Introduction to turbulence/Statistical analysis/Multivariate random variables
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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. | + | 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> | ||
| + | :<math> | ||
| + | u^{'} = u - U | ||
| + | </math> | ||
| + | </td><td width="5%">(2)</td></tr></table> | ||
| + | |||
| + | <table width="100%"><tr><td> | ||
| + | :<math> | ||
| + | v^{'} = v - V | ||
| + | </math> | ||
| + | </td><td width="5%">(2)</td></tr></table> | ||
=== The bi-variate normal (or Gaussian) distribution === | === The bi-variate normal (or Gaussian) distribution === | ||
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Revision as of 18:43, 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. For convenience the fluctuating variables 
and 
can be defined as
|
| (2) |
|
| (2) |
The bi-variate normal (or Gaussian) distribution
dssd
