對於一般線性模型而言，其A-最適設計不會受到參數的影響；但是對於廣義線性模型而言，其A-最適設計則會跟參數有所關聯性，而此緣故會對尋找A-最適設計有一定的困難之處。目前有Yang (2008) 有做相關的研究並且已經解決一部分的模型，主要解決的模型的共同點在於權重函數為對稱，而在本論文主要討論權重函數為非對稱的模型，所考慮模型Poisson 迴歸模型、Gamma 迴歸模型跟Inverse Gaussian 迴歸模型。我們利用粒子群最佳化演算法找出A-最適設計規律，再使用A-等價定理跟數學手法來找出一部分廣義線性模型的A-最適設計。

SUMMARY
The A-optimal design for common linear models is in general independent of its parameters. However, when finding an A-optimal design for generalized linear models, parameters must be taken into consideration. Previous work citep{Yang} discussed models of which weight function is symmetric. In this thesis, we consider Poisson, Gamma and Inverse Gaussian regression models, which holds an asymetric weight function. Particle swarm optimization is utilized to find rules of the A-optimal design while A-equivalence theorem and other mathematical approaches are applied to solve some part of the problem.
INTRODUCTION
Recently, it is usually of concern to choose a good design before an experiment is con-
ducted. The A-, D- and E- optimality criteria are commonly used. In this thesis, we dis-
cuss the A-optimal design that is the best design in terms of the A-optimality criteria.
The A-optimality criterion was introduced by Chernoff (1953). It is defined as minimiz-
ing the average variance of the parameter estimates. In other words, an A-optimal design
should minimize the trace of inverse of Fisher’s information matrix. For common linear
models, the problem of finding the A-optimal design is relatively easy, because the infor-
mation matrix is independent of the unknown parameters. For generalized linear models
(GLMs), on the other hand, the information matrix depends on the unknown parameters.
When finding the A-optimal design for generalized linear models, parameters must be
taken into consideration.
Yang (2008) discussed A-optimal design for generalized linear models with two parame-
ters and solved A-optimal design problem of Logistic regression model, Probit regression
model and Double exponential regression model. For generalized linear models, Fisher’s
information matrix of design contain the support points, weights and weight function. If
the weight function follows some conditions, Yang (2008) can prove the A-optimal design.
In this thesis, we discuss the A-optimal design for Poisson regression model with log link,
Gamma regression model with inverse link and Inverse Gaussian regression model with
inverse link. Weight functions for these models do not follow such the conditions, there-
fore, we can not use the method by Yang (2008) to achieve A-optimal design. Then, we try
to find A-optimal design.
METHODS
We can decide whether a design is A-optimal by A-equivalence theorem. In simple terms,
if the function F(x,») of A-equivalence theorem is less than or equal zero, the deign » will
be the A-optimal design. To find the rule or pattern of A-optimal designs for generalized
linear models with two parameters, we utilize the particle swarmoptimization(PSO).
Particle swarmoptimization is a popular method and is utilized to find the solution of optimization.
Finding A-optimal design can be regarded as a problem of optimization. The
PSO results are summarzied in table 1 and it shows that we can find out designs by PSO
generated well and we can observe the pattern of design. By A-equivalence theorem, we
can ensure that these designs by PSO generated are A-optimal designs.
RESULTS AND DISCUSSION
We apply mathematical approaches and A-equivalence theorem to find the general form
about A-optimal design. We guess the formof A-optimal design by PSO result at first sight
and check the design that we guess by A-equivalence theorem.
CONCLUSION
By the table 1, there are two types of A-optimal design. The first type design is that the
support points are at the boundary of design space. The second is that one of support
points is at the boundary of design space. So far, the existing result about the first type Aoptimal
design for Gamma, Inverse Gaussian and Poisson regression model. If we know
the model, parameters and conform some condition, then we can deduce the A-optimal
design.
IV

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