# separating stationary and non stationary parts of univariate signal

Does anyone know if there is a procedure as to separate the stationary and non stationary parts of a univariate signal. I have seen signal source separation and blind separation algorithms (all of them assuming that the signal is a linear superpossition of underlying hidden source signals where the signals are statistically independent - linearly independent - not sure if colored noises are linearly independent) where different signals are mixed in a single combined signal. For this several methods exist as to separate the sources mostly based on either orthogonality, correlation, variance and distances, however all of them based on having already multivariate variables:

Principal components analysis
Singular value decomposition
Independent component analysis
Dependent component analysis
Non-negative matrix factorization
Low-complexity coding and decoding
Stationary subspace analysis
Common spatial pattern


My question is that if you could treat a single signal which is the combination of signal + uncorrelated stationary noise colors (white) + correlated noise colors (pink,brown,etc) as a source problem (implying that there is a mix of signals formed by the stationary noise, non stationary noise and signal) and thus apply a specific blind signal separation algorithm as to separate the sources statistically (for example as in stationary subspace analysis there is identification of the stationary and non stationary parts).

All blind signal separation algos I have seen can only be applied to multiple variables, my question is if there are there any blind separation algorithms that can be applied to univariate series where stationary and non stationary components can be separated as individual signal sources.

EDIT

I know the signal is spread across a wide range of frequencies from the Evoluionary power spectrum. I know uncorrelated stationary white noise affects the signal and that other unidentifed stationary correlated colors do as well but ignore their mix. My idea was to run a wavelet at one level decomposition as to have two range of frequencies and use this as to run one of the analysis above as to determine the stationary and non stationary parts. As the stationary parts will correspond to correlated and uncorrelated noise colors which I will assume to be non gausian and additive. As the nature of the signal is non stationary I would effectively have separated the signal from noise. Then from noise I will run wavelet thresholding as to eliminate white noise based on a MA MAD and remain with correlated noises which via FFT and their power relation understand the type dominating noise color or identify their mix.

All these algorithms are based on the determination of a new basis (i.e. a new set of basis vectors), chosen according to a given constrain (to maximize the variance of each component in PCA for instance).

If you have only one signal, you have only one basis vector. So you cannot use these algorithms because you don't have enough information.

In lower dimensions, you can think of it as the calculation of the arithmetic mean from a set of noisy values corresponding to the same measurement. You need multiple measurement to be sure to get close to the "true" value of the measurement. What you are asking is how to get the "true" value from only one measurement without any repetition. That cannot be done because you don't have enough information even to do anything.

However, depending on you signal and its specific noise, there might be other ways to achieve what you want. Actually, all you have to do is to identify a way to differentiate noise from signal.

Their distributions are obviously different, but these are mere projections, so again, you have too few data to use them.

One working possibility is to filter your data to remove high frequency components (to remove most of the noise) and very low frequency components (to remove pink noise, but usually pink noise is really not a problem). But that would work only if you have an idea of the bandwidth of your signal (the maximum frequency beyond which your signal has no meaning anymore).

What you describe won't work. If I continue my analogy of the average over noisy measurement data, it is like saying that you will split the result of you unique measurement (say, 15 = 12 + 3) and use that result (12 and 3) to calculate the mean (7.5). I hope you understand that this makes no sense (the mean of 15 is at best undefined, or at worst 15). And even if it did, your decomposition would be quite arbitrary (why not use 10+5 or 20-5 or 13264585120-13264585105?). Again, you won't get more information by decomposition of your signal.

Final word:

If you really want us to help you or to give you more details about why this won't work as you expect, please open a new question, provide actual data and explain your setup.

• If you want to go into details or even try what you suggest, please open a new question and provide data. – user13706 Jul 13 '15 at 17:07
• Will do. In any case the wavelet splits the signal into two parts equaly with 50% of the frequencies each (different frequencies each). However I understand that I will get no more information by decomposing the signal. – Barnaby Jul 13 '15 at 19:34
• I did describe my work but no answer has been provided so I will deleat it – Barnaby Jul 20 '15 at 23:56