Introduction to turbulence/Statistical analysis/Multivariate random variables

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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

$u' = u - U$

$v' = v - V$

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:

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

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

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

Property 3: $\int^{\infty}_{ - \infty} \int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{1} dc_{2} = 1$

Property 4: $\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{2} = B_{u}\left( c_{1} \right)$ , where $B_{u}$ is a function of $c_{1}$ only

Property 5: $\int^{\infty}_{ - \infty} B_{uv}\left( c_{1}c_{2} \right) dc_{1} = B_{v}\left( c_{2} \right)$ , where $B_{v}$ is a function of $c_{2}$ only

The functions $B_{u}$ and $B_{v}$ are called the marginal probability density functions and they are simply the single variable pdf's defined earlier. The subscript is used to indicate which variable is left after the others are integrated out. Note that $B_{u}\left( c_{1} \right)$ is not the same as $B_{uv}\left( c_{1},0 \right)$ . The latter is only a slice through the $c_{2}$ - axis, whale the marginal distribution is weighted by the integral of the distribution of the other variable. Figure 2.6. illustrates these differences.

If the joint probability density function is known, the joint moments of all orders can be determined. Thus the $m,n$ -th joint moment is

$\left\langle \left( u- U \right)^{m} \left( v - V \right)^n \right\rangle = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \left( c_{1} - U \right)^{m} \left( c_{2} - V \right)^{n} B_{uv}\left( c_{1} , c_{2} \right) dc_{1} dc_{2}$

In the preceding discussions, only two random variables have been considered. The definitions, however, can easily be geberalized to accomodate any number of random variables. In addition, the joint statistics of a single random at different times or at different points in space could be considered. This will be done later when stationary and homogeneous random processes are considered.

The bi-variate normal (or Gaussian) distribution

If $u$ and $v$ are normally distributed random variables with standard deviations given by $\sigma_{u}$ and $\sigma_{v}$ respectively , with correlation coefficient $\rho_{uv}$ , then their joint probability density function is given by

$B_{uvG} \left(c_{1},c_{2} \right) = \frac{1}{2 \pi \sigma_{u} \sigma_{v} }exp \left[ \frac{ \left( c_{1} - U \right)^{2} }{ 2\sigma^{2}_{u} } + \frac{ \left( c_{2}-V \right)^{2}}{2\sigma^{2}_{v} } - \rho_{uv}\frac{c_{1}c_{2}}{\sigma_{u} \sigma_{v}} \right]$

This distribution is plotted in Figure 2.7. for several values of $\rho_{uv}$ where $u$ and $v$ are assumed to be identically distributed (i.e., $\left\langle u^{2} \right\rangle = \left\langle v^{2} \right\rangle$ ).

It is straightforward to show (by completing the square and integrating) that this yields the single variable Gaussian distribution for the marginal distributions. It is also possible to write a multivariate Gaussian probability density function for any number of random variables.

Statistical independence and lack of correlation

Definition: Statistical Independence Two random variables are said to be statistically independent if their joint probability density is equal to the product of their marginal probability density functions. That is,

$B_{uv}\left(c_{1}c_{2} \right) = B_{u}\left(c_{1} \right) B_{v} \left( c_{2} \right)$

It is easy to see that statistical independence implies a complete lack of correlation; i.e., $\rho_{uv} \equiv 0$ . From the definition of the cross-correlation

$\begin{matrix} \left\langle \left(u-U \right) \left( v - V \right) \right\rangle & = & \int ^{\infty}_{-\infty} \int ^{\infty}_{-\infty} \left( c_{1} - U \right) \left( c_{2} - V \right) B_{uv} \left( c_{1} , c_{2} \right) dc_{1} dc_{2} \\ & = & \int ^{\infty}_{-\infty} \int ^{\infty}_{-\infty} \left( c_{1} - U \right) \left( c_{2} - V \right) B_{u}\left(c_{1} \right) B_{v} \left( c_{2} \right) dc_{1} dc_{2} \\ & = & \int ^{\infty}_{-\infty} \left(c_{1} - U \right) B_{u}\left(c_{1} \right) dc_{1} \int ^{\infty}_{-\infty} \left( c_{2} - V \right) B_{v} \left( c_{2} \right) dc_{2} \\ & = & 0 \end{matrix}$

where we have used equation 2.39 since the first central moments are zero by definiion.

It is important to note that the inverse is not true - lack of correlation does not imply statistical independence! To see this consider two identically distributed random variables, $u'$ and $v'$ , which have zero means and non-zero correlation $\left\langle u'v' \right\rangle$ . From these two correlated random variables two other random variables $x$ and $y$ , can be formed as

$x = u' + v'$

$y = u' - v'$

Clearly $x$ and $y$ are not statistically independent. They are, however, uncorrelated because:

$\begin{matrix} \left\langle xy \right\rangle & = & \left\langle \left( u'+ v' \right) \left( u' - v' \right) \right\rangle \\ & = & \left\langle u'^{2} \right\rangle + \left\langle u'v' \right\rangle - \left\langle u'v' \right\rangle - \left\langle v'^{2} \right\rangle \\ & = & 0 \\ \end{matrix}$

since $u'$ and $v'$ are identically distributed (and as a consequence $\left\langle u'^{2} \right\rangle = \left\langle v'^{2} \right\rangle$ ).

Figure 2.8 illustrates the change of variables carried out above. The jpdf resulting from the transformation is symmetric about both axes, thereby eliminating the correlation. Transformation, however, does not insure that the distribution is separable, i.e., $B_{x,y} \left( a_{1},a_{2} \right) = B_{x} \left( a_{1} \right) B_{y} \left( a_{2} \right)$ , as required for statistical independence.

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Credits

This text was based on "Lectures in Turbulence for the 21st Century" by Professor William K. George, Professor of Turbulence, Chalmers University of Technology, Gothenburg, Sweden.

Introduction to turbulence/Statistical analysis/Multivariate random variables

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Joint pdfs and joint moments

The bi-variate normal (or Gaussian) distribution

Statistical independence and lack of correlation

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