Editing Covariance (section)

==Applications==

=== In genetics and molecular biology ===
Covariance is an important measure in [[biology]]. Certain sequences of [[DNA]] are conserved more than others among species, and thus to study secondary and tertiary structures of [[protein]]s, or of [[RNA]] structures, sequences are compared in closely related species. If sequence changes are found or no changes at all are found in [[noncoding RNA]] (such as [[microRNA]]), sequences are found to be necessary for common structural motifs, such as an RNA loop. In genetics, covariance serves a basis for computation of Genetic Relationship Matrix (GRM) (aka kinship matrix), enabling inference on population structure from sample with no known close relatives as well as inference on estimation of heritability of complex traits.

In the theory of [[evolution]] and [[natural selection]], the [[price equation]] describes how a [[genetic trait]] changes in frequency over time. The equation uses a covariance between a trait and [[fitness (biology)|fitness]], to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the proportion of genes within each new generation of a population.<ref name="Price1970">{{cite journal |last1= Price | first1=George |year=1970 |title=Selection and covariance |journal=Nature |volume=227 |issue=5257 |pages=520–521 | doi=10.1038/227520a0 |pmid=5428476| bibcode=1970Natur.227..520P | s2cid=4264723 }}</ref><ref name="Harman2020">{{cite journal |last1= Harman |first1=Oren |year=2020 | title=When science mirrors life: on the origins of the Price equation |publisher=royalsocietypublishing.org |journal= Philosophical Transactions of the Royal Society B: Biological Sciences|volume=375 |issue=1797 |pages=1–7 | doi=10.1098/rstb.2019.0352 |pmid=32146891 |pmc=7133509 |doi-access=free }}</ref>

===In financial economics===

Covariances play a key role in [[financial economics]], especially in [[modern portfolio theory]] and in the [[capital asset pricing model]]. Covariances among various assets' returns are used to determine, under certain assumptions, the relative amounts of different assets that investors should (in a [[Normative economics|normative analysis]]) or are predicted to (in a [[Positive economics|positive analysis]]) choose to hold in a context of [[Diversification (finance)|diversification]].

===In meteorological and oceanographic data assimilation===
{{unsourced section|date=May 2025}}
The covariance matrix is important in estimating the initial conditions required for running weather forecast models, a procedure known as [[data assimilation]]. The "forecast error covariance matrix" is typically constructed between perturbations around a mean state (either a climatological or ensemble mean). The "observation error covariance matrix" is constructed to represent the magnitude of combined observational errors (on the diagonal) and the correlated errors between measurements (off the diagonal). This is an example of its widespread application to [[Kalman filtering]] and more general [[state estimation]] for time-varying systems.

===In micrometeorology ===
The [[eddy covariance]] technique is a key atmospherics measurement technique where the covariance between instantaneous deviation in vertical wind speed from the mean value and instantaneous deviation in gas concentration is the basis for calculating the vertical turbulent fluxes.

===In signal processing===
The covariance matrix is used to capture the spectral variability of a signal.<ref>{{cite journal|last=Sahidullah|first=Md.|author2=Kinnunen, Tomi|title=Local spectral variability features for speaker verification|journal=Digital Signal Processing|date=March 2016|volume=50|pages=1–11|doi=10.1016/j.dsp.2015.10.011|bibcode=2016DSP....50....1S |url=https://erepo.uef.fi/handle/123456789/4375}}</ref>

===In statistics and image processing ===
The covariance matrix is used in [[principal component analysis]] to reduce feature dimensionality in [[data preprocessing]].