| OPTIMAL DESIGN FUNDAMENTALS |
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What is a design space?
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This is the region that the experimenter is allowed to set
the values of the independent variable x to observe the response. This region
is decided in advance and is usually an interval but this need not be the case.
The design space in theory can be multi-dimensional and irregularly shaped. |
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I frequently see optimality criterion specified by p. What
is p? |
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Kiefer proposed a class of optimality criteria indexed
famously by the value of p. Different values of p lead to different optimality
criteria. These optimality criteria seek to make the confidence ellipsoid for
the parameters of interest small in various ways by choice of design. For
example, D-optimality corresponds to p=0 and a D-optimal design assures the
volume of the ellipsoid is as small as possible; A-optimality corresponds to
p=-1 and a A-optimal design makes the ellipsoid small by minimizing the sum of
the lengths of the major axes in the ellipsoid. Further details and explanation
for other criteria are given in design monographs and texts, for instance in
Pukelsheim (1993, chapter 6).
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What is k-point optimal design?
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An optimal design is one that provides the minimum (or
maximum) value of the criterion function and the minimization (or maximization)
is taken over all designs on the given design space. A k-point optimal design
is one that is only optimum within the class of all k-point designs on the
given design space. Usually the the value of k is equal to the number of
parameters in the mean response function. For example, a minimally supported
design for a polynomial regression model of degree n has k= n+1.
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What do I look for in the plot of the derivative function?
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If the generated design is optimal over all designs on the
user-selected design space, the plot over the design space should show a
graph that has the same maximum value at all the design points of the optimal
design. This maximum value is usually 0 but not necessarily, depending on an
additive constant in the equivalence theorem. You will also see that in some
cases the plot does not have this property even though it is optimal. This
happens where we want a minimally supported optimal design. The minimally
supported optimal design is only optimal among designs supported at a fixed
number of points and so this design may not remain optimal when we search among
all designs on the design space. So the plot of the derivative function may not
have the required property. If it does, this means the minimally supported
optimal design is also optimal over all designs on the design space.
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What are optional parameters? |
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Sometimes an optimal design does not depend on all the
parameters in the mean function E(y). These parameters are therefore not needed
to generate the optimal design and we call them optional parameters. However,
we require all parameters in the mean function E(y) to be specified to draw the
mean function.
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Why is it that the plot of the derivative function is
available for some cases only?
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This is because the derivative function can be very
complicated to work with, as in the case of minimax or maximin optimal
designs.
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Why is it that some programs do not calculate the efficiency
of a user-selected design?
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This is because we feel that this feature may have limited
value or relevance for the setup under consideration.
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Why are we sometimes asked to re-run the algorithm several
times to obtain the optimal design?
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The checking condition is only available for a convex
optimality criterion. It is used to confirm the optimality of the design. If
the criterion is not convex as a function of the design or the information
matrix, no checking condition exists and the problem becomes a general
optimization problem. The design generated from the algorithm depends on the
starting design and may be only locally optimal. It is therefore advisable to
run the algorithm several times using different starting designs and hopefully
each time the same optimal design is obtained.
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Why is it that some programs terminate without finding the
optimal design? |
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This is because the program takes too long to find the
optimal design and there is a default time limit imposed on each program. This
is especially so for programs for finding minimax or maximin optimal designs.
We are currently working to extend the default time limit imposed on each
program.
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| GENERAL QUESTIONS REGARDING THIS WEBSITE |
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What web browsers may I use? What is the best browser to use
to run the web based programs?
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You could use any type of web browsers to view this
website. e.g. Internet Explorer, Netscape Communicator, Firefox, etc. However,
we recommend Internet Explorer 6.0+ to get best view when you run the web based
program.
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What are the core technologies underlying this website?
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We integrated several cutting age technologies to found
this website. Such as Microsoft® ASP.Net framework for the user's
front end and Matlab® at the back end for statistical computing.
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How does the web-based program work? Why web-based ? |
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In short, the mechanism is straightforward. Users input
their data through dedicated web interfaces, the ASP.Net web application then
takes the input and send them to the Matlab® computing facilities.
Web based technology has been widely used for years. It can provide a
distributed computing platform without requiring the end users to install any
client software on local computers. We employ web based technologies to provide
readily accesible optimal experimental design programs. |
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Is the current web-based programs all you can provide?
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This website is a work in progress. We are developing
optimal design theories and algorithms for various practical models and will
upload these new design programs when they are ready. We have much more to
offer and we would like to know your opinion. Please email us at
info@optimal-design.org. Your interest and input are greatly
appreciated. .
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