Two-Dimensional Random Variables

1 # OVERVIEW OF THE FOURTH MAIN CHAPTER #
2 Properties and examples
3 Joint probability density function
4 Two-dimensional cumulative distribution function
5 PDF for statistically independent components
6 PDF for statistically dependent components
7 Expected values of two-dimensional random variables
8 Correlation coefficient
9 Regression line
10 Exercises for the chapter

# OVERVIEW OF THE FOURTH MAIN CHAPTER #

Now random variables with statistical bindings are treated and illustrated by typical examples.

After the general description of two-dimensional random variables, we turn to

the "auto-correlation function",
the "cross-correlation function"
and the associated spectral functions $($"power-spectral density", "cross power-spectral density"$)$.

Specifically, this chapter covers:

the statistical description of »two-dimensional random variables« using the »joint PDF«,
the difference between »statistical dependence« and »correlation«,
the classification features »stationarity« and »ergodicity« of stochastic processes,
the definitions of »auto-correlation function« $\rm (ACF)$ and »power-spectral density« $\rm (PSD)$,
the definitions of »cross-correlation function« $\rm (CCF)$ and »cross power-spectral density« $\rm (C–PSD)$,
the numerical determination of all these variables in the two- and multi-dimensional case.

Properties and examples

As a transition to the $\text{correlation functions}$ we now consider two random variables $x$ and $y$, between which statistical dependences exist.

Each of these two random variables can be described on its own with the introduced characteristic variables corresponding

to the second main chapter ⇒ "Discrete Random Variables"
and the third main chapter ⇒ "Continuous Random Variables".

$\text{Definition:}$ To describe the statistical dependences between two variables $x$ and $y$, it is convenient to combine the two components
into one »two-dimensional random variable« or »2D random variable« $(x, y)$.

The individual components can be signals such as the real and imaginary parts of a phase modulated signal.
But there are a variety of two-dimensional random variables in other domains as well, as the following example will show.

$\text{Example 1:}$ The left diagram is from the random experiment "Throwing two dice".

Two examples of statistically dependent random variables

Plotted to the right is the number of the first die $(W_1)$,
plotted to the top is the sum $S$ of both dice.

The two components here are each discrete random variables between which there are statistical dependencies:

If $W_1 = 1$, then the sum $S$ can only take values between $2$ and $7$, each with equal probability.
In contrast, for $W_1 = 6$ all values between $7$ and $12$ are possible, also with equal probability.

In the right diagram, the maximum temperatures of the $31$ days in May 2002 of Munich (to the top) and the mountain "Zugspitze" (to the right) are contrasted. Both random variables are continuous in value:

Although the measurement points are about $\text{100 km}$ apart, and on the Zugspitze, it is on average about $20$ degrees colder than in Munich due to the different altitudes $($nearly $3000$ versus $520$ meters$)$, one recognizes nevertheless a certain statistical dependence between the two random variables ${\it Θ}_{\rm M}$ and ${\it Θ}_{\rm Z}$.
If it is warm in Munich, then pleasant temperatures are also more likely to be expected on the Zugspitze. However, the relationship is not deterministic: The coldest day in May 2002 was a different day in Munich than the coldest day on the Zugspitze.

Joint probability density function

We restrict ourselves here mostly to continuous valued random variables.

However, sometimes the peculiarities of two-dimensional discrete random variables are discussed in more detail.
Most of the characteristics previously defined for one-dimensional random variables can be easily extended to two-dimensional variables.

$\text{Definition:}$ The probability density function $\rm (PDF)$ of the two-dimensional random variable at the location $(x_\mu,\hspace{0.1cm} y_\mu)$ ⇒ »joint PDF« or »2D–PDF«
is an extension of the one-dimensional PDF $(∩$ denotes logical "and" operation$)$:

$$f_{xy}(x_\mu, \hspace{0.1cm}y_\mu) = \lim_{\left.{\Delta x\rightarrow 0 \atop {\Delta y\rightarrow 0} }\right.}\frac{ {\rm Pr}\big [ (x_\mu - {\rm \Delta} x/{\rm 2} \le x \le x_\mu + {\rm \Delta} x/{\rm 2}) \cap (y_\mu - {\rm \Delta} y/{\rm 2} \le y \le y_\mu +{\rm \Delta}y/{\rm 2}) \big] }{ {\rm \Delta} \ x\cdot{\rm \Delta} y}.$$

$\rm Note$:

If the two-dimensional random variable is discrete, the definition must be slightly modified:
For the lower range limits, the "less-than-equal" sign must then be replaced by "less-than" according to the section "CDF for discrete-valued random variables".

Using this joint PDF $f_{xy}(x, y)$, statistical dependencies within the two-dimensional random variable $(x,\ y)$ are also fully captured in contrast to the two one-dimensional density functions ⇒ »marginal probability density functions« $($or "edge probability density functions"$)$:

$$f_{x}(x) = \int _{-\infty}^{+\infty} f_{xy}(x,y) \,\,{\rm d}y ,$$

$$f_{y}(y) = \int_{-\infty}^{+\infty} f_{xy}(x,y) \,\,{\rm d}x .$$

These two marginal probability density functions $f_x(x)$ and $f_y(y)$

provide only statistical information about the individual components $x$ and $y$, resp.
but not about the statistical bindings between them.

Two-dimensional cumulative distribution function

$\text{Definition:}$ Like the "2D–PDF", the »2D cumulative distribution function« is merely a useful extension of the $\text{one-dimensional distribution function}$ $\rm (CDF)$:

$$F_{xy}(r_{x},r_{y}) = {\rm Pr}\big [(x \le r_{x}) \cap (y \le r_{y}) \big ] .$$

The following similarities and differences between the "1D–CDF" and the 2D–CDF" emerge:

The functional relationship between two-dimensional PDF and two-dimensional CDF is given by integration as in the one-dimensional case, but now in two dimensions. For continuous valued random variables:

$$F_{xy}(r_{x},r_{y})=\int_{-\infty}^{r_{y}} \int_{-\infty}^{r_{x}} f_{xy}(x,y) \,\,{\rm d}x \,\, {\rm d}y .$$

Inversely, the probability density function can be given from the cumulative distribution function by partial differentiation to $r_{x}$ and $r_{y}$:

$$f_{xy}(x,y)=\frac{{\rm d}^{\rm 2} F_{xy}(r_{x},r_{y})}{{\rm d} r_{x} \,\, {\rm d} r_{y}}\Bigg|_{\left.{r_{x}=x \atop {r_{y}=y}}\right.}.$$

Relative to the two-dimensional cumulative distribution function $F_{xy}(r_{x}, r_{y})$ the following limits apply:

$$F_{xy}(-\infty,-\infty) = 0,$$

$$F_{xy}(r_{\rm x},+\infty)=F_{x}(r_{x} ),$$

$$F_{xy}(+\infty,r_{y})=F_{y}(r_{y} ) ,$$

$$F_{xy} (+\infty,+\infty) = 1.$$

From the last equation $($infinitely large $r_{x}$ and $r_{y})$ we obtain the »normalization condition« for the "2D– PDF":

$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f_{xy}(x,y) \,\,{\rm d}x \,\,{\rm d}y=1 . $$

$\text{Conclusion:}$ Note the significant difference between one-dimensional and two-dimensional random variables:

For one-dimensional random variables, the area under the PDF always yields the value $1$.
For two-dimensional random variables, the PDF volume is always equal to $1$.

PDF for statistically independent components

For statistically independent components $x$, $y$ the following holds for the joint probability according to the elementary laws of statistics if $x$ and $y$ are continuous in value:

$${\rm Pr} \big[(x_{\rm 1}\le x \le x_{\rm 2}) \cap( y_{\rm 1}\le y\le y_{\rm 2})\big] ={\rm Pr} (x_{\rm 1}\le x \le x_{\rm 2}) \cdot {\rm Pr}(y_{\rm 1}\le y\le y_{\rm 2}) .$$

For this, in the case of independent components can also be written:

$${\rm Pr} \big[(x_{\rm 1}\le x \le x_{\rm 2}) \cap(y_{\rm 1}\le y\le y_{\rm 2})\big] =\int _{x_{\rm 1}}^{x_{\rm 2}}f_{x}(x) \,{\rm d}x\cdot \int_{y_{\rm 1}}^{y_{\rm 2}} f_{y}(y) \, {\rm d}y.$$

$\text{Definition:}$ It follows that for »statistical independence« the following condition must be satisfied with respect to the »two-dimensional probability density function«:

$$f_{xy}(x,y)=f_{x}(x) \cdot f_y(y) .$$

$\text{Example 2:}$ In the graph, the instantaneous values of a two-dimensional random variable are plotted as points in the $(x,\, y)$–plane.

Ranges with many points, which accordingly appear dark, indicate large values of the two-dimensional PDF $f_{xy}(x,\, y)$.
In contrast, the random variable $(x,\, y)$ has relatively few components in rather bright areas.

Statistically independent components: $f_{xy}(x, y)$, $f_{x}(x)$ and $f_{y}(y)$

The graph can be interpreted as follows:

The marginal probability densities $f_{x}(x)$ and $f_{y}(y)$ already indicate that both $x$ and $y$ are Gaussian and zero mean, and that the random variable $x$ has a larger standard deviation than $y$.
$f_{x}(x)$ and $f_{y}(y)$ do not provide information on whether or not statistical bindings exist for the random variable $(x,\, y)$.
However, using the "2D-PDF" $f_{xy}(x,\, y)$ one can see that here there are no statistical bindings between the two components $x$ and $y$.
With statistical independence, any cut through $f_{xy}(x, y)$ parallel to $y$–axis yields a function that is equal in shape to the marginal PDF $f_{y}(y)$. Similarly, all cuts parallel to $x$–axis are equal in shape to $f_{x}(x)$.
This fact is equivalent to saying that in this example $f_{xy}(x,\, y)$ can be represented as the product of the two marginal probability densities:

$$f_{xy}(x,\, y)=f_{x}(x) \cdot f_y(y) .$$

PDF for statistically dependent components

If there are statistical bindings between $x$ and $y$, then different cuts parallel to $x$– and $y$–axis, resp., yield different (non-shape equivalent) functions. In this case, of course, the joint PDF cannot be described as a product of the two (one-dimensional) marginal probability densities functions either.

$\text{Example 3:}$ The graph shows the instantaneous values of a two-dimensional random variable in the $(x, y)$–plane.

Statistically dependent components: $f_{xy}(x, y)$, $f_{x}(x)$, $f_{y}(y)$

Now, unlike $\text{Example 2}$ there are statistical bindings between $x$ and $y$.

The two-dimensional random variable takes all "2D" values with equal probability in the parallelogram drawn in blue.
No values are possible outside the parallelogram.

One recognizes from this representation:

Integration over $f_{xy}(x, y)$ parallel to the $x$–axis leads to the triangular marginal PDF $f_{y}(y)$, integration parallel to $y$–axis to the trapezoidal PDF $f_{x}(x)$.
From the joint PDF $f_{xy}(x, y)$ it can already be guessed that for each $x$–value on statistical average, a different $y$–value is to be expected.
This means that the components $x$ and $y$ are statistically dependent on each other.

Expected values of two-dimensional random variables

A special case of statistical dependence is "correlation".

$\text{Definition:}$ Under »correlation« one understands a "linear dependence" between the individual components $x$ and $y$.

Correlated random variables are thus always also statistically dependent.
But not every statistical dependence implies correlation at the same time.

To quantitatively capture correlation, one uses various expected values of the two-dimensional random variable $(x, y)$.

These are defined analogously to the one-dimensional case,

according to "Chapter 2" (for discrete valued random variables).
and "Chapter 3" (for continuous valued random variables):

$\text{Definition:}$ For the (non-centered) »moments« the following relation holds:

$$m_{kl}={\rm E}\big[x^k\cdot y^l\big]=\int_{-\infty}^{+\infty}\hspace{0.2cm}\int_{-\infty}^{+\infty} x\hspace{0.05cm}^{k} \cdot y\hspace{0.05cm}^{l} \cdot f_{xy}(x,y) \, {\rm d}x\, {\rm d}y.$$

Thus, the two linear means are $m_x = m_{10}$ and $m_y = m_{01}.$

$\text{Definition:}$ The »central moments« $($related to $m_x$ and $m_y)$ are:

$$\mu_{kl} = {\rm E}\big[(x-m_{x})\hspace{0.05cm}^k \cdot (y-m_{y})\hspace{0.05cm}^l\big] .$$

In this general definition equation, the variances $σ_x^2$ and $σ_y^2$ of the two individual components are included by $\mu_{20}$ and $\mu_{02}$, resp.

$\text{Definition:}$ Of particular importance is the »covariance« $(k = l = 1)$, which is a measure of the "linear statistical dependence" between the variables $x$ and $y$:

$$\mu_{11} = {\rm E}\big[(x-m_{x})\cdot(y-m_{y})\big] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} (x-m_{x}) \cdot (y-m_{y})\cdot f_{xy}(x,y) \,{\rm d}x \, {\rm d}y .$$

In the following, we also denote the covariance $\mu_{11}$ in part by "$\mu_{xy}$", if the covariance refers to the random variables $x$ and $y$.

Notes:

The covariance $\mu_{11}=\mu_{xy}$ is related to the non-centered moment $m_{11} = m_{xy} = {\rm E}\big[x \cdot y\big]$ as follows:

$$\mu_{xy} = m_{xy} -m_{x }\cdot m_{y}.$$

This equation is enormously advantageous for numerical evaluations, since $m_{xy}$, $m_x$ and $m_y$ can be found from the sequences $〈x_v〉$ and $〈y_v〉$ in a single run.
On the other hand, if one were to calculate the covariance $\mu_{xy}$ according to the above definition equation, one would have to find the mean values $m_x$ and $m_y$ in a first run and could then only calculate the expected value ${\rm E}\big[(x - m_x) \cdot (y - m_y)\big]$ in a second run.

$\text{Example 4:}$ In the first two rows of the table, the first elements of two random sequences $〈x_ν〉$ and $〈y_ν〉$ are entered. In the last row, the respective products $x_ν - y_ν$ are given.

Example for two-dimensional expected values

By averaging over ten sequence elements in each case, one obtains

$$m_x =0.5,\ \ m_y = 1, \ \ m_{xy} = 0.69.$$

This directly results in the value for the covariance:

$$\mu_{xy} = 0.69 - 0.5 · 1 = 0.19.$$

Without knowledge of the equation $\mu_{xy} = m_{xy} - m_x\cdot m_y$ one would have had to first determine the means $m_x$ and $m_y$ in the first run, and then determine the covariance $\mu_{xy}$ as the expected value of the product of the zero mean variables in a second run.

Correlation coefficient

With statistical independence of the two components $x$ and $y$ the covariance $\mu_{xy} \equiv 0$. This case has already been considered in $\text{Example 2}$ in the section "PDF for statistically independent components".

But the result $\mu_{xy} = 0$ is also possible for statistically dependent components $x$ and $y$ namely when they are uncorrelated, i.e. "linearly independent".
The statistical dependence is then not of first order, but of higher order, for example corresponding to the equation $y=x^2.$

One speaks of »complete correlation« when the (deterministic) dependence between $x$ and $y$ is expressed by the equation $y = K · x$. Then the covariance is given by:

$\mu_{xy} = σ_x · σ_y$ with positive $K$ value,
$\mu_{xy} = - σ_x · σ_y$ with negative $K$ value.

Therefore, instead of the "covariance" one often uses the so-called "correlation coefficient" as descriptive quantity.

$\text{Definition:}$ The »correlation coefficient« is the quotient of the covariance $\mu_{xy}$ and the product of the standard deviations $σ_x$ and $σ_y$ of the two components:

$$\rho_{xy}=\frac{\mu_{xy} }{\sigma_x \cdot \sigma_y}.$$

The correlation coefficient $\rho_{xy}$ has the following properties:

Because of normalization, $-1 \le ρ_{xy} ≤ +1$ always holds.
If the two random variables $x$ and $y$ are uncorrelated, then $ρ_{xy} = 0$.
For strict linear dependence between $x$ and $y$ ⇒ $ρ_{xy}= ±1$ ⇒ complete correlation.
A positive correlation coefficient means that when $x$ is larger, on statistical average, $y$ is also larger than when $x$ is smaller.
In contrast, a negative correlation coefficient expresses that $y$ becomes smaller on average as $x$ increases.

Two-dimensional Gaussian PDF with correlation

$\text{Example 5:}$ The following conditions apply:

The considered components $x$ and $y$ each have a Gaussian PDF.
The two standard deviations are different $(σ_y < σ_x)$.
The correlation coefficient is $ρ_{xy} = 0.8$.

Unlike $\text{Example 2}$ with statistically independent components ⇒ $ρ_{xy} = 0$ $($even though $σ_y < σ_x)$ one recognizes that here

with larger $x$–value, on statistical average, $y$ is also larger
than with a smaller $x$–value.

Regression line

$\text{Definition:}$ The »regression line« – sometimes called "correlation line" – is the straight line $y = K(x)$ in the $(x, y)$–plane through the "midpoint" $(m_x, m_y)$.

Two-dimensional Gaussian PDF with regression line $\rm (RL)$

The regression line has the following properties:

The mean square deviation from this straight line - viewed in $y$–direction and averaged over all $N$ points - is minimal:

$$\overline{\varepsilon_y^{\rm 2} }=\frac{\rm 1}{N} \cdot \sum_{\nu=\rm 1}^{N}\; \;\big [y_\nu - K(x_{\nu})\big ]^{\rm 2}={\rm minimum}.$$

The regression line can be interpreted as a kind of "statistical symmetry axis". The equation of the straight line is:

$$y=K(x)=\frac{\sigma_y}{\sigma_x}\cdot\rho_{xy}\cdot(x - m_x)+m_y.$$

The angle taken by the regression line to the $x$–axis is:

$$\theta_{y\hspace{0.05cm}\rightarrow \hspace{0.05cm}x}={\rm arctan}\ (\frac{\sigma_{y} }{\sigma_{x} }\cdot \rho_{xy}).$$

By this nomenclature it should be made clear that we are dealing here with the regression of $y$ on $x$.

The regression in the opposite direction – that is, from $x$ to $y$ – on the other hand, means the minimization of the mean square deviation in $x$–direction.

The (German language) applet "Korrelation und Regressionsgerade" ⇒ "Correlation Coefficient and Regression Line" illustrates
that in general $($if $σ_y \ne σ_x)$ for the regression of $x$ on $y$ will result in a different angle and thus a different regression line:

$$\theta_{x\hspace{0.05cm}\rightarrow \hspace{0.05cm} y}={\rm arctan}\ (\frac{\sigma_{x}}{\sigma_{y}}\cdot \rho_{xy}).$$