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Chapters

Spectral Audio Signal Processing
    Beginning Statistical Signal Processing
       Random Variables & Stochastic Processes
          Sample Variance

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Sample Variance



Definition: The sample variance of a set of $ N$ samples from a particular realization of a stationary stochastic process $ v$ is defined as average squared magnitude after removing the known mean:

$\displaystyle \hat{\sigma}^2_{v} \isdef {\cal E}_N\{\left\vert v(n)-\mu_v\right... ...ght\vert^2 = \frac{1}{N}\sum_{n=0}^{N-1} \left\vert v(n)\right\vert^2 -\mu_v^2 $

The sample variance is a unbiased estimator of the true variance when the mean is known, i.e.,

$\displaystyle E\{\hat{\sigma}^2_{v}\} = \sigma^2_v. $

This is easy to show by taking the expected value:
$\displaystyle E\{\hat{\sigma}^2_{v}\}$ $\displaystyle =$ $\displaystyle E{\cal E}_N\{\left\vert v(n)-\mu_v\right\vert^2\} = {\cal E}_N\{E\left\vert v(n)-\mu_v\right\vert^2\}$  
  $\displaystyle =$ $\displaystyle {\cal E}_N\{E\left\vert v(n)\right\vert^2-E\overline{v(n)}\mu_v-Ev(n)\overline{\mu_v}+\left\vert\mu_v\right\vert^2\}$  
  $\displaystyle =$ $\displaystyle {\cal E}_N\{\sigma_v^2+\left\vert\mu_v\right\vert^2-\overline{\mu_v}\mu_v-\mu_v\overline{\mu_v}+\left\vert\mu_v\right\vert^2\}$  
  $\displaystyle =$ $\displaystyle {\cal E}_N\{\sigma_v^2\} = \sigma^2_v. \protect$ (D.2)

When the mean is unknown, the sample mean is used in its place:

$\displaystyle \hat{\sigma}^2_{v} \isdef \frac{1}{N-1}\sum_{n=0}^{N-1} \left\vert v(n)-\hat{\mu}_v\right\vert^2 $

The normalization by $ N-1$ instead of $ N$ is necessary to make the sample variance be an unbiased estimator of the true variance. This adjustment is necessary because the sample mean is correlated with the term $ v(n)$ in the sample variance expression. This is revealed by replacing $ \mu_v$ with $ \hat{\mu}_v$ in the calculation of (D.2).

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Next: Correlation Analysis
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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