# Introduction to turbulence/Statistical analysis/Multivariate random variables

### From CFD-Wiki

(→Joint pdfs and joint moments) |
(→Joint pdfs and joint moments) |
||

Line 5: | Line 5: | ||

For example if <math>u</math> and <math>v</math> are two random variables, there are three second-order moments which can be defined <math>\left\langle u^{2} \right\rangle </math> , <math>\left\langle v^{2} \right\rangle </math> , and <math>\left\langle uv \right\rangle </math>. The product moment <math>\left\langle uv \right\rangle </math> is called the ''cross-correlation'' or ''cross-covariance''. The moments <math>\left\langle u^{2} \right\rangle </math> and <math>\left\langle v^{2} \right\rangle </math> are referred to as the ''covariances'', or just simply the ''variances''. Sometimes <math>\left\langle uv \right\rangle </math> is also referred to as the ''correlation''. | For example if <math>u</math> and <math>v</math> are two random variables, there are three second-order moments which can be defined <math>\left\langle u^{2} \right\rangle </math> , <math>\left\langle v^{2} \right\rangle </math> , and <math>\left\langle uv \right\rangle </math>. The product moment <math>\left\langle uv \right\rangle </math> is called the ''cross-correlation'' or ''cross-covariance''. The moments <math>\left\langle u^{2} \right\rangle </math> and <math>\left\langle v^{2} \right\rangle </math> are referred to as the ''covariances'', or just simply the ''variances''. Sometimes <math>\left\langle uv \right\rangle </math> is also referred to as the ''correlation''. | ||

- | In a manner similar to that used to build-up the probabilility density function from its measurable counterpart, the histogram, a '''joint probability density function''' (or '''jpdf'''),<math>B_{uv}</math> , can be built-up from the ''joint histogram''. Figure 2.5 illustrates several examples of jpdf's which have different cross correlations. For convenience the fluctuating variables <math>u | + | In a manner similar to that used to build-up the probabilility density function from its measurable counterpart, the histogram, a '''joint probability density function''' (or '''jpdf'''),<math>B_{uv}</math> , can be built-up from the ''joint histogram''. Figure 2.5 illustrates several examples of jpdf's which have different cross correlations. For convenience the fluctuating variables <math>u'</math> and <math>v'</math> can be defined as |

<table width="100%"><tr><td> | <table width="100%"><tr><td> | ||

:<math> | :<math> | ||

- | u | + | u' = u - U |

</math> | </math> | ||

</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> | ||

Line 15: | Line 15: | ||

<table width="100%"><tr><td> | <table width="100%"><tr><td> | ||

:<math> | :<math> | ||

- | v | + | v' = v - V |

</math> | </math> | ||

</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> | ||

Line 21: | Line 21: | ||

where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities <math>u'</math> and <math>v'</math> are random variables with zero man. | where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities <math>u'</math> and <math>v'</math> are random variables with zero man. | ||

- | A psitive value of <math>\left\langle u | + | A psitive value of <math>\left\langle u'v' \right\rangle </math> indicates that <math>u'</math> and <math>v'</math> tend to vary together. A negative value indicates |

=== The bi-variate normal (or Gaussian) distribution === | === The bi-variate normal (or Gaussian) distribution === | ||

dssd | dssd |

## Revision as of 08:17, 2 June 2006

### Joint pdfs and joint moments

Often it is importamt to consider more than one random variable at a time. For example, in turbulence the three components of the velocity vector are interralated and must be considered together. In addition to the *marginal* (or single variable) statistical moments already considered, it is necessary to consider the **joint** statistical moments.

For example if and are two random variables, there are three second-order moments which can be defined , , and . The product moment is called the *cross-correlation* or *cross-covariance*. The moments and are referred to as the *covariances*, or just simply the *variances*. Sometimes is also referred to as the *correlation*.

In a manner similar to that used to build-up the probabilility density function from its measurable counterpart, the histogram, a **joint probability density function** (or **jpdf**), , can be built-up from the *joint histogram*. Figure 2.5 illustrates several examples of jpdf's which have different cross correlations. For convenience the fluctuating variables and can be defined as

| (2) |

| (2) |

where as before capital letters are usd to represent the mean values. Clearly the fluctuating quantities and are random variables with zero man.

A psitive value of indicates that and tend to vary together. A negative value indicates

### The bi-variate normal (or Gaussian) distribution

dssd