I am confused with Random walk and Red noise. Some people said that random walk is red noise. Both of them can be generated by integrating white noise. However, Random walk is a nonstationary process, which doesn't have PSD. But red noise has its own PSD, which is 1/f^2. How should I understand the difference between them? Is there anybody has some comments on this?
Although random walk is non-stationary without PSD and the PSD of red noise exists, in the sense of sample PSD, random walk with large window length has the same PSD as the red noise, which indicates the red noise can be treated a segment of random walk. Can I understand like this?
Thank you very much for your information. The random walk is nonstationary, which doesn't have PSD. But red noise has its PSD. How can I understand this difference?
Random walks and noises are very different stochastic processes.
White (or red, or pink or whatever colour) noise have values that are independent: the value of the noise at time t is a random variable that is independent of the value at time s, provided t and s are not equal. In fact this is an approximation because the notion of value here is not well defined, but this is the idea. Your PSD is the distribution function, that is the same at any time (stationary process). Again this is not a proper distribution function but this is not the point.
A random walk is a Markov process, and the most classical is the Brownian motion. This means that the values at times t and s are not independent. But the increases are independent. To make it simple, a Brownian motion is the integral of white noise, and more generally random walks are the integral of some "noise". Since this integral defines the random walk, there is a relationship (integration/derivation) between "noises" and random walks, but they really differ. E.g. a random walk is continuous while a noise is discontinuous.
I hope it helps.
Prof. Arnaud de La Fortelle, thank you very much for your answering. I learned a lot from your introduction. However, I still have a small question. The red noise by definition is also an integration of a white noise. How can I understand the difference between the it and random walk? Could you give some hints? Thank you very much again.
Dear Yingwei Zhao,
In fact I read that red noise = brown noise = Brownian motion. So you are right and my answer is not completely exact. Red noise (for me) is not a real noise since the process goes away from zero (meaning that physically the pressure - if it is a sound - is not constant in average): this is typical for a random walk (non-stationary). However you can produce signals that look like Brownian motion hence the red noise as a limit model.
There is quite a difference in approach between the signal processing community and the stochastic processes community. In signal processing you are interested by frequency analysis that is linked to cross-correlation in stochastic processes. If processes are stationary and Gaussian, there is a bijection between the two representations, but stochastic processes can be much richer (and signals also). But it would probably lead the discussion too far from the original question.
Keep in mind that a signal that has long cross-correlation should not be considered as a noise: it is probably a meaningful signal. Noise is typically uncorrelated. Usually.
Yes, "red noise = brown noise = Brownian motion".
Additional correction: "White (or red, or pink or whatever colour) noise have values that are independent: the value of the noise at time t is a random variable that is independent of the value at time s, provided t and s are not equal." -- TRUE ONLY FOR WHITE NOISE. See this paper for methods to generate coloured noise from white noise, methods that cause correlation beyond t=s.
http://hal.in2p3.fr/in2p3-00024797v3/document
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