On computing Gaussian curvature of some well known distribution

Transcription

1 Theoretical Mathematics & Applications, ol.3, no.4, 03, ISSN: (print), (online) Scienpress Ltd, 03 On computing Gaussian curature of some well known distribution William W.S. Chen Abstract The objectie of this paper is to proide a deeper and broader understanding of the meaning of Gaussian curature. It also suggests some more general alternatie computational methods than Kass, R.E. and Vos, P.W. [, ]. It is not intended to compare the superiority of the four formulas to be used in computing the Gaussian curature. We define the coefficients of the expected Fisher Information Matrix as the coefficients of the first fundamental form see Lehmann, E.L. [3]. Four different formulas adopted from Struik, D.J. [4] are used, and labeled here as (A), (B), (C), and (D). It has been found that all four of these formulas can compute the Gaussian curature effectiely and successfully. This is demonstrated by four commonly used examples. Also it would be interested to compare our methods with Kass, R.E. and Vos, P.W. approach [, ]. Mathematics Subject Classification: 60E05 Department of Statistics, The George Washington Uniersity, Washington D.C Williamwschen@gmail.com Article Info: Receied: July, 03. Reised: September 9, 03. Published online : December 6, 03.

2 86 On computing Gaussian curature of some well known distribution Keywords: Cauchy; Christoffel Symbols; Expected Fisher Information Matrix; Gaussian Curature; Metric tensor; Normal; t; Trinomial Introduction The Gaussian curature, K, of the surface, at the point p, is the product of the extreme alues of curatures in the family. If p is a point on a regular surface in R 3 and K(p) is positie, then the two curatures hae the same sign and the point p is called an elliptic point of the surface. Example is one instance of this case. If K(p) is negatie, then the two curatures hae opposite signs and the point p is called the hyperbolic point of the surface. Examples,3 and 4 demonstrate these cases. If exactly one curature equals zero, then the point p is a parabolic point of the surface. If the Gaussian curature equals zero, then the surface is either planar or deeloper. Computing the Gaussian curature plays a central role in determining the shape of the surface. It is also a well known fact that two surfaces which hae the same Gaussian curature are always isometric and bending inariant. For instance, Struik, D.J. [4] on p0 proided an excellent example that demonstrated a correspondence between the points of a catenoid and that of a right helicoid, such that at corresponding points the coefficients of the first fundamental form and the Gaussian curatures are identical. In fact, one surface can pass into the other by a continuous bending. This has been demonstrated by the deformation of six different stages. Howeer, if the Gaussian curature is different, then the two surfaces will not be isometric. For example, a sphere and plane are not locally isometric because the Gaussian curature of a sphere is nonzero while the Gaussian curature of a plane is zero. This is why any map of a portion of the earth must distort distances. In this paper, we define the coefficients of the expected Fisher Information Matrix as equal to the coefficients of the first fundamental form. There are numerous authors who hae used this

3 William W.S. Chen 87 concept, including Lehmann,E.L. [3] on p6, Barndorff-Nielsen O.E.,et.al. [5] and Kass R.E. []. Chen, W.W.S. [7] has compared the Gaussian or Riemann curature in his McMaster Uniersity talk. Using these defined metric tensors, we can then adopt the same notation and apply the formulas listed in Struik, D.J. [4]. The Gaussian curature then becomes a function of the coefficients of the first fundamental form and their first and second deriaties. In this paper, we suggest the following four systematic steps to compute the Gaussian curature: Step - compute the coefficients of the expected Fisher Information Matrix or coefficients of the first fundamental form, namely, E,F and G; Step -compute the needed first or second deriatie of E,F and G, and thus the six Christoffel symbols; Step 3- apply formula (D), which necessitates in the computation of the mixed Riemann curature tensors R and R ; the subsequent computation of the inner product of this tensor with the metric tensor, F or G, results in the coariant Riemann curature tensor R ; Step 4-obsere that the Gaussian curature has a ery simple relation to Riemann symbols of the second kind. By adhearing to this procedure, the correct Gaussian curature will be calculated. In the case where F 0 or the parametric lines on the surface are not orthogonal, the computational procedure can be extremely tedious. It is always prudent to find a proper transformation to form an orthogonal system of parametric lines in order to simplify the computational procedures. Notation and Terminology In this section, we define the basic notations and terminologies that will apply in the next two sections. These notations and symbols can also be found in Struik, D.J.[4] or Gray, A.[6]. First and foremost, we define the coefficients of the first fundamental form or metric tensors as;

4 88 On computing Gaussian curature of some well known distribution ln f ln f ln f E E( ), F E( ), and G E( ) u u (.) where f(u,) are two parameters of the probability density functions. It is clear that E, F and G are functions of the parameters u and. The expectations apply to the whole sample space where the random ariables are defined. We also assume that the regular conditions of the information metrics tensors are all satisfied. The details of these fie conditions are summerized in Kass R.E. and Vos, P.W. [] on p85. Since E, F and G are functions of parameters (u,) and are continuously twice differentiable, Eu, E, Fu, F, G u and G all exists and are all well defined. Because F 0, formula (A) turns out to be a simplified form of Gauss Equation. In the next section, we will demonstrate that formulas (C) and (D) are heaily dependent on the six Christoffel symbols. Additionally, no assumption is made regarding F 0, and so the parametric lines are not necessarily orthogonal. Howeer, if F 0, the six Christoffel symbols can be greatly simplified. The four distributions discussed here belong to this case. 3 Formula and Tables In this section, we select four formulas that can be used to compute the Gaussian curature. (A) G E ( ( ) + ( )) EG u E u G E F G Gu F Fu E Eu Fu Gu 4( EG F ) EG F u EG F EG F E F G (B) D D D D (C) ( Γ) ( Γ ) ( Γ) ( Γ) E u E D u G G

5 William W.S. Chen 89 where (D) where D EG F. R (,), EG F EG F m m (,) R R g, m R Γ Γ +Γ Γ Γ Γ l l l m l m l ijk ik jk ik mj jk mi uj ui where the quantities of, sum on m, l R ijk are components of a tensor of the fourth order. This tensor is called the mixed Riemann curature tensor. Notice that g, g and g are simply tensor notation for E,F and G. Formula (B) was deeloped by G. Frobenius while formula (C ) was deried by J. Liouille. Clearly, formula (A) is a special case that is alid only when the parametric lines are orthogonal. Formula (D) is a general form represented in Riemann symbols of the first and second kind, respectiely. In formula (D), R, the inner product of the mixed Riemann curature tensor and the metric tensor, is called the coariant Riemann curature tensor; it is a coariant tensor of the fourth order. The components l R ijk and R are also known as Riemann symbols of the first and second kind, respectiely. Notice that Riemann symbols of the second kind will satisfy the relation R R R R, the well-known property of skew-symmetry with respect to the last two indices. It is useful to be aware of the fact that the Christoffel symbols depend only on the coefficients of the first fundamental form and their deriaties. The same holds true for the mixed Riemann curature tensor. From this point of iew, as long as we can find the coefficients of the first fundamental form of a gien distribution and their first and second deriaties, we can uniquely define the corresponding Christoffel symbols and hence mixed Riemann curature tensors. Thus, the process of computing the coariant Riemann curature tensor and Gaussian curature is simplified. From a different

6 90 On computing Gaussian curature of some well known distribution perspectie, we know that the mixed Riemann curature tensor will link with the coefficient of the second fundamental form; namely e, f, and g, by R g eg f, where n n ( ) G F E g, g, g. EG F EG F EG F (3.) The aboe relation can then be easily used to derie R, and the eg f result will coincide with equation (7-3) of Struik, D.J. [4] on p.83, the original fundamental definition of Gaussian curature. These points conince us that formulas (A) and (D) basically define the same quantity, but only in different forms. The reason why only formula (D) was selected for presentation is due to the following two facts:. to aoid repetition of Kass R.E [] on p.89;. when F 0, formulas (B) and (C) are triially similar to formula (A). For example, in formula (C), we may substitute the following equation on the left hand side: D G E E D G Gu Gu Γ ( ) or Γ. (3.) E E G EG E E G EG We can immediately calculate the same results as found from formula (A) while formula (D) results in a Riemann representation. In this way, we hae supplied some more general alternatie methods to compute the Gaussian curature, including the case when F 0. Next, we summarize the computational results of the four selected distributions into two tables. In Table, we tabulate the coefficients of the first fundamental form, their deriaties and their Gaussian curature. In Table, assuming F 0, we tabulate the six Christoffel symbols needed when we applied formulas (C) and (D)

7 William W.S. Chen 9 Table : E E F G E E G G EG K u u EG n(-) n -n(-)(-u) -n -n(-) n - u(-u) (-) u ( u) u(-u) ( ) u( u) 4n Trinomial 0 0 u(-u) Normal Cauchy r+ t 0 ( r + 3) - 3(r + ) ( - ) + r 3 - (r + ) -(r + 3) -(r + ) -4r r (r + ) r r ( r+ 3) ( r+ 3) ( r+ 3)

8 9 On computing Gaussian curature of some well known distribution Table : E -E E G -G G Γ u Γ Γ u u Γ Γ Γ E G E G E G Trinomial u- (-) u(-u) u(-u) (-) (-) - - Normal Cauchy t r r

9 William W.S. Chen 93 4 Four examples In this section, we gie the details of four examples and demonstrate how we could apply formula (D) and Table, from the preious section to compute our Gaussian curature. In example, we deal with trinomial families. We are gien that the Gaussian curature is equal to a sphere of radius can be calculated with a proper transformation. 4n, a fact that Example : Let ℵ be the trinomial family with one obseration. The region of ℵ may be represented as a simplex, that is, in { 3 3 ℵ ( π, π, π ) π + π + π, π i 0, i,, 3 3 R. We can write the trinomial densities as follows : f( y π, π ) π π π, y y y3 3 where π + π + π 3, and y + y + y 3. If we consider a proper } transformation zi π, i,,3, then we could map this simplex into the i positie portion of the sphere with radius. The geometrical interpretation of this distribution may be seen from the spherical representation of the trinomial family. Let us introduce spherical polar coordinates ( θϕ, ), defined by θ cos ( ), ϕ u, where π, 3 sin ( ) π u π + π or we can represent π, i,,3 as a i function of u,. We obtain π 3, π u( ), π ( u)( ) and the trinomial likelihood becomes l( y π, π ) u ( u) ( ) y y y y y+ y (4.) Based on the likelihood equation (4.), we can routinely compute our coefficient of the first fundamental form E,F and G. It is straightforward to write the logarithm of the likelihood and their first two partial deriaties with respect to u and as follows:

10 94 On computing Gaussian curature of some well known distribution ln l y ln u+ y ln( u) + ( y y ) ln + ( y + y ) ln( ) (4.) From (4.), we derie, ln l y y, u u u ln l y y y + y ln l y y ( + ), u u ( u) ln l y3 y+ y ( + ). ( ), (4.3) It is well-known that the trinomial random ariables y, y, y 3 hae expected alues n( u)( ), nu( ) and n, respectiely. Hence, we know that E( y+ y) n( ). Taking the expected alues of both sides of (4.3), and substituting the expected alues of y, y, and y 3 we finally get: ln l n( ) ln l ln l n E E( ), F E( ) 0, G E( ). u u( u) u ( ) (4.4) It is important to know that E,F and G themseles are still functions of u and. Therefore, it is possible to make a further computation on these coefficients: n n( )( u) E, G 0, E, u( u) u u u ( u) n( ) and G ( ) (4.5) If we want to apply formula (A) or (B), then it is by a straightforward substitution that these results are calculated and simplified. If formula (C) is selected, then it would be necessary to compute the six Christoffel symbols. Howeer, due to the reasons that we stated before, only formula(d) is presented. Since g F 0, we only need to find R.

11 William W.S. Chen 95 R Γ Γ + ( Γ Γ Γ Γ ) m m m m u m Γ ΓΓ +ΓΓ ( ) ( ) ( ) ( ) ( )( ) + ( )( ). u( u) ( ) u( u) u( u) ( ) 4 u( u) (4.6) Notice that the Christoffel symbols in this case satisfied : Γ Γ Γ 0 m m m R R g R g +R g R G K n n 4 u( u) ( ) 4 u( u) R n u( u) EG 4 u( u) n 4n (4.7) This result shows that the trinomial model space is locally isometric to the sphere of radius 4 n. In our four selected examples, only the trinomial model has a positie Gaussian curature. The remaining three examples will deal with the location-scale family of densities and the methods of finding those with negatie Gaussian curature. Kass, R.E. [], gae the general form of a location-scale manifold of density: x u px ( ) f( ) (u,) R R +, for some density function f. Then, the information metric of the Riemannian geometry space has constant negatie curature. We proide the deriation of the formula for the Gaussian curature of normal distribution in Example, Cauchy distribution in Example 3 and t family distribution in Example 4. Example : Let Ω be a location scale manifold of density that has the following general form:

12 96 On computing Gaussian curature of some well known distribution ( x u) Ω f( x) exp( ) (u,) R R, + π where u is the location parameter and is the scale parameter. We also assume that the regular conditions of the information metric are satisfied. The first and second partial deriatie, with respect to parametric lines u and, are gien as: ln f x u, u ln f u ln f ( x u) ln f 3( x u) +, and. 3 4 (4.8) It is commonly known that the expected alue and ariance of the random ariable x are u and, respectiely. From this, we could easily derie the coefficient of the first fundamental form E, F 0, G, as well as their corresponding deriaties with respect to the parametric lines u and : 4 E 0, E, G 0, G, EG, u 3 u 3 4 and. EG EG (4.9) Substituting the listed results into formula (A), (B) or (C), it should be easy to compute the Gaussian curature, obtaining. Again, we present the details for formula (D) only. Similar to equation (4.6), howeer, this time we hae

13 William W.S. Chen 97 Γ Γ Γ 0. R Γ ΓΓ +ΓΓ ( ) ( )( ) + ( )( ). R R G 4. 4 R Κ 4. EG (4.0) Example 3: Let Ω be the location scale manifold of density which has the following general form: Ω f( x) x R, (u,) R R, + π + ( x u) where u is the location parameter and is the scale parameter. The logarithm of the likelihood function of Cauchy density with one obseration can be written as f + x u. π ln ln ln( ( ) ) As before, we can derie the first two partial deriaties with respect to the parametric lines u and. ln f ( x u) u + ( x u) ln f (( x u) ), u (( x u) + ) ln f + ( x u) ln f ( ( x u) ) +, ( + ( x u)) ln f 4 x ( u) u ( + ( x u)),, (4.) Taking the expected alues of equations (4.), we finally get the following results:

14 98 On computing Gaussian curature of some well known distribution ln f ( x u) E E( ) E( ) ( ), u (( x u) + ) 4 ln f 4 x ( u) F E( ) E( ) 0, u ( + ( x u)) ln f ( x u) G E( ) ( + E( ) ( + ) ( + ( x u)) 4 (4.) The details of deriing the expected alues of (4.) can be found in the appendix. The deriaties of the coefficients of the first fundamental form and six Christoffel symbols are all straightforward computations. We list only their results in Table,. Due to the fact that the Cauchy distribution is the same as the normal distribution, that is, Gaussian curature. Γ Γ Γ 0. We use formula (D) to derie the R Γ ΓΓ +ΓΓ ( ) ( )( ) + ( )( ). EG 4. 4 R R R ( )( ). gm G 4 m R EG 4 Κ 4 ( ). 4 (4.3) Example 4: Let Ω 3 be the location-scale manifold of density that has the student t distribution and generally has the form: r + ( ) r+ Γ x u Ω 3 f( x) ( + [( ) ] ) x R, ( u, ) R R +, r π rγ( ) r where u is location parameter and is scale parameter. Let us define the following ariables to simplify the notation:

15 William W.S. Chen 99 r + Γ( ) r + a, b, c. r r r π rγ( ) Then the logarithm of likelihood function of family t, can be written as follows: x u f x c b + a (4.4) ln ( ) ln r ln( ( ) ) ln. From equation (4.4), we can derie the first and second partial deriaties : ln f ab( x u) u + ax ( u) 3 ln f ab( x u) + ax ( u), ln f ab( a( x u) ) u ( ax ( u) + ), ln f 6 ab( x u) ( + a( x u) ) + 4 a b( x u) ( + ax ( u) ) ln f 4 ab( x u) u ( ax ( u) + ). ] (4.5) We can now take the expected alues of (4.5) and leae the details of computing the expectations to the appendix, ln f r r+ E E( ) ab( ), u r+ 3) ( r+ 3) ln f F E( ) 0, (4.6) u ln f r+ G E( ) ( 3 ). r+ 3 It now becomes a routine procedure to compute the deriatie of the coefficient of the first fundamental form and six Christoffel symbols. Compute the Riemann symbols of the first and second kind, respectiely. Thus, the Gaussian curature is calculated.

16 00 On computing Gaussian curature of some well known distribution r+ r+ r+ ( r+ ) R Γ ΓΓ +ΓΓ ( ) ( )( ) + ( )( ) r r r r ( r+ ) r ( r+ ) R R R (4.7) r ( r+ 3) ( r+ 3) G 4 m R EG ( r + 3) r( r + ) r 4 ( r ) ( r 3) ( r 3) Κ 4. 5 Concluding remark One of the most important theorems of the 9 th century is Theorema Egregium. Many mathematicians at the end of the 8 th century, including Euler and Monge, had used the Gaussian curature, but only when defined as the product of the principal curatures. Since each principal curature of a surface 3 depends on the particular way the surface is defined in R, there is no obious reason for the product of the principal curatures to be intrinsic to that particular surface. Gauss published in 87 that the product of the principal curatures depends only on the intrinsic geometry of the surface reolutionized differential geometry. Gauss wrote The Gaussian curature of a surface is a bending inariant, a most excellent theorem, This is a Theorema egregium. In this theorem, Gauss proed that the Gaussian curature, K, of a surface, depends only on the coefficient of the first fundamental form and their first and second deriaties. This important geometric fact will link the concepts of bending and isometric mapping. By bending inariant, we mean that it is unchanged by such deformations of the surface when restricted to a limited region that does not inole stretching, shrinking, or tearing. When measured along a cure on the surface, the distance between two points on the surface is unchanged. The angle of the two tangent directions at the point is also unchanged. This property of surfaces expressible as bending inariant is called the intrinsic property. We

17 William W.S. Chen 0 would like to conclude this study by repeating Kass (989,997) faorite and most interesting piece of triia: Suppose we ask which distribution in the t family is half way between Normal and Cauchy on the statistical curature scale, the scale of sufficiency loss of the Maximum Likelihood Estimator. For Normal, γ 0 and for Cauchy, γ 5. Thus, we seek γ such that 5 γ. There is no reason why γ should turn out to be an integer; it merely has to be a number greater than. Since γ for Cauchy and γ for Normal, the answer is γ 3. Thus, in the sense of the insufficiency of the MLE, as measured by statistical curature, the t, on 3 degrees of freedom, is halfway between Normal and Cauchy. This means that the statistical curature of the t 3 distribution is the arithmetic mean of the statistical curatures for the Cauchy and Normal distribution. From the Gaussian curature that we deried in this paper, we showed that in Normal distribution we obtain Κ, and in Cauchy distribution we obtain Κ, while in t family distribution with r degrees of r + 3 freedom, we get Κ. In other words, the Gaussian curature of the t 3 r distribution is the geometric mean of the curatures for the Cauchy and Normal distribution. Thus, we conclude that whether one uses statistical or geometric mean curature, the t 3 may be considered half way in between a Normal and Cauchy distribution.

18 0 On computing Gaussian curature of some well known distribution 6 Appendix In this section we show some expected alue computations. Taking the expected alue of equation (4.5) with respect to the random ariable x, where x is defined, and obtain n: Let y x u y, ay w ( b + ), r α (( b+ ) ) a] ( ), then r + 4 αw. In general, the lth moments of the t distribution with r degrees of freedom is defined as µ lr, CC r r r+ 4 l 3 ( l ), ( r l)( r l+ ) ( r ) and r+ r+ 4 Γ( ) Γ( ) r+ 4 r+ 4 rr ( + ). r r 5 ( r 3)( r ) ( ) ( r ) r + + Γ Γ + a( x u) a( x u) x u r + b E( ) C ( )( a( ) ) dx ( ax ( u) + ) ( ax ( u) + ) Cr 3 { a, r 4 C r 4 C αµ + + α r+ 4} r+ 4 r+ 4 rr ( + ) 4 r+ r ( ) ( ) r r ( r+ 3)( r+ ) r r+ r+ 4 r ( r+ 3) (6.) (6.) E 6 ab( x u) ( + a( x u) ) + 4 a b( x u) + ( + ax ( u) ) 4 4 Cr 3 5 C r abc r 4(3, r 4, a 4, r 4) + αµ + + αµ + r+ r+ 4 3 ( ) ( ) Γ Γ r+ r+ 4 3 ( ) ( ) r a+ r r r r+ 5 Γ( ) Γ ( ) ( r+ 4) ( r+ ) r + ( 3 ) r + 3 (6.3)

19 William W.S. Chen 03 x ( u) x ( u) abe abc a x u dx ( ax ( u) + ) ( ax ( u) + ) b 4 ( ) 4 ( ( ) ) r + 4abCr y + ay dy ( b+ ) ( ) 0. (6.4) Next, we define the six well known Christoffel symbols see Struik D.J. [4], or Gray A. GEu FFu + FE EGu FE EFu EE FEu Γ, Γ, Γ, ( EG F ) ( EG F ) ( EG F ) (6.5) GF GGu FG GE FGu EG FF + FGu Γ, Γ, Γ. ( EG F ) ( EG F ) ( EG F )

20 04 On computing Gaussian curature of some well known distribution References [] R.E. Kass and P.W. Vos, Geometrical Foundations of Asymptotic Inference, John Wiley & Sons, Inc, 997. [] R.E. Kass, The Geometry of Asymptotic Inference (with discussion), Statistical Science, 4(3), (August, 989), [3] E.L. Lehmann, Theory of Point Estimation, John Wiley & Sons, Inc, New York, 983. [4] D.J. Struik, Lectures on Classical Differential Geometry, Second Edition, Doer Publications, Inc, 96. [5] O.E. Barndorff-Nielsen, D.R. Cox and N. Reid, The Role of Differential Geometry in Statistical Theory, International Statistical Reiew, 54(), (986), [6] A. Gray, Modern Differential Geometry of Cures and Surfaces, CRC Press, Inc, 983. [7] W.W.S. Chen, Curature: Gaussian or Riemann, International Conference (IISA) held at McMaster Uniersity, Hamilton, Canada, (998). [8] B. Efron, Defining the Curature of a Statistical Problem (With Applications To Second Order Efficiency), Annual Statistics, 3, (975), 89-7.