# Question about linear combination of non-stationary signals

Suppose $$f_1,\ f_2,\ ...\ f_{n-1},\ f_n$$ are $$n$$ non-stationary discrete signals of same length and linearly independent. Each $$f_i$$ is an intrinsic mode function (IMF) of a seismic signal evaluated by empirical mode decomposition (EMD). That is, they are amplitude-modulated, frequency-modulated signals. Linear combination of these signals can be expressed as follow:

$$F(t) = \sum_{j=1}^{j=n} \alpha _j f_j(t)$$

where, $$\alpha_js$$ are real numbers and they are to be selected such that $$\sum_{j=0}^{j=n} \alpha_j\ne0$$. Is it true to claim that $$F(t)$$ is also a non-stationary signal?

No, try with $$\alpha_j = 0$$, $$\forall j$$.

I suspect this to be a tricky question, to deceive fast thinking. For instance, having the "same length" does not mean a lot, for a continuous variable $$t$$. Misleading fake clues to make the question look serious.

Without more hypotheses on the notion of stationarity (in law, wide-sense) and the statistical interrelations between the so-called signals (it would be wiser to talk about processes). See Marcus' answer for more examples.

• Let me to clarify the question. Suppose $f_1, ..., f_n$ be $n$ non-stationary discrete signals of same length and linearly independent. Also, $\alpha_js$ are to be selected such that $\sum_{j=1}^{j=n}\alpha_j\ne0$. With these conditions, what is the answer of my question? Commented May 18, 2020 at 6:12
• Do you have more hypotheses about the type of non-stationary in $f_i$? Commented May 18, 2020 at 6:40
• Each $f_i$ is an intrinsic mode function (IMF) of a seismic signal evaluated by empirical mode decomposition technique (EMD, developed by Prof. Huang). That is, they are amplitude-modulated, frequency-modulated signals. Commented May 18, 2020 at 12:10

And to extend Laurent's excellent, since on-point, answer:

non-stationarity doesn't say anything about correlation.

For example, a totally valid scenario would be:

$$f_{j} = -f_{j-1}\,\text{for } j>1$$

and then, for any even $$n$$ and a lot of potential sequences $$(\alpha_i)$$, the result is a constant, too.

Whenever you see combinations of multiple stochastic entities, you should intuitively start wondering about correlations.

• Great addition, I answered a bit too fast before lunch Commented May 17, 2020 at 13:23
• no, don't think so, the question was "is A?" and you proved by counter example that NOT A is true. Commented May 17, 2020 at 13:24
• While you were answering, I added some bits about unstated hypotheses Commented May 17, 2020 at 13:24
• can't double-upvote your answer :) Commented May 17, 2020 at 13:25
• In econometrics, when the linear combination of multiple non-stationary series is stationary, the series are said to be cointegrated. Google for "johansen's cointegration methdology" if interested. In the two series case, cointegration is what engle and granger received the nobel prize in econometrics for. johansen later extended the concept to more than two series. Commented May 17, 2020 at 14:06