Talk:Autocorrelation

Unclear passage

This passage:

"For a weak-sense stationarity, wide-sense stationarity (WSS) process, the definition is

$$\rho _{XX}(\tau )={\frac {\operatorname {K} _{XX}(\tau )}{\sigma ^{2}}}={\frac {\operatorname {E} \left[(X_{t+\tau }-\mu ){\overline {(X_{t}-\mu )}}\right]}{\sigma ^{2}}}$$

where

$$\operatorname {K} _{XX}(0)=\sigma ^{2}.$$"

seems unclear to me, because to the uninitiated (like me), the phrase:

"For a weak-sense stationarity, wide-sense stationarity (WSS) process"

seems to be self-contradictory nonsense.

I hope someone knowledgeable about this subject will please rewrite this so that it does not come across as nonsense.

Please note: I am not saying that it is nonsense. But since it appears to be nonsense, this would benefit from some clarification.

Is there supposed to be an and between "weak-sense stationarity" and "wide-sense stationarity" ??? An or ??? Or what??? 2601:200:C000:1A0:300E:BD77:DEE5:AA45 (talk) 03:27, 17 August 2022 (UTC)

The passage you quoted contains a link to a section entitles "Weak or wide-sense stationarity" of the Wikipedia article on "Stationary process". That article begins by saying, "a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time." That subsection says that a process is "weak-sense stationarity, wide-sense stationarity (WSS), or covariance stationarity" if the first two moments and autocovariance are finite and do not change over time.

From that it's clear that we could rewrite that phrase as 'For a weak-sense stationary (WSS) process (also called a "wide-sense stationary" or "covariance stationary") process'. Please feel free to make that change. (Please excuse: It's 1:16 AM where I am right now, and I'm going back to bed. I believe that I tend to be more productive if I check my email in the middle of the night and spend a few minutes doing things like this. However, I don't feel an urgency to change this myself, but if you wanted to make this change, I would encourage you to do so. You may know that almost anyone can change almost anything on Wikipedia. What stays tends to be written from a neutral point of view citing credible sources. (For the policy regarding articles that can NOT be changed by anyone see see "Wikipedia:Protection policy". Wikipedia also has a policy encouraging people to make changes like this. The policy is summarized as "be bold but not reckless", i.e., just do it. ;-) Users who repeatedly make changes that are obvious vandalism can be blocked. This clearly isn't. If a change you is not quite right for some reason, some other user will likely fix it. As an aside, you may know that if you have an account, you can "watch" articles. I get several emails a day telling me that articles I'm "watching" have changed and inviting me to look at the change. That's how I saw your question.) DavidMCEddy (talk) 06:44, 17 August 2022 (UTC)

If it's obvious vadalism

DavidMCEddy (talk) 06:44, 17 August 2022 (UTC)

Contradictory definition

The section Auto-correlation of stochastic processes begins as follows:

"In statistics, the autocorrelation of a real or complex random process is the Pearson correlation between values of the process at different times, as a function of the two times or of the time lag."

The definition is then presented as

$\operatorname {R} _{XX}(t_{1},t_{2})=\operatorname {E} \left[X_{t_{1}}{\overline {X}}_{t_{2}}\right]$ Eq.1

(where $\operatorname {E}$ is the expected value operator and the bar represents complex conjugation.)

However, this does not use Pearson correlation, but instead uses covariance.

Later in the article it is explained that there are two conventions, one for using Pearson correlation and another for using covriance (and calllling it "correlation" anyway).

Very bad idea. The passage I quote above contradicts itself.

And regardless of whether some people misuse the word "correlation" to mean "covariance", that should not be what Wikipedia does. 2601:200:C000:1A0:1841:4827:BAAA:F4DF (talk) 15:52, 17 August 2022 (UTC)

Agreed. (Eq. 1) is wrong.

Can someone check the two references? If they contain that expression, those references are wrong -- or at least using non-standard definitions.

I do not have time now to fix this, but I would support others doing it. E.g., what about deleting (Eq. 1). I suggest also deleting all equation numbers unless there are actual references. Simple searches yielded nothing.

FYI: Almost anyone can change almost anything in Wikipedia. Others change new contributions they find inappropriate. What stays tends to be written from a neutral point of view citing credible sources. We encourage users to Wikipedia:Be bold but not reckless.

Thanks for the observation. DavidMCEddy (talk) 16:18, 17 August 2022 (UTC)

@RaviGaaDu: Can you please check a reference for the formula you changed?

I just reverted it for two reasons:

• The n or (n - 1) or (n - k) in the denominator would seem to refer to the number of terms in the summation, NOT the number of degrees of freedom in an estimate of $\sigma$.
• The line above says it's "For a discrete process with known mean and variance", BUT this formula does NOT subtract the known mean.

Beyond this, I think all the math in this article should be reviewed carefully and check with published sources, as suggested in my comment above from almost two years ago: I think there are other errors in this article, but I am not prepared to spend the time myself to check and fix them. DavidMCEddy (talk) 12:41, 24 July 2024 (UTC)

Potentially incorrect passage

At this line: https://en.wikipedia.org/wiki/Autocorrelation#:~:text=This%20gives%20the%20more%20familiar%20forms%20for%20the%20autocorrelation%20function you get that final formula of the auto-correlation function by writing tau = t2 - t1. But I think that you get that result by writing tau = t1 - t2. If you write tau = t2 - t1, then the conjugate should be on X_(t+tau) and not on X_t. As a reference, on my book you get that final formula that you see on Wikipedia by writing tau = t1-t2 and NOT tau = t2-t1. Actually, I don't know if they are equivalent, but starting from the definition of autocorrelation and writing tau = t2 - t1 leads me to a different result than the one shown on Wikipedia. 79.18.185.77 (talk) 17:48, 9 April 2025 (UTC)

I believe that an autocovariance should be the covariance of a stochastic process at one point in time with itself at a different point in time, and an autocorrelation should be the correlation of a stochastic process at one point in time with itself at a different point in time. The current formulae do not represent that, especially for the alleged "autocorrelation": I think it should be a number between (-1) and (+1); it's not.

If we talk about a complex vector stochastic process, then I think the autocovariance should be the covariance of the vector at one point in time with the transpose of the complex conjugate of that vector at a different point in time.

However, I just found a reference that suggests that what I just wrote may not be honored in the literature on this subject:

Correlation between Two Complex-Valued Random Variables by Prof. Joon Ho Cho

I just consulted 9 different time series books in my personal library. I could not quickly find general definition of autocovariance and autocorrelation that I could cite here ... and I don't feel I can afford the time to research this futher.

I wish you luck in recruiting someone who could help with this.

Thanks for raising the question. DavidMCEddy (talk) 18:57, 9 April 2025 (UTC)

I could not easily access the re