ℹ️ Skipped - page is already crawled
| Filter | Status | Condition | Details |
|---|---|---|---|
| HTTP status | PASS | download_http_code = 200 | HTTP 200 |
| Age cutoff | FAIL | download_stamp > now() - 6 MONTH | 7.9 months ago |
| History drop | PASS | isNull(history_drop_reason) | No drop reason |
| Spam/ban | PASS | fh_dont_index != 1 AND ml_spam_score = 0 | ml_spam_score=0 |
| Canonical | PASS | meta_canonical IS NULL OR = '' OR = src_unparsed | Not set |
| Property | Value |
|---|---|
| URL | https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes |
| Last Crawled | 2025-08-21 19:35:36 (7 months ago) |
| First Indexed | not set |
| HTTP Status Code | 200 |
| Meta Title | probability - Karhunen-Loeve expansion of non-centered processes - Mathematics Stack Exchange |
| Meta Description | null |
| Meta Canonical | null |
| Boilerpipe Text | The typical form of Karhunen-Loeve expansion is on a detrended stochastic process. E.g., let $Y(t)$ be a stochastic process on $[0,T]$ , and let $X(t) = Y(t)-\mathbb{E}Y(t)$ with a continuous covariance function $R_X(s,t)$ , then there exists a series of orthonormal functions $\phi_k(t)$ on $[0,T]$ and independent zero-mean and normalized random variables $Z_k$ such that $$
X(t) = \sum_{k=1}^{\infty} Z_k \sqrt{\lambda_k} \phi_k(t)
$$ where $\lambda_k$ and $\phi_k(t)$ are the solutions of the following eigensystems equation: $$
\int_0^T R_X(s,t)\phi_k(t)dt=\lambda_k\phi_k(s)
$$ My question is, if we do not subtract the mean from the process, and just focus on the raw autocorrelation function $R_Y(s,t)$ , as $R_Y(s,t)$ is also positive definite symmetric functions in $s,t$ , it has a spectral decomposition (Mercer's theorem) $$
R_Y(s,t) = \sum_k \mu_k \psi_k(s)\psi_k(t)
$$ Can we expand the original process $Y(t)$ in $\psi(t)$ ? And does this decomposition enjoy some similar properties of the original K-L decomposition, such as the optimal compaction of energies of the signal in the eigenfunctions? $$
Y(t)=\sum_k \tilde{Z}_k \sqrt{\mu_k} \psi_k(t)
$$ The empirical reason is that sometimes we are interested in the information contained in the mean as well, not only in the fluctuation around the mean. While the original K-L expansion optimally compacts the total variance into the first few eigenfunctions $\phi_k(t)$ , the direct decomposition of $Y(t)$ may optimally compact the total fluctuation around $0$ , i.e., the raw energy of the signal: $\mathbb{E}\int_0^T |Y(t)|^2dt$ into $\psi_k(t)$ . Has it been shown to be true? And if true, is the result trivial in the sense that $\psi_k(t)$ and $\phi_k(t)$ are connected in some simple way? |
| Markdown | [Skip to main content](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes#content)
#### Stack Exchange Network
Stack Exchange network consists of 183 Q\&A communities including [Stack Overflow](https://stackoverflow.com/), the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
[Visit Stack Exchange](https://stackexchange.com/)
1. - [Tour Start here for a quick overview of the site](https://math.stackexchange.com/tour)
- [Help Center Detailed answers to any questions you might have](https://math.stackexchange.com/help)
- [Meta Discuss the workings and policies of this site](https://math.meta.stackexchange.com/)
- [About Us Learn more about Stack Overflow the company, and our products](https://stackoverflow.co/)
2. ### [current community](https://math.stackexchange.com/)
- [Mathematics](https://math.stackexchange.com/)
[help](https://math.stackexchange.com/help) [chat](https://chat.stackexchange.com/?tab=site&host=math.stackexchange.com)
- [Mathematics Meta](https://math.meta.stackexchange.com/)
### your communities
[Sign up](https://math.stackexchange.com/users/signup?ssrc=site_switcher&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F4099490%2Fkarhunen-loeve-expansion-of-non-centered-processes) or [log in](https://math.stackexchange.com/users/login?ssrc=site_switcher&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F4099490%2Fkarhunen-loeve-expansion-of-non-centered-processes) to customize your list.
### [more stack exchange communities](https://stackexchange.com/sites)
[company blog](https://stackoverflow.blog/)
3. [Log in](https://math.stackexchange.com/users/login?ssrc=head&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F4099490%2Fkarhunen-loeve-expansion-of-non-centered-processes)
4. [Sign up](https://math.stackexchange.com/users/signup?ssrc=head&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F4099490%2Fkarhunen-loeve-expansion-of-non-centered-processes)
[](https://math.stackexchange.com/)
1. 1. [Home](https://math.stackexchange.com/)
2. [Questions](https://math.stackexchange.com/questions)
3. [Unanswered](https://math.stackexchange.com/unanswered)
4. [AI Assist Labs](https://stackoverflow.ai/)
5. [Tags](https://math.stackexchange.com/tags)
6. [Chat](https://chat.stackexchange.com/rooms/158962/stack-exchange-lobby)
7. [Users](https://math.stackexchange.com/users)
2. Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
[Try Teams for free](https://stackoverflowteams.com/teams/create/free/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams) [Explore Teams](https://stackoverflow.co/teams/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams)
3. [Teams]()
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. [Explore Teams](https://stackoverflow.co/teams/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams-compact)
**Teams**
Q\&A for work
Connect and share knowledge within a single location that is structured and easy to search.
[Learn more about Teams](https://stackoverflow.co/teams/)
# [Karhunen-Loeve expansion of non-centered processes](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes)
[Ask Question](https://math.stackexchange.com/questions/ask)
Asked
4 years, 4 months ago
Modified [3 years, 3 months ago](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes?lastactivity "2022-04-26 07:52:24Z")
Viewed 445 times
7
\$\\begingroup\$
The typical form of Karhunen-Loeve expansion is on a detrended stochastic process. E.g., let \$Y(t)\$ be a stochastic process on \$\[0,T\]\$, and let \$X(t) = Y(t)-\\mathbb{E}Y(t)\$ with a continuous covariance function \$R\_X(s,t)\$, then there exists a series of orthonormal functions \$\\phi\_k(t)\$ on \$\[0,T\]\$ and independent zero-mean and normalized random variables \$Z\_k\$ such that
\$\$ X(t) = \\sum\_{k=1}^{\\infty} Z\_k \\sqrt{\\lambda\_k} \\phi\_k(t) \$\$
where \$\\lambda\_k\$ and \$\\phi\_k(t)\$ are the solutions of the following eigensystems equation:
\$\$ \\int\_0^T R\_X(s,t)\\phi\_k(t)dt=\\lambda\_k\\phi\_k(s) \$\$
My question is, if we do not subtract the mean from the process, and just focus on the raw autocorrelation function \$R\_Y(s,t)\$, as \$R\_Y(s,t)\$ is also positive definite symmetric functions in \$s,t\$, it has a spectral decomposition (Mercer's theorem)
\$\$ R\_Y(s,t) = \\sum\_k \\mu\_k \\psi\_k(s)\\psi\_k(t) \$\$
Can we expand the original process \$Y(t)\$ in \$\\psi(t)\$? And does this decomposition enjoy some similar properties of the original K-L decomposition, such as the optimal compaction of energies of the signal in the eigenfunctions?
\$\$ Y(t)=\\sum\_k \\tilde{Z}\_k \\sqrt{\\mu\_k} \\psi\_k(t) \$\$
The empirical reason is that sometimes we are interested in the information contained in the mean as well, not only in the fluctuation around the mean. While the original K-L expansion optimally compacts the total **variance** into the first few eigenfunctions \$\\phi\_k(t)\$, the direct decomposition of \$Y(t)\$ may optimally compact the total fluctuation around \$0\$, i.e., the raw energy of the signal: \$\\mathbb{E}\\int\_0^T \|Y(t)\|^2dt\$ into \$\\psi\_k(t)\$. Has it been shown to be true? And if true, is the result trivial in the sense that \$\\psi\_k(t)\$ and \$\\phi\_k(t)\$ are connected in some simple way?
- [probability](https://math.stackexchange.com/questions/tagged/probability "show questions tagged 'probability'")
- [stochastic-processes](https://math.stackexchange.com/questions/tagged/stochastic-processes "show questions tagged 'stochastic-processes'")
- [spectral-theory](https://math.stackexchange.com/questions/tagged/spectral-theory "show questions tagged 'spectral-theory'")
- [signal-processing](https://math.stackexchange.com/questions/tagged/signal-processing "show questions tagged 'signal-processing'")
[Share](https://math.stackexchange.com/q/4099490 "Short permalink to this question")
Cite
Follow
asked Apr 12, 2021 at 15:52
[](https://math.stackexchange.com/users/525589/yixianshuiesuan)
[yixianshuiesuan](https://math.stackexchange.com/users/525589/yixianshuiesuan)yixianshuiesuan
9755 bronze badges
\$\\endgroup\$
1
- \$\\begingroup\$ isn't \$R\_X\$ the same as \$R\_Y\$? \$\\endgroup\$
– [G. Gare](https://math.stackexchange.com/users/568973/g-gare "1,555 reputation")
Commented Apr 25, 2022 at 7:54
[Add a comment](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes "Use comments to ask for more information or suggest improvements. Avoid answering questions in comments.") \|
## 1 Answer 1
Sorted by:
[Reset to default](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes?answertab=scoredesc#tab-top)
2
\$\\begingroup\$
It is trivial to see that \\begin{align} R\_X(t,s)&=E\[(X(t) - E\[X(t)\])(X(s)-E\[X(s)\])\] \\\\ &= E\[(Y(t) - E\[Y(t)\])(Y(s)-E\[Y(s)\])\] = R\_Y(t,s). \\end{align} Since both Mercer's theorem and the KL decomposition use an eigendecomposition of \$R\_X\$ (and thus \$R\_Y\$), we have that \$\\psi\_k\$ and \$\\phi\_k\$ are actually the same vectors. In order to find a decomposition of the non-centered process \$Y(t)\$ it suffices to notice that \$\\phi\_k\$ is a complete orthonormal basis of some Lebesgue space on \$\[0,T\]\$ (e.g. \$L^2\$) - in most cases - and therefore denoting \$y(t) = E\[Y(t)\]\$ there exist scalars \$\\{y\_k\\}\$ such that \$\$ y(t) = \\sum\_{k} y\_k \\sqrt{\\lambda\_k} \\phi\_k(t). \$\$ You can then write the process \$Y(t)\$ as \$\$ Y(t) = X(t) + y(t) =\\sum\_k (Z\_k+y\_k)\\sqrt{\\lambda\_k}\\phi\_k. \$\$
[Share](https://math.stackexchange.com/a/4436519 "Short permalink to this answer")
Cite
Follow
answered Apr 26, 2022 at 7:52
[](https://math.stackexchange.com/users/568973/g-gare)
[G. Gare](https://math.stackexchange.com/users/568973/g-gare)G. Gare
1,5551010 silver badges2828 bronze badges
\$\\endgroup\$
[Add a comment](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \|
## You must [log in](https://math.stackexchange.com/users/login?ssrc=question_page&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F4099490) to answer this question.
Start asking to get answers
Find the answer to your question by asking.
[Ask question](https://math.stackexchange.com/questions/ask)
Explore related questions
- [probability](https://math.stackexchange.com/questions/tagged/probability "show questions tagged 'probability'")
- [stochastic-processes](https://math.stackexchange.com/questions/tagged/stochastic-processes "show questions tagged 'stochastic-processes'")
- [spectral-theory](https://math.stackexchange.com/questions/tagged/spectral-theory "show questions tagged 'spectral-theory'")
- [signal-processing](https://math.stackexchange.com/questions/tagged/signal-processing "show questions tagged 'signal-processing'")
See similar questions with these tags.
- Featured on Meta
- [Upcoming initiatives on Stack Overflow and across the Stack Exchange network...](https://meta.stackexchange.com/questions/411578/upcoming-initiatives-on-stack-overflow-and-across-the-stack-exchange-network-j "Upcoming initiatives on Stack Overflow and across the Stack Exchange network - July 2025")
- [Community help needed to clean up goo.gl links (by August 25)](https://math.meta.stackexchange.com/questions/39468/community-help-needed-to-clean-up-goo-gl-links-by-august-25)
#### Related
[0](https://math.stackexchange.com/questions/1001484/karhunen-lo%C3%A8ve-expansion-finding-the-eigenfunctions "Question score (upvotes - downvotes)")
[Karhunen-Loève Expansion (Finding the Eigenfunctions)](https://math.stackexchange.com/questions/1001484/karhunen-lo%C3%A8ve-expansion-finding-the-eigenfunctions)
[3](https://math.stackexchange.com/questions/1361936/generating-cross-correlated-stochastic-processes "Question score (upvotes - downvotes)")
[Generating cross-correlated stochastic processes](https://math.stackexchange.com/questions/1361936/generating-cross-correlated-stochastic-processes)
[1](https://math.stackexchange.com/questions/3525088/convergence-in-karhunen-loeve-expansion "Question score (upvotes - downvotes)")
[Convergence in Karhunen-Loeve expansion](https://math.stackexchange.com/questions/3525088/convergence-in-karhunen-loeve-expansion)
[1](https://math.stackexchange.com/questions/3819785/properties-of-autocorrelation-of-a-convolution "Question score (upvotes - downvotes)")
[Properties of autocorrelation of a convolution](https://math.stackexchange.com/questions/3819785/properties-of-autocorrelation-of-a-convolution)
[3](https://math.stackexchange.com/questions/4528709/pathwise-limit-of-karhunen-loeve-expansion "Question score (upvotes - downvotes)")
[Pathwise Limit of Karhunen-Loeve Expansion](https://math.stackexchange.com/questions/4528709/pathwise-limit-of-karhunen-loeve-expansion)
#### [Hot Network Questions](https://stackexchange.com/questions?tab=hot)
- [Will NASA send Juno to investigate 3i/Atlas?](https://space.stackexchange.com/questions/69778/will-nasa-send-juno-to-investigate-3i-atlas)
- [Curve deformed object won't shade smooth](https://blender.stackexchange.com/questions/338301/curve-deformed-object-wont-shade-smooth)
- [Which version of Hosea 11:4 is closest to the original?](https://hermeneutics.stackexchange.com/questions/107724/which-version-of-hosea-114-is-closest-to-the-original)
- [For us to see an exoplanet transit, what is the maximum angle between that planet's orbital plane and our line of sight?](https://astronomy.stackexchange.com/questions/61577/for-us-to-see-an-exoplanet-transit-what-is-the-maximum-angle-between-that-plane)
- [Why is the heliopause so hot?](https://physics.stackexchange.com/questions/857914/why-is-the-heliopause-so-hot)
- [Pennies to Dollars](https://codegolf.stackexchange.com/questions/283193/pennies-to-dollars)
- [Does the French toll tag bip & go invoice show all tolls or just the total per month?](https://travel.stackexchange.com/questions/200000/does-the-french-toll-tag-bip-go-invoice-show-all-tolls-or-just-the-total-per-m)
- [Who was Rav Abun of Narbonne](https://judaism.stackexchange.com/questions/151120/who-was-rav-abun-of-narbonne)
- [How to prove an example of uncorrelated but not independent?](https://stats.stackexchange.com/questions/669694/how-to-prove-an-example-of-uncorrelated-but-not-independent)
- [What's an example of valuative criterion of properness needing all DVRs?](https://mathoverflow.net/questions/499391/whats-an-example-of-valuative-criterion-of-properness-needing-all-dvrs)
- [Exam Question: Bank angle/Turn radius change with identical plane with reduction in weight](https://aviation.stackexchange.com/questions/111232/exam-question-bank-angle-turn-radius-change-with-identical-plane-with-reduction)
- [Can an AI model license determine which governing law and jurisdiction apply to the users of this model?](https://law.stackexchange.com/questions/110658/can-an-ai-model-license-determine-which-governing-law-and-jurisdiction-apply-to)
- [Why does my resistance in the LC tank circuit appear different?](https://electronics.stackexchange.com/questions/754131/why-does-my-resistance-in-the-lc-tank-circuit-appear-different)
- [What happens to a Stalker of Kharash who starts as a non-caster ranger?](https://rpg.stackexchange.com/questions/216363/what-happens-to-a-stalker-of-kharash-who-starts-as-a-non-caster-ranger)
- [Practical advice on using common diodes in series for voltage regulation](https://electronics.stackexchange.com/questions/754087/practical-advice-on-using-common-diodes-in-series-for-voltage-regulation)
- [What kind of instrument is my door?](https://music.stackexchange.com/questions/141244/what-kind-of-instrument-is-my-door)
- [Spectral sequences every mathematician should know](https://mathoverflow.net/questions/499399/spectral-sequences-every-mathematician-should-know)
- [Book with an alien world whose lifeforms are made of reeds](https://scifi.stackexchange.com/questions/298798/book-with-an-alien-world-whose-lifeforms-are-made-of-reeds)
- [Why wrap fruitcake in grease-proof paper?](https://cooking.stackexchange.com/questions/132990/why-wrap-fruitcake-in-grease-proof-paper)
- [Should I apply to volunteer while actively looking for a job, if it means I won't be volunteering for long?](https://workplace.stackexchange.com/questions/202455/should-i-apply-to-volunteer-while-actively-looking-for-a-job-if-it-means-i-won)
- [What is the principle of non-liability in the European Union wrt the US Section 230 and social media?](https://law.stackexchange.com/questions/110669/what-is-the-principle-of-non-liability-in-the-european-union-wrt-the-us-section)
- [Perspectives on a crumpled word](https://puzzling.stackexchange.com/questions/132953/perspectives-on-a-crumpled-word)
- [What is meant by fill power steering with cold engine?](https://mechanics.stackexchange.com/questions/99363/what-is-meant-by-fill-power-steering-with-cold-engine)
- [Is "-ledge" a suffix in the word knowledge?](https://english.stackexchange.com/questions/632815/is-ledge-a-suffix-in-the-word-knowledge)
[more hot questions](https://math.stackexchange.com/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes)
[Question feed](https://math.stackexchange.com/feeds/question/4099490 "Feed of this question and its answers")
# Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
##### [Mathematics](https://math.stackexchange.com/)
- [Tour](https://math.stackexchange.com/tour)
- [Help](https://math.stackexchange.com/help)
- [Chat](https://chat.stackexchange.com/?tab=site&host=math.stackexchange.com)
- [Contact](https://math.stackexchange.com/contact)
- [Feedback](https://math.meta.stackexchange.com/)
##### [Company](https://stackoverflow.co/)
- [Stack Overflow](https://stackoverflow.com/)
- [Teams](https://stackoverflow.co/teams/)
- [Advertising](https://stackoverflow.co/advertising/)
- [Talent](https://stackoverflow.co/advertising/employer-branding/)
- [About](https://stackoverflow.co/)
- [Press](https://stackoverflow.co/company/press/)
- [Legal](https://stackoverflow.com/legal)
- [Privacy Policy](https://stackoverflow.com/legal/privacy-policy)
- [Terms of Service](https://stackoverflow.com/legal/terms-of-service/public)
- Cookie Settings
- [Cookie Policy](https://policies.stackoverflow.co/stack-overflow/cookie-policy)
##### [Stack Exchange Network](https://stackexchange.com/)
- [Technology](https://stackexchange.com/sites#technology)
- [Culture & recreation](https://stackexchange.com/sites#culturerecreation)
- [Life & arts](https://stackexchange.com/sites#lifearts)
- [Science](https://stackexchange.com/sites#science)
- [Professional](https://stackexchange.com/sites#professional)
- [Business](https://stackexchange.com/sites#business)
- [API](https://api.stackexchange.com/)
- [Data](https://data.stackexchange.com/)
- [Blog](https://stackoverflow.blog/?blb=1)
- [Facebook](https://www.facebook.com/officialstackoverflow/)
- [Twitter](https://twitter.com/stackoverflow)
- [LinkedIn](https://linkedin.com/company/stack-overflow)
- [Instagram](https://www.instagram.com/thestackoverflow)
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under [CC BY-SA](https://stackoverflow.com/help/licensing) . rev 2025.8.21.33163 |
| Readable Markdown | null |
| Shard | 18 (laksa) |
| Root Hash | 8045678284012640218 |
| Unparsed URL | com,stackexchange!math,/questions/4099490/karhunen-loeve-expansion-of-non-centered-processes s443 |