Gaussian Probability Density Functions: Properties and Error Characterization

Transcription

1 Gaussian Probabilit Densit Functions: Properties and Error Characterization Maria Isabel Ribeiro Institute for Sstems and Robotics Instituto Superior Tcnico Av. Rovisco Pais, Lisboa PORTUGAL {mir@isr.ist.utl.pt} c M. Isabel Ribeiro, 4 Februar 4

2 Contents 1 Normal random variables Normal random vectors 7.1 Particularization for second order Locus of constant probabilit Properties 4 Covariance matrices and error ellipsoid 4 1

3 Chapter 1 Normal random variables A random variable X is said to be normall distributed with mean µ and variance σ if its probabilit densit function (pdf) is f X () = 1 ] ( µ) ep [, < <. (1.1) πσ σ Whenever there is no possible confusion between the random variable X and the real argument,, of the pdf this is simpl represented b f() omitting the eplicit reference to the random variable X in the subscript. The Normal or Gaussian distribution of X is usuall represented b, or also, X N (µ, σ ), X N ( µ, σ ). The Normal or Gaussian pdf (1.1) is a bell-shaped curve that is smmetric about 1 the mean µ and that attains its maimum value of πσ.399 at = µ as σ represented in Figure 1.1 for µ = and σ = 1.. The Gaussian pdf N (µ, σ ) is completel characterized b the two parameters µ and σ, the first and second order moments, respectivel, obtainable from the pdf as µ = E[X] = σ = E[(X µ) ] = f()d, (1.) ( µ) f()d (1.3)

4 .3.3. f() Figure 1.1: Gaussian or Normal pdf, N(, 1. ) The mean, or the epected value of the variable, is the centroid of the pdf. In this particular case of Gaussian pdf, the mean is also the point at which the pdf is maimum. The variance σ is a measure of the dispersion of the random variable around the mean. The fact that (1.1) is completel characterized b two parameters, the first and second order moments of the pdf, renders its use ver common in characterizing the uncertaint in various domains of application. For eample, in robotics, it is common to use Gaussian pdf to statisticall characterize sensor measurements, robot locations, map representations. The pdfs represented in Figure 1. have the same mean, µ =, and σ 1 > σ > σ 3 showing that the larger the variance the greater the dispersion around the mean σ 3 f().. σ.1 σ Figure 1.: Gaussian pdf with different variances (σ 1 = 3, σ =, σ 3 = 1) 3

5 Definition 1.1 The square-root of the variance, σ, is usuall known as standard deviation. Given a real number a R, the probabilit that the random variable X N (µ, σ ) takes values less or equal a is given b a a ] 1 ( µ) P r{x a } = f()d = ep [ d, (1.4) πσ σ represented b the shaded area in Figure a Figure 1.3: Probabilit evaluation using pdf To evaluate the probabilit in (1.4) the error function, erf(), which is related with N (, 1), erf() = 1 ep / d (1.) π plas a ke role. In fact, with a change of variables, (1.4) ma be rewritten as. erf( µ a ) for σ a µ P r{x a } =. + erf( a µ ) for σ a µ stating the importance of the error function, whose values for various are displaed in Table 1.1 In various aspects of robotics, in particular when dealing with uncertaint in mobile robot localization, it is common the evaluation of the probabilit that a 4

6 erf erf erf erf Table 1.1: erf - Error function random variable Y (more generall a random vector representing the robot location) lies in an interval around the mean value µ. This interval is usuall defined in terms of the standard deviation, σ, or its multiples. Using the error function, (1.),the probabilit that the random variable X lies in an interval whose width is related with the standard deviation, is P r{ X µ σ} =.erf(1) =.6868 (1.6) P r{ X µ σ} =.erf() =.94 (1.7) P r{ X µ 3σ} =.erf(3) =.9973 (1.8) In other words, the probabilit that a Gaussian random variable lies in the interval [µ 3σ, µ + 3σ] is equal to Figure 1.4 represents the situation (1.6)corresponding to the probabilit of X ling in the interval [µ σ, µ + σ]. Another useful evaluation is the locus of values of the random variable X where the pdf is greater or equal a given pre-specified value K 1, i.e., 1 πσ ep [ ( µ) σ ] K 1 ( µ) σ K (1.9)

7 .3.3 σ.. f() Figure 1.4: Probabilit of X taking values in the interval [µ σ, µ + σ], µ =, σ = 1. with K = ln( πσk 1 ). This locus is the line segment as represented in Figure 1.. µ σ K µ + σ K f() K 1 Figure 1.: Locus of where the pdf is greater or equal than K 1 6

8 Chapter Normal random vectors A random vector X = [X 1, X,... X n ] T R n is Gaussian if its pdf is { 1 f X () = ep 1 } (π) n/ Σ 1/ ( m X) T Σ 1 ( m X ) (.1) where m X = E(X) is the mean vector of the random vector X, Σ X = E[(X m X )(X m X ) T ] is the covariance matri, n = dimx is the dimension of the random vector, also represented as X N (m X, Σ X ). In (.1), it is assumed that is a vector of dimension n and that Σ 1 eists. If Σ is simpl non-negative definite, then one defines a Gaussian vector through the characteristic function, []. The mean vector m X is the collection of the mean values of each of the random variables X i, m X = E X 1 X. X n = m X1 m X. m Xn. 7

9 The covariance matri is smmetric with elements, Σ X = Σ T X = E(X 1 m X1 ) E(X 1 m X1 )(X m X )... E(X 1 m X1 )(X n m Xn ) E(X m X )(X 1 m X1 ) E(X m X )... E(X m X )(X n m Xn ) =... E(X n m Xn )(X 1 m X1 ) E(X n m Xn ) The diagonal elements of Σ are the variance of the random variables X i and the generic element Σ ij = E(X i m Xi )(X j m Xj ) represents the covariance of the two random variables X i and X j. Similarl to the scalar case, the pdf of a Gaussian random vector is completel characterized b its first and second moments, the mean vector and the covariance matri, respectivel. This ields interesting properties, some of which are listed in Chapter 3. When studing the localization of autonomous robots, the random vector X plas the role of the robot s location. Depending on the robot characteristics and on the operating environment, the location ma be epressed as: a two-dimensional vector with the position in a D environment, a three-dimensional vector (d-position and orientation) representing a mobile robot s location in an horizontal environment, a si-dimensional vector (3 positions and 3 orientations) in an underwater vehicle When characterizing a D-laser scanner in a statistical framework, each range measurement is associated with a given pan angle corresponding to the scanning mechanism. Therefore the pair (distance, angle) ma be considered as a random vector whose statistical characterization depends on the phsical principle of the sensor device. The above eamples refer quantities, (e.g., robot position, sensor measurements) that are not deterministic. To account for the associated uncertainties, we consider them as random vectors. Moreover, we know how to deal with Gaussian random vectors that show a number of nice properties; this (but not onl) pushes us to consider these random variables as been governed b a Gaussian distribution. In man cases, we have to deal with low dimension Gaussian random vectors (second or third dimension), and therefore it is useful that we particularize 8

10 the n-dimensional general case to second order and present and illustrate some properties. The following section particularizes some results for a second order Gaussian pdf..1 Particularization for second order In the first two above referred cases, the Gaussian random vector is of order two or three. In this section we illustrate the case when n=. Let [ ] X Z =, Y be a second-order Gaussian random vector, with mean, [ ] [ ] X mx E[Z] = E = Y and covariance matri, [ σ Σ = X σ XY σ XY σ Y m Y ] (.) (.3) where σ X and σ Y are the variances of the random variables X and Y and σ XY is the covariance of X and Y, defined below. Definition.1 The covariance σ XY of the two random variables X and Y is the number σ XY = E[(X m X )(Y m Y )] (.4) where m X = E(X) and m Y = E(Y ). Epanding the product (.4), ields, σ XY = E(XY ) m X E(Y ) m Y E(X) + m X m Y (.) = E(XY ) E(X)E(Y ) (.6) = E(X) m X m Y. (.7) Definition. The correlation coefficient of the variables X and Y is defined as ρ = σ XY σ X σ Y (.8) 9

11 Result.1 The correlation coefficient and the covariance of the variables X and Y satisf the following inequalities, Proof: [] Consider the mean value of ρ 1, σ XY σ X σ Y. (.9) E[a(X m X ) + (Y m Y )] = a σ X + aσ XY + σ Y which is a positive quadratic for an a, and hence, the discriminant is negative, i.e., σ XY σ Xσ Y from where (.9) results. According to the previous definitions, the covariance matri (.3) is rewritten as [ ] σx Σ = ρσ X σ Y. (.1) ρσ X σ Y For this second-order case, the Gaussian pdf particularizes as, with z = [ ] T R, 1 f(z) = [ π detσ ep 1 ] [ m X m Y ]Σ 1 [ m X m Y ] T (.11) [ ( 1 = πσ X σ ep 1 ( mx ) Y 1 ρ (1 ρ ) σx ρ( m X)( m Y ) + ( m Y ) )] σ X σ Y σy where the role plaed b the correlation coefficient ρ is evident. At this stage we present a set of definitions and properties that, even though being valid for an two random variables, X and Y, also appl to the case when the random variables (rv) are Gaussian. Definition.3 Independence Two random variables X and Y are called independent if the joint pdf, f(, ) equals the product of the pdf of each random variable, f(), f(), i.e., σ Y f(, ) = f()f() In the case of Gaussian random variables, clearl X and Y are independent when ρ =. This issue will be further eplored later. 1

12 Definition.4 Uncorrelatedness Two random varibles X and Y are called uncorrelated if their covariance is zero, i.e., σ XY = E[(X m X )(Y m Y )] =, which can be written in the following equivalent forms: Note that ρ =, E(XY ) = E(X)E(Y ). E(X + Y ) = E(X) + E(Y ) but, in general, E(XY ) E(X)E(Y ). However, when X and Y are uncorrelated, E(XY ) = E(X)E(Y ) according to Definition.4. Definition. Orhthogonalit Two random variables X and Y are called orthognal if E(XY ) =, which is represented as X Y Propert.1 If X and Y are uncorrelated, then X m X Y m Y. Propert. If X and Y are uncorrelated and m X = and m Y =, then X Y. Propert.3 If two random variables X and Y are independent, then the are uncorrelated, i.e., f(, ) = f()f() E(XY ) = E(X)E(Y ) but the converse is not, in general, true. Proof: From the definition of mean value, E(XY ) = = f()d f()dd f()d = E(X)E(Y ). 11

13 Propert.4 If two Gaussian random variables X and Y are uncorrelated, the are also independent, i.e., for Normal or Gaussian random variables,independenc is equivalent to uncorrelatedness. If X N (µ X, Σ X ) and Y N (µ Y, Σ Y ) f() = f()f() E(XY ) = E(X)E(Y ) ρ =. Result. Variance of the sum of two random variables Let X and Y be two random variables, jointl distributed, with mean m X and m Y and correlation coefficient ρ and let Z = X + Y. Then, E(Z) = m Z = E(X) + E(Y ) = m X + m Y (.1) σ Z = E[(Z m Z ) )] = σ X + ρσ X σ Y + σ Y. (.13) Proof: Evaluating the second term in (.13) ields: σ Z = E[(X m X ) + (Y m Y )] = E[(X m X ) ] + E[(X m X )(Y m Y )] + E[(Y m Y ) ] from where the result immediatel holds. Result.3 Variance of the sum of two uncorrelated random variables Let X and Y be two uncorrelated random variables, jointl distributed, with mean m X and m Y and let Z = X + Y. Then, σ Z = σ X + σ Y (.14) i.e., if two random variables are uncorrelated, then the variance of their sum equals the sum of their variances. We regain the case of two jointl Gaussian random varaibles, X and Y, with pdf represented b (.11) to analze, in the plots of Gaussian pdfs, the influence of the correlation coefficient ρ in the bell-shaped pdf. Figure.1 represents four distinct situations with zero mean and null correlation between X and Y, i.e., ρ =, but with different values of the standard deviations σ X and σ Y. It is clear that, in all cases, the maimum of the pdf is obtained for the mean value. As ρ =, i.e., the random variables are uncorrelated, 1

14 E(X)=, E(Y)=, σ X =1, σ Y =1, ρ= E(X)=, E(Y)=, σ X =1., σ Y =1., ρ=.1.1 f(,) f(,).. a) b) E(X)=, E(Y)=, σ X =1, σ Y =., ρ= E(X)=, E(Y)=, σ X =, σ Y =1., ρ=.1.1 f(,) f(,).. c) d) Figure.1: Second-order Gaussian pdfs, with m X = m Y =, ρ = a) σ X = 1, σ Y = 1, b) σ X = 1., σ Y = 1., c) σ X = 1, σ Y =., d) σ X =, σ Y = 1 13

15 the change in the standard deviations σ X and σ Y has independent effects in each of the components. For eample, in Figure.1-d) the spread around the mean is greater along the coordinate. Moreover, the locus of constant value of the pdf is an ellipse with its ais parallel to the and ais. This ellipse has equal ais length, i.e, is a circumference, when both random variables, X and Y have the same standard deviation, σ X = sigma Y. The eamples in Figure. show the influence of the correlation coefficient on the shape of the pdf. What happens is that the ais of the ellipse referred before will no longer be parallel to the ais and. The greater the correlation coefficient, the larger the misalignment of these ais. When ρ = 1 or ρ = 1 the ais of the ellipse has an angle of π/4 relative to the -ais of the pdf.. Locus of constant probabilit Similarl to what was considered for a Gaussian random variable, it is also useful for a variet of applications and for a second order Gaussian random vector, to evaluate the locus (, ) for which the pdf is greater or equal a specified constant, K 1, i.e., { (, ) : 1 [ π detσ ep 1 ] } [ m X m Y ]Σ 1 [ m X m Y ] T K 1 which is equivalent to { (, ) : [ m X m Y ]Σ 1 [ mx m Y with K = ln(πk 1 detσ). ] } K (.1) (.16) Figures.3 and.4 represent all the pairs (, ) for which the pdf is less or equal a given specified constant K 1. The locus of constant value is an ellipse with the ais parallel to the and coordinates when ρ =, i.e., when the random variables X and Y are uncorrelated. When ρ the ellipse ais are not parallel with the and ais. The center of the ellipse coincides in all cases with (m X, m Y ). The locus in (.16) is the border and the inner points of an ellipse, centered in (m X, m Y ). The length of the ellipses ais and the angle the do with the ais and are a function of the constant K, of the eigenvalues of the covariance matri 14

16 E(X)=1, E(Y)=, σ X =1., σ Y =1., ρ=.8 E(X)=1, E(Y)=, σ X =1., σ Y =1., ρ= f(,) f(,) a) b) E(X)=1, E(Y)=, σ X =1., σ Y =1., ρ=. E(X)=1, E(Y)=, σ X =1., σ Y =1., ρ= f(,) c) d) E(X)=1, E(Y)=, σ X =1., σ Y =1., ρ=.4 E(X)=1, E(Y)=, σ X =1., σ Y =1., ρ= f(,) f(,) e) f) Figure.: Second-order Gaussian pdfs, with m X = 1, m Y =, σ X = 1., σ Y = 1. 1

17 E()=1, E()=, sigma =1, sigma =1, rho= E()=1, E()=, sigma =, sigma =1, rho= a) b) E()=1, E()=, sigma =, sigma =1, rho= c) 1 Figure.3: Locus of (, ) such that the Gaussian pdf is less than a constant K for uncorrelated Gaussian random variables with m X = 1, m Y = - a) σ X = σ Y = 1, b) σ X =, σ Y = 1, c) σ X = 1, σ Y = 16

18 E(X)=1, E(Y)=, σ X =1., σ Y =, ρ=. E(X)=1, E(Y)=, σ X =1., σ Y =, ρ= a) c) 1 E(X)=1, E(Y)=, σ X =1., σ Y =, ρ=.7 E(X)=1, E(Y)=, σ X =1., σ Y =, ρ= b) d) 1 Figure.4: Locus of (, ) such that the Gaussian pdf is less than a constant K for correlated variables. m X = 1, m Y =, σ X = 1., σ Y = - a) ρ =., b) ρ =.7, c) ρ =., d) ρ =.7 17

19 Σ and of the correlation coefficient. We will demonstrate this statement in two different steps. We show that: 1. Case 1 - if Σ in (.16) is a diagonal matri, which happens when ρ =, i.e., X and Y are uncorrelated, the ellipse ais are parallel to the frame ais.. Case - if Σ in (.16) is non-diagonal, i.e, ρ, the ellipse ais are not parallel to the frame ais. In both cases, the length of the ellipse ais is related with the eigenvalues of the covariance matri Σ in (.3) given b: λ 1 = 1 [ ] σx + σy + (σ X + 4σ X σy ρ, (.17) λ = 1 [ ] σx + σy (σ X + 4σ X σy ρ. (.18) Case 1 - Diagonal covariance matri When ρ =, i.e., the variables X and Y are uncorrelated, the covariance matri is diagonal, [ ] σ Σ = X σy (.19) and the eigenvalues particularize to λ 1 = σx and λ = σy. In this particular case, illustrated in Figure., the locus (.16) ma be written as {(, ) : ( m X) σ X or also, { (, ) : ( m X) + ( m Y ) σ Y } K (.) } 1. (.1) + ( m Y ) KσX KσY Figure. represents the ellipse that is the border of the locus in (.1) having: -ais with length σ X K -ais with length σ Y K. Case - Non-diagonal covariance matri When the covariance matri Σ in (.3) is non-diagonal, the ellipse that borders the locus (.16) has center in (m X, m Y ) but its ais are not aligned with the 18

20 3. 3 \sigma_x \sqrt{k}. \sigma_y \sqrt{k} 1. 1 (m X, m Y ) Figure.: Locus of constant pdf: ellipse with ais parallel to the frame ais coordinate frame. In the sequel we evaluate the angle between the ellipse ais and those of the coordinate frame. With no loss of generalit we will consider that m X = m Y =, i.e., the ellipse is centered in the coordinated frame. Therefore, the locus under analsis is given b { [ ] } (, ) : [ ]Σ 1 K (.) where Σ is the matri in (.3). As it is a smmetric matri, the eigenvectors corresponding to distinct eigenvalues are orthogonal. When σ X σ Y the eigenvalues (.17) and (.18) are distinct, the corresponding eigenvectors are orthogonal and therefore Σ has simple structure which means that there eists a non-singular and unitar coordinate transformation T such that where Σ = T D T 1 (.3) T = [ v 1 v ], D = diag(λ 1, λ ) and v 1, v are the unit-norm eigenvectors of Σ associated with λ 1 and λ. Replacing (.3) in (.) ields { [ ] } (, ) : [ ]T D 1 T 1 K. (.4) 19

21 Denoting [ w1 w ] [ ] = T 1 and given that T T = T 1, it is immediate that (.4) can be epressed as { [ ] 1 [ ] } λ1 w1 (w 1, w ) : [w 1 w ] K λ w (.) (.6) that corresponds, in the new coordinate sstem defined b the ais w 1 and w, to the locus bordered b an ellipse aligned with those ais. Given that v 1 and v are unit-norm orthogonal vectors, the coordinate transformation defined b (.) corresponds to a rotation of the coordinate sstem (, ), around its origin b an angle α = 1 ( ) ρσx σ Y tan 1 σx, π σ Y 4 α π 4, σ X σ Y. (.7) Evaluating (.6), ields, { (w 1, w ) : } w1 + w 1 Kλ 1 Kλ (.8) that corresponds to an ellipse having w 1 -ais with length Kλ 1 w -ais with length Kλ as represented in Figure.6.

22 w w Figure.6: Ellipses non-aligned with the coordinate ais and. 1

23 Chapter 3 Properties Let X and Y be two jointl distributed Gaussian random vectors, of dimension n and m, respectivel, i.e, X N (m X, Σ X ) Y N (m Y, Σ Y ) and Σ X a square matri of dimension n and Σ Y a square matri of dimension m. Result 3.1 The conditional pdf of X and Y is given b 1 f( ) = [ (π)n detσ ep 1 ] ( m)t Σ 1 ( m) N (m, Σ) (3.1) with m = E[X Y ] = m X + Σ XY Σ 1 Y (Y m Y ) (3.) Σ = Σ X Σ XY Σ 1 Y Σ Y X (3.3) The previous result states that, when X and Y are jointl Gaussian, f( ) is also Gaussian with mean and covariance matri given b (3.1) and (3.3), respectivel. Result 3. Let X R n, Y R m and Z R r be jointl distributed Gaussian random vectors. If Y and Z are independent, then where E[X] = m X. E[X Y, Z] = E[X Y ] + E[X Z] m X (3.4)

24 Result 3.3 Let X R n, Y R m and Z R r be jointl distributed Gaussian random vectors. If Y and Z are not necessaril independent, then E[X Y, Z] = E[X Y, Z] (3.) where ildeing Z = Z E[Z Y ] E[X Y, Z] = E[X Y ] + E[X Z] m X 3

25 Chapter 4 Covariance matrices and error ellipsoid Let X be a n-dimensional Gaussian random vector, with X N (m X, Σ X ) and consider a constant, K 1 R. The locus for which the pdf f() is greater or equal a specified constant K 1,i.e., { [ 1 : ep 1 ] } (π) n/ Σ 1/ [ m X] T Σ 1 X [ m X] K 1 (4.1) which is equivalent to { : [ mx ] T Σ 1 X [ m X] K } (4.) with K = ln((π) n/ K 1 Σ 1/) is an n-dimensional ellipsoid centered at the mean m X and whose ais are onl aligned with the cartesian frame if the covariance matri Σ is diagonal. The ellipsoid defined b (4.) is the region of minimum volume that contains a given probabilit mass under the Gaussian assumption. When in (4.) rather than having an inequalit there is an equalit, (4.), i.e., { : [ mx ] T Σ 1 X [ m X] = K } this locus ma be interpreted as the contours of equal probabilit. Definition 4.1 Mahalanobis distance The scalar quantit [ m X ] T Σ 1 X [ m X] = K (4.3) is known as the Mahalanobis distance of the vector to the mean m X. 4

26 The Mahalanobis distance, is a normalized distance where normalization is achieved through the covariance matri. The surfaces on which K is constant are ellipsoids that are centered about the mean m X, and whose semi-ais are K times the eigenvalues of Σ X, as seen before. In the special case where the random variables that are the components of X are uncorrelated and with the same variance, i.e., the covariance matri Σ is a diagonal matri with all its diagonal elements equal, these surfaces are spheres, and the Mahalanobis distance becomes equivalent to the Euclidean distance (m X,m Y ) Figure 4.1: Contours of equal Mahalanobis and Euclidean distance around (m X, m Y ) for a second order Gaussian random vector Figure 4.1 represents the contours of equal Mahalanobis and Euclidean distance around (m X, m Y ) for a second order Gaussian random vector. In oher words, an point (, ) in the ellipse is at the same Mahalanobis distance to the center of the ellipses. Also, an point (, ) in the circumference is at the same Euclidean distance to the center. This plot enhances the fact that the Mahalanobis distance is weighted b the covariance matri. For decision making purposes (e.g., the field-of-view, a validation gate), and given m X and Σ X, it is necessar to determine the probabilit that a given vector will lie within, sa, the 9% confidence ellipse or ellipsoid given b (4.3). For a given K, the relationship between K and the probabilit of ling within the

27 ellipsoid specified b K is, [3], n = 1; P r{ inside the ellipsoid} = 1 π + erf( K) n = ; P r{inside the ellipsoid} = 1 e K/ n = 3; P r{ inside the ellipsoid} = 1 π + erf( K) π Ke K/ (4.4) where n is the dimension of the random vector. Numeric values of the above epression for n = are presented in the following table Probabilit K % % %.48 8% % 4.6 For a given K the ellispoid ais are fied. The probabilit that a given value of the random vector X lies within the ellipsoid centered in the mean value, increases with the increase of K. This problem can be stated the other wa around. In the case where we specif a fied probabilit value, the question is the value of K that ields an ellipsoid satisfing that probabilit. To answer the question the statistics of K has to be analzed. The scalar random variable (4.3) has a known random distribution, as stated in the following result. Result 4.1 Given the n-dimensional Gaussian random vector X, with mean m X and covariance matri Σ X, the scalar random variable K defined b the quadratic form [ m X ] T Σ 1 X [ m X] = K (4.) has a chi-square distribution with n degrees of freedom. Proof: see, p.e., in [1]. The pdf of K in (4.), i.e., the chi-square densit with n degrees of freedom is, (see, p.e., [1]) 1 n f(k) = n Γ( n ep k )k 6

28 where the gamma function satisfies, Γ( 1 ) = π, Γ(1) = 1 Γ(n + 1) = Γ(n). The probabilit that the scalar random variable, K in (4.) is less or equal a given constant, χ p P r{k χ p} = P r{[ m X ] T Σ 1 [ m X ] χ p} = p is given in the following table where n is the number of degrees of freedom and the sub-indice p in χ p represents the corresponding probabilit under evaluation. n χ.99 χ.99 χ.97 χ.9 χ.9 χ.7 χ. χ. χ.1 χ From this table we can conclude, for eample, that for a third-order Gaussian random vector, n = 3, P r{k 6.} = P r{[ m X ] T Σ 1 [ m X ] 6.} =.9 Eample 4.1 Mobile robot localization and the error ellipsoid This eample illustrates the use of the error ellipses and ellipsoids in a particular application, the localization of a mobile robot operating in a given environment. Consider a mobile platform, moving in an environment and let P R be the position of the platform relative to a world frame. P has two components, P = [ X Y The eact value of P is not known and we have to use an particular localization algorithm to evaluate P. The most common algorithms combine internal and eternal sensor measurements to ield an estimated value of P. The uncertaint associated with all the quantities involved in this procedure, namel vehicle model, sensor measurements, environment map representation, leads to consider P as a random vector. Gaussianit is assumed for simplicit. Therefore, the localization algorithm provides an estimated value of P, denoted 7 ].

29 as ˆP, which is the mean value of the Gaussian pdf, and the associated covariance matri, i.e., P N ( ˆP, Σ P ) At each time step of the algorithm we do not know the eact value of P, but we have an estimated value, ˆP and a measure of the uncertaint of this estimate, given b Σ P. The evident question is the following: Where is the robot?, i.e., What is the eact value of P? It is not possible to give a direct answer to this question, but rather a probabilistic one. We ma answer, for eample: Given ˆP and Σ P, with 9% of probabilit, the robot is located in an ellipse centered in ˆP and whose border is defined according to the Mahalanobis distance. In this case the value of K in (4.) will be K = Someone ma sa that, for the involved application, a probabilit of 9% is small and ask to have an answer with an higher probabilit, for eample 99%. The answer will be similar but, in this case, the error ellipse, will be defined for K = 9.1, i.e.,the ellipse will be larger than the previous one. 8

30 Bibliograph [1] Yaakov Bar-Shalom, X. Rong Li, Thiagalingam Kirubarajan, Estimation with Applications to Tracking and Navigation, John Wile & Sons, 1. [] A. Papoulis, Probabilit, Random Variables and Stochastic Processes, McGraw-Hill, 196. [3] Randall C. Smith, Peter Cheeseman, On the Representation and Estimation of Spatial Uncertaint, the International Journal of Robotics Research, Vol., No.4, Winter