your preference.
I need to find the frequency of a sample, stored (in vb) as an array of byte. Sample is a sine wave, known frequency, so I can check), but the numbers are a bit odd, and my maths-foo is weak.
Full range of values 0-255. 99% of numbers are in range 235 to 245, but there are some outliers down to 0 and 1, and up to 255 in the remaining 1%.
How do I normalise this to remove outliers, (calculating the 235-245 interval as it may change with different samples), and how do I then calculate zero-crossings to get the frequency?
Apologies if this description is rubbish!
The FFT is probably the best answer, but if you really want to do it by your method, try this:
To normalize, first make a histogram to count how many occurrances of each value from 0 to 255. Then throw out X percent of the values from each end with something like:
for (i=lower=0;i< N*(X/100); lower++)
i+=count[lower];
//repeat in other direction for upper
Now normalize with
A[i] = 255*(A[i]-lower)/(upper-lower)-128
Throw away results outside the -128..127 range.
Now you can count zero crossings. To make sure you are not fooled by noise, you might want to keep track of the slope over the last several points, and only count crossings when the average slope is going the right way.
The standard method to attack this problem is to consider one block of data, hopefully at least twice the actual frequency (taking more data isn't bad, so it's good to overestimate a bit), then take the FFT and guess that the frequency corresponds to the largest number in the resulting FFT spectrum.
By the way, very similar problems have been asked here before - you could search for those answers as well.
Use the Fourier transform, it's much more noise insensitive than counting zero crossings
Edit: #WaveyDavey
I found an F# library to do an FFT: From here
As it turns out, the best free
implementation that I've found for F#
users so far is still the fantastic
FFTW library. Their site has a
precompiled Windows DLL. I've written
minimal bindings that allow
thread-safe access to FFTW from F#,
with both guru and simple interfaces.
Performance is excellent, 32-bit
Windows XP Pro is only up to 35%
slower than 64-bit Linux.
Now I'm sure you can call F# lib from VB.net, C# etc, that should be in their docs
If I understood well from your description, what you have is a signal which is a combination of a sine plus a constant plus some random glitches. Say, like
x[n] = A*sin(f*n + phi) + B + N[n]
where N[n] is the "glitch" noise you want to get rid of.
If the glitches are one-sample long, you can remove them using a median filter which has to be bigger than the glitch length. On both sides of the glitch. Glitches of length 1, mean you will have enough with a median of 3 samples of length.
y[n] = median3(x[n])
The median is computed so: Take the samples of x you want to filter (x[n-1],x[n],x[n+1]), sort them, and your output is the middle one.
Now that the noise signal is away, get rid of the constant signal. I understand the buffer is of a limited and known length, so you can just compute the mean of the whole buffer. Substract it.
Now you have your single sinus signal. You can now compute the fundamental frequency by counting zero crossings. Count the amount of samples above 0 in which the former sample was below 0. The period is the total amount of samples of your buffer divided by this, and the frequency is the oposite (1/x) of the period.
Although I would go with the majority and say that it seems like what you want is an fft solution (fft algorithm is pretty quick), if fft is not the answer for whatever reason you may want to try fitting a sine curve to the data using a fitting program and reading off the fitted frequency.
Using Fityk, you can load the data, and fit to a*sin(b*x-c) where 2*pi/b will give you the frequency after fitting.
Fityk can be used from a gui, from a command-line for scripting and has a C++ API so could be included in your programs directly.
I googled for "basic fft". Visual Basic FFT Your question screams FFT, but be careful, using FFT without understanding even a little bit about DSP can lead results that you don't understand or don't know where they come from.
get the Frequency Analyzer at http://www.relisoft.com/Freeware/index.htm and run it and look at the code.