Theory of Stochastic Signals/Two-Dimensional Random Variables

1. Theory of Stochastic Signals
2. Random Variables with Statistical Dependence
3. Two-Dimensional Random Variables

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# 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 autocorrelation function  (ACF),  the cross correlation function  (CCF)  and the associated spectral functions  (PSD, CPSD) .

Specifically, it covers:

• the statistical description of 2D 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 autocorrelation function  (ACF) and power spectral density  (PSD),
• the definitions of cross correlation function  and cross power spectral density, and
• the numerical determination of all these quantities in the two- and multi-dimensional cases.

For more information on Two-Dimensional Random Variables, as well as tasks, simulations, and programming exercises, see

• Chapter 5:   Two-dimensional random variables (program "zwd")
• Chapter 9:   Stochastic Processes (program "sto")

of the practical course "Simulation Methods in Communications Engineering".  This (former) LNT course at the TU Munich is based on

• the teaching software package  LNTsim   ⇒   Link refers to the German ZIP–version of the program,
•   Internship Guide – Part A   ⇒   Link refers to the German PDF–version with chapter 5:  pages 81-97,
• the  Internship Guide – Part B   ⇒   Link refers to the German PDF–version with chapter 9:  pages 207-228.

Properties and examples

As a transition to the  correlation functions  we now consider two random variables  $x$  and  $y$,  between which statistical bindings(???) exist.  Each of the two random variables can be described on its own with the introduced characteristic quantities

• corresponding to the second main chapter   ⇒  Discrete Random Variables  
• but the third main chapter   ⇒   Continuous Random Variables.

$\text{Definition:}$  To describe the correlations between two variables  $x$  and  $y$  it is convenient to combine the two components into one  two-dimensional 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 2D–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".  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  $S$  can only take values between  $2$  and  $7$  and each with equal probability.
• In contrast, for  $W_1 = 6$  all values between  $7$  and  $12$  are possible, also with equal probability.

Two examples of statistically dependent random variables

In the right graph, the maximum temperatures of the  $31$ days in May 2002 of Munich (to the top) and the 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, due to the different altitudes  $($nearly  $3000$  versus  $520$  meters$)$  is on average about  $20$  degrees colder than in Munich, 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 PDF

We restrict ourselves here mostly to continuous 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 of the two-dimensional random variable at the location  $(x_\mu, y_\mu)$   ⇒   joint 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 2D–random variable is discrete, the definition must be slightly modified:
• For the lower range limits in each case, the "≤" sign must then be replaced by the "<" sign according to the page  CDF for discrete random variables 

.

Using this (joint) WDF  $f_{xy}(x, y)$  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:

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

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

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

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

Two-dimensional CDF

$\text{Definition:}$  The  2D distribution function  like the 2D WDF, is merely a useful extension of the  one-dimensional distribution function  (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 VTF is given by integration as in the one-dimensional case, but now in two dimensions.  For continuous 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 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 distribution function  $F_{xy}(r_{x}, r_{y})$  the following limits apply:

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

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

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

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

• In the limiting case  $($infinitely large  $r_{x}$  and  $r_{y})$  Thus, for the 2D VTF, the value  $1$.  From this, we obtain the  normalization condition  for the 2D WDF:

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

$\text{Conclusion:}$  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  $1$.

PDF and CDF for statistically independent components

For statistically independent components  $x$  and  $y$  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, 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 2D–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 2D–WDF  $f_{xy}(x, y)$.
• 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  $f_{y}(y)$  however, do not provide information on whether or not statistical bindings exist for the random variable  $(x, y)$ .
• However, using the 2D WDF  $f_{xy}(x,y)$  one can see that there are no statistical bindings between the two components  $x$  and  $y$  here.
• With statistical independence, any cut through  $f_{xy}(x, y)$  parallel to  $y$-axis yields a function that is equal in shape to the edge–WDF  $f_{y}(y)$.  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 and CDF for statistically dependent components

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

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

$\text{Example 3:}$  The graph shows the instantaneous values of a two-dimensional random variable in the  $(x, y)$–plane, where now, unlike  $\text{Example 2}$  there are statistical bindings between  $x$  and  $y$  .

• The 2D–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  $x$–axis leads to the triangular marginal density $f_{y}(y)$, integration parallel to  $y$–axis to the trapezoidal WDF $f_{x}(x)$.
• From the 2D-WDF $f_{xy}(x, y)$  it can already be guessed that for each  $x$–value on statistical average a different  $y$–value is to be expected.
• This means that here 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 2D random variable  $(x, y)$.

These are defined analogously to the one-dimensional case.

• according to  Chapter 2  (for discrete value random variables).
• bzw.  Chapter 3  (for continuous value random variables):

$\text{Definition:}$  For the (non-centered)  moments  the 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  $m_x$  and  $m_y$  related  central moments  respectively 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  $\mu_{02}$  respectively.

$\text{Definition:}$  Of particular importance is the  covariance  $(k = l = 1)$, which is a measure of the linear statistical dependence  between the random 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.

Example 2D expected values

$\text{Example 4:}$  In the first two rows of the table, the respective 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.

• By averaging over the 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 mean values  $m_x$  and  $m_y$  in the first run,
in order to then determine the covariance  $\mu_{xy}$  as the expected value of the product of the mean-free variables in a second run.

Korrelationskoeffizient

Bei statististischer Unabhängigkeit der beiden Komponenten  $x$  und  $y$  ist die Kovarianz  $\mu_{xy} \equiv 0$.  Dieser Fall wurde bereits im  $\text{Beispiel 2}$  auf der Seite  WDF und VTF bei statistisch unabhängigen Komponenten  betrachtet.

• Das Ergebnis  $\mu_{xy} = 0$  ist aber auch bei statistisch abhängigen Komponenten  $x$  und  $y$  möglich, nämlich dann, wenn diese unkorreliert, also  linear unabhängig  sind.
• Die statistische Abhängigkeit ist dann nicht von erster, sondern von höherer Ordnung, zum Beispiel entsprechend der Gleichung  $y=x^2.$

Man spricht von  vollständiger Korrelation, wenn die (deterministische) Abhängigkeit zwischen  $x$  und  $y$  durch die Gleichung  $y = K · x$  ausgedrückt wird. Dann ergibt sich für die Kovarianz:

• $\mu_{xy} = σ_x · σ_y$  bei positivem Wert von  $K$,
• $\mu_{xy} = - σ_x · σ_y$  bei negativem  $K$–Wert.

Deshalb verwendet man häufig als Beschreibungsgröße anstelle der Kovarianz den so genannten Korrelationskoeffizienten.

$\text{Definition:}$  Der  Korrelationskoeffizient  ist der Quotient aus der Kovarianz  $\mu_{xy}$  und dem Produkt der Effektivwerte  $σ_x$  und  $σ_y$  der beiden Komponenten:

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

Der Korrelationskoeffizient  $\rho_{xy}$  weist folgende Eigenschaften auf:

• Aufgrund der Normierung gilt stets  $-1 \le ρ_{xy} ≤ +1$.
• Sind die beiden Zufallsgrößen  $x$  und  $y$  unkorreliert, so ist  $ρ_{xy} = 0$.
• Bei strenger linearer Abhängigkeit zwischen  $x$  und  $y$  ist  $ρ_{xy}= ±1$   ⇒   vollständige Korrelation.
• Ein positiver Korrelationskoeffizient bedeutet, dass bei größerem  $x$–Wert im statistischen Mittel auch  $y$  größer ist als bei kleinerem  $x$.
• Dagegen drückt ein negativer Korrelationskoeffizient aus, dass  $y$  mit steigendem  $x$  im Mittel kleiner wird.

Gaußsche 2D-WDF mit Korrelation

$\text{Beispiel 5:}$  Es gelten folgende Voraussetzungen:

• Die betrachteten Komponenten  $x$  und  $y$  besitzen jeweils eine gaußförmige WDF.
• Die beiden Streuungen sind unterschiedlich  $(σ_y < σ_x)$.
• Der Korrelationskoeffizient beträgt  $ρ_{xy} = 0.8$.

Im Unterschied zum  Beispiel 2  mit statistisch unabhängigen Komponenten   ⇒   $ρ_{xy} = 0$  $($trotz  $σ_y < σ_x)$  erkennt man, dass hier bei größerem  $x$–Wert im statistischen Mittel auch  $y$  größer ist als bei kleinerem  $x$.

Korrelationsgerade

Gaußsche 2D-WDF mit Korrelationsgerade

$\text{Definition:}$  Als  Korrelationsgerade  bezeichnet man die Gerade  $y = K(x)$  in der  $(x, y)$–Ebene durch den „Mittelpunkt”  $(m_x, m_y)$. Manchmal wird diese Gerade auch  Regressionsgerade  genannt.

Die Korrelationsgerade besitzt folgende Eigenschaften:

• Die mittlere quadratische Abweichung von dieser Geraden – in  $y$–Richtung betrachtet und über alle  $N$  Punkte gemittelt – ist minimal:

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

• Die Korrelationsgerade kann als eine Art  „statistische Symmetrieachse“  interpretiert werden. Die Geradengleichung lautet:

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

Der Winkel, den die Korrelationsgerade zur  $x$–Achse einnimmt, beträgt:

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

Durch diese Nomenklatur soll deutlich gemacht werden, dass es sich hier um die Regression von  $y$  auf  $x$  handelt.

• Die Regression in Gegenrichtung – also von  $x$  auf  $y$ – bedeutet dagegen die Minimierung der mittleren quadratischen Abweichung in  $x$–Richtung.

• Das interaktive Applet  Korrelationskoeffizient und Regressionsgerade  verdeutlicht, dass sich im Allgemeinen  $($falls  $σ_y \ne σ_x)$  für die Regression von  $x$  auf  $y$  ein anderer Winkel und damit auch eine andere Regressionsgerade ergeben wird:

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

Aufgaben zum Kapitel

Aufgabe 4.1: Dreieckiges (x, y)-Gebiet

Aufgabe 4.1Z: Verabredung zum Frühstück

Aufgabe 4.1: Wieder Dreieckgebiet

Aufgabe 4.2Z: Korrelation zwischen $x$ und $e^x$

Aufgabe 4.3: Algebraische und Modulo-Summe

Aufgabe 4.3Z: Diracförmige 2D-WDF