# Conversion from stationarity to non-stationarity

Is there any way to convert a non-stationary signal to a stationary one, perform operations on it meant for a stationary signal and then convert it back to the non-stationary one?

• differencing the series can sometimes work. that's pretty much the only one if the non-stationarity is in the mean. if the non-stationarity is only in the variance, the transformations such as log or sine can sometimes help. – mark leeds Sep 24 '18 at 5:49
• What sort of signal are you dealing with? – A_A Sep 24 '18 at 5:59
• Its a time series data for a country's GDP for about 100 years. – Curiosity Sep 24 '18 at 6:35
• Best way is to partition the non-stationary signal to the duration you can assume its stationary and perform the operation you require on it. But depends on what you want to do? – learner Sep 24 '18 at 7:53
• I want to use the discrete Wiener estimation method of Levinson to predict the next value in the series using the available ones. – Curiosity Sep 24 '18 at 8:47

## 2 Answers

Strict sense stationarity relates to hypothesis that the statistical law $$f_{\omega(t)}$$ behind the process does not change over time. Even in that case, it is difficult to access to the law from a few realizations, unless you add an ergodicity assumption (possibility to estimate sample properties from past and future data).

Strict sense non-stationarity can offer such a variety of behaviors that a stationarization is unlikely: all $$f_{\omega(t)}$$ should be cast to a single $$f_{\omega}$$. Would it exist, non-stationary processing would be a paradise, as we could apply all stationary methods.

In a lighter sense, one can consider wide-sense non-stationarity, where only the first and second moments (expectation and variance) are allowed to vary. In Nonstationary Models for Time Series, one can explore either what is "Nonstationary in the Variance" and "Nonstationarity in the Mean".

In the mean, several methods can be applied for deterministic trends and stochastic trends. Differentiation, parametric modeling or ARMA, ARIMA, random walk techniques can be used.

In the variance, one can look for Variance-stabilizing transformations (Anscombe, Box-Cox) aiming at:

creating new values such that the variability of the values y is not related to their mean value

• Thanks Laurent: Note that the ARIMA class of models still require that a difference of some order ( which could be greater than 1 but generally 1 is common ) be stationary ( in the mean ) and it requires stationary variance. – mark leeds Sep 25 '18 at 5:20
• Nice, I will be more specific – Laurent Duval Sep 25 '18 at 8:19

You've asked, if it's possible to convert a non-stationary signal into a stationary one and vice versa; the answer is yes it's possible to some extend as outlined below.

Indeed it forms the basis of a family of transform based signal compression algorithms. Specifically, for transform based signal compression, a block of time domain samples $$x=\left[x_0,x_1,...,x_{N-1} \right]$$ are taken, and transformed into a block of frequency domain coefficients $$c=\left[c_0,c_1,...,c_{N-1}\right]$$ practically via a DCT (discrete cosine transform) or similar and are re-quantized for furher processing.

Note that the time domain signal in the block is assumed to be WSS and correlated, as each sample $$x[n]$$ has the same variance $$\sigma_x^2$$, whereas the frequency domain coefficients are non-stationary with each coefficient $$c[k]$$ having a different variance, $$\sigma_{c_k}^2$$. This change in variance results in change in bit-resolution of the quantizer, hence compression. Furthermore, the compression efficieny is a direct consequence of the strength of the correlation in the time domain samples.

This non-stationary frequency transform coefficients $$c[k]$$ are made use of opimal quantization of those coefficients with larger variances (typically the lower frequency coefficients) assisted by the human visual system perceptual characteristics as well. That's a lossy compression. However, you don't have to compress anything, if all you need to get is non-stationary signal.

So this provides an answer for your question, but whether a sutiable method for your financial signal exists is up to you.

Have a look at the following books:

A Primer for Financial Engineering: Financial Signal Processing and Electronic Trading_AKANSU

Financial Signal Processing and Machine Learning_AKANSU

• In your answer, you that each block in the time domain is WSS so stationary. Is this an assumption or do you figure out where the "stationaryness" of the block ends ? Thanks for references you gave. I'll check them out because the field I work in ( although I'm trying to learn DSP, albeit at a snail's pace ) is financial econometrics. – mark leeds Sep 24 '18 at 17:59
• @markleeds Hi mark! The time domain samples $x[n]$ are assumed to be WSS (not strict sense) and correlated to eachother within their own block. This is an engineering assumption and is typically observed for engineering signals of interest such as speech, image, video over short lengths of signal blocks. Otherwise it wont hold for large signal blocks and you will not get compression. – Fat32 Sep 24 '18 at 18:02
• Gotcha. But when you say "within their own block", it sounds kind of like an assumption but do you figure out the block length based on the signal type ( speech, image, video etc ). I ask because, in financial series, the block length decision would be a big problem. – mark leeds Sep 24 '18 at 18:09
• In practice of engineering, the block length is generally fixed (due standards after exhaustive testing of available possibilities). And whether the samples inside that fixed block are really correlated or not, is about how that particular realization confirms to the theoretical assumptions and if that does not happen to confirm you get less efficient compression; that's why signal compression efficiency depends on the input statistics... That's why some images compress more and some do less ;-) – Fat32 Sep 24 '18 at 18:16
• This could be another answer to your question.. mesasoftware.com If you really had something that worked, would you teach it to everyone else ? – mark leeds Sep 25 '18 at 22:13