Editing Pink noise (section)

== Occurrence ==
Pink noise has been discovered in the [[statistical fluctuations]] of an extraordinarily diverse number of physical and biological systems (Press, 1978;<ref name="Press-1978">{{cite journal
      | author = Press, W. H.
      | year = 1978
      | title = Flicker noises in astronomy and elsewhere
      | journal = Comments in Astrophysics
      | volume = 7
      | pages = 103–119
      | issue = 4
      | bibcode = 1978ComAp...7..103P
 }}</ref> see articles in Handel & Chung, 1993,<ref name="Handel-1993">{{cite book
      | author = Handel, P. H. |author2=Chung, A. L.
      | year = 1993
      | title = Noise in Physical Systems and 1/"f" Fluctuations
      | publisher = American Institute of Physics 
      | location = New York 
      }}</ref> and references therein). Examples of its occurrence include fluctuations in [[tide]] and river heights, [[quasar]] light emissions, heart beat, firings of single [[neuron]]s, [[resistivity]] in [[solid-state electronics]] and single-molecule conductance signals<ref>{{Cite journal|last1=Adak|first1=Olgun|last2=Rosenthal|first2=Ethan|last3=Meisner|first3=Jeffery|last4=Andrade|first4=Erick F.|last5=Pasupathy|first5=Abhay N.|last6=Nuckolls|first6=Colin|last7=Hybertsen|first7=Mark S.|last8=Venkataraman|first8=Latha|date=2015-05-07|title=Flicker Noise as a Probe of Electronic Interaction at Metal–Single Molecule Interfaces|url=https://pubs.acs.org/doi/pdf/10.1021/acs.nanolett.5b01270?rand=4a25iu4o|journal=Nano Letters|volume=15|issue=6|pages=4143–4149|doi=10.1021/acs.nanolett.5b01270|pmid=25942441|bibcode=2015NanoL..15.4143A|issn=1530-6984|url-access=subscription}}</ref> resulting in [[flicker noise]]. Pink noise describes the [[Scene statistics|statistical structure of many natural images]].<ref name="Field-1987">{{Cite journal
 | volume = 4
 | issue = 12
 | pages = 2379–2394
 | last = Field
 | first = D. J.
 | title = Relations between the statistics of natural images and the response properties of cortical cells
 | journal = J. Opt. Soc. Am. A
 | year = 1987
 | doi = 10.1364/JOSAA.4.002379
 | pmid = 3430225
 |bibcode = 1987JOSAA...4.2379F
| url = http://redwood.psych.cornell.edu/papers/field_87.pdf
 | citeseerx = 10.1.1.136.1345
 }}</ref> 

General 1/''f''<sup>&nbsp;α</sup> noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.<ref name="Bak-1987">{{cite journal
      | author = Bak, P. |author2=Tang, C. |author3=Wiesenfeld, K.
      |year= 1987
      | title = Self-Organized Criticality: An Explanation of 1/''ƒ'' Noise
      | journal = [[Physical Review Letters]]
      | volume = 59
      | pages = 381–384
      | doi = 10.1103/PhysRevLett.59.381
      |pmid= 10035754
      | issue = 4
      | bibcode=1987PhRvL..59..381B
|s2cid=7674321 }}</ref> In physical systems, they are present in some [[meteorological]] data series, the [[electromagnetic radiation]] output of some astronomical bodies. In biological systems, they are present in, for example, [[cardiac cycle|heart beat]] rhythms, neural activity, and the statistics of [[DNA sequence]]s, as a generalized pattern.<ref>Josephson, Brian D. (1995). "A trans-human source of music?" in (P. Pylkkänen and P. Pylkkö, eds.) ''New Directions in Cognitive Science'', Finnish Artificial Intelligence Society, Helsinki; pp. 280–285.</ref>

An accessible introduction to the significance of pink noise is one given by [[Martin Gardner]] (1978) in his ''Scientific American'' column "Mathematical Games".<ref name="Gardner-1978">{{cite journal
      | author = Gardner, M.
      | year = 1978
      | title = Mathematical Games—White and brown music, fractal curves and one-over-f fluctuations
      | journal = Scientific American
      | volume = 238
      | issue = 4
      | pages = 16–32
      | doi = 10.1038/scientificamerican0478-16
      }}</ref> In this column, Gardner asked for the sense in which music imitates nature. Sounds in nature are not musical in that they tend to be either too repetitive (bird song, insect noises) or too chaotic (ocean surf, wind in trees, and so forth). The answer to this question was given in a statistical sense by Voss and Clarke (1975, 1978), who showed that pitch and loudness fluctuations in speech and music are pink noises.<ref name="Voss-1975">{{cite journal
      | author = Voss, R. F. |author2=Clarke, J.
      | year = 1975
      | title = '1/f Noise' in Music and Speech
      | journal = Nature
      | volume = 258
      | issue = 5533
      | pages = 317–318
      | doi=10.1038/258317a0
      |bibcode = 1975Natur.258..317V |s2cid=4182664
 }}</ref><ref name="Voss-1978">{{cite journal
      | author = Voss, R. F. |author2=Clarke, J.
      | year = 1978
      | title = 1/f noise" in music: Music from 1/f noise
      | journal = Journal of the Acoustical Society of America
     | volume = 63
      | issue = 1
      | pages = 258–263
      | doi=10.1121/1.381721
      |bibcode = 1978ASAJ...63..258V }}</ref> So music is like tides not in terms of how tides sound, but in how tide heights vary.

=== Precision timekeeping ===
{{Main page|Allan variance}}
The ubiquitous 1/f noise poses a "noise floor" to precision timekeeping.<ref name="Press-1978" /> The derivation is based on.<ref>{{Cite book |last=Voss |first=R.F. |title=33rd Annual Symposium on Frequency Control |chapter=1/F (Flicker) Noise: A Brief Review |date=May 1979 |chapter-url=https://ieeexplore.ieee.org/document/1537237  |pages=40–46 |doi=10.1109/FREQ.1979.200297|s2cid=37302662 }}</ref>
[[File:AllanDeviation.svg|right|thumb|300x300px|A clock is most easily tested by comparing it with a ''far more accurate'' reference clock. During an interval of time ''τ'', as measured by the reference clock, the clock under test advances by ''τy'', where ''y'' is the average (relative) clock frequency over that interval.]]
Suppose that we have a timekeeping device (it could be anything from [[Crystal oscillator|quartz oscillators]], [[Atomic clock|atomic clocks]], and [[Hourglass|hourglasses]]<ref>{{Cite journal |last1=Schick |first1=K. L. |last2=Verveen |first2=A. A. |date=October 1974 |title=1/f noise with a low frequency white noise limit |url=https://www.nature.com/articles/251599a0 |journal=Nature |language=en |volume=251 |issue=5476 |pages=599–601 |doi=10.1038/251599a0 |bibcode=1974Natur.251..599S |s2cid=4200003 |issn=1476-4687|url-access=subscription }}</ref>). Let its readout be a real number <math>x(t)</math> that changes with the actual time <math>t</math>. For concreteness, let us consider a quartz oscillator. In a quartz oscillator, <math>x(t)</math> is the number of oscillations, and <math>\dot x(t)</math> is the rate of oscillation. The rate of oscillation has a constant component <math>\dot x_0</math>and a fluctuating component <math>\dot x_f</math>, so <math display="inline">\dot x(t) = \dot x_0 + \dot x_f(t)</math>. By selecting the right units for <math>x</math>, we can have <math>\dot x_0 = 1</math>, meaning that on average, one second of clock-time passes for every second of real-time.

The stability of the clock is measured by how many "ticks" it makes over a fixed interval. The more stable the number of ticks, the better the stability of the clock. So, define the average clock frequency over the interval <math>[k\tau, (k+1)\tau]</math> as<math display="block">y_k = \frac{1}{\tau}\int_{k\tau}^{(k+1)\tau}\dot x(t)dt = \frac{x( (k+1 ) \tau) - x(k\tau)}{\tau}</math>Note that <math>y_k</math> is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock{{NoteTag|Though in practice, since there are no ideal clocks, <math>t</math> is actually the ticks of a much more accurate clock.}}.

The [[Allan variance]] of the clock frequency is half the mean square of change in average clock frequency:<math display="block">\sigma^2(\tau) = \frac 12 \overline{(y_{k} - y_{k-1})^2} = \frac{1}{K}\sum_{k=1}^K \frac 12 (y_{k} - y_{k-1})^2</math>where <math>K</math> is an integer large enough for the averaging to converge to a definite value.
For example, a 2013 atomic clock<ref>{{Cite journal |last1=Hinkley |first1=N. |last2=Sherman |first2=J. A. |last3=Phillips |first3=N. B. |last4=Schioppo |first4=M. |last5=Lemke |first5=N. D. |last6=Beloy |first6=K. |last7=Pizzocaro |first7=M. |last8=Oates |first8=C. W. |last9=Ludlow |first9=A. D. |date=2013-09-13 |title=An Atomic Clock with 10 –18 Instability |url=https://www.science.org/doi/10.1126/science.1240420 |journal=Science |language=en |volume=341 |issue=6151 |pages=1215–1218 |doi=10.1126/science.1240420 |pmid=23970562 |arxiv=1305.5869 |bibcode=2013Sci...341.1215H |s2cid=206549862 |issn=0036-8075}}</ref> achieved <math>\sigma(25000\text{ seconds}) = 1.6 \times 10^{-18}</math>, meaning that if the clock is used to repeatedly measure intervals of 7 hours, the standard deviation of the actually measured time would be around 40 [[Femtosecond|femtoseconds]].

Now we have<math display="block">y_{k} - y_{k-1} = \int_\R g(k\tau - t) \dot x_f(t) dt = (g\ast \dot x_f)(k\tau) </math>where <math>g(t) = \frac{-1_{[0, \tau]}(t) + 1_{[-\tau, 0]}(t)}{\tau}</math> is one packet of a [[Square wave (waveform)|square wave]] with height <math>1/\tau</math> and wavelength <math>2\tau</math>. Let <math>h(t)</math> be a packet of a square wave with height 1 and wavelength 2, then <math>g(t) = h(t/\tau)/\tau</math>, and its Fourier transform satisfies <math>\mathcal F[g](\omega) = \mathcal F[h](\tau\omega)</math>.

The Allan variance is then <math>\sigma^2(\tau) = \frac 12 \overline{(y_{k} - y_{k-1})^2} = \frac 12 \overline{(g\ast \dot x_f)(k\tau)^2}
</math>, and the discrete averaging can be approximated by a continuous averaging: <math>\frac{1}{K}\sum_{k=1}^K \frac 12 (y_{k} - y_{k-1})^2 \approx \frac{1}{K\tau}\int_0^{K\tau} \frac 12(g\ast \dot x_f)(t)^2 dt</math>, which is the total power of the signal <math>(g\ast \dot x_f)</math>, or the integral of its [[Spectral density|power spectrum]]:
[[File:Illustration for Allan variance of 1-f noise.png|thumb|315x315px|<math>\sigma^2(1)</math> is approximately the area under the green curve; when <math>\tau</math> increases, <math>S[g](\omega) </math> shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis; when <math>S[\dot x_f](\omega) \propto \omega^{-\alpha}</math>, the combined effect of both is that <math>\sigma^2(\tau) \propto \tau^{\alpha-1}</math>]]
<math display="block">\sigma^2(\tau) \approx \int_0^\infty S[g\ast \dot x_f](\omega) d\omega = \int_0^\infty S[g](\omega) \cdot S[\dot x_f](\omega)  d\omega = \int_0^\infty S[h](\tau \omega) \cdot S[\dot x_f](\omega) d\omega</math>In words, the Allan variance is approximately the power of the fluctuation after [[Band-pass filter|bandpass filtering]] at <math>\omega \sim 1/\tau</math> with bandwidth <math>\Delta\omega \sim 1/\tau </math>.


For <math>1/f^\alpha</math> fluctuation, we have <math>S[\dot x_f](\omega) = C/\omega^\alpha</math> for some constant <math>C</math>, so <math>\sigma^2(\tau) \approx \tau^{\alpha-1} \sigma^2(1) \propto \tau^{\alpha-1}</math>. In particular, when the fluctuating component <math>\dot x_f</math> is a 1/f noise, then <math>\sigma^2(\tau)</math> is independent of the averaging time <math>\tau</math>, meaning that the clock frequency does not become more stable by simply averaging for longer. This contrasts with a white noise fluctuation, in which case <math>\sigma^2(\tau) \propto \tau^{-1}</math>, meaning that doubling the averaging time would improve the stability of frequency by <math>\sqrt 2</math>.<ref name="Press-1978" />

The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.<ref>{{Citation |last=Vessot |first=Robert F. C. |title=5.4. Frequency and Time Standards††This work was supported in part by contract NSR 09-015-098 from the National Aeronautics and Space Administration. |date=1976-01-01 |url=https://www.sciencedirect.com/science/article/pii/S0076695X08607103 |work=Methods in Experimental Physics |volume=12 |pages=198–227 |editor-last=Meeks |editor-first=M. L. |access-date=2023-07-17 |series=Astrophysics |publisher=Academic Press |doi=10.1016/S0076-695X(08)60710-3 |language=en|url-access=subscription }}</ref>

=== Humans ===
In [[brains]], pink noise has been widely observed across many temporal and physical scales from [[ion channel]] gating to [[EEG]] and [[Magnetoencephalography|MEG]] and [[Local field potential|LFP]] recordings in humans.<ref>{{Citation |last1=Destexhe |first1=Alain |title=Local Field Potentials: LFP |date=2020 |url=https://doi.org/10.1007/978-1-4614-7320-6_548-2 |encyclopedia=Encyclopedia of Computational Neuroscience |pages=1–12 |editor-last=Jaeger |editor-first=Dieter |access-date=2023-07-26 |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-1-4614-7320-6_548-2 |isbn=978-1-4614-7320-6 |last2=Bédard |first2=Claude |s2cid=243735998 |editor2-last=Jung |editor2-first=Ranu|url-access=subscription }}</ref> In clinical EEG, deviations from this 1/f pink noise can be used to identify [[epilepsy]], even in the absence of a [[seizure]], or during the interictal state.<ref name="Kerr-2012">{{Cite journal
 | volume = 53
 | issue = 11
 | pages = e189–e192
 | author = Kerr, W.T. 
|display-authors=etal
 | title = Automated diagnosis of epilepsy using EEG power spectrum
 | journal = Epilepsia
 | year = 2012
 | doi=10.1111/j.1528-1167.2012.03653.x |pmc=3447367
| pmid = 22967005
 }}</ref> Classic models of EEG generators suggested that dendritic inputs in [[gray matter]] were principally responsible for generating the 1/f power spectrum observed in EEG/MEG signals.  However, recent computational models using [[cable theory]] have shown that [[action potential]] transduction along [[white matter]] tracts in the brain also generates a 1/f spectral density.  Therefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings, <ref name="Douglas-2019">{{Cite book
 | author = Douglas, PK
|title=2019 7th International Winter Conference on Brain-Computer Interface (BCI)
 |chapter=Reconsidering Spatial Priors in EEG Source Estimation : Does White Matter Contribute to EEG Rhythms?
 |display-authors=etal
 | publisher = IEEE 
 |year = 2019
 |pages=1–12
 | doi =10.1109/IWW-BCI.2019.8737307|arxiv=2111.08939
 |isbn=978-1-5386-8116-9
 |s2cid=195064621
 }}</ref>
particularly if the effects of ephaptic coupling are taken into consideration.<ref name="Douglas-2024">{{Cite book
 | author = Douglas, PK |author2= Blair, G.
 |title=2024 12th International Winter Conference on Brain-Computer Interface (BCI)
 |chapter=Towards a white matter ephaptic coupling model of 1/f spectra
 |display-authors=etal
 | publisher = IEEE 
 |year = 2024
 |pages=1–3
 | doi =10.1109/BCI60775.2024.10480498
 }}</ref> 

It has also been successfully applied to the modeling of [[mental representation|mental states]] in [[psychology]],<ref name="cognitive_2003">{{Cite journal
 | volume = 132
 | issue = 3
 | pages = 331–350
 | author = Van Orden, G.C. |author2=Holden, J.G. |author3=Turvey, M.T.
 | title = Self-organization of cognitive performance
 | journal = Journal of Experimental Psychology: General
 | year = 2003
 | doi = 10.1037/0096-3445.132.3.331
| pmid = 13678372
 }}</ref> and used to explain stylistic variations in music from different cultures and historic periods.<ref>Pareyon, G. (2011). ''On Musical Self-Similarity'', International Semiotics Institute & University of Helsinki. {{cite web
     | title = On Musical Self-Similarity
     | url = https://helda.helsinki.fi/bitstream/handle/10138/29824/Pareyon_Dissertation.pdf?sequence=2
}}</ref> Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of [[pitch (music)|pitches]], will tend towards a pink noise spectrum.<ref name="Kuittinen">{{cite web| url = http://mlab.uiah.fi/~eye/mediaculture/noise.html| title = Noise in Man-generated Images and Sound}}</ref> Similarly, a generally pink distribution pattern has been observed in [[Shot (filmmaking)|film shot]] length by researcher [[James E. Cutting]] of [[Cornell University]], in the study of 150 popular movies released from 1935 to 2005.<ref>Anger, Natalie (March 1, 2010). [https://www.nytimes.com/2010/03/02/science/02angi.html "Bringing New Understanding to the Director's Cut"]. ''The New York Times''. Retrieved on March 3, 2010. See also [http://pss.sagepub.com/content/early/2010/02/04/0956797610361679.full.pdf+html original study] {{Webarchive|url=https://web.archive.org/web/20130124170244/http://pss.sagepub.com/content/early/2010/02/04/0956797610361679.full.pdf+html |date=2013-01-24 }}</ref>

Pink noise has also been found to be endemic in human response. Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.<ref name="Gilden-1995">{{cite journal
      | author = Gilden, David L |author2=Thornton, T |author3=Mallon, MW
      | year = 1995
      | title = 1/''ƒ'' Noise in Human Cognition
      | journal = Science
      | volume = 267
      | pages = 1837–1839
      | doi = 10.1126/science.7892611
      | pmid = 7892611
 | issn = 0036-8075
      | issue = 5205
      |bibcode = 1995Sci...267.1837G }}</ref> Later, Gilden (1997) and Gilden (2001) found that time series formed from [[reaction time]] measurement and from iterated two-alternative forced choice also produced pink noises.<ref name="Gilden-1997">{{cite journal
      | author = Gilden, D. L. 
      | year = 1997
      | title = Fluctuations in the time required for elementary decisions
      | journal = Psychological Science
      | volume = 8
      | pages = 296–301
      | doi = 10.1111/j.1467-9280.1997.tb00441.x
      | issue = 4
      | s2cid = 145051976
 }}</ref><ref name="Gilden-2001">{{cite journal
      | author = Gilden, David L
      | year = 2001
      | title = Cognitive Emissions of 1/''ƒ'' Noise
      | journal = [[Psychological Review]]
      | volume = 108
      | pages = 33–56
      | doi = 10.1037/0033-295X.108.1.33
      | issn = 0033-295X
      | issue = 1
      | pmid = 11212631
 | citeseerx = 10.1.1.136.1992
      }}</ref>

===Electronic devices===
{{Main|Flicker noise}}

The principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices. In many cases the specific sources of the fluctuations are known. These include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials.<ref name="Kogan-1996" /><ref name="Weissman-1988">{{cite journal
      | author = Weissman, M. B.
      |year= 1988
      | title = 1/''ƒ'' Noise and other slow non-exponential kinetics in condensed matter
      | journal = [[Reviews of Modern Physics]]
      | volume = 60
      | pages = 537–571
      | doi = 10.1103/RevModPhys.60.537
      | issue = 2
      | bibcode=1988RvMP...60..537W
}}</ref> The explanation for the approximately pink spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes.<ref name="Dutta-1981">{{cite journal
      |author1=Dutta, P. |author2=Horn, P. M.
      |name-list-style=amp |year= 1981
      | title = Low-frequency fluctuations in solids: 1/''f'' noise
      | journal = [[Reviews of Modern Physics]]
      | volume = 53
      | pages = 497–516
      | doi = 10.1103/RevModPhys.53.497
      | issue = 3
      | bibcode=1981RvMP...53..497D
}}</ref> Since the frequency range of the typical noise experiment (e.g., 1&nbsp;Hz – 1&nbsp;kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 10<sup>14</sup>&nbsp;Hz), the exponential factors in the [[Arrhenius equation]] for the rates are large. Relatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates. In the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because <math>\textstyle \frac{d}{df}\ln f = \frac{1}{f}.</math>

There is no known lower bound to background pink noise in electronics. Measurements made down to 10<sup>−6</sup>&nbsp;Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.<ref name="Kleinpenning-1988">{{cite journal
      |author1=Kleinpenning, T. G. M. |author2=de Kuijper, A. H.
       |name-list-style=amp |year= 1988
      | title = Relation between variance and sample duration of 1/f Noise signals
      | journal = [[Journal of Applied Physics]]
      | volume = 63
      |issue=1
       | pages = 43
      | doi = 10.1063/1.340460
|bibcode = 1988JAP....63...43K }}</ref> (Kleinpenning, de Kuijper, 1988)<ref>{{Cite journal |last1=Kleinpenning |first1=T. G. M. |last2=de Kuijper |first2=A. H. |date=1988-01-01 |title=Relation between variance and sample duration of 1/ f noise signals |url=https://pubs.aip.org/jap/article/63/1/43/174101/Relation-between-variance-and-sample-duration-of-1 |journal=Journal of Applied Physics |language=en |volume=63 |issue=1 |pages=43–45 |doi=10.1063/1.340460 |bibcode=1988JAP....63...43K |issn=0021-8979|url-access=subscription }}</ref> measured the resistance in a noisy carbon-sheet resistor, and found 1/f noise behavior over the range of <math>[10^{-5.5} \mathrm{Hz}, 10^4 \mathrm{Hz}]</math>, a range of 9.5 decades.

A pioneering researcher in this field was [[Aldert van der Ziel]].<ref>Aldert van der Ziel, (1954), ''Noise'', Prentice–Hall</ref>

Flicker noise is commonly used for the reliability characterization of electronic devices.<ref>{{cite journal|url=https://doi.org/10.1016/S0026-2714(02)00347-5|title=Low-frequency noise study in electron devices: review and update|journal= Microelectronics Reliability|author=Hei Wong|date=2003 |volume=43 |issue=4 |pages=585–599 |doi=10.1016/S0026-2714(02)00347-5 |bibcode=2003MiRe...43..585W |url-access=subscription }}</ref> It is also used for gas detection in chemoresistive sensors <ref>{{cite journal|url=https://doi.org/10.1038/nnano.2013.144|title=Low-frequency 1/f noise in graphene devices|journal= Nature Nanotechnology|author=Alexander A. Balandin|date=2013 |volume=8 |issue=8 |pages=549–555 |doi=10.1038/nnano.2013.144|pmid=23912107 |arxiv=1307.4797 |bibcode=2013NatNa...8..549B |s2cid=16030927 }}</ref> by dedicated measurement setups.<ref>{{cite journal|title=Flicker Noise in Resistive Gas Sensors—Measurement Setups and Applications for Enhanced Gas Sensing|journal=Sensors  |date=2024 |volume=24 |issue=2 |page=405 |doi=10.3390/s24020405|doi-access=free |last1=Smulko |first1=Janusz |last2=Scandurra |first2=Graziella |last3=Drozdowska |first3=Katarzyna |last4=Kwiatkowski |first4=Andrzej |last5=Ciofi |first5=Carmine |last6=Wen |first6=He |pmid=38257498 |pmc=10821460 |bibcode=2024Senso..24..405S }}</ref>

===In gravitational wave astronomy===

[[Image:Gravitational-wave detector sensitivities and astrophysical gravitational-wave sources.png|thumb|upright=2|Noise curves for a selection of [[gravitational-wave detector]]s as a function of frequency]]
1/''f''<sup>&nbsp;α</sup> noises with α near 1 are a factor in [[gravitational-wave astronomy]]. The noise curve at very low frequencies affects [[pulsar timing array]]s, the [[European Pulsar Timing Array]] (EPTA) and the future [[International Pulsar Timing Array]] (IPTA); at low frequencies are space-borne detectors, the formerly proposed [[Laser Interferometer Space Antenna]] (LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initial [[LIGO|Laser Interferometer Gravitational-Wave Observatory]] (LIGO) and its advanced configuration (aLIGO). The characteristic strain of potential astrophysical sources are also shown. To be detectable the characteristic strain of a signal must be above the noise curve.<ref>{{cite web|title=Gravitational Wave Detectors and Sources|url=http://rhcole.com/apps/GWplotter/|access-date=17 April 2014|author=Moore, Christopher|author2=Cole, Robert |author3=Berry, Christopher |date=19 July 2013}}</ref>

===Climate dynamics===
Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the [[climate system]].<ref>{{cite web|url=https://news.yale.edu/2018/09/04/think-pink-better-view-climate-change|title=Think pink for a better view of climate change|work=YaleNews|access-date=5 September 2018|author=Jim Shelton|date=2018-09-04}}</ref><ref>{{Cite journal |last1=Moon |first1=Woosok |last2=Agarwal |first2=Sahil |last3=Wettlaufer |first3=J. S. |date=2018-09-04 |title=Intrinsic Pink-Noise Multidecadal Global Climate Dynamics Mode |url=https://link.aps.org/doi/10.1103/PhysRevLett.121.108701 |journal=Physical Review Letters |volume=121 |issue=10 |pages=108701 |doi=10.1103/PhysRevLett.121.108701|pmid=30240245 |arxiv=1802.00392 |bibcode=2018PhRvL.121j8701M |s2cid=52243763 }}</ref>

=== Diffusion processes ===
Many time-dependent stochastic processes are known to exhibit 1/''f''<sup>&nbsp;α</sup> noises with α between 0 and 2. In particular [[Brownian motion]] has a [[Spectral density|power spectral density]] that equals 4''D''/''f''<sup>&nbsp;2</sup>,<ref>{{Cite book|title=Fundamentals of noise and vibration analysis for engineers|last=Norton, M. P. |date=2003|publisher=Cambridge University Press|others=Karczub, D. G. (Denis G.)|isbn=9780511674983|edition= 2nd|location=Cambridge, UK|oclc=667085096}}</ref> where ''D'' is the [[Mass diffusivity|diffusion coefficient]]. This type of spectrum is sometimes referred to as [[Brownian noise]]. The analysis of individual Brownian motion trajectories also show 1/''f''<sup>&nbsp;2</sup> spectrum, albeit with random amplitudes.<ref>{{Cite journal|last1=Krapf|first1=Diego|last2=Marinari|first2=Enzo|last3=Metzler|first3=Ralf|last4=Oshanin|first4=Gleb|last5=Xu|first5=Xinran|last6=Squarcini|first6=Alessio|date=2018-02-09|title=Power spectral density of a single Brownian trajectory: what one can and cannot learn from it|journal=New Journal of Physics|volume=20|issue=2|pages=023029|doi=10.1088/1367-2630/aaa67c|arxiv=1801.02986|bibcode=2018NJPh...20b3029K|issn=1367-2630|doi-access=free}}</ref> [[Fractional Brownian motion]] with [[Hurst exponent]] ''H'' also show 1/''f''<sup>&nbsp;α</sup> power spectral density with α=2''H''+1 for subdiffusive processes (''H''<0.5) and α=2 for superdiffusive processes (0.5<''H''<1).<ref>{{Cite journal|last1=Krapf|first1=Diego|last2=Lukat|first2=Nils|last3=Marinari|first3=Enzo|last4=Metzler|first4=Ralf|last5=Oshanin|first5=Gleb|last6=Selhuber-Unkel|first6=Christine|last7=Squarcini|first7=Alessio|last8=Stadler|first8=Lorenz|last9=Weiss|first9=Matthias|last10=Xu|first10=Xinran|date=2019-01-31|title=Spectral Content of a Single Non-Brownian Trajectory|journal=Physical Review X|language=en|volume=9|issue=1|pages=011019|doi=10.1103/PhysRevX.9.011019| arxiv=1902.00481 |bibcode=2019PhRvX...9a1019K|issn=2160-3308|doi-access=free}}</ref>