# Introduction to turbulence/Statistical analysis/Multivariate random variables

### From CFD-Wiki

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

Line 3: | Line 3: | ||

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. | 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 <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 | + | 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. | ||

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

dssd | dssd |

## Revision as of 18:35, 1 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.

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

dssd