Ferguson-Dirichlet過程在N維空間上的隨機函數分配

行政院國家科學委員會專題研究計畫 成果報告

Ferguson-Dirichlet 過程在 n 維空間上的隨機函數分配

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計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 97-2118-M-004-003- 執 行 期 間 ： 97 年 08 月 01 日至 98 年 09 月 30 日 執 行 單 位 ： 國立政治大學應用數學學系 計 畫 主 持 人 ： 姜志銘 計畫參與人員： 碩士班研究生-兼任助理人員：曾琬甯 大專生-兼任助理人員：陳玫芳 報 告 附 件 ： 出席國際會議研究心得報告及發表論文 處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

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Ferguson-Dirichlet 過程在 n 維空間上的隨機函數分配

On the distribution of the random functional of a Ferguson-Dirichlet

process over the n-dimensional space

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計畫主持人：姜志銘

計畫參與人員：陳玫芳 曾琬甯

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執行單位：國立政治大學應用數學系

中 華 民 國 九十八 年 十二 月 十八 日

行政

院國家科學委員會專題研究計畫成果報告

Ferguson-Dirichlet過程

_{在n維空間上的隨機函數分配}

On the distribution of the random functional of a

Ferguson-Dirichlet process over the n-dimensional space

計畫編號:NSC-97-2118-M-004-003

執行期限:97年8月1日至98年9月30日

主持人:姜志銘 國立政治大學應用數學系

jiangt@math.nccu.edu.tw

計畫參

_{與人員:陳玫芳 曾琬甯}

國立政治大學應用數學系

中

中

中文

文

文摘

摘

摘要

要

要

自從Ferguson於1973年提出 Ferguson-Dirichlet 過程後，就一直有很多的研究者探討它的隨機 函數，但是大部份的研究者都專注於低維度問題的探討。本研究中，我們探討並且提出用於得到 Ferguson-Dirichlet 過程_{在任何n維球體表面的隨機函數之機率密度函數的一套理論，這對目前僅有低} 維度的結果而言，是一個重要的擴充。

關鍵詞: Ferguson-Dirichlet過程；對稱分配；隨機函數；c-特徵函數

Abstract

Ferguson-Dirichlet process was first introduced by Ferguson (1973). Since then, many

re-searchers have studied its random functional. However, most of them focus on the low

dimen-sions. In this research, we study and give a unified theory to find the probability density functions

of the random functional of the FergusoDirichlet process over the spherical surface of any

n-dimensional ball. This would be a very important generalization of the current low n-dimensional

results.

Keywords: Ferguson-Dirichlet process, symmetrical distribution, random functional, c-characteristic

function.

1

Introduction

The study of distribution of random functional of the Ferguson-Dirichlet process has been

in active research in the recent decades, e.g., Hannum, Hollander, and Langberg (1981), Yamato

(1984), Jiang (1988), Diaconis and Kemperman (1996), Muliere and Tardella (1998), Nomachi

and Yamato (1999), and Jiang, Dickey, and Kuo (2004). Most of these papers study the cases on

real line. Although Jiang, Dickey, and Kuo (2004) and Jiang and Kuo (2008a) study the cases in

higher dimensions, the results are still restricted to the cases in two or three dimensions only. We

give a unify theory and many interesting results in the spherical surface of any n-dimensional ball

2

Random function of a Ferguson-Dirichlet process with

parameter measure over the surface of a n-dimensional

ball.

Let µ be a finite non-null measure on(Ω, B), where B is the σ-field of Borel subsets of Euclidean

space Ω, and let U be a stochastic process indexed by elements of B. We say U is a

Ferguson-Dirichlet process with parameter µ, denoted by U ∼ D(µ) on Ω, if for every finite measurable

partition B1, ..., Bm of Ω (i.e., the B0is are measurable, disjoint, and m

[

i=1

Bi = Ω), the random

vector (U (B1), . . . , U (Bm)) has a Dirichlet distribution with parameter vector(µ(B1), ..., µ(Bm)),

where

m

X

i=1

U (Bi) = 1.

Let X be a N-dimensional random vector defined as

X = Z Ω `(y)dU (y) = ( Z Ω `1(y)dU (y), ..., Z Ω `N(y)dU (y))0, (2.1)

where U ∼ D(µ) on Ω, µ(Ω) = c , y = (y1, ..., yL)0 , `(y) = (`1(y), ..., `N(y))0, and the `n(y)0s are

bounded measurable real-valued functions defined on Ω.

The following lemma gives the c-characteristic function expression for X.

Lemma 2.1. The c-characteristic function of X, as in Eq.(2.1), can be expressed by

g(t; X, c) = exp[− Z

Ω

ln(1 − it · `(y))dµ(y)],

where t = (t1, . . . , tN)0.

The following corollary can be obtained by applying the above Lemma 2.1 and Jiang,Dickey,and

Kuo(2004).

Eq.(2.1). Then E(Xn) = 1 c Z Ω `n(y)dµ(y) f or 1 6 n 6 N and Cov(Xn, Xm) = 1 c + 1[ 1 c Z Ω `n(y)`m(y)dµ(y) − ( 1 c Z Ω `n(y)dµ(y))( 1 c Z Ω `m(y)dµ(y))] f or 1 6 n 6 N

The following corollary is obtained immediately from Corollary 2.2.

Corollary 2.3. Let Y be a random vector associated with the probability measure µ/c.Then the

mean vector and the variance-covariance matrix of X, Eq.(2.1), can be expressed as

E(X) = E(`(Y )) and Cov(X) = 1

1 + cCov(`(Y ))

respectively.

In the following, we want to study the distributions of the random functionals,

e Sn,r =

Z

Sn,r

ydUn,r(y) Un,r ∼ D(µn, r) (2.2)

where n is any positive integer, Sn,r = {y ∈ Rn||y| = r}, and r > 0, is the spherical surface of

the n-dimensional ball, and where µn,r is the usual Lebesgue measure(i.e., usual rotation-invariant

measure) on Sn,r with total measure c.

First, we give the c-characteristic function expression for eSn,r in the next theorem.

Theorem 2.4. The c-characteristic functions of eSn,r in Eq.(2.2) can be expressed as

g(t; eSn,r, c) = exp{ ∞ X k=1 c(1/2, k) 2k(n/2, k)[−r 2_(t2 1+ ... + t 2 n)] k_} where t = (t1, . . . , tn)0.

Theorem 2.4 shows that the c-characteristic function of eSn,r is a function of |t| and c, only. It,

of any one-dimensional margin of eSn,r, say eSn,r, is expressed as g(t; eSn,r, c) = exp{ ∞ X k=1 c(1/2, k) 2k(n/2, k)(−r 2_t2₎k_{} (2.3)}

Our aim now is to find the probability density function of eSn,r. Lord (1954) demonstrated

that a spherical symmetric distribution can be determined by its marginal distribution. Therefore,

if the probability density function of the marginal random variable eSn,r is known, then we can

obtain the probability density function of the random vector eSn,r. Before giving the probability

density function of eSn,r, we show that an interesting random mean of a Ferguson-Dirichlet process

and another related random mean have the same c-characteristic function, and hence the same

distribution, as the one-dimensional marginal eSn,r.

Lemma 2.5. Let n be any positive integer and c > 0. Suppose that eRn,r =

Z r

−r

ydWn,r(y) where

Wn,r ∼ D(αn,r) and let αn,r be the measure on {−r, r} with α1,r({−r}) = α1,r({r}) = c/2 if n =

1, and be the measure on (−r, r) with density

dαn,r(y) =

c(r2_{− y}2₎(n−3)/2

B(1/2, (n − 1)/2)rn−2dy

if n > 1. Then the c-characteristic function of eRn,r can be expressed as

g(t; eRn,r, c) = exp{ ∞ X k=1 c(1/2, k) 2k(n/2, k)(−r 2 t2)k} In particular, g(t; eR1,r, c) = (1 + r2t2)−c/2

Corollary 2.6. Let eSn,r be any one-dimensional margin of eSn,r in Theorem 2.4. Then eSn,r has

the same distribution as eRn,r in Lemma 2.5.

Next, we provide the probability density functions of eRn,r, which can be derived by using the

inversion formula of the c-characteristic function (Jiang and Kuo(2008b)), when c = 1.

Lemma 2.7. Let f(x; n, r) be the probability density function of eRn,r.

(a) f (x; 1, r) = 1 π√r2_{− x}2,

that is, eR1,r is distributed as −ru1 + ru2 where (u1, u2) has a Dirichlet distribution with

parameter vector (1/2, 1/2),and fR2

1,r/r2 is distributed as Beta(1/2, 1/2); (b) f (x; 2, r) = 2 √ r2_{− x}2 r2_π , that is, fR 2 2,r/r2 is distributed as Beta(1/2, 3/2); (c) f (x; 3, r) = e π(r + x) −(r+x)/(2r)_{(r − x)}−(r−x)/(2r)_cosπx 2r; (d) f (x; 4, r) = 2 rπe 1/2−x2/r2_sin(x √ r2_{− x}2 r2 + 2 arcsin r r + x 2r ); More generally, we have

(e) when n is an odd integer greater than 3,

f (x; n, r) = 1 rπexp ( −1 B(1/2, (n − 1)/2) (n−3)/2 X k=0     n − 3 2 k     (−1)k 2k + 1 ×   1 + x 2k+1 r2k+1  ln  1 + x r  +  1 − x 2k+1 r2k+1  ln  1 −x r  −2 k X m=0 x2k−2m (2m + 1)r2k−2m # ) sin  Z x −r πdαn,r(y)  , − r < x < r;

(f ) when n is an even integer greater than 4,

f (x; n, r) = 2 rπexp ( 23−nπ B(1/2, (n − 1)/2) (n−4)/2 X k=0     n − 2 k     cos[(n − 2 − 2k) arcsin(x/r)] n − 2 − 2k ) × sin  Z x −r πdαn,r(y)  , − r < x < r.

By Lemma 2.7 and Lord (1954), we have the following theorem.

Theorem 2.8. Let h_Se(x; n, r) be the probability density functions of eSn,r =

Z

Sn

ydUn,r(y) defined

(a) When n is odd, h e S(x; n, r) =  −1 2πt d dt m f (t; n, r) |x| < r with t = q x2 1+ ... + x2n and m = (n − 1)/2 (b) When n is even, h_Se(x; n, r) =  −1 2πs d ds m −1 π Z r s f0(t; n, r) √ t2_{− s}2  |x| < r with s = q x2 1+ ... + x2n and m = (n − 1)/2,

where f(t; n, r) is given in Lemma 2.7 and

 −1 2πt d dt m f (t; n, r) = −1 2πt d dt   −1 2πt d dt m−1 f (t; n, r)  for m ≥ 1,where  −1 2πt d dt 0 ≡ 1.

3

Conclusion

Through the c-characteristic function, we have given a new approach for studying spherical

distributions. The c-characteristic function expression of any functional of any Ferguson-Dirichlet

process has also been given. In addition, we have given the exact distribution in n dimensions

of the random mean of a symmetric Ferguson-Dirichlet process with parameter measure over any

spherical surface.

References

[1] Diaconis, P. and Kemperman, J. (1996) Some new tools for Dirichlet priors. In J.M. Bernardo,

J.O. Berger, A.P. Dawid, and A.F.M. Smith (eds.), Bayesian Statistics 5, pp. 97-106. Oxford

[2] Dickey, J. M. and Jiang, T. J. (1998).Filtered-Variate prior distribution for histogram

smooth-ing. Journal of the American Statistical Association, 93, 651-662.

[3] Ferguson, T. S. (1973), A Bayesian analysis of some nonparametric problems, Annals of

Statistics, 1, 209-230.

[4] Hannum, R., Hollander, M., and Langberg, N. (1981), Distributional results for random

functionals of a Dirichlet process, Annals of probability, 9, 665-670.

[5] Jiang, J. (1988) Starlike functions and linear functions of a Dirichlet distributed vector. SIAM

J. Math. Anal., 19, 390-397.

[6] Jiang, T.J., Dickey, J.M., and Kuo, K.-L. (2004) A new multivariate transform and the

distribution of a random functional of a Ferguson-Dirichlet process. Stochastic Process. Appl.,

111, 77-95.

[7] Jiang, T.J. and Kuo, K.-L. (2008a), Distribution of a random functional of a

Ferguson-Dirichlet process over the unit sphere. To appear in Electron. Comm. Probab..

[8] Jiang, T.J. and Kuo, K.-L. (2008b), The inversion formula of the c-characteristic function

and its applications. Technical report No. NCCU 701-08-T11-01, Department of Mathematical

Sciences, National Chengchi University.

[9] Nomachi, T.; Yamato, H. The expectation of random functionals with the Dirichlet process

and its application s. Bull. Inf. Cybernet. 31, 165-178 (1999).

[10] Yamato. M. (1984), Characteristic functions of means of distributions chosen from a Dirichlet

Some new application results of the c-characteristic

function

Thomas J. Jiang

a

_{and Kun-Lin Kuo}

b

a_{Department of Mathematical Sciences, National Chengchi University, 64 Chih-Nan Road, Section 2,} Wen-Shan, Taipei 11605, Taiwan

b_{Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan}

1

Introduction

The c-characteristic function, which is in a special form of the generalized Stieltjes transforma-tion, has been shown to be a useful alternative tool for cases that the traditional characteristic function is hard or too complicated to use. First, we give its definition and properties. Then, we use it to give new results on random moments of Ferguson-Dirichlet processes with some interesting parameter measures.

2

Definition and properties of c-characteristic function

Definition 1. The c-characteristic function of a bounded random variable X (with a support

in [−a, a], a > 0) is defined as

g(t; X, c) = E(1 − itX)−c, where |t| < a−1 _{and c > 0.}

Definition 2. A random vector u = (u1, . . . , un)0 is said to have a Dirichlet distribution with

parameter b = (b1, . . . , bn)0, denoted by u ∼ Dir(b), if its PDF has the form

f (u; b) = 1 B(b) n Y j=1 ubj−1 j ,

for all u in the probability simplex {u | each uj ≥ 0, u+ = 1}, where each bj > 0, u+=

P_n

j=1uj,

and B(b) =Qn_j=1Γ(bj)/Γ(b+).

Properties:

(a) If u ∼ Dir(b), then g(t; a0_{u, b}

+) =

Q_n

j=1(1 − itaj)−bj, where a0 = (a1, . . . , an). 1

(b) For any c, there is a one-to-one correspondence between g(t; X, c) and X.

(c) For any c, if limn→∞g(t; Xn, c) = g(t; X, c) for all |t| < a−1, then Xn→ X in distribution. (d) Inversion formulas.

Let X be a random variable on (a, b). Suppose that g(t; X, c) is the univariate c-characteristic function of X. Set G(z) = z−c_{g(i/z; X, c). Then the related PDF and CDF can be} ex-pressed as (i) fX(x+) + fX(x−) 2 = lim²→0+ −1 2πi Z D²,x (z + x)c−1_G0_{(z) dz, and} (ii) FX(x+) + FX(x−) 2 = lim²→0+ 1 2πi Z D²,x (z + x)c−1_{G(z) dz,}

where D²,x is the contour which starts at the point −x − i², proceeds along the straight line Im z = −² to the point −a − i², then along the semi-circle |z + a| = ², Re z ≥ −a, to the point −a + i², and finally along the line Im z = ² to the point −x + i². In particular, when c = 1, (i) can be simplied as

(iii) fX(x

+_{) + f}

X(x−)

2 = lim²→0+

1

2πi[G(−x − i²) − G(−x + i²)] .

3

Random moments of a Ferguson-Dirichlet process

Definition 3. Let µ be a finite non-null measure on (Ω, A), where A is the σ-field of Borel

subsets of Euclidean space Ω, and let U be a stochastic process indexed by elements of A. We say that U is a Ferguson-Dirichlet process with parameter µ, denoted by U ∼ D(µ) on Ω, if for every finite measurable partition {A1, . . . , Am} of Ω, the random vector (U(A1), . . . , U (Am))

has a Dirichlet distribution with parameter (µ(A1), . . . , µ(Am)).

Define ξµ(h) =

R

Ωh(y) dU(y), where U ∼ D(µ) on Ω and h(y) is a bounded measurable

function on Ω.

This random functional can be applied in many areas such as

nonparametric density estimation: see, e.g. Lo (1984) and Dickey, Garthwaite, and Bian (1995) smoothness prior distribution: see, e.g. Dickey and Jiang (1998)

quality control problems: see, e.g. Epifani, Guglielmi, and Melilli (2006)

One possible derivation of the random functional ξµ(h) is through the limit of Xn =

P_n

j=1dnjuj as n approaches ∞, where dnj = h(anj), (u1, . . . , un) ∼ Dir(b1, . . . , bn), anj ∈ Anj,

{An1, . . . , Ann} is a partition of Ω, bnj = µ(Anj), and max1≤j≤nµ(Anj) → 0 as n approaches

∞. The traditional characteristic function (the Fourier transform) of Xn is

φ(t; Xn) = eitdnn ∞ X mj=0 1≤j≤n−1 (bn1, m1) · · · (bn,n−1, mn−1) (bn,+, m1+ · · · + mn−1)m1! · · · mn−1! × (i(dn1− dnn)t)m1· · · (i(dn,n−1− dnn)t)mn−1, 2

where (bn1, m1) = bn1(bn1+ 1) · · · (bn1 + m1 − 1) and bn,+ =

P_n

j=1bnj. (See also Exton (1976, p. 233).) However, the c(= bn,+)-characteristic function of Xn has the simple form

g(t; Xn, bn,+) = n

Y

j=1

(1 − itdnj)−bnj.

One can easily use Theorem 2.5 of Jiang, Dickey, and Kuo (2004) to obtain any moment of Xn. For example, E(Xn) = P_n j=1bnjdnj bn,+ and E(X2 n) = (Pn_j=1bnjdnj)2+ P_n j=1bnjd2nj bn,+(bn,++ 1) . In addition, g(t; ξµ(h), c) = exp · − Z Ωln[1 − ith(y)] dµ(y) ¸ .

Definition 4. Let v = (a2− a1)u + a1, a linear transformation of u from [0, 1] to [a1, a2] where

u has a beta distribution with parameters b1 and b2, then v is said to have a generalized beta

distribution on [a1, a2], denoted by Gbeta(b1, b2; a1, a2), with parameters b1 and b2. The PDF of

v has the form

1

B(b1, b2)

(v − a1)b1−1(a2− v)b2−1

(a2− a1)b1+b2−1

, for a1 ≤ v ≤ a2.

Theorem 5. Let L be any integer greater than 1, and XL =

R₁

−1y dUL(y) where UL ∼ D(µL)

on (−1, 1) and µL is a probability measure corresponding to Gbeta((L − 1)/2, (L − 1)/2; −1, 1).

Then, for −1 < x < 1, (i) fX2(x) = 2√1 − x2 π ; (ii) fX3(x) = e π(1 + x) −(1+x)/2_{(1 − x)}−(1−x)/2_cosπx 2 ; (iii) fX4(x) = 2 πe 1/2−x2 cos(x√1 − x2_{+ arcsin x).} In general, for −1 < x < 1,

(iv) when L is an odd integer and L ≥ 4,

fXL(x) = sin³R₋₁x π dµL(y) ´ π exp    −1 B(1/2, (L − 1)/2) (L−3)/2_X k=0 Ã_L−3 2 k ! (−1)k (2k + 1) × " (1 + x2k+1_{) ln(1 + x) + (1 − x}2k+1_{) ln(1 − x) − 2} Xk m=0 x2k−2m 2m + 1 #_   ;

(v) when L is an even integer and L ≥ 5,

fXL(x) = 2 sin³R₋₁x π dµL(y) ´ π exp    2 3−L_π B(1/2, (L − 1)/2) (L−4)/2_X k=0 Ã L − 2 k ! cos[(L − 2 − 2k) arcsin x] L − 2 − 2k   . 3

4

Conclusions

The study of the random moments of the Ferguson-Dirichlet process has drawn the attention of statisticians for decades. The use of c-characteristic function together with its inversion formulas provide a useful method to study the random moments of the Ferguson-Dirichlet process. Here, we provide the PDF, which are new in the literature, of the random moments of a Ferguson-Dirichlet process with some interesting parameter measures.

References

B.C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. D.M. Cifarelli, E. Regazzini, Distribution functions of means of a Dirichlet process, Ann. Statist.

18 (1990) 429–442. Correction: Ann. Statist. 22 (1994) 1633–1634.

D.M. Cifarelli, E. Melilli, Some new results for Dirichlet priors, Ann. Statist. 28 (2000) 1390– 1413.

P. Diaconis, J. Kemperman, Some new tools for Dirichlet priors, in: J.M. Bernardo, J.O. Berger, A.P. Dawid, A.F.M. Smith (Eds.), Bayesian Statistics 5, Oxford University Press, 1996, pp. 97–106.

J.M. Dickey, P.H. Garthwaite, G. Bian, An elementary continuous-type nonparametric distri-bution estimate, Int. J. Math. Stat. Sci. 4 (1995) 193–247.

J.M. Dickey, T.J. Jiang, Filtered-variate prior distributions for histogram smoothing, J. Amer. Statist. Assoc. 93 (1998) 651–662.

J.M. Dickey, T.J. Jiang, K.-L. Kuo, Distribution of functionals of a Ferguson-Dirichlet process (2009). To be published.

I. Epifani, A. Guglielmi, E. Melilli, A stochastic equation for the law of the random Dirichlet variance, Statist. Probab. Lett. 76 (2006) 495–502.

H. Exton, Multiple Hypergeometric Functions and Applications, Wiley, New Yprk, 1976. T.S. Ferguson, A Bayesian analysis of some nonparametric problems, Ann. Statist. 1 (1973)

209–230.

D.A. Freedman, On the asymptotic beheavior of Bayes estimate in the discrete case, Ann. Math. Statist. 34 (1963) 1386–1403.

I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, New York, 2000.

A. Guglielmi and R.L. Tweedie, Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process, Bernoulli 7 (2001) 573–592.

R.C. Hannum, M. Hollander, N.A. Langberg, Distributional results for random functionals of a Dirichlet process, Ann. Probab. 9 (1981) 665–670.

N.L. Hjort, A. Ongaro, Exact inference for random Dirichlet means, Stat. Inference Stoch. Process. 8 (2005) 227–254.

J. Jiang, Starlike functions and linear functions of a Dirichlet distributed vector, SIAM J. Math. Anal. 19 (1988) 390–397.

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表 Y04

行政院國家科學委員會補助國內專家學者出席國際學術會議報告

2009 年 9 月 8 日 報告人姓名 姜志銘 服務機構 及職稱 國立政治大學應用數學系教授 時間 會議 地點 2009 年 8 月 1-6 日 _本會核定 補助文號 NSC-97-2118-M-004-003 會議 名稱 (中文)2009 聯合統計季會議

(英文)2009Joint Statistical Meetings 發表

論文 題目

(中文)

(英文)Some new application results of the c-characteristic functions 報告內容應包括下列各項： 一、參加會議經過 由美國統計學會及其他多個國際知名的統計學會，如 IMS、ENAR、WNAR 及 SSC 聯合 主辦的 2009 年聯合統計會議，為全球最大型的統計會議，今年在美國首府(DC)舉 辦，吸引來自全球各地約六千名的統計專家學者們，共同為追求及交換新的統計理 論與方法而來參加，本人亦抱持著這種態度，希望藉由這個機會與其他的統計學者 專家作學術上的交流。 二、與會心得

我的演講題目為「Some new application results of the c-characteristic functions」。 c-特徵函數可以解決某些不容易利用傳統特徵函數來解決的機率或 統計問題。在這次演講中，除了介紹 c-特徵函數一些特別性質外，並也提供了一些 它在應用上的有趣新結果，演講中也引起與會者的興趣與發問。跟往年一樣，2009 年聯合統計會議涵蓋的領域非常廣泛，從理論到計算，從貝式統計到生物統計，甚 至目前大家所關心的區域性甚至國際性的議題，如溫度變化等等，都可以看到與會 者所展現的最新研究成果。透過參與這些會議，以及會議期間與所遇到的部份學者、 專家，互相交換最近一些研究領域方向的看法，使得參加這次會議得益不少。最後， 謝謝國科會給予這次機會參加這個有意義的會議。 三、考察參觀活動(無是項活動者省略) 四、建議 無 五、攜回資料名稱及內容 JSM 2009 Abstracts(CD)-研討會摘要 六、其他 附件三