Chapter 7R andom P rocesses7.1 Corr elation in Rando m VariablesArandomvariableX takes on num erical values as the resu lt of an ex peri-ment. Suppose that the experiment also produces anoth er ran do m variable,Y. W hat can we sa y about the relationship between X and Y ?.One of the best wa ys to visualize the possible relationship is to plot the(X, Y ) pair that is produced by sev er al trials of the experiment. An exam p leof correlated samples is show n in Figure 7.1. The poin ts fall within a some-what elliptical contou r, slanting downward, and centered at approxima tely(4,0). The poin ts were created w ith a r ando m number generator using acorrelation coefficien t of ρ = −0.5,E[X]=4,E[Y ]=0. The mean v aluesare the coordinates of the cluster center. The negative correlation coefficientindicates that an increase in X abo ve its mean value generally correspond stoadecreaseinY belo w its mean value. Th is tendency makes it possible tomake predictions about the value that one variable will tak e giv en the valueof the other, somethin g which can be useful in many settings.The joint behavior of X and Y is fully captured in the joint probab ilitydistribution. If the random variables are con tin uous then it is appropriate touse a probabilit y densit y function, fXY(x, y). We will presum e that the pdfis kno wn or can be estimated. Com pu tation of the usual expected values isthen straigh tfo rward .E[XmYn]=ZZ∞−∞xmynfXY(x, y)dxdy (7.1)123124 CHAPTER 7. RANDOM PROCESSESFigure 7.1: Scatter plot of random variables X and Y. These random variablesha ve a correlation of ρ = −0.5.7.1.1 Covariance FunctionThe covariance function is a n umber that measures the common variation ofX and Y. It is defined ascov(X, Y )=E[(X − E[X])(Y − E[Y ])] (7.2)= E[XY ] − E[X]E[Y ] (7.3)The co variance is determined by the difference in E[XY] and E[X]E[Y ]. IfX and Y were statistically independent then E[XY ] would equal E[X]E[Y ]and the co variance wou ld be zero. Hence, the covariance, as its name im-plies, measures the common variation. T he covariance can be normalized toproducewhatisknownasthecorrelationcoefficien t, ρ.ρ =cov(X, Y)pvar(X)var(Y)(7.4)The correlation coefficient is bounded b y −1 ≤ ρ ≤ 1. It will have value ρ =0when the covariance is zero and value ρ = ±1 when X and Y are perfectly7.2. LIN EA R ESTIM ATION 125correlated or anti-correlated.7.1.2 Autocorrelation FunctionThe autocorrelation1function is very similar to the covariance function. Itis defined asR(X, Y )=E[XY ]=cov(X, Y )+E[X]E[Y ] (7.5)It retains the mean valu es in the calculation of the value. The rando mvariables are orthogonal if R(X, Y )=0.7.1.3 Joint Norm al DistributionIf X and Y hav e a joint norm al distribution then the proba b ility den sityfunctio n isfXY(x, y)=12πσxσyp1 − ρ2exp−³x−µxσx´2− 2ρ³x−µxσx´³y−µyσy´+³y−µyσy´22(1 − ρ2)(7.6)The con tour s of equal probability are ellipses, as shown in Figure 7.2. Theprobab ility ch ang es muc h more rapidly along the minor axis of the ellipsesthan along the major axes. T he orientation of the elliptical contours is alon gthe line y = x if ρ > 0 and along the line y = −x if ρ < 0. The contours area circle, and the variables are uncorrelated, if ρ =0. The center of the ellipseis¡µx,µy¢.7.2 Linea r E st ima tionIt is often the case that one would lik e to estimate or predict the value ofone random variable based on an observation of the other. If the randomvariables are correlated then this should yield a better result, on the a verag e,than just guessing. We will see this is indeed the case.1Be careful to not confuse the term “autocorrelation function” with “correlation coef-ficient”.126 CHAPTER 7. RANDOM PROCESSESFigure 7.2: The norma l probability distribution sho w n as a surface plot onthe left and a contou r plot in the cen ter. A nu mber of sa m ple poin ts areshownoverlaidonthecontourplotintherightframe. Thelinearpredictorline is drawn in the righ t frame. ρ = −0.7, σx= σy=1,µx= µy=0.ThetaskistoconstructaruleforthepredictionofY basedonanobser-vation of X. We will call the predictio nˆY, and compute its value with thesimple linear equationˆY = aX + b (7.7)where a and b are parameters to be c hosen to pro vid e the best results. Weareencouragedtoselectthislinearrulewhenwenotethatthesamplepointstend to fall about a sloping line. We wou ld expect a to correspond to theslope and b to the intercep t.To find a means of calculating the coefficien ts from a set of sample points,construct the predictor errorε = E[(Y −ˆY )2] (7.8)We want to cho ose a and b to minim ize ε. Therefore, compute the appropriatederiv a tives and set them to zero.∂ε∂a= −2E[(Y −ˆY )∂ˆY∂a]=0 (7.9)∂ε∂b= −2E[(Y −ˆY )∂ˆY∂b]=0 (7.10)7.3. RA N D O M PR O C E SSE S 127upon substitution ofˆY = aX + b and rearrangement we get the pair ofequationsE[XY ]=aE[X2]+bE[X] (7.11)E[Y ]=aE[X]+b (7.12)These can be solv ed for a and b in terms of the expected values. The expectedvalues can be themselves estimated from the samp le set.a =cov(X, Y)var(X)(7.13)b = E[Y ] −cov(X, Y)var(X)E[X] (7.14)The prediction error with these parameter values isε =(1− ρ2)var(Y) (7.15)Wh en the correlation coefficien t ρ = ±1 the error is zero, meaning thatperfect prediction can be made. When ρ =0thevarianceinthepredictionis as large as the variatio n in Y, and the predicto r is of no help at all. Forin termediate values of ρ, whether positive or negative, the predictor reducesthe error.7.3 Ran dom ProcessesWe ha ve seen that a random variable X is a rule which assigns a n u mber toevery outcome e of an experimen t. The random variable is a function X(e)that maps the set of experiment outcomes to the set of nu mbers. A randomprocess is a rule that maps every outcome e of an experim ent to a functionX(t, e). A random process is usually conceived of as a function of time, butthere is no reason to not consider random processes that are functions of otherindependen t variables, such as spatial coordinates. T he function X(u, v, e)w ould be a function whose value depended on the location (u, v) and theoutcome e, and could be used in represen ting random variations in an image.In the follo w ing w e will deal with one-dimensional random processes todev elop a n umber of basic concepts. Having them in hand, we can then goon to multiple dimensions.128 CHAPTER 7. RANDOM PROCESSESThe domain of e