Editing White noise (section)

==Mathematical applications==

===Time series analysis and regression===
In [[statistics]] and [[econometrics]] one often assumes that an observed series of data values is the sum of the values generated by a [[deterministic]] [[linear model|linear process]], depending on certain [[Dependent and independent variables|independent (explanatory) variables]], and on a series of random noise values. Then [[regression analysis]] is used to infer the parameters of the model process from the observed data, e.g. by [[ordinary least squares]], and to [[hypothesis testing|test the null hypothesis]] that each of the parameters is zero against the alternative hypothesis that it is non-zero. Hypothesis testing typically assumes that the noise values are mutually uncorrelated with zero mean and have the same Gaussian probability distribution{{snd}}in other words, that the noise is Gaussian white (not just white). If there is non-zero correlation between the noise values underlying different observations then the estimated model parameters are still [[bias of an estimator|unbiased]], but estimates of their uncertainties (such as [[confidence interval]]s) will be biased (not accurate on average). This is also true if the noise is [[heteroskedastic]]{{snd}}that is, if it has different variances for different data points.

Alternatively, in the subset of regression analysis known as [[time series analysis]] there are often no explanatory variables other than the past values of the variable being modeled (the [[dependent variable]]). In this case the noise process is often modeled as a [[Moving average model|moving average]] process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.

===Random vector transformations===
These two ideas are crucial in applications such as [[channel estimation]] and [[Mixing console#Channel equalization|channel equalization]] in [[telecommunications|communications]] and [[sound reproduction|audio]]. These concepts are also used in [[data compression]].

<!-- This does not seem to be incorrect but seems to be original research, sort of. Needs to be trimmed to the bare essentials. -->
In particular, by a suitable linear transformation (a [[coloring transformation]]), a white random vector can be used to produce a non-white random vector (that is, a list of random variables) whose elements have a prescribed [[covariance matrix]]. Conversely, a random vector with known covariance matrix can be transformed into a white random vector by a suitable [[whitening transformation]].