Introduction to turbulence/Statistical analysis/Multivariate random variables

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Revision as of 12:34, 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 $u$ and $v$ are two random variables, there are three second-order moments which can be defined $\left\langle u^{2} \right\rangle$ , $\left\langle v^{2} \right\rangle$ , and $\left\langle uv \right\rangle$ . The product moment $\left\langle uv \right\rangle$ is called the cross-correlation or cross-covariance. The moments $\left\langle u^{2} \right\rangle$ and $\left\langle v^{2} \right\rangle$ are referred to as the covariances, or just simply the variances. Sometimes $\left\langle uv \right\rangle$ 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), $B_{uv}$ , 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 $u'$ and $v'$ can be defined as

$u' = u - U$

(2)

$v' = v - V$

(2)

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

A positive value of $\left\langle u'v' \right\rangle$ indicates that $u'$ and $v'$ tend to vary together. A negative value indicates value indicates that when one variable is increasing the other tends to be decreasing. A zero value of $\left\langle u'v' \right\rangle$ indicates that there is no correlation between $u'$ and $v'$ . As will be seen below, it does not mean that they are statistically independent.

It is sometimes more convinient to deal with values of the cross-variances which have ben normalized by the appropriate variances. Thus the correlation coefficient is defined as:

$\rho_{uv}\equiv \frac{ \left\langle u'v' \right\rangle}{ \left[ \left\langle u'^{2} \right\rangle \left\langle v'^{2} \right\rangle \right]^{1/2}}$

(2)

The correlation coefficient is bounded by plus or minus one, the former representing perfect correlation and the latter perfect anti-correlation.

As with the single-variable pdf, there are certain conditions the joint probability density function must satisfy. If $B_{uv}\left( c_{1}c_{2} \right)$ indicates the jpdf of the random variables $u$ and $v$ , then:

Property 1

$B_{uv}\left( c_{1}c_{2} \right) > 0$

(2)

Property 2

$Prob \left\{ c_{1} < u < c_{1} + dc_{1} , c_{2} < v < c_{2} + dc_{2} \right\} = B_{uv}\left( c_{1}c_{2} \right) dc_{1}, dc_{2}$

(2)

Property 3

$\int^{\infty}_{ - \infty} \int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{1} dc_{2} = 1$

(2)

Property 4

$\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{2} = B_{u}\left( c_{1} \right)$

(2)

Property 5

$\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{1} = B_{v}\left( c_{2} \right)$

(2)

The bi-variate normal (or Gaussian) distribution

dssd

@@ Line 59: / Line 59: @@
 <table width="100%"><tr><td>
 :<math>
-zfds
+\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{2} = B_{u}\left( c_{1} \right)
 </math>
 </td><td width="5%">(2)</td></tr></table>
@@ Line 66: / Line 66: @@
 <table width="100%"><tr><td>
 :<math>
-zfds
+\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{1} = B_{v}\left( c_{2} \right)
 </math>
 </td><td width="5%">(2)</td></tr></table>

Introduction to turbulence/Statistical analysis/Multivariate random variables

From CFD-Wiki

Revision as of 12:34, 2 June 2006

Joint pdfs and joint moments

The bi-variate normal (or Gaussian) distribution

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